Beginning of Section Topology

Definition. We define topology_on to be λX T ⇒ T ⊆ 𝒫 X ∧ Empty ∈ T ∧ X ∈ T ∧ (∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T) ∧ (∀U ∈ T, ∀V ∈ T, U ∩ V ∈ T) of type set → set → prop.

In Proofgold the corresponding term root is bb1861... and object id is 137515...

Theorem. (topology_sub_Power)

∀X T : set, topology_on X T → T ⊆ 𝒫 X

In Proofgold the corresponding term root is bb2d54... and proposition id is 9a1f4b...

Proof:
Proof not loaded.

Definition. We define open_in to be λX T U ⇒ topology_on X T ∧ U ∈ T of type set → set → set → prop.

In Proofgold the corresponding term root is eedad1... and object id is 8d51ca...

Theorem. (open_in_topology)

∀X T U : set, open_in X T U → topology_on X T

In Proofgold the corresponding term root is 654d75... and proposition id is ceaa8f...

Proof:
Proof not loaded.

Theorem. (open_in_elem)

∀X T U : set, open_in X T U → U ∈ T

In Proofgold the corresponding term root is ddb8d5... and proposition id is 199323...

Proof:
Proof not loaded.

Theorem. (open_inI)

∀X T U : set, topology_on X T → U ∈ T → open_in X T U

In Proofgold the corresponding term root is ff965d... and proposition id is e27662...

Proof:
Proof not loaded.

Theorem. (Eps_i_unique)

∀P : set → prop, ∀x : set, P x → (∀y : set, P y → y = x) → Eps_i P = x

In Proofgold the corresponding term root is dee77b... and proposition id is 021741...

Proof:
Proof not loaded.

Theorem. (open_in_subset)

∀X T U : set, open_in X T U → U ⊆ X

In Proofgold the corresponding term root is 7f87ef... and proposition id is 9ae8df...

Proof:
Proof not loaded.

Theorem. (topology_elem_subset)

∀X T U : set, topology_on X T → U ∈ T → U ⊆ X

In Proofgold the corresponding term root is 8ae492... and proposition id is 1125cb...

Proof:
Proof not loaded.

Theorem. (topology_has_empty)

∀X T : set, topology_on X T → Empty ∈ T

In Proofgold the corresponding term root is ab166b... and proposition id is 4c567a...

Proof:
Proof not loaded.

Theorem. (topology_has_X)

∀X T : set, topology_on X T → X ∈ T

In Proofgold the corresponding term root is df1876... and proposition id is e2e114...

Proof:
Proof not loaded.

Theorem. (topology_subset_axiom)

∀X T : set, topology_on X T → T ⊆ 𝒫 X

In Proofgold the corresponding term root is bb2d54... and proposition id is 9a1f4b...

Proof:
Proof not loaded.

Theorem. (topology_union_closed)

∀X T UFam : set, topology_on X T → UFam ⊆ T → ⋃ UFam ∈ T

In Proofgold the corresponding term root is 2314e1... and proposition id is 87e3f4...

Proof:
Proof not loaded.

Theorem. (topology_union_closed_pow)

∀X T UFam : set, topology_on X T → UFam ∈ 𝒫 T → ⋃ UFam ∈ T

In Proofgold the corresponding term root is 20b69b... and proposition id is 46a086...

Proof:
Proof not loaded.

Theorem. (topology_union_axiom)

∀X T : set, topology_on X T → ∀UFam ∈ 𝒫 T, ⋃ UFam ∈ T

In Proofgold the corresponding term root is 1a2824... and proposition id is bfa94e...

Proof:
Proof not loaded.

Theorem. (topology_binintersect_closed)

∀X T U V : set, topology_on X T → U ∈ T → V ∈ T → U ∩ V ∈ T

In Proofgold the corresponding term root is 367ad1... and proposition id is 4e5488...

Proof:
Proof not loaded.

Theorem. (topology_binintersect_axiom)

∀X T : set, topology_on X T → ∀U ∈ T, ∀V ∈ T, U ∩ V ∈ T

In Proofgold the corresponding term root is 8f8ae7... and proposition id is b0fe2d...

Proof:
Proof not loaded.

Theorem. (Empty_is_open)

∀X T : set, topology_on X T → open_in X T Empty

In Proofgold the corresponding term root is 980da4... and proposition id is be5dc8...

Proof:
Proof not loaded.

Theorem. (X_is_open)

∀X T : set, topology_on X T → open_in X T X

In Proofgold the corresponding term root is 16a70b... and proposition id is 59db52...

Proof:
Proof not loaded.

Theorem. (union_open)

∀X T UFam : set, topology_on X T → (∀U ∈ UFam, open_in X T U) → open_in X T (⋃ UFam)

In Proofgold the corresponding term root is 1e7945... and proposition id is c2c8ce...

Proof:
Proof not loaded.

Theorem. (binintersect_open)

∀X T U V : set, open_in X T U → open_in X T V → open_in X T (U ∩ V)

In Proofgold the corresponding term root is 1c479f... and proposition id is 706661...

Proof:
Proof not loaded.

Definition. We define closed_in to be λX T C ⇒ topology_on X T ∧ (C ⊆ X ∧ ∃U ∈ T, C = X ∖ U) of type set → set → set → prop.

In Proofgold the corresponding term root is 6211a5... and object id is 1cef68...

Theorem. (closed_in_topology)

∀X T C : set, closed_in X T C → topology_on X T

In Proofgold the corresponding term root is 973b56... and proposition id is 03d73d...

Proof:
Proof not loaded.

Theorem. (closed_in_package)

∀X T C : set, closed_in X T C → C ⊆ X ∧ ∃U ∈ T, C = X ∖ U

In Proofgold the corresponding term root is 002055... and proposition id is 31373f...

Proof:
Proof not loaded.

Theorem. (closed_inI)

∀X T C : set, topology_on X T → C ⊆ X → (∃U ∈ T, C = X ∖ U) → closed_in X T C

In Proofgold the corresponding term root is f42869... and proposition id is 3385e5...

Proof:
Proof not loaded.

Theorem. (tuple_2_setprod_by_pair_Sigma)

∀X Y : set, ∀x y : set, x ∈ X → y ∈ Y → (x,y) ∈ setprod X Y

In Proofgold the corresponding term root is 7f845b... and proposition id is 8ae2b0...

Proof:
Proof not loaded.

Theorem. (closed_in_exists_open_complement)

∀X T C : set, closed_in X T C → ∃U ∈ T, C = X ∖ U

In Proofgold the corresponding term root is 78a12f... and proposition id is 29a082...

Proof:
Proof not loaded.

Theorem. (closed_in_subset)

∀X T C : set, closed_in X T C → C ⊆ X

In Proofgold the corresponding term root is 8de107... and proposition id is 962493...

Proof:
Proof not loaded.

Theorem. (closed_of_open_complement)

∀X T U : set, topology_on X T → U ∈ T → closed_in X T (X ∖ U)

In Proofgold the corresponding term root is fe3e1f... and proposition id is 3864ac...

Proof:
Proof not loaded.

Theorem. (X_is_closed)

∀X T : set, topology_on X T → closed_in X T X

In Proofgold the corresponding term root is 9ebffc... and proposition id is ea3315...

Proof:
Proof not loaded.

Theorem. (Empty_is_closed)

∀X T : set, topology_on X T → closed_in X T Empty

In Proofgold the corresponding term root is b914ec... and proposition id is e6cbc6...

Proof:
Proof not loaded.

Theorem. (closed_binintersect)

∀X T C D : set, closed_in X T C → closed_in X T D → closed_in X T (C ∩ D)

In Proofgold the corresponding term root is 1bac09... and proposition id is 7ac0ff...

Proof:
Proof not loaded.

Theorem. (closed_binunion)

∀X T C D : set, closed_in X T C → closed_in X T D → closed_in X T (C ∪ D)

In Proofgold the corresponding term root is de0012... and proposition id is 2cfa97...

Proof:
Proof not loaded.

Theorem. (open_of_closed_complement)

∀X T C : set, closed_in X T C → open_in X T (X ∖ C)

In Proofgold the corresponding term root is 900d49... and proposition id is 7b196a...

Proof:
Proof not loaded.

Theorem. (binunion_eq_Union_pair)

∀X Y : set, X ∪ Y = ⋃ {X,Y}

In Proofgold the corresponding term root is b94f4c... and proposition id is ed8d87...

Proof:
Proof not loaded.

Theorem. (topology_binunion_closed)

∀X T U V : set, topology_on X T → U ∈ T → V ∈ T → U ∪ V ∈ T

In Proofgold the corresponding term root is 9163f4... and proposition id is 2f5926...

Proof:
Proof not loaded.

Theorem. (binunion_open)

∀X T U V : set, open_in X T U → open_in X T V → open_in X T (U ∪ V)

In Proofgold the corresponding term root is 765afe... and proposition id is 9a4c37...

Proof:
Proof not loaded.

Definition. We define finer_than to be λT' T ⇒ T ⊆ T' of type set → set → prop.

In Proofgold the corresponding term root is dc6cac... and object id is fd51eb...

Definition. We define coarser_than to be λT' T ⇒ T' ⊆ T of type set → set → prop.

In Proofgold the corresponding term root is 81c0ef... and object id is 317007...

Definition. We define discrete_topology to be λX ⇒ 𝒫 X of type set → set.

In Proofgold the corresponding term root is 2a38db... and object id is 0a6206...

Definition. We define indiscrete_topology to be λX ⇒ {Empty,X} of type set → set.

In Proofgold the corresponding term root is b81793... and object id is 7c7f62...

Definition. We define finite_complement_topology to be λX ⇒ {U ∈ 𝒫 X|finite (X ∖ U) ∨ U = Empty} of type set → set.

In Proofgold the corresponding term root is fe593d... and object id is 9ef7ff...

Definition. We define countable to be λX ⇒ atleastp X ω of type set → prop.

In Proofgold the corresponding term root is 9bb8ca... and object id is bdda90...

Theorem. (finite_countable)

∀X : set, finite X → countable X

In Proofgold the corresponding term root is be49c8... and proposition id is f5a333...

Proof:
Proof not loaded.

Theorem. (countable_Empty)

countable Empty

In Proofgold the corresponding term root is d9321d... and proposition id is 55ec9d...

Proof:
Proof not loaded.

Theorem. (Subq_countable)

∀X Y : set, countable Y → X ⊆ Y → countable X

In Proofgold the corresponding term root is f92860... and proposition id is 5584a3...

Proof:
Proof not loaded.

Theorem. (binintersect_countable_left)

∀X Y : set, countable X → countable (X ∩ Y)

In Proofgold the corresponding term root is 40c6ae... and proposition id is 8fa13c...

Proof:
Proof not loaded.

Theorem. (setminus_countable)

∀X A : set, countable X → countable (X ∖ A)

In Proofgold the corresponding term root is 3fc51c... and proposition id is 234e92...

Proof:
Proof not loaded.

Theorem. (binunion_countable)

∀X Y : set, countable X → countable Y → countable (X ∪ Y)

In Proofgold the corresponding term root is b04050... and proposition id is 563e19...

Proof:
Proof not loaded.

Theorem. (Sigma_countable)

∀X : set, countable X → ∀Y : set → set, (∀x : set, x ∈ X → countable (Y x)) → countable (∑x ∈ X, Y x)

In Proofgold the corresponding term root is 5dba94... and proposition id is 89820a...

Proof:
Proof not loaded.

Theorem. (setprod_countable)

∀X Y : set, countable X → countable Y → countable (X ⨯ Y)

In Proofgold the corresponding term root is 5e5b16... and proposition id is 6d97b5...

Proof:
Proof not loaded.

Theorem. (Union_Power)

∀X Fam : set, Fam ⊆ 𝒫 X → ⋃ Fam ⊆ X

In Proofgold the corresponding term root is 8d8901... and proposition id is 0ea506...

Proof:
Proof not loaded.

Theorem. (binintersect_Power)

∀X U V : set, U ∈ 𝒫 X → V ∈ 𝒫 X → U ∩ V ∈ 𝒫 X

In Proofgold the corresponding term root is d4976b... and proposition id is 6ea6a1...

Proof:
Proof not loaded.

Theorem. (binintersect_Empty_left)

∀A : set, Empty ∩ A = Empty

In Proofgold the corresponding term root is 19837a... and proposition id is 66dfca...

Proof:
Proof not loaded.

Theorem. (binintersect_Empty_right)

∀A : set, A ∩ Empty = Empty

In Proofgold the corresponding term root is 0e7e2f... and proposition id is 347af5...

Proof:
Proof not loaded.

Theorem. (setminus_Power)

∀X U : set, U ∈ 𝒫 X → X ∖ U ∈ 𝒫 X

In Proofgold the corresponding term root is 2a61ee... and proposition id is e07cce...

Proof:
Proof not loaded.

Definition. We define countable_complement_topology to be λX ⇒ {U ∈ 𝒫 X|countable (X ∖ U) ∨ U = Empty} of type set → set.

In Proofgold the corresponding term root is 58878d... and object id is 921044...

Theorem. (countable_complement_topology_on)

∀X, topology_on X (countable_complement_topology X)

In Proofgold the corresponding term root is 74c81d... and proposition id is f9a8bf...

Proof:
Proof not loaded.

Theorem. (discrete_topology_on)

∀X, topology_on X (discrete_topology X)

In Proofgold the corresponding term root is 6b73db... and proposition id is 262383...

Proof:
Proof not loaded.

Theorem. (indiscrete_topology_on)

∀X, topology_on X (indiscrete_topology X)

In Proofgold the corresponding term root is ecf1ef... and proposition id is 8ce7dc...

Proof:
Proof not loaded.

Theorem. (finite_complement_topology_on)

∀X, topology_on X (finite_complement_topology X)

In Proofgold the corresponding term root is 6f0b3b... and proposition id is a98a95...

Proof:
Proof not loaded.

Theorem. (finer_than_refl)

∀T : set, finer_than T T

In Proofgold the corresponding term root is 0f302a... and proposition id is afdc02...

Proof:
Proof not loaded.

Theorem. (finer_than_trans)

∀A B C : set, finer_than B A → finer_than C B → finer_than C A

In Proofgold the corresponding term root is f1260a... and proposition id is 2f9d8f...

Proof:
Proof not loaded.

Theorem. (finer_coarser_dual)

∀T T' : set, finer_than T' T → coarser_than T T'

In Proofgold the corresponding term root is ddfa5b... and proposition id is 7592c5...

Proof:
Proof not loaded.

Definition. We define comparable_topologies to be λT1 T2 ⇒ finer_than T1 T2 ∨ finer_than T2 T1 of type set → set → prop.

In Proofgold the corresponding term root is 310f6a... and object id is 209d03...

Definition. We define topology_eq to be λX T1 T2 ⇒ topology_on X T1 ∧ topology_on X T2 ∧ T1 = T2 of type set → set → set → prop.

In Proofgold the corresponding term root is 2e5f76... and object id is 6221c3...

Theorem. (topology_eq_sym)

∀X T1 T2 : set, topology_eq X T1 T2 → topology_eq X T2 T1

In Proofgold the corresponding term root is fbcc7e... and proposition id is c77487...

Proof:
Proof not loaded.

Theorem. (topology_eq_trans)

∀X T1 T2 T3 : set, topology_eq X T1 T2 → topology_eq X T2 T3 → topology_eq X T1 T3

In Proofgold the corresponding term root is 32f299... and proposition id is 0f3a7d...

Proof:
Proof not loaded.

Theorem. (topology_eq_refl)

∀X T : set, topology_on X T → topology_eq X T T

In Proofgold the corresponding term root is 4dc43c... and proposition id is 671873...

Proof:
Proof not loaded.

Definition. We define strictly_finer_than to be λT' T ⇒ finer_than T' T ∧ ¬ finer_than T T' of type set → set → prop.

In Proofgold the corresponding term root is 391972... and object id is f17d93...

Definition. We define strictly_coarser_than to be λT' T ⇒ coarser_than T' T ∧ ¬ coarser_than T T' of type set → set → prop.

In Proofgold the corresponding term root is a159db... and object id is 351212...

Definition. We define discrete_topology_alt to be discrete_topology of type set → set.

In Proofgold the corresponding term root is 2a38db... and object id is 0a6206...

Definition. We define trivial_topology to be indiscrete_topology of type set → set.

In Proofgold the corresponding term root is b81793... and object id is 7c7f62...

Definition. We define finer_than_topology to be λX T' T ⇒ topology_on X T' ∧ topology_on X T ∧ finer_than T' T of type set → set → set → prop.

In Proofgold the corresponding term root is 3d83ff... and object id is 96fc54...

Theorem. (finer_than_def)

∀T T' : set, finer_than T' T ↔ coarser_than T T'

In Proofgold the corresponding term root is c0f0c4... and proposition id is 8003f0...

Proof:
Proof not loaded.

Theorem. (discrete_topology_finest)

∀X T : set, topology_on X T → finer_than (discrete_topology X) T

In Proofgold the corresponding term root is 26b184... and proposition id is ba4c69...

Proof:
Proof not loaded.

Theorem. (indiscrete_topology_coarsest)

∀X T : set, topology_on X T → coarser_than (indiscrete_topology X) T

In Proofgold the corresponding term root is b1e55a... and proposition id is c5eae3...

Proof:
Proof not loaded.

Theorem. (discrete_open_all)

∀X U : set, U ⊆ X → U ∈ discrete_topology X

In Proofgold the corresponding term root is 7143d7... and proposition id is 3c48d3...

Proof:
Proof not loaded.

Theorem. (indiscrete_open_iff)

∀X U : set, U ∈ indiscrete_topology X ↔ (U = Empty ∨ U = X)

In Proofgold the corresponding term root is a58418... and proposition id is 1d6de6...

Proof:
Proof not loaded.

Theorem. (finite_complement_topology_open_criterion)

∀X U : set, open_in X (finite_complement_topology X) U → finite (X ∖ U) ∨ U = Empty

In Proofgold the corresponding term root is 76de7d... and proposition id is 66e844...

Proof:
Proof not loaded.

Theorem. (finite_complement_topology_contains_empty)

∀X : set, Empty ∈ finite_complement_topology X

In Proofgold the corresponding term root is cf3756... and proposition id is 19683c...

Proof:
Proof not loaded.

Theorem. (finite_complement_topology_contains_full)

∀X : set, X ∈ finite_complement_topology X

In Proofgold the corresponding term root is a1ea79... and proposition id is 7ab81b...

Proof:
Proof not loaded.

Theorem. (countable_complement_topology_open_iff)

∀X U : set, open_in X (countable_complement_topology X) U → countable (X ∖ U) ∨ U = Empty

In Proofgold the corresponding term root is 75a7c2... and proposition id is c14c86...

Proof:
Proof not loaded.

Theorem. (countable_complement_topology_contains_empty)

∀X : set, Empty ∈ countable_complement_topology X

In Proofgold the corresponding term root is 0ee93f... and proposition id is edeb90...

Proof:
Proof not loaded.

Theorem. (countable_complement_topology_contains_full)

∀X : set, X ∈ countable_complement_topology X

In Proofgold the corresponding term root is 5588a4... and proposition id is 91e743...

Proof:
Proof not loaded.

Theorem. (countable_complement_finer_than_finite_complement)

∀X : set, finer_than (countable_complement_topology X) (finite_complement_topology X)

In Proofgold the corresponding term root is 894d65... and proposition id is 7f6600...

Proof:
Proof not loaded.

Theorem. (finite_complement_coarser_than_discrete)

∀X : set, coarser_than (finite_complement_topology X) (discrete_topology X)

In Proofgold the corresponding term root is a66c14... and proposition id is 8e9d83...

Proof:
Proof not loaded.

Theorem. (indiscrete_coarser_than_countable_complement)

∀X : set, coarser_than (indiscrete_topology X) (countable_complement_topology X)

In Proofgold the corresponding term root is 8189cf... and proposition id is af77a9...

Proof:
Proof not loaded.

Definition. We define finer_than_topology_by_inclusion to be finer_than_topology of type set → set → set → prop.

In Proofgold the corresponding term root is 3d83ff... and object id is 96fc54...

Theorem. (finer_than_topology_by_inclusion_eq)

∀X T' T : set, finer_than_topology X T' T ↔ finer_than_topology_by_inclusion X T' T

In Proofgold the corresponding term root is 0ba07c... and proposition id is 86ae76...

Proof:
Proof not loaded.

Theorem. (lemma_union_of_topology_family_open)

∀X T UFam : set, topology_on X T → UFam ∈ 𝒫 T → ⋃ UFam ∈ T

In Proofgold the corresponding term root is 20b69b... and proposition id is 46a086...

Proof:
Proof not loaded.

Theorem. (lemma_intersection_two_open)

∀X T U V : set, topology_on X T → U ∈ T → V ∈ T → U ∩ V ∈ T

In Proofgold the corresponding term root is 367ad1... and proposition id is 4e5488...

Proof:
Proof not loaded.

Definition. We define topological_space to be topology_on of type set → set → prop.

In Proofgold the corresponding term root is bb1861... and object id is 137515...

Definition. We define open_set_family to be λ_ T ⇒ T of type set → set → set.

In Proofgold the corresponding term root is 104bd1... and object id is 4b1310...

Definition. We define open_set to be open_in of type set → set → set → prop.

In Proofgold the corresponding term root is eedad1... and object id is 8d51ca...

Definition. We define basis_on to be λX B ⇒ B ⊆ 𝒫 X ∧ (∀x ∈ X, ∃b ∈ B, x ∈ b) ∧ (∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2) of type set → set → prop.

In Proofgold the corresponding term root is 4c6355... and object id is 2d1756...

Theorem. (basis_on_sub_Power)

∀X B : set, basis_on X B → B ⊆ 𝒫 X

In Proofgold the corresponding term root is 40d5a6... and proposition id is 3f27bf...

Proof:
Proof not loaded.

Theorem. (basis_on_cover)

∀X B : set, basis_on X B → ∀x ∈ X, ∃b ∈ B, x ∈ b

In Proofgold the corresponding term root is 19aeac... and proposition id is 91324e...

Proof:
Proof not loaded.

Theorem. (basis_on_refine)

∀X B : set, basis_on X B → ∀b1 ∈ B, ∀b2 ∈ B, ∀x : set, x ∈ b1 → x ∈ b2 → ∃b3 ∈ B, x ∈ b3 ∧ b3 ⊆ b1 ∩ b2

In Proofgold the corresponding term root is c07e68... and proposition id is 6f04e5...

Proof:
Proof not loaded.

Definition. We define generated_topology to be λX B ⇒ {U ∈ 𝒫 X|∀x ∈ U, ∃b ∈ B, x ∈ b ∧ b ⊆ U} of type set → set → set.

In Proofgold the corresponding term root is 4231ae... and object id is d0de89...

Theorem. (generated_topology_local_refine)

∀X B U x : set, U ∈ generated_topology X B → x ∈ U → ∃b ∈ B, x ∈ b ∧ b ⊆ U

In Proofgold the corresponding term root is 9f1c47... and proposition id is c8ef60...

Proof:
Proof not loaded.

Theorem. (generated_topology_contains_elem)

∀X B U : set, U ∈ 𝒫 X → U ∈ B → U ∈ generated_topology X B

In Proofgold the corresponding term root is 7cdd29... and proposition id is 6f5acf...

Proof:
Proof not loaded.

Theorem. (basis_elem_subset)

∀X B b : set, basis_on X B → b ∈ B → b ⊆ X

In Proofgold the corresponding term root is f6b3e3... and proposition id is 11dfc0...

Proof:
Proof not loaded.

Theorem. (basis_in_generated)

∀X B b : set, basis_on X B → b ∈ B → b ∈ generated_topology X B

In Proofgold the corresponding term root is fc5f07... and proposition id is 7d53ab...

Proof:
Proof not loaded.

Theorem. (generated_topology_subset)

∀X B U : set, U ∈ generated_topology X B → U ⊆ X

In Proofgold the corresponding term root is a7055b... and proposition id is 4e1fa4...

Proof:
Proof not loaded.

Theorem. (lemma_topology_from_basis)

∀X B : set, basis_on X B → topology_on X (generated_topology X B)

In Proofgold the corresponding term root is d7132e... and proposition id is 26d89d...

Proof:
Proof not loaded.

Theorem. (generated_topology_contains_basis)

∀X B : set, basis_on X B → ∀b : set, b ∈ B → b ∈ generated_topology X B

In Proofgold the corresponding term root is 644e5a... and proposition id is 4df7b9...

Proof:
Proof not loaded.

Theorem. (generated_topology_finer_weak)

∀X B T : set, topology_on X T → (∀b ∈ B, b ∈ T) → finer_than T (generated_topology X B)

In Proofgold the corresponding term root is f42b1c... and proposition id is f8f478...

Proof:
Proof not loaded.

Definition. We define basis_generates to be λX B T ⇒ basis_on X B ∧ generated_topology X B = T of type set → set → set → prop.

In Proofgold the corresponding term root is 6cce2e... and object id is 8c2cd7...

Definition. We define basis_refines to be λX B T ⇒ topology_on X T ∧ (∀U ∈ T, ∀x ∈ U, ∃b ∈ B, x ∈ b ∧ b ⊆ U) of type set → set → set → prop.

In Proofgold the corresponding term root is 2cf9a4... and object id is e74588...

Theorem. (lemma_generated_topology_characterization)

∀X B : set, basis_on X B → generated_topology X B = {U ∈ 𝒫 X|∀x ∈ U, ∃b ∈ B, x ∈ b ∧ b ⊆ U}

In Proofgold the corresponding term root is c8432c... and proposition id is 063588...

Proof:
Proof not loaded.

Theorem. (open_sets_as_unions_of_basis)

∀X B : set, basis_on X B → ∀U : set, open_in X (generated_topology X B) U → ∃Fam ∈ 𝒫 B, ⋃ Fam = U

In Proofgold the corresponding term root is c1e863... and proposition id is d2093b...

Proof:
Proof not loaded.

Theorem. (basis_generates_open_sets)

∀X B : set, basis_on X B → ∀U : set, (∃Fam ∈ 𝒫 B, ⋃ Fam = U) → open_in X (generated_topology X B) U

In Proofgold the corresponding term root is adf14f... and proposition id is ab7b94...

Proof:
Proof not loaded.

Theorem. (open_as_union_of_basis_elements)

∀X B : set, basis_on X B → ∀U : set, open_in X (generated_topology X B) U → U = ⋃ {b ∈ B|b ⊆ U}

In Proofgold the corresponding term root is 3b6dba... and proposition id is 0c265f...

Proof:
Proof not loaded.

Theorem. (basis_refines_topology)

∀X T C : set, topology_on X T → (∀c ∈ C, c ∈ T) → (∀U ∈ T, ∀x ∈ U, ∃Cx ∈ C, x ∈ Cx ∧ Cx ⊆ U) → basis_on X C ∧ generated_topology X C = T

In Proofgold the corresponding term root is 8a26a8... and proposition id is ced1f4...

Proof:
Proof not loaded.

Theorem. (lemma13_2_basis_from_open_subcollection)

∀X T C : set, topology_on X T → (∀c ∈ C, c ∈ T) → (∀U ∈ T, ∀x ∈ U, ∃c ∈ C, x ∈ c ∧ c ⊆ U) → basis_on X C ∧ generated_topology X C = T

In Proofgold the corresponding term root is 8a26a8... and proposition id is ced1f4...

Proof:
Proof not loaded.

Theorem. (finer_via_basis)

∀X B B' : set, basis_on X B → basis_on X B' → (∀x ∈ X, ∀b : set, b ∈ B → x ∈ b → ∃b' ∈ B', x ∈ b' ∧ b' ⊆ b) → finer_than (generated_topology X B') (generated_topology X B)

In Proofgold the corresponding term root is f5076c... and proposition id is 339bed...

Proof:
Proof not loaded.

Theorem. (basis_finer_equiv_condition)

∀X B B' : set, basis_on X B → basis_on X B' → ((∀x ∈ X, ∀b ∈ B, x ∈ b → ∃b' ∈ B', x ∈ b' ∧ b' ⊆ b) ↔ finer_than (generated_topology X B') (generated_topology X B))

In Proofgold the corresponding term root is 9667d1... and proposition id is f5499f...

Proof:
Proof not loaded.

Theorem. (generated_topology_finer)

∀X B T : set, basis_on X B → topology_on X T → (∀b ∈ B, b ∈ T) → finer_than T (generated_topology X B)

In Proofgold the corresponding term root is 06d86b... and proposition id is 2d0172...

Proof:
Proof not loaded.

Theorem. (topology_generated_by_basis_is_smallest)

∀X B T : set, basis_on X B → topology_on X T → (∀b ∈ B, b ∈ T) → finer_than T (generated_topology X B)

In Proofgold the corresponding term root is 06d86b... and proposition id is 2d0172...

Proof:
Proof not loaded.

Theorem. (union_of_basis_equals_open)

∀X B : set, basis_on X B → generated_topology X B = {⋃ Fam|Fam ∈ 𝒫 B}

In Proofgold the corresponding term root is 3ee001... and proposition id is 05b8bc...

Proof:
Proof not loaded.

Definition. We define singleton_basis to be λX ⇒ {{x}|x ∈ X} of type set → set.

In Proofgold the corresponding term root is ab4e07... and object id is 61ee14...

Theorem. (singleton_basis_is_basis)

∀X : set, basis_on X (singleton_basis X)

In Proofgold the corresponding term root is e75d9b... and proposition id is 4d1b5a...

Proof:
Proof not loaded.

Theorem. (generated_topology_singletons_discrete)

∀X : set, generated_topology X (singleton_basis X) = discrete_topology X

In Proofgold the corresponding term root is 047e37... and proposition id is 334958...

Proof:
Proof not loaded.

Definition. We define R to be real of type set.

In Proofgold the corresponding term root is e26ffa... and object id is 5277ef...

Definition. We define Q to be rational of type set.

In Proofgold the corresponding term root is efaf16... and object id is 6ea078...

Definition. We define Rlt to be λa b ⇒ a ∈ R ∧ b ∈ R ∧ a < b of type set → set → prop.

In Proofgold the corresponding term root is d991c7... and object id is ff36c7...

Definition. We define Rle to be λa b ⇒ a ∈ R ∧ b ∈ R ∧ ¬ (Rlt b a) of type set → set → prop.

In Proofgold the corresponding term root is ab5cbe... and object id is 1e2bc8...

Theorem. (RleI)

∀a b : set, a ∈ R → b ∈ R → ¬ (Rlt b a) → Rle a b

In Proofgold the corresponding term root is a90251... and proposition id is 96469e...

Proof:
Proof not loaded.

Theorem. (RleE_left)

∀a b : set, Rle a b → a ∈ R

In Proofgold the corresponding term root is 221568... and proposition id is b6f1b3...

Proof:
Proof not loaded.

Theorem. (RleE_right)

∀a b : set, Rle a b → b ∈ R

In Proofgold the corresponding term root is faaf00... and proposition id is 0000c8...

Proof:
Proof not loaded.

Theorem. (RleE_nlt)

∀a b : set, Rle a b → ¬ (Rlt b a)

In Proofgold the corresponding term root is 156e7c... and proposition id is 152324...

Proof:
Proof not loaded.

Theorem. (Rle_neq_implies_Rlt)

∀a b : set, Rle a b → ¬ (a = b) → Rlt a b

In Proofgold the corresponding term root is 756775... and proposition id is 92c8c4...

Proof:
Proof not loaded.

Theorem. (Rle_minus_nonneg)

∀t : set, t ∈ R → ¬ (Rlt t 0) → Rle (minus_SNo t) 0

In Proofgold the corresponding term root is ff52f6... and proposition id is c41178...

Proof:
Proof not loaded.

Theorem. (RltI)

∀a b : set, a ∈ R → b ∈ R → a < b → Rlt a b

In Proofgold the corresponding term root is 9462e8... and proposition id is 69254e...

Proof:
Proof not loaded.

Theorem. (RltE_left)

∀a b : set, Rlt a b → a ∈ R

In Proofgold the corresponding term root is 4df610... and proposition id is ca8a93...

Proof:
Proof not loaded.

Theorem. (RltE_right)

∀a b : set, Rlt a b → b ∈ R

In Proofgold the corresponding term root is 7c89a8... and proposition id is fed97e...

Proof:
Proof not loaded.

Theorem. (RltE_lt)

∀a b : set, Rlt a b → a < b

In Proofgold the corresponding term root is 2aac0c... and proposition id is 5fd2fc...

Proof:
Proof not loaded.

Theorem. (Rlt_tra)

∀a b c : set, Rlt a b → Rlt b c → Rlt a c

In Proofgold the corresponding term root is cf2ace... and proposition id is 24191a...

Proof:
Proof not loaded.

Theorem. (not_Rlt_refl)

∀a : set, a ∈ R → ¬ (Rlt a a)

In Proofgold the corresponding term root is b0079e... and proposition id is 3f1203...

Proof:
Proof not loaded.

Theorem. (not_Rlt_sym)

∀a b : set, Rlt a b → ¬ (Rlt b a)

In Proofgold the corresponding term root is 00663c... and proposition id is a5d7bc...

Proof:
Proof not loaded.

Theorem. (Rlt_0_diff_of_lt)

∀a b : set, Rlt a b → Rlt 0 (add_SNo b (minus_SNo a))

In Proofgold the corresponding term root is 23eb55... and proposition id is 036420...

Proof:
Proof not loaded.

Theorem. (Rlt_implies_Rle)

∀a b : set, Rlt a b → Rle a b

In Proofgold the corresponding term root is 0d7630... and proposition id is ff9f07...

Proof:
Proof not loaded.

Theorem. (Rle_refl)

∀a : set, a ∈ R → Rle a a

In Proofgold the corresponding term root is 11ae98... and proposition id is 339c06...

Proof:
Proof not loaded.

Theorem. (R_eq_of_not_Rlt)

∀a b : set, a ∈ R → b ∈ R → ¬ (Rlt a b) → ¬ (Rlt b a) → a = b

In Proofgold the corresponding term root is f9d48c... and proposition id is 4e7ead...

Proof:
Proof not loaded.

Theorem. (Rle_antisym)

∀a b : set, Rle a b → Rle b a → a = b

In Proofgold the corresponding term root is cf620d... and proposition id is 5b0775...

Proof:
Proof not loaded.

Theorem. (SNoLe_of_Rle)

∀a b : set, Rle a b → a ≤ b

In Proofgold the corresponding term root is 1076b7... and proposition id is 2e5cc9...

Proof:
Proof not loaded.

Theorem. (Rlt_0_1)

Rlt 0 1

In Proofgold the corresponding term root is 851ab2... and proposition id is 908609...

Proof:
Proof not loaded.

Theorem. (minus_1_lt_0)

minus_SNo 1 < 0

In Proofgold the corresponding term root is 71f8fd... and proposition id is a05177...

Proof:
Proof not loaded.

Definition. We define EuclidPlane to be setprod R R of type set.

In Proofgold the corresponding term root is e12063... and object id is 4b4e01...

Definition. We define R2_xcoord to be λp ⇒ p 0 of type set → set.

In Proofgold the corresponding term root is f0bb69... and object id is 9211e4...

Definition. We define R2_ycoord to be λp ⇒ p 1 of type set → set.

In Proofgold the corresponding term root is 24bd9a... and object id is 84977b...

Theorem. (EuclidPlane_xcoord_in_R)

∀p : set, p ∈ EuclidPlane → R2_xcoord p ∈ R

In Proofgold the corresponding term root is a68713... and proposition id is 768fae...

Proof:
Proof not loaded.

Theorem. (EuclidPlane_ycoord_in_R)

∀p : set, p ∈ EuclidPlane → R2_ycoord p ∈ R

In Proofgold the corresponding term root is 243e0d... and proposition id is 96c65e...

Proof:
Proof not loaded.

Theorem. (R2_xcoord_tuple)

∀x y : set, R2_xcoord (x,y) = x

In Proofgold the corresponding term root is 474d9a... and proposition id is 054aa7...

Proof:
Proof not loaded.

Theorem. (R2_ycoord_tuple)

∀x y : set, R2_ycoord (x,y) = y

In Proofgold the corresponding term root is b72ff1... and proposition id is 358e60...

Proof:
Proof not loaded.

Theorem. (tuple_eq_coords)

∀x1 y1 x2 y2 : set, (x1,y1) = (x2,y2) → x1 = x2 ∧ y1 = y2

In Proofgold the corresponding term root is 5a5be1... and proposition id is 8feace...

Proof:
Proof not loaded.

Theorem. (tuple_coords_eq)

∀x1 y1 x2 y2 : set, x1 = x2 → y1 = y2 → (x1,y1) = (x2,y2)

In Proofgold the corresponding term root is a77749... and proposition id is cbdc18...

Proof:
Proof not loaded.

Theorem. (coords_eq_tuple)

∀x1 y1 x2 y2 : set, x1 = x2 → y1 = y2 → (x1,y1) = (x2,y2)

In Proofgold the corresponding term root is a77749... and proposition id is cbdc18...

Proof:
Proof not loaded.

Theorem. (EuclidPlane_eta)

∀p : set, p ∈ EuclidPlane → (R2_xcoord p,R2_ycoord p) = p

In Proofgold the corresponding term root is 99bc04... and proposition id is 33672e...

Proof:
Proof not loaded.

Theorem. (tuple_eq_coords_R2)

∀x1 y1 x2 y2 : set, (x1,y1) = (x2,y2) → x1 = x2 ∧ y1 = y2

In Proofgold the corresponding term root is 5a5be1... and proposition id is 8feace...

Proof:
Proof not loaded.

Definition. We define distance_R2 to be λp c ⇒ sqrt_SNo_nonneg (add_SNo (mul_SNo (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord c))) (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord c)))) (mul_SNo (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord c))) (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord c))))) of type set → set → set.

In Proofgold the corresponding term root is ab28eb... and object id is 0e8f75...

Theorem. (distance_R2_in_R)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → distance_R2 p c ∈ R

In Proofgold the corresponding term root is d17503... and proposition id is 2bd1db...

Proof:
Proof not loaded.

Theorem. (distance_R2_nonneg)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → 0 ≤ distance_R2 p c

In Proofgold the corresponding term root is cd739c... and proposition id is f61b41...

Proof:
Proof not loaded.

Theorem. (distance_R2_sqr)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → mul_SNo (distance_R2 p c) (distance_R2 p c) = add_SNo (mul_SNo (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord c))) (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord c)))) (mul_SNo (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord c))) (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord c))))

In Proofgold the corresponding term root is 2fdcb6... and proposition id is 1634a9...

Proof:
Proof not loaded.

Theorem. (abs_SNo_sqr_eq)

∀x : set, SNo x → mul_SNo (abs_SNo x) (abs_SNo x) = mul_SNo x x

In Proofgold the corresponding term root is 1322c2... and proposition id is 1ef087...

Proof:
Proof not loaded.

Theorem. (SNoLe_add_nonneg_right)

∀x y : set, SNo x → SNo y → 0 ≤ y → x ≤ add_SNo x y

In Proofgold the corresponding term root is c61b9a... and proposition id is dad414...

Proof:
Proof not loaded.

Theorem. (SNo_sqr_nonneg)

∀x : set, SNo x → 0 ≤ mul_SNo x x

In Proofgold the corresponding term root is d25bf0... and proposition id is 37e7dc...

Proof:
Proof not loaded.

Theorem. (dx2_le_distance_R2_sqr)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → mul_SNo (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord c))) (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord c))) ≤ mul_SNo (distance_R2 p c) (distance_R2 p c)

In Proofgold the corresponding term root is 95d121... and proposition id is 21da8c...

Proof:
Proof not loaded.

Theorem. (dy2_le_distance_R2_sqr)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → mul_SNo (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord c))) (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord c))) ≤ mul_SNo (distance_R2 p c) (distance_R2 p c)

In Proofgold the corresponding term root is 31151a... and proposition id is bb0404...

Proof:
Proof not loaded.

Theorem. (abs_dx2_le_distance_R2_sqr)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → mul_SNo (abs_SNo (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord c)))) (abs_SNo (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord c)))) ≤ mul_SNo (distance_R2 p c) (distance_R2 p c)

In Proofgold the corresponding term root is 89cce3... and proposition id is 25969b...

Proof:
Proof not loaded.

Theorem. (abs_dy2_le_distance_R2_sqr)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → mul_SNo (abs_SNo (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord c)))) (abs_SNo (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord c)))) ≤ mul_SNo (distance_R2 p c) (distance_R2 p c)

In Proofgold the corresponding term root is b58c15... and proposition id is 79243f...

Proof:
Proof not loaded.

Theorem. (abs_SNo_nonneg)

∀x : set, SNo x → 0 ≤ abs_SNo x

In Proofgold the corresponding term root is fa46c8... and proposition id is 6e9d4c...

Proof:
Proof not loaded.

Theorem. (SNo_sqr_lt_of_lt_nonneg)

∀x y : set, SNo x → SNo y → 0 ≤ x → x < y → mul_SNo x x < mul_SNo y y

In Proofgold the corresponding term root is 86b95a... and proposition id is 60b9f7...

Proof:
Proof not loaded.

Theorem. (abs_dx_le_distance_R2)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → abs_SNo (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord c))) ≤ distance_R2 p c

In Proofgold the corresponding term root is 122485... and proposition id is d21e22...

Proof:
Proof not loaded.

Theorem. (abs_dy_le_distance_R2)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → abs_SNo (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord c))) ≤ distance_R2 p c

In Proofgold the corresponding term root is 37a63f... and proposition id is e5f0dc...

Proof:
Proof not loaded.

Theorem. (abs_SNo_lt_imp_lt)

∀t r : set, SNo t → SNo r → 0 < r → abs_SNo t < r → t < r

In Proofgold the corresponding term root is 9d9e33... and proposition id is b8db0d...

Proof:
Proof not loaded.

Theorem. (abs_SNo_lt_imp_neg_lt)

∀t r : set, SNo t → SNo r → 0 < r → abs_SNo t < r → minus_SNo t < r

In Proofgold the corresponding term root is 11fa28... and proposition id is 31b0c4...

Proof:
Proof not loaded.

Theorem. (SNo_lt_of_abs_diff_and_margin)

∀a x y r : set, SNo a → SNo x → SNo y → SNo r → 0 < r → a < add_SNo x (minus_SNo r) → abs_SNo (add_SNo x (minus_SNo y)) < r → a < y

In Proofgold the corresponding term root is 811640... and proposition id is 0839df...

Proof:
Proof not loaded.

Theorem. (abs_xcoord_lt_of_distance_lt)

∀p x r : set, p ∈ EuclidPlane → x ∈ EuclidPlane → r ∈ R → Rlt (distance_R2 p x) r → abs_SNo (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord x))) < r

In Proofgold the corresponding term root is f7a6d2... and proposition id is 90c768...

Proof:
Proof not loaded.

Theorem. (abs_ycoord_lt_of_distance_lt)

∀p x r : set, p ∈ EuclidPlane → x ∈ EuclidPlane → r ∈ R → Rlt (distance_R2 p x) r → abs_SNo (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord x))) < r

In Proofgold the corresponding term root is f24d1d... and proposition id is dd7405...

Proof:
Proof not loaded.

Theorem. (distance_R2_refl_0)

∀p : set, p ∈ EuclidPlane → distance_R2 p p = 0

In Proofgold the corresponding term root is fbcc07... and proposition id is 087cf1...

Proof:
Proof not loaded.

Theorem. (distance_R2_sym)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → distance_R2 p c = distance_R2 c p

In Proofgold the corresponding term root is 4121a0... and proposition id is f30845...

Proof:
Proof not loaded.

Theorem. (SNo_nonneg_ne0_pos)

∀x : set, SNo x → 0 ≤ x → x ≠ 0 → 0 < x

In Proofgold the corresponding term root is 5f7d6a... and proposition id is 2b5df4...

Proof:
Proof not loaded.

Theorem. (SNoLt_sqr_nonneg)

∀x y : set, SNo x → SNo y → 0 ≤ x → 0 ≤ y → x < y → mul_SNo x x < mul_SNo y y

In Proofgold the corresponding term root is f3b5e4... and proposition id is 553308...

Proof:
Proof not loaded.

Theorem. (SNo_nonneg_sqr_Le_imp_Le)

∀x y : set, SNo x → SNo y → 0 ≤ x → 0 ≤ y → mul_SNo x x ≤ mul_SNo y y → x ≤ y

In Proofgold the corresponding term root is 78afe4... and proposition id is ae7361...

Proof:
Proof not loaded.

Theorem. (CauchySchwarz2_sq)

∀a b c d : set, SNo a → SNo b → SNo c → SNo d → mul_SNo (add_SNo (mul_SNo a c) (mul_SNo b d)) (add_SNo (mul_SNo a c) (mul_SNo b d)) ≤ mul_SNo (add_SNo (mul_SNo a a) (mul_SNo b b)) (add_SNo (mul_SNo c c) (mul_SNo d d))

In Proofgold the corresponding term root is cccf1b... and proposition id is 525849...

Proof:
Proof not loaded.

Theorem. (distance_R2_triangle_sqr_Le)

∀x y z : set, x ∈ EuclidPlane → y ∈ EuclidPlane → z ∈ EuclidPlane → mul_SNo (distance_R2 x z) (distance_R2 x z) ≤ mul_SNo (add_SNo (distance_R2 x y) (distance_R2 y z)) (add_SNo (distance_R2 x y) (distance_R2 y z))

In Proofgold the corresponding term root is 4c2eda... and proposition id is b2d6fa...

Proof:
Proof not loaded.

Theorem. (distance_R2_triangle_Rle)

∀x y z : set, x ∈ EuclidPlane → y ∈ EuclidPlane → z ∈ EuclidPlane → Rle (distance_R2 x z) (add_SNo (distance_R2 x y) (distance_R2 y z))

In Proofgold the corresponding term root is b8b8c6... and proposition id is cf4327...

Proof:
Proof not loaded.

Theorem. (distance_R2_eq0)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → distance_R2 p c = 0 → p = c

In Proofgold the corresponding term root is 8a203e... and proposition id is 84d7ca...

Proof:
Proof not loaded.

Theorem. (distance_R2_not_Rlt_0)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → ¬ (Rlt (distance_R2 p c) 0)

In Proofgold the corresponding term root is c375c7... and proposition id is 29fde9...

Proof:
Proof not loaded.

Definition. We define circular_regions to be {U ∈ 𝒫 EuclidPlane|∃c : set, ∃r : set, c ∈ EuclidPlane ∧ Rlt 0 r ∧ U = {p ∈ EuclidPlane|Rlt (distance_R2 p c) r}} of type set.

In Proofgold the corresponding term root is 615c7e... and object id is 52fb7a...

Theorem. (circular_regionI)

∀c r : set, c ∈ EuclidPlane → Rlt 0 r → {p ∈ EuclidPlane|Rlt (distance_R2 p c) r} ∈ circular_regions

In Proofgold the corresponding term root is c8c484... and proposition id is 11bf57...

Proof:
Proof not loaded.

Definition. We define rectangular_regions to be {U ∈ 𝒫 EuclidPlane|∃a b c d : set, a ∈ R ∧ b ∈ R ∧ c ∈ R ∧ d ∈ R ∧ Rlt a b ∧ Rlt c d ∧ U = {p ∈ EuclidPlane|∃x y : set, p = (x,y) ∧ Rlt a x ∧ Rlt x b ∧ Rlt c y ∧ Rlt y d}} of type set.

In Proofgold the corresponding term root is e4a77a... and object id is 49ac18...

Theorem. (rectangular_regionI)

∀a b c d : set, a ∈ R → b ∈ R → c ∈ R → d ∈ R → Rlt a b → Rlt c d → {p ∈ EuclidPlane|∃x y : set, p = (x,y) ∧ Rlt a x ∧ Rlt x b ∧ Rlt c y ∧ Rlt y d} ∈ rectangular_regions

In Proofgold the corresponding term root is 966fa5... and proposition id is 430355...

Proof:
Proof not loaded.

Theorem. (exists_eps_lt_pos_Euclid)

∀d : set, d ∈ R → Rlt 0 d → ∃N ∈ ω, eps_ N < d

In Proofgold the corresponding term root is 0307bb... and proposition id is 69e182...

Proof:
Proof not loaded.

Theorem. (exists_eps_lt_two_pos_Euclid)

∀a b : set, a ∈ R → b ∈ R → Rlt 0 a → Rlt 0 b → ∃r3 : set, r3 ∈ R ∧ Rlt 0 r3 ∧ Rlt r3 a ∧ Rlt r3 b

In Proofgold the corresponding term root is 320e56... and proposition id is adb7af...

Proof:
Proof not loaded.

Theorem. (exists_eps_lt_four_pos_Euclid)

∀a b c d : set, a ∈ R → b ∈ R → c ∈ R → d ∈ R → Rlt 0 a → Rlt 0 b → Rlt 0 c → Rlt 0 d → ∃r3 : set, r3 ∈ R ∧ Rlt 0 r3 ∧ Rlt r3 a ∧ Rlt r3 b ∧ Rlt r3 c ∧ Rlt r3 d

In Proofgold the corresponding term root is b9b5b6... and proposition id is 8e4f70...

Proof:
Proof not loaded.

Theorem. (Rle_Rlt_tra_Euclid)

∀a b c : set, Rle a b → Rlt b c → Rlt a c

In Proofgold the corresponding term root is 5f7cc8... and proposition id is 5935d4...

Proof:
Proof not loaded.

Theorem. (ball_refine_two_balls)

∀x c1 c2 r1 r2 : set, x ∈ EuclidPlane → c1 ∈ EuclidPlane → c2 ∈ EuclidPlane → Rlt 0 r1 → Rlt 0 r2 → Rlt (distance_R2 x c1) r1 → Rlt (distance_R2 x c2) r2 → ∃r3 : set, Rlt 0 r3 ∧ (∀p : set, p ∈ EuclidPlane → Rlt (distance_R2 p x) r3 → Rlt (distance_R2 p c1) r1 ∧ Rlt (distance_R2 p c2) r2)

In Proofgold the corresponding term root is 352dc4... and proposition id is dcdf3e...

Proof:
Proof not loaded.

Theorem. (abs_diff_lt_of_between)

∀x y r : set, x ∈ R → y ∈ R → r ∈ R → Rlt 0 r → Rlt (add_SNo y (minus_SNo r)) x → Rlt x (add_SNo y r) → abs_SNo (add_SNo x (minus_SNo y)) < r

In Proofgold the corresponding term root is 59565f... and proposition id is d467fc...

Proof:
Proof not loaded.

Theorem. (distance_R2_le_absdx_plus_absdy)

∀p c : set, p ∈ EuclidPlane → c ∈ EuclidPlane → distance_R2 p c ≤ add_SNo (abs_SNo (add_SNo (R2_xcoord p) (minus_SNo (R2_xcoord c)))) (abs_SNo (add_SNo (R2_ycoord p) (minus_SNo (R2_ycoord c))))

In Proofgold the corresponding term root is 74bfa4... and proposition id is 8ec76d...

Proof:
Proof not loaded.

Theorem. (rectangle_inside_ball)

∀x c r0 : set, x ∈ EuclidPlane → c ∈ EuclidPlane → Rlt 0 r0 → Rlt (distance_R2 x c) r0 → ∃r ∈ rectangular_regions, x ∈ r ∧ r ⊆ {p ∈ EuclidPlane|Rlt (distance_R2 p c) r0}

In Proofgold the corresponding term root is 76bf36... and proposition id is 23a3a7...

Proof:
Proof not loaded.

Theorem. (ball_inside_rectangle)

∀b x : set, b ∈ rectangular_regions → x ∈ EuclidPlane → x ∈ b → ∃r3 : set, Rlt 0 r3 ∧ (∀p : set, p ∈ EuclidPlane → Rlt (distance_R2 p x) r3 → p ∈ b)

In Proofgold the corresponding term root is bfbf94... and proposition id is d57e3c...

Proof:
Proof not loaded.

Theorem. (circular_regions_basis_plane)

basis_on EuclidPlane circular_regions

In Proofgold the corresponding term root is 2998dd... and proposition id is 1d9c14...

Proof:
Proof not loaded.

Theorem. (rectangular_regions_basis_plane)

basis_on EuclidPlane rectangular_regions

In Proofgold the corresponding term root is 818430... and proposition id is c24ac9...

Proof:
Proof not loaded.

Theorem. (rectangular_refines_circular_plane)

∀b ∈ circular_regions, ∀x : set, x ∈ b → ∃r ∈ rectangular_regions, x ∈ r ∧ r ⊆ b

In Proofgold the corresponding term root is bf693f... and proposition id is c93501...

Proof:
Proof not loaded.

Theorem. (circular_refines_rectangular_plane)

∀b ∈ rectangular_regions, ∀x : set, x ∈ b → ∃u ∈ circular_regions, x ∈ u ∧ u ⊆ b

In Proofgold the corresponding term root is 2187b1... and proposition id is df34a6...

Proof:
Proof not loaded.

Theorem. (circular_rectangular_same_topology_plane)

generated_topology EuclidPlane circular_regions = generated_topology EuclidPlane rectangular_regions

In Proofgold the corresponding term root is 4ad238... and proposition id is d324d8...

Proof:
Proof not loaded.

Theorem. (lemma_finer_if_basis_refines)

∀X B B' : set, basis_on X B → basis_refines X B' (generated_topology X B) → finer_than (generated_topology X B') (generated_topology X B)

In Proofgold the corresponding term root is 92c83c... and proposition id is 6c089e...

Proof:
Proof not loaded.

Definition. We define subbasis_on to be λX S ⇒ S ⊆ 𝒫 X ∧ ⋃ S = X of type set → set → prop.

In Proofgold the corresponding term root is df7da0... and object id is 36b113...

Definition. We define intersection_of_family to be λX Fam ⇒ {x ∈ X|∀U : set, U ∈ Fam → x ∈ U} of type set → set → set.

In Proofgold the corresponding term root is 6d7088... and object id is 443aac...

Definition. We define finite_subcollections to be λS ⇒ {F ∈ 𝒫 S|finite F} of type set → set.

In Proofgold the corresponding term root is b6350a... and object id is b7ce2a...

Theorem. (intersection_of_familyI)

∀X Fam x : set, x ∈ X → (∀U : set, U ∈ Fam → x ∈ U) → x ∈ intersection_of_family X Fam

In Proofgold the corresponding term root is 784781... and proposition id is 48f144...

Proof:
Proof not loaded.

Theorem. (intersection_of_familyE1)

∀X Fam x : set, x ∈ intersection_of_family X Fam → x ∈ X

In Proofgold the corresponding term root is 685008... and proposition id is c30f1e...

Proof:
Proof not loaded.

Theorem. (intersection_of_familyE2)

∀X Fam x : set, x ∈ intersection_of_family X Fam → ∀U : set, U ∈ Fam → x ∈ U

In Proofgold the corresponding term root is 1f39e5... and proposition id is 81b1ad...

Proof:
Proof not loaded.

Definition. We define finite_intersections_of to be λX S ⇒ {intersection_of_family X F|F ∈ finite_subcollections S} of type set → set → set.

In Proofgold the corresponding term root is e339f5... and object id is 0d2a40...

Definition. We define basis_of_subbasis to be λX S ⇒ {b ∈ finite_intersections_of X S|b ≠ Empty} of type set → set → set.

In Proofgold the corresponding term root is fd0e02... and object id is 2ece32...

Theorem. (finite_intersection_in_basis)

∀X S F : set, F ∈ finite_subcollections S → intersection_of_family X F ≠ Empty → intersection_of_family X F ∈ basis_of_subbasis X S

In Proofgold the corresponding term root is 1053b7... and proposition id is 3ffa5b...

Proof:
Proof not loaded.

Theorem. (intersection_of_family_empty_eq)

∀X : set, intersection_of_family X Empty = X

In Proofgold the corresponding term root is 7b25d3... and proposition id is c6179c...

Proof:
Proof not loaded.

Theorem. (intersection_of_family_singleton_eq)

∀X s : set, s ⊆ X → intersection_of_family X {s} = s

In Proofgold the corresponding term root is a62c21... and proposition id is 699af6...

Proof:
Proof not loaded.

Theorem. (subbasis_elem_in_basis)

∀X S s : set, subbasis_on X S → s ∈ S → s ≠ Empty → s ∈ basis_of_subbasis X S

In Proofgold the corresponding term root is 4295ba... and proposition id is 950ebc...

Proof:
Proof not loaded.

Theorem. (X_in_basis_of_subbasis)

∀X S : set, X ≠ Empty → X ∈ basis_of_subbasis X S

In Proofgold the corresponding term root is 1c9562... and proposition id is 78f684...

Proof:
Proof not loaded.

Theorem. (intersection_of_family_adjoin)

∀X F U : set, intersection_of_family X (F ∪ {U}) = (intersection_of_family X F) ∩ U

In Proofgold the corresponding term root is 22d9dd... and proposition id is 6f3045...

Proof:
Proof not loaded.

Theorem. (finite_intersection_in_topology)

∀X T F : set, topology_on X T → F ∈ 𝒫 T → finite F → intersection_of_family X F ∈ T

In Proofgold the corresponding term root is cd54fc... and proposition id is ad6f1a...

Proof:
Proof not loaded.

Definition. We define generated_topology_from_subbasis to be λX S ⇒ generated_topology X (basis_of_subbasis X S) of type set → set → set.

In Proofgold the corresponding term root is 0071fe... and object id is 2fc0aa...

Theorem. (finite_intersections_basis_of_subbasis)

∀X S : set, subbasis_on X S → basis_on X (basis_of_subbasis X S)

In Proofgold the corresponding term root is 1e46ce... and proposition id is 62fada...

Proof:
Proof not loaded.

Theorem. (topology_from_subbasis_is_topology)

∀X S : set, subbasis_on X S → topology_on X (generated_topology_from_subbasis X S)

In Proofgold the corresponding term root is d400f4... and proposition id is be9d85...

Proof:
Proof not loaded.

Theorem. (topology_generated_by_basis_is_minimal)

∀X S T : set, subbasis_on X S → topology_on X T → S ⊆ T → finer_than T (generated_topology_from_subbasis X S)

In Proofgold the corresponding term root is d24278... and proposition id is 21af8f...

Proof:
Proof not loaded.

Theorem. (ex13_1_local_open_subset)

∀X T A : set, topology_on X T → (∀x ∈ A, ∃U ∈ T, x ∈ U ∧ U ⊆ A) → open_in X T A

In Proofgold the corresponding term root is b3939f... and proposition id is 083354...

Proof:
Proof not loaded.

Definition. We define a_elt to be Empty of type set.

In Proofgold the corresponding term root is e2a833... and object id is dee93d...

Definition. We define b_elt to be {Empty} of type set.

In Proofgold the corresponding term root is c95c26... and object id is 4848d8...

Definition. We define c_elt to be {{Empty}} of type set.

In Proofgold the corresponding term root is 4145a1... and object id is 07be9b...

Definition. We define abc_set to be UPair a_elt b_elt ∪ {c_elt} of type set.

In Proofgold the corresponding term root is 8bfd35... and object id is f05de3...

Definition. We define top_abc_1 to be UPair Empty abc_set of type set.

In Proofgold the corresponding term root is 329562... and object id is 98cd61...

Definition. We define top_abc_2 to be 𝒫 abc_set of type set.

In Proofgold the corresponding term root is eefeb4... and object id is 794b4f...

Definition. We define top_abc_3 to be UPair Empty {a_elt} ∪ {abc_set} of type set.

In Proofgold the corresponding term root is d0fa8a... and object id is cba783...

Definition. We define top_abc_4 to be UPair Empty {b_elt} ∪ {abc_set} of type set.

In Proofgold the corresponding term root is c04d9e... and object id is 38339c...

Definition. We define top_abc_5 to be UPair Empty {c_elt} ∪ {abc_set} of type set.

In Proofgold the corresponding term root is 8024ec... and object id is fe6d7a...

Definition. We define top_abc_6 to be UPair Empty (UPair a_elt b_elt) ∪ {abc_set} of type set.

In Proofgold the corresponding term root is 7f9944... and object id is 71e452...

Definition. We define top_abc_7 to be UPair Empty (UPair a_elt c_elt) ∪ {abc_set} of type set.

In Proofgold the corresponding term root is 5396fd... and object id is 26a32b...

Definition. We define top_abc_8 to be UPair Empty (UPair b_elt c_elt) ∪ {abc_set} of type set.

In Proofgold the corresponding term root is bfff3e... and object id is 9fa20b...

Definition. We define top_abc_9 to be (UPair Empty {a_elt} ∪ {UPair a_elt b_elt}) ∪ {abc_set} of type set.

In Proofgold the corresponding term root is bc21f0... and object id is 1b37b9...

Theorem. (topology_three_sets)

∀X A : set, A ⊆ X → topology_on X (UPair Empty A ∪ {X})

In Proofgold the corresponding term root is f853a3... and proposition id is 3bda03...

Proof:
Proof not loaded.

Theorem. (topology_chain_four_sets)

∀X A B : set, A ⊆ B → B ⊆ X → topology_on X ((UPair Empty A ∪ {B}) ∪ {X})

In Proofgold the corresponding term root is 10825e... and proposition id is def69e...

Proof:
Proof not loaded.

Theorem. (ex13_2_compare_nine_topologies)

In Proofgold the corresponding term root is 5b2267... and proposition id is e70d52...

Proof:
Proof not loaded.

Theorem. (top_abc_3_finer_than_top_abc_1)

finer_than top_abc_3 top_abc_1

In Proofgold the corresponding term root is 3f826d... and proposition id is fbda8d...

Proof:
Proof not loaded.

Theorem. (top_abc_2_finer_than_top_abc_3)

finer_than top_abc_2 top_abc_3

In Proofgold the corresponding term root is 086801... and proposition id is 5ffdee...

Proof:
Proof not loaded.

Theorem. (top_abc_9_finer_than_top_abc_3)

finer_than top_abc_9 top_abc_3

In Proofgold the corresponding term root is 25238c... and proposition id is 5c67fc...

Proof:
Proof not loaded.

Theorem. (top_abc_9_finer_than_top_abc_6)

finer_than top_abc_9 top_abc_6

In Proofgold the corresponding term root is 9a2900... and proposition id is 223fec...

Proof:
Proof not loaded.

Definition. We define Intersection_Fam to be λX Fam ⇒ {U ∈ 𝒫 X|∀T : set, T ∈ Fam → U ∈ T} of type set → set → set.

In Proofgold the corresponding term root is fca4cb... and object id is 00a09b...

Theorem. (Intersection_Fam_empty_eq)

∀X Fam : set, ¬ (∃T : set, T ∈ Fam) → Intersection_Fam X Fam = 𝒫 X

In Proofgold the corresponding term root is 3f7408... and proposition id is e67d2d...

Proof:
Proof not loaded.

Theorem. (intersection_of_family_sub_X)

∀X Fam : set, intersection_of_family X Fam ⊆ X

In Proofgold the corresponding term root is f10922... and proposition id is eae651...

Proof:
Proof not loaded.

Theorem. (intersection_of_family_empty)

∀Fam : set, (∀T : set, T ∈ Fam → Empty ∈ T) → ∀X : set, Empty ∈ Intersection_Fam X Fam

In Proofgold the corresponding term root is 050153... and proposition id is fc4145...

Proof:
Proof not loaded.

Definition. We define infinite_complement_family to be λX ⇒ {U ∈ 𝒫 X|infinite (X ∖ U) ∨ U = Empty ∨ U = X} of type set → set.

In Proofgold the corresponding term root is 7809a4... and object id is 66ca72...

Theorem. (ex13_3a_Tc_topology)

∀X : set, topology_on X (countable_complement_topology X)

In Proofgold the corresponding term root is 74c81d... and proposition id is f9a8bf...

Proof:
Proof not loaded.

Theorem. (ex13_3a_countable_complement_open)

∀X : set, ∀U ∈ countable_complement_topology X, U ≠ Empty → countable (X ∖ U)

In Proofgold the corresponding term root is 2307f0... and proposition id is 850fad...

Proof:
Proof not loaded.

Theorem. (ex13_3a_union_helper)

∀X : set, ∀UFam ∈ 𝒫 (countable_complement_topology X), ⋃ UFam ∈ countable_complement_topology X

In Proofgold the corresponding term root is c81519... and proposition id is 6627e7...

Proof:
Proof not loaded.

Theorem. (infinite_setminus_finite)

∀X F : set, infinite X → finite F → infinite (X ∖ F)

In Proofgold the corresponding term root is 0b6da5... and proposition id is 6d6ecb...

Proof:
Proof not loaded.

Theorem. (infinite_nonempty)

∀X : set, infinite X → ∃x : set, x ∈ X

In Proofgold the corresponding term root is 4b521d... and proposition id is 2ac55c...

Proof:
Proof not loaded.

Theorem. (finite_UPair)

∀a b : set, finite (UPair a b)

In Proofgold the corresponding term root is 820004... and proposition id is 92c817...

Proof:
Proof not loaded.

Theorem. (infinite_remove2)

∀X a b : set, infinite X → infinite (X ∖ UPair a b)

In Proofgold the corresponding term root is 91d11f... and proposition id is b4d2b3...

Proof:
Proof not loaded.

Theorem. (infinite_two_distinct)

∀X : set, infinite X → ∃a b : set, a ∈ X ∧ b ∈ X ∧ ¬ (a = b)

In Proofgold the corresponding term root is d0a7f4... and proposition id is 25ae74...

Proof:
Proof not loaded.

Theorem. (infinite_setminus_finite_nonempty)

∀X F : set, infinite X → finite F → ∃x : set, x ∈ X ∖ F

In Proofgold the corresponding term root is ab379e... and proposition id is 2d81c4...

Proof:
Proof not loaded.

Theorem. (infinite_remove1_top)

∀X y : set, infinite X → infinite (X ∖ {y})

In Proofgold the corresponding term root is 99772d... and proposition id is d7e048...

Proof:
Proof not loaded.

Theorem. (ex13_3b_Tinfty_not_topology)

∀X : set, infinite X → ¬ topology_on X (infinite_complement_family X)

In Proofgold the corresponding term root is 0f4730... and proposition id is c2f72d...

Proof:
Proof not loaded.

Theorem. (ex13_3b_witness_outline)

∀X : set, infinite X → ∃U V : set, U ∈ infinite_complement_family X ∧ V ∈ infinite_complement_family X

In Proofgold the corresponding term root is 7179da... and proposition id is 9c15e6...

Proof:
Proof not loaded.

Theorem. (ex13_4a_intersection_topology)

∀X Fam : set, (∃T : set, T ∈ Fam) → (∀T ∈ Fam, topology_on X T) → topology_on X (Intersection_Fam X Fam)

In Proofgold the corresponding term root is 5850be... and proposition id is 87921d...

Proof:
Proof not loaded.

Theorem. (ex13_4b_smallest_largest)

∀X Fam : set, (∀T ∈ Fam, topology_on X T) → ∃Tmin, topology_on X Tmin ∧ (∀T ∈ Fam, T ⊆ Tmin) ∧ (∀T', topology_on X T' ∧ (∀T ∈ Fam, T ⊆ T') → Tmin ⊆ T') ∧ ∃Tmax, topology_on X Tmax ∧ (∀T ∈ Fam, Tmax ⊆ T) ∧ (∀T', topology_on X T' ∧ (∀T ∈ Fam, T' ⊆ T) → T' ⊆ Tmax)

In Proofgold the corresponding term root is 485583... and proposition id is 865f6e...

Proof:
Proof not loaded.

Theorem. (ex13_4c_specific_topologies)

∃Tsmall Tall : set, topology_on abc_set Tsmall ∧ topology_on abc_set Tall

In Proofgold the corresponding term root is 06247a... and proposition id is f89d79...

Proof:
Proof not loaded.

Theorem. (ex13_5_basis_intersection)

∀X A : set, basis_on X A → generated_topology X A = Intersection_Fam X {T ∈ 𝒫 (𝒫 X)|topology_on X T ∧ A ⊆ T}

In Proofgold the corresponding term root is 05221c... and proposition id is 1b298b...

Proof:
Proof not loaded.

Definition. We define rational_numbers to be Q of type set.

In Proofgold the corresponding term root is efaf16... and object id is 6ea078...

Theorem. (rational_numbers_Subq_R)

rational_numbers ⊆ R

In Proofgold the corresponding term root is 0691ee... and proposition id is 2749c2...

Proof:
Proof not loaded.

Theorem. (rational_numbers_in_R)

∀q ∈ rational_numbers, q ∈ R

In Proofgold the corresponding term root is 6bb080... and proposition id is bbbbc8...

Proof:
Proof not loaded.

Definition. We define open_interval to be λa b ⇒ {x ∈ R|Rlt a x ∧ Rlt x b} of type set → set → set.

In Proofgold the corresponding term root is 58cccb... and object id is 7f9904...

Theorem. (open_interval_left_in_R)

∀a b x : set, x ∈ open_interval a b → a ∈ R

In Proofgold the corresponding term root is dbcca7... and proposition id is c5ca4b...

Proof:
Proof not loaded.

Theorem. (open_interval_right_in_R)

∀a b x : set, x ∈ open_interval a b → b ∈ R

In Proofgold the corresponding term root is 1a80b7... and proposition id is 4af20c...

Proof:
Proof not loaded.

Definition. We define halfopen_interval_left to be λa b ⇒ {x ∈ R|¬ (Rlt x a) ∧ Rlt x b} of type set → set → set.

In Proofgold the corresponding term root is da334e... and object id is f7a3d7...

Definition. We define halfopen_interval_right to be λa b ⇒ {x ∈ R|Rlt a x ∧ ¬ (Rlt b x)} of type set → set → set.

In Proofgold the corresponding term root is 94dc48... and object id is 6ac391...

Theorem. (halfopen_interval_right_Subq_R)

∀a b : set, halfopen_interval_right a b ⊆ R

In Proofgold the corresponding term root is e401c1... and proposition id is c060ab...

Proof:
Proof not loaded.

Theorem. (halfopen_interval_right_rightmem)

∀a b : set, Rlt a b → b ∈ halfopen_interval_right a b

In Proofgold the corresponding term root is 0aaa1a... and proposition id is d56e13...

Proof:
Proof not loaded.

Theorem. (open_interval_Subq_R)

∀a b : set, open_interval a b ⊆ R

In Proofgold the corresponding term root is ec5937... and proposition id is 6ef95f...

Proof:
Proof not loaded.

Theorem. (real_in_open_interval_minus1_plus1)

∀x : set, x ∈ R → x ∈ open_interval (add_SNo x (minus_SNo 1)) (add_SNo x 1)

In Proofgold the corresponding term root is e79253... and proposition id is 74b32a...

Proof:
Proof not loaded.

Theorem. (halfopen_interval_left_Subq_R)

∀a b : set, halfopen_interval_left a b ⊆ R

In Proofgold the corresponding term root is 1bc53a... and proposition id is f2f2a5...

Proof:
Proof not loaded.

Theorem. (halfopen_interval_left_leftmem)

∀a b : set, Rlt a b → a ∈ halfopen_interval_left a b

In Proofgold the corresponding term root is aed46c... and proposition id is 8fa7a0...

Proof:
Proof not loaded.

Theorem. (halfopen_interval_left_leftmonotone)

∀a x b : set, a ∈ R → x ∈ R → b ∈ R → Rle a x → halfopen_interval_left x b ⊆ halfopen_interval_left a b

In Proofgold the corresponding term root is b90bc5... and proposition id is 4af32d...

Proof:
Proof not loaded.

Definition. We define R_standard_basis to be ⋃_{a ∈ R}{open_interval a b|b ∈ R} of type set.

In Proofgold the corresponding term root is 6c64ea... and object id is e0fd50...

Definition. We define R_standard_topology to be generated_topology R R_standard_basis of type set.

In Proofgold the corresponding term root is f35291... and object id is 9f1b53...

Theorem. (R_standard_topology_def_eq)

R_standard_topology = generated_topology R R_standard_basis

In Proofgold the corresponding term root is def984... and proposition id is aa0c23...

Proof:
Proof not loaded.

Theorem. (R_standard_basis_is_basis_local)

basis_on R R_standard_basis

In Proofgold the corresponding term root is fd011d... and proposition id is d4e838...

Proof:
Proof not loaded.

Theorem. (open_interval_in_R_standard_topology)

∀a b : set, Rlt a b → open_interval a b ∈ R_standard_topology

In Proofgold the corresponding term root is b00a78... and proposition id is 6aab92...

Proof:
Proof not loaded.

Theorem. (R_standard_topology_is_topology_local)

topology_on R R_standard_topology

In Proofgold the corresponding term root is 5b5a68... and proposition id is bd3695...

Proof:
Proof not loaded.

Theorem. (R_standard_open_refine_interval)

∀U x : set, U ∈ R_standard_topology → x ∈ U → ∃a b : set, a ∈ R ∧ b ∈ R ∧ x ∈ open_interval a b ∧ open_interval a b ⊆ U ∧ Rlt a x ∧ Rlt x b

In Proofgold the corresponding term root is 658b7c... and proposition id is 47f251...

Proof:
Proof not loaded.

Definition. We define R_lower_limit_basis to be ⋃_{a ∈ R}{halfopen_interval_left a b|b ∈ R} of type set.

In Proofgold the corresponding term root is cc0dae... and object id is 99cdb3...

Definition. We define R_lower_limit_topology to be generated_topology R R_lower_limit_basis of type set.

In Proofgold the corresponding term root is ce1277... and object id is fb0317...

Theorem. (R_lower_limit_basis_is_basis_local)

basis_on R R_lower_limit_basis

In Proofgold the corresponding term root is d4c71e... and proposition id is 3cda89...

Proof:
Proof not loaded.

Theorem. (R_lower_limit_topology_is_topology)

topology_on R R_lower_limit_topology

In Proofgold the corresponding term root is f37d53... and proposition id is dd9206...

Proof:
Proof not loaded.

Theorem. (halfopen_interval_left_in_R_lower_limit_topology)

∀a b : set, a ∈ R → b ∈ R → halfopen_interval_left a b ∈ R_lower_limit_topology

In Proofgold the corresponding term root is 0c743f... and proposition id is d570fb...

Proof:
Proof not loaded.

Theorem. (open_interval_in_R_lower_limit_topology)

∀a b : set, a ∈ R → b ∈ R → open_interval a b ∈ R_lower_limit_topology

In Proofgold the corresponding term root is 228759... and proposition id is d659e5...

Proof:
Proof not loaded.

Theorem. (R_lower_limit_finer_than_R_standard)

finer_than R_lower_limit_topology R_standard_topology

In Proofgold the corresponding term root is d99160... and proposition id is dffb1e...

Proof:
Proof not loaded.

Definition. We define R_upper_limit_basis to be ⋃_{a ∈ R}{halfopen_interval_right a b|b ∈ R} of type set.

In Proofgold the corresponding term root is 819f82... and object id is f6ef78...

Theorem. (R_upper_limit_basis_is_basis_local)

basis_on R R_upper_limit_basis

In Proofgold the corresponding term root is 8f0a28... and proposition id is 9a85b1...

Proof:
Proof not loaded.

Definition. We define R_upper_limit_topology to be generated_topology R R_upper_limit_basis of type set.

In Proofgold the corresponding term root is d07fcd... and object id is 6262c6...

Theorem. (R_upper_limit_topology_is_topology_local)

topology_on R R_upper_limit_topology

In Proofgold the corresponding term root is d67efe... and proposition id is c91b2e...

Proof:
Proof not loaded.

Definition. We define inv_nat to be recip_SNo of type set → set.

In Proofgold the corresponding term root is aa96a2... and object id is 0d39b7...

Theorem. (inv_nat_real)

∀n : set, n ∈ ω → inv_nat n ∈ R

In Proofgold the corresponding term root is 26bb10... and proposition id is a015d5...

Proof:
Proof not loaded.

Definition. We define K_set to be {inv_nat n|n ∈ ω ∖ {0}} of type set.

In Proofgold the corresponding term root is bfedd3... and object id is 968aec...

Theorem. (zero_not_in_K_set)

0 ∉ K_set

In Proofgold the corresponding term root is c29e1a... and proposition id is 67dae5...

Proof:
Proof not loaded.

Theorem. (one_in_K_set)

1 ∈ K_set

In Proofgold the corresponding term root is 5aae9b... and proposition id is e931be...

Proof:
Proof not loaded.

Theorem. (not_in_K_set_implies_neq1)

∀x : set, ¬ (x ∈ K_set) → x ≠ 1

In Proofgold the corresponding term root is a31702... and proposition id is 59465c...

Proof:
Proof not loaded.

Theorem. (inv_nat_not_in_open_interval_of_empty_cap_K_set)

∀a b : set, open_interval a b ∩ K_set = Empty → ∀n : set, n ∈ ω ∖ {0} → ¬ (inv_nat n ∈ open_interval a b)

In Proofgold the corresponding term root is a2fac0... and proposition id is 91a52a...

Proof:
Proof not loaded.

Theorem. (K_set_Subq_R)

K_set ⊆ R

In Proofgold the corresponding term root is 90b916... and proposition id is 7fb059...

Proof:
Proof not loaded.

Theorem. (inv_nat_pos)

∀n : set, n ∈ ω ∖ {0} → Rlt 0 (inv_nat n)

In Proofgold the corresponding term root is 79a8cd... and proposition id is b903e2...

Proof:
Proof not loaded.

Definition. We define R_K_basis to be ⋃_{a ∈ R}{open_interval a b ∖ K_set|b ∈ R} of type set.

In Proofgold the corresponding term root is f56cac... and object id is 16a950...

Definition. We define R_K_topology to be generated_topology R (R_standard_basis ∪ R_K_basis) of type set.

In Proofgold the corresponding term root is 9c75c6... and object id is fa9c3d...

Theorem. (R_standard_plus_K_basis_is_basis_local)

basis_on R (R_standard_basis ∪ R_K_basis)

In Proofgold the corresponding term root is b70b01... and proposition id is 654916...

Proof:
Proof not loaded.

Theorem. (R_K_topology_is_topology_local)

topology_on R R_K_topology

In Proofgold the corresponding term root is 0206ad... and proposition id is 4575df...

Proof:
Proof not loaded.

Theorem. (K_set_meets_lower_limit_neighborhood_0)

∀a b : set, a ∈ R → b ∈ R → ¬ (Rlt 0 a) → Rlt 0 b → ∃y : set, y ∈ halfopen_interval_left a b ∧ y ∈ K_set

In Proofgold the corresponding term root is 55433e... and proposition id is 76461e...

Proof:
Proof not loaded.

Theorem. (ex13_6_Rl_RK_not_comparable)

¬ finer_than R_lower_limit_topology R_K_topology ∧ ¬ finer_than R_K_topology R_lower_limit_topology

In Proofgold the corresponding term root is d4264c... and proposition id is 15744d...

Proof:
Proof not loaded.

Definition. We define R_finite_complement_topology to be finite_complement_topology R of type set.

In Proofgold the corresponding term root is 9986fb... and object id is 678b6e...

Definition. We define R_ray_topology to be {U ∈ 𝒫 R|U = Empty ∨ U = R ∨ (∃a ∈ R, U = {x ∈ R|Rlt x a})} of type set.

In Proofgold the corresponding term root is 63bd30... and object id is 2e1444...

Theorem. (open_ray_in_R_standard_topology)

∀a : set, a ∈ R → {x ∈ R|Rlt a x} ∈ R_standard_topology

In Proofgold the corresponding term root is 47e548... and proposition id is fbb58b...

Proof:
Proof not loaded.

Theorem. (open_left_ray_in_R_standard_topology)

∀b : set, b ∈ R → {x ∈ R|Rlt x b} ∈ R_standard_topology

In Proofgold the corresponding term root is 0e6ab9... and proposition id is c05731...

Proof:
Proof not loaded.

Theorem. (binunion_in_R_standard_topology)

∀U V : set, U ∈ R_standard_topology → V ∈ R_standard_topology → U ∪ V ∈ R_standard_topology

In Proofgold the corresponding term root is e3d9c9... and proposition id is ab6b1c...

Proof:
Proof not loaded.

Theorem. (R_minus_singleton_eq_rays_union)

∀a : set, a ∈ R → R ∖ {a,a} = {x ∈ R|Rlt x a} ∪ {x ∈ R|Rlt a x}

In Proofgold the corresponding term root is 54028f... and proposition id is 21e7f3...

Proof:
Proof not loaded.

Theorem. (R_minus_singleton_in_R_standard_topology)

∀a : set, a ∈ R → R ∖ {a,a} ∈ R_standard_topology

In Proofgold the corresponding term root is 8b7f1f... and proposition id is 648db4...

Proof:
Proof not loaded.

Theorem. (setminus_binunion_eq_binintersect)

∀X A B : set, X ∖ (A ∪ B) = (X ∖ A) ∩ (X ∖ B)

In Proofgold the corresponding term root is db9010... and proposition id is 326899...

Proof:
Proof not loaded.

Theorem. (setminus_binintersect_eq_binunion)

∀X A B : set, X ∖ (A ∩ B) = (X ∖ A) ∪ (X ∖ B)

In Proofgold the corresponding term root is 2625c6... and proposition id is f76412...

Proof:
Proof not loaded.

Theorem. (setminus_setminus_eq)

∀X U : set, U ⊆ X → X ∖ (X ∖ U) = U

In Proofgold the corresponding term root is 4cead2... and proposition id is 2f2b8f...

Proof:
Proof not loaded.

Theorem. (binintersect_closed)

∀X T C D : set, closed_in X T C → closed_in X T D → closed_in X T (C ∩ D)

In Proofgold the corresponding term root is 1bac09... and proposition id is 7ac0ff...

Proof:
Proof not loaded.

Theorem. (binunion_closed)

∀X T C D : set, closed_in X T C → closed_in X T D → closed_in X T (C ∪ D)

In Proofgold the corresponding term root is de0012... and proposition id is 2cfa97...

Proof:
Proof not loaded.

Theorem. (Sing_eq_UPair)

∀x : set, {x} = {x,x}

In Proofgold the corresponding term root is 364566... and proposition id is 8e626e...

Proof:
Proof not loaded.

Theorem. (finite_complement_open_in_R_standard_topology)

∀F : set, finite F → F ⊆ R → R ∖ F ∈ R_standard_topology

In Proofgold the corresponding term root is b6be4e... and proposition id is 56f63b...

Proof:
Proof not loaded.

Theorem. (ex13_7_R_topology_containments)

finer_than R_upper_limit_topology R_standard_topology ∧ finer_than R_K_topology R_standard_topology ∧ finer_than R_standard_topology R_finite_complement_topology ∧ finer_than R_standard_topology R_ray_topology

In Proofgold the corresponding term root is dc2081... and proposition id is d81011...

Proof:
Proof not loaded.

Definition. We define rational_open_intervals_basis to be ⋃_{q1 ∈ rational_numbers}{open_interval q1 q2|q2 ∈ rational_numbers} of type set.

In Proofgold the corresponding term root is fee8d4... and object id is 0be9ad...

Theorem. (rational_open_intervals_basis_Subq_R_standard_basis)

rational_open_intervals_basis ⊆ R_standard_basis

In Proofgold the corresponding term root is 9522ce... and proposition id is 74c09d...

Proof:
Proof not loaded.

Theorem. (rational_dense_between_reals)

∀a b : set, a ∈ R → b ∈ R → Rlt a b → ∃q ∈ rational_numbers, Rlt a q ∧ Rlt q b

In Proofgold the corresponding term root is 19b8a3... and proposition id is 0956ce...

Proof:
Proof not loaded.

Theorem. (rational_interval_refines_real_interval)

∀a b x : set, a ∈ R → b ∈ R → x ∈ R → x ∈ open_interval a b → ∃q1 ∈ rational_numbers, ∃q2 ∈ rational_numbers, x ∈ open_interval q1 q2 ∧ open_interval q1 q2 ⊆ open_interval a b

In Proofgold the corresponding term root is 21df2c... and proposition id is c21af4...

Proof:
Proof not loaded.

Theorem. (ex13_8a_rational_intervals_basis_standard)

basis_on R rational_open_intervals_basis ∧ generated_topology R rational_open_intervals_basis = R_standard_topology

In Proofgold the corresponding term root is e8ee52... and proposition id is ee0f18...

Proof:
Proof not loaded.

Definition. We define rational_halfopen_intervals_basis to be ⋃_{q1 ∈ rational_numbers}{halfopen_interval_left q1 q2|q2 ∈ rational_numbers} of type set.

In Proofgold the corresponding term root is cb2ef2... and object id is fca3a7...

Theorem. (ex13_8b_halfopen_rational_basis_topology)

basis_on R rational_halfopen_intervals_basis ∧ generated_topology R rational_halfopen_intervals_basis ≠ R_lower_limit_topology

In Proofgold the corresponding term root is 5900a8... and proposition id is 3b94f4...

Proof:
Proof not loaded.

Definition. We define order_rel to be λX a b ⇒ (X = R ∧ Rlt a b) ∨ (X = rational_numbers ∧ Rlt a b) ∨ (X = ω ∧ a ∈ b) ∨ (X = ω ∖ {0} ∧ a ∈ b) ∨ (X = setprod 2 ω ∧ ∃i m j n : set, (i ∈ 2 ∧ m ∈ ω ∧ j ∈ 2 ∧ n ∈ ω ∧ a = (i,m) ∧ b = (j,n) ∧ (i ∈ j ∨ (i = j ∧ m ∈ n)))) ∨ (X = setprod R R ∧ ∃a1 a2 b1 b2 : set, a = (a1,a2) ∧ b = (b1,b2) ∧ (Rlt a1 b1 ∨ (a1 = b1 ∧ Rlt a2 b2))) ∨ (ordinal X ∧ X ≠ R ∧ X ≠ rational_numbers ∧ X ≠ setprod 2 ω ∧ X ≠ setprod R R ∧ a ∈ b) of type set → set → set → prop.

In Proofgold the corresponding term root is b314c3... and object id is a2a63f...

Definition. We define simply_ordered_set to be λX ⇒ X = R ∨ X = rational_numbers ∨ X = ω ∨ X = ω ∖ {0} ∨ X = setprod 2 ω ∨ X = setprod R R ∨ ordinal X of type set → prop.

In Proofgold the corresponding term root is 8d096a... and object id is 477c00...

Theorem. (simply_ordered_set_R)

simply_ordered_set R

In Proofgold the corresponding term root is f16eba... and proposition id is c0b4a5...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_2_omega_00_10)

order_rel (setprod 2 ω) (0,0) (1,0)

In Proofgold the corresponding term root is a8d518... and proposition id is 3b81ce...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_2_omega_00_1n)

∀n : set, n ∈ ω → order_rel (setprod 2 ω) (0,0) (1,n)

In Proofgold the corresponding term root is 303c57... and proposition id is 3dc6e6...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_2_omega_0k_1n)

∀k n : set, k ∈ ω → n ∈ ω → order_rel (setprod 2 ω) (0,k) (1,n)

In Proofgold the corresponding term root is d80fe9... and proposition id is 14d8e6...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_2_omega_0k_0succk)

∀k : set, k ∈ ω → order_rel (setprod 2 ω) (0,k) (0,ordsucc k)

In Proofgold the corresponding term root is a34805... and proposition id is 2c4b04...

Proof:
Proof not loaded.

Theorem. (eps_1_not_in_2)

eps_ 1 ∉ 2

In Proofgold the corresponding term root is 68aa3e... and proposition id is a0f021...

Proof:
Proof not loaded.

Theorem. (setprod_R_R_neq_setprod_2_omega)

setprod R R ≠ setprod 2 ω

In Proofgold the corresponding term root is bad39a... and proposition id is 0e36db...

Proof:
Proof not loaded.

Definition. We define tag_topology to be λy ⇒ SetAdjoin y {1} of type set → set.

In Proofgold the corresponding term root is 3003d0... and object id is ad2c1f...

Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag_topology.

Theorem. (Sing1_not_SNo)

¬ SNo {1}

In Proofgold the corresponding term root is c0cb9c... and proposition id is 690afc...

Proof:
Proof not loaded.

Theorem. (Inj1_0_eq_1)

Inj1 0 = 1

In Proofgold the corresponding term root is 69c7ce... and proposition id is 9aac1d...

Proof:
Proof not loaded.

Theorem. (tuple_0_1_eq_Sing1)

(0,1) = {1}

In Proofgold the corresponding term root is 22a932... and proposition id is 511be5...

Proof:
Proof not loaded.

Theorem. (setprod_R_R_neq_R)

setprod R R ≠ R

In Proofgold the corresponding term root is c44f14... and proposition id is 99ff2b...

Proof:
Proof not loaded.

Theorem. (setprod_R_R_neq_rational_numbers)

setprod R R ≠ rational_numbers

In Proofgold the corresponding term root is c54f1a... and proposition id is c1ed5b...

Proof:
Proof not loaded.

Theorem. (R_neq_omega)

R ≠ ω

In Proofgold the corresponding term root is dff405... and proposition id is 059b10...

Proof:
Proof not loaded.

Theorem. (R_neq_omega_nonzero)

R ≠ (ω ∖ {0})

In Proofgold the corresponding term root is 70cb85... and proposition id is 6e1cb1...

Proof:
Proof not loaded.

Theorem. (R_neq_rational_numbers)

R ≠ rational_numbers

In Proofgold the corresponding term root is cc0fb9... and proposition id is b57cfa...

Proof:
Proof not loaded.

Theorem. (R_neq_setprod_2_omega)

R ≠ setprod 2 ω

In Proofgold the corresponding term root is c99a7d... and proposition id is 2c16e8...

Proof:
Proof not loaded.

Theorem. (Rlt_implies_order_rel_R)

∀a b : set, Rlt a b → order_rel R a b

In Proofgold the corresponding term root is 440dfd... and proposition id is b9f689...

Proof:
Proof not loaded.

Theorem. (order_rel_R_implies_Rlt)

∀a b : set, order_rel R a b → Rlt a b

In Proofgold the corresponding term root is 49c7ed... and proposition id is da5e0d...

Proof:
Proof not loaded.

Theorem. (Rlt_implies_order_rel_Q)

∀a b : set, Rlt a b → order_rel rational_numbers a b

In Proofgold the corresponding term root is 63e708... and proposition id is 737b04...

Proof:
Proof not loaded.

Theorem. (mem_implies_order_rel_omega)

∀a b : set, a ∈ b → order_rel ω a b

In Proofgold the corresponding term root is 736d7c... and proposition id is 53cd62...

Proof:
Proof not loaded.

Theorem. (order_rel_omega_implies_mem)

∀a b : set, order_rel ω a b → a ∈ b

In Proofgold the corresponding term root is d94c59... and proposition id is 1bd8a2...

Proof:
Proof not loaded.

Theorem. (order_rel_omega_nonzero_implies_mem)

∀a b : set, order_rel (ω ∖ {0}) a b → a ∈ b

In Proofgold the corresponding term root is cb9af8... and proposition id is 8c7f36...

Proof:
Proof not loaded.

Theorem. (mem_implies_order_rel_omega_nonzero)

∀a b : set, a ∈ b → order_rel (ω ∖ {0}) a b

In Proofgold the corresponding term root is 83ab03... and proposition id is 623c91...

Proof:
Proof not loaded.

Theorem. (omega_nonzero_neq_omega)

(ω ∖ {0}) ≠ ω

In Proofgold the corresponding term root is 13b7b4... and proposition id is b79364...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_2_omega_intro)

∀i m j n : set, i ∈ 2 → m ∈ ω → j ∈ 2 → n ∈ ω → (i ∈ j ∨ (i = j ∧ m ∈ n)) → order_rel (setprod 2 ω) (i,m) (j,n)

In Proofgold the corresponding term root is 7d1e83... and proposition id is ce1f32...

Proof:
Proof not loaded.

Theorem. (order_rel_trichotomy_or_impred)

∀X a b : set, simply_ordered_set X → a ∈ X → b ∈ X → ∀p : prop, (order_rel X a b → p) → (a = b → p) → (order_rel X b a → p) → p

In Proofgold the corresponding term root is aa2d24... and proposition id is cb1e9f...

Proof:
Proof not loaded.

Theorem. (order_rel_trans)

∀X a b c : set, simply_ordered_set X → a ∈ X → b ∈ X → c ∈ X → order_rel X a b → order_rel X b c → order_rel X a c

In Proofgold the corresponding term root is c200f7... and proposition id is 042855...

Proof:
Proof not loaded.

Theorem. (order_rel_irref)

∀X x : set, ¬ (order_rel X x x)

In Proofgold the corresponding term root is bf81e9... and proposition id is 15c76a...

Proof:
Proof not loaded.

Definition. We define order_topology_basis to be λX ⇒ (({I ∈ 𝒫 X|∃a ∈ X, ∃b ∈ X, I = {x ∈ X|order_rel X a x ∧ order_rel X x b}} ∪ {I ∈ 𝒫 X|∃b ∈ X, I = {x ∈ X|order_rel X x b}} ∪ {I ∈ 𝒫 X|∃a ∈ X, I = {x ∈ X|order_rel X a x}}) ∪ {X}) of type set → set.

In Proofgold the corresponding term root is 412003... and object id is 330416...

Theorem. (order_topology_basis_cases)

∀X I : set, I ∈ order_topology_basis X → (∃a ∈ X, ∃b ∈ X, I = {x ∈ X|order_rel X a x ∧ order_rel X x b}) ∨ (∃b ∈ X, I = {x ∈ X|order_rel X x b}) ∨ (∃a ∈ X, I = {x ∈ X|order_rel X a x}) ∨ I = X

In Proofgold the corresponding term root is 1745c2... and proposition id is 2cf0da...

Proof:
Proof not loaded.

Theorem. (order_topology_interval_refine)

∀X a1 b1x a2 b2x b1 b2 x : set, simply_ordered_set X → a1 ∈ X → b1x ∈ X → a2 ∈ X → b2x ∈ X → b1 = {x0 ∈ X|order_rel X a1 x0 ∧ order_rel X x0 b1x} → b2 = {x0 ∈ X|order_rel X a2 x0 ∧ order_rel X x0 b2x} → x ∈ b1 → x ∈ b2 → ∃b3 ∈ (order_topology_basis X), x ∈ b3 ∧ b3 ⊆ b1 ∩ b2

In Proofgold the corresponding term root is 885e13... and proposition id is 501664...

Proof:
Proof not loaded.

Definition. We define is_least_element to be λX a0 ⇒ a0 ∈ X ∧ ∀x : set, x ∈ X → ¬ (order_rel X x a0) of type set → set → prop.

In Proofgold the corresponding term root is 7481d4... and object id is 0bacb4...

Definition. We define is_largest_element to be λX b0 ⇒ b0 ∈ X ∧ ∀x : set, x ∈ X → ¬ (order_rel X b0 x) of type set → set → prop.

In Proofgold the corresponding term root is 00bba6... and object id is 636802...

Definition. We define order_topology_basis_tex to be λX ⇒ ({I ∈ 𝒫 X|∃a ∈ X, ∃b ∈ X, order_rel X a b ∧ I = {x ∈ X|order_rel X a x ∧ order_rel X x b}} ∪ {I ∈ 𝒫 X|∃a0 b : set, is_least_element X a0 ∧ b ∈ X ∧ order_rel X a0 b ∧ I = {x ∈ X|(x = a0 ∨ order_rel X a0 x) ∧ order_rel X x b}} ∪ {I ∈ 𝒫 X|∃b0 a : set, is_largest_element X b0 ∧ a ∈ X ∧ order_rel X a b0 ∧ I = {x ∈ X|order_rel X a x ∧ (x = b0 ∨ order_rel X x b0)}}) of type set → set.

In Proofgold the corresponding term root is f17055... and object id is b2efee...

Definition. We define order_topology to be λX ⇒ generated_topology X (order_topology_basis X) of type set → set.

In Proofgold the corresponding term root is 041322... and object id is 449e8f...

Theorem. (order_topology_basis_is_basis)

∀X : set, simply_ordered_set X → basis_on X (order_topology_basis X)

In Proofgold the corresponding term root is 805550... and proposition id is c94d49...

Proof:
Proof not loaded.

Theorem. (order_topology_is_topology)

∀X : set, simply_ordered_set X → topology_on X (order_topology X)

In Proofgold the corresponding term root is e5ec8f... and proposition id is aa6d5d...

Proof:
Proof not loaded.

Definition. We define open_ray_upper to be λX a ⇒ {x ∈ X|order_rel X a x} of type set → set → set.

In Proofgold the corresponding term root is 5472b5... and object id is 32a232...

Definition. We define open_ray_lower to be λX a ⇒ {x ∈ X|order_rel X x a} of type set → set → set.

In Proofgold the corresponding term root is 5a6ac3... and object id is 47d996...

Definition. We define open_rays_subbasis to be λX ⇒ (({I ∈ 𝒫 X|∃a ∈ X, I = open_ray_upper X a} ∪ {I ∈ 𝒫 X|∃b ∈ X, I = open_ray_lower X b}) ∪ {X}) of type set → set.

In Proofgold the corresponding term root is 836991... and object id is 0ad7eb...

Theorem. (open_rays_subbasis_sub_Power)

∀X : set, open_rays_subbasis X ⊆ 𝒫 X

In Proofgold the corresponding term root is df849d... and proposition id is 1a7612...

Proof:
Proof not loaded.

Theorem. (open_rays_subbasis_is_subbasis)

∀X : set, subbasis_on X (open_rays_subbasis X)

In Proofgold the corresponding term root is 9094f4... and proposition id is a1cd8f...

Proof:
Proof not loaded.

Theorem. (open_ray_upper_in_order_topology_basis)

∀X a : set, a ∈ X → open_ray_upper X a ∈ order_topology_basis X

In Proofgold the corresponding term root is a3b130... and proposition id is a742f6...

Proof:
Proof not loaded.

Theorem. (open_ray_lower_in_order_topology_basis)

∀X b : set, b ∈ X → open_ray_lower X b ∈ order_topology_basis X

In Proofgold the corresponding term root is dc2929... and proposition id is f3e221...

Proof:
Proof not loaded.

Theorem. (order_topology_basis_covers_omega)

∀x : set, x ∈ ω → ∃b ∈ (order_topology_basis ω), x ∈ b

In Proofgold the corresponding term root is 0fa895... and proposition id is f04a53...

Proof:
Proof not loaded.

Theorem. (open_ray_lower_omega_1_eq_Sing0)

open_ray_lower ω 1 = {0}

In Proofgold the corresponding term root is 8e569e... and proposition id is c1e173...

Proof:
Proof not loaded.

Theorem. (singleton_in_order_topology_basis_omega)

∀x : set, x ∈ ω → {x} ∈ order_topology_basis ω

In Proofgold the corresponding term root is ff75a3... and proposition id is cd618d...

Proof:
Proof not loaded.

Theorem. (order_topology_basis_omega_is_basis)

basis_on ω (order_topology_basis ω)

In Proofgold the corresponding term root is 130bc9... and proposition id is 844bf4...

Proof:
Proof not loaded.

Theorem. (order_topology_omega_is_topology)

topology_on ω (order_topology ω)

In Proofgold the corresponding term root is 1c6543... and proposition id is 9135a7...

Proof:
Proof not loaded.

Theorem. (open_rays_subbasis_sub_order_topology)

∀X : set, simply_ordered_set X → open_rays_subbasis X ⊆ order_topology X

In Proofgold the corresponding term root is 6161a3... and proposition id is 22ad7c...

Proof:
Proof not loaded.

Theorem. (subbasis_elem_open_in_generated_from_subbasis)

∀X S s : set, subbasis_on X S → s ∈ S → s ∈ generated_topology_from_subbasis X S

In Proofgold the corresponding term root is 6af98e... and proposition id is 185cf4...

Proof:
Proof not loaded.

Theorem. (open_ray_upper_in_open_rays_subbasis)

∀X a : set, a ∈ X → open_ray_upper X a ∈ open_rays_subbasis X

In Proofgold the corresponding term root is aa64bb... and proposition id is c4501e...

Proof:
Proof not loaded.

Theorem. (open_ray_lower_in_open_rays_subbasis)

∀X b : set, b ∈ X → open_ray_lower X b ∈ open_rays_subbasis X

In Proofgold the corresponding term root is 260f6d... and proposition id is 888da2...

Proof:
Proof not loaded.

Theorem. (open_interval_eq_rays_intersection)

∀X a b : set, {x ∈ X|order_rel X a x ∧ order_rel X x b} = (open_ray_upper X a) ∩ (open_ray_lower X b)

In Proofgold the corresponding term root is edbe1d... and proposition id is c821e2...

Proof:
Proof not loaded.

Theorem. (order_topology_basis_sub_generated_from_open_rays)

∀X : set, order_topology_basis X ⊆ generated_topology_from_subbasis X (open_rays_subbasis X)

In Proofgold the corresponding term root is ece8cc... and proposition id is b892fc...

Proof:
Proof not loaded.

Theorem. (open_rays_subbasis_for_order_topology)

∀X : set, simply_ordered_set X → generated_topology_from_subbasis X (open_rays_subbasis X) = order_topology X

In Proofgold the corresponding term root is a302b1... and proposition id is 2b8900...

Proof:
Proof not loaded.

Theorem. (R_standard_basis_is_basis)

basis_on R R_standard_basis

In Proofgold the corresponding term root is fd011d... and proposition id is d4e838...

Proof:
Proof not loaded.

Theorem. (R_standard_topology_is_topology)

topology_on R R_standard_topology

In Proofgold the corresponding term root is 5b5a68... and proposition id is bd3695...

Proof:
Proof not loaded.

Theorem. (open_interval_in_order_topology_basis_R)

∀a b : set, a ∈ R → b ∈ R → open_interval a b ∈ order_topology_basis R

In Proofgold the corresponding term root is 4f2b7d... and proposition id is b8aab9...

Proof:
Proof not loaded.

Theorem. (order_topology_basis_R_refines_standard_basis)

∀b ∈ order_topology_basis R, ∀x : set, x ∈ b → ∃I ∈ R_standard_basis, x ∈ I ∧ I ⊆ b

In Proofgold the corresponding term root is 998c8a... and proposition id is 88cc8e...

Proof:
Proof not loaded.

Theorem. (standard_topology_is_order_topology)

order_topology R = R_standard_topology

In Proofgold the corresponding term root is c30d33... and proposition id is 5ff3fc...

Proof:
Proof not loaded.

Theorem. (order_topology_R_is_topology)

topology_on R (order_topology R)

In Proofgold the corresponding term root is b1326f... and proposition id is 32f153...

Proof:
Proof not loaded.

Theorem. (order_topology_basis_R_is_basis)

basis_on R (order_topology_basis R)

In Proofgold the corresponding term root is 72f1aa... and proposition id is 7ba10e...

Proof:
Proof not loaded.

Theorem. (open_rays_subbasis_sub_order_topology_R)

open_rays_subbasis R ⊆ order_topology R

In Proofgold the corresponding term root is 76cfce... and proposition id is 94e78b...

Proof:
Proof not loaded.

Theorem. (open_rays_subbasis_for_order_topology_R)

generated_topology_from_subbasis R (open_rays_subbasis R) = order_topology R

In Proofgold the corresponding term root is 36704c... and proposition id is 83f19f...

Proof:
Proof not loaded.

Definition. We define R2_dictionary_order_topology to be generated_topology_from_subbasis (setprod R R) (open_rays_subbasis (setprod R R)) of type set.

In Proofgold the corresponding term root is 731158... and object id is 4f98b3...

Theorem. (dictionary_order_topology_is_topology)

topology_on (setprod R R) R2_dictionary_order_topology

In Proofgold the corresponding term root is ef0658... and proposition id is c77949...

Proof:
Proof not loaded.

Theorem. (rectangles_basis_for_R2)

∃B : set, basis_on (setprod R R) B ∧ generated_topology (setprod R R) B = R2_dictionary_order_topology

In Proofgold the corresponding term root is eebbd6... and proposition id is 4489a3...

Proof:
Proof not loaded.

Definition. We define Zplus to be ω ∖ {0} of type set.

In Proofgold the corresponding term root is c4823e... and object id is c277a3...

Theorem. (zero_not_in_Zplus)

0 ∉ Zplus

In Proofgold the corresponding term root is bf8234... and proposition id is 30d6c8...

Proof:
Proof not loaded.

Theorem. (one_in_Zplus)

1 ∈ Zplus

In Proofgold the corresponding term root is cdfa49... and proposition id is bd1cc3...

Proof:
Proof not loaded.

Theorem. (two_in_Zplus)

2 ∈ Zplus

In Proofgold the corresponding term root is b5bea5... and proposition id is e0e8c2...

Proof:
Proof not loaded.

Theorem. (zero_in_rational_numbers)

0 ∈ rational_numbers

In Proofgold the corresponding term root is 340262... and proposition id is 7422d5...

Proof:
Proof not loaded.

Theorem. (one_in_rational_numbers)

1 ∈ rational_numbers

In Proofgold the corresponding term root is 4a5681... and proposition id is dea2e4...

Proof:
Proof not loaded.

Theorem. (minus_one_in_rational_numbers)

minus_SNo 1 ∈ rational_numbers

In Proofgold the corresponding term root is c41631... and proposition id is d862e8...

Proof:
Proof not loaded.

Theorem. (minus_one_not_in_omega)

minus_SNo 1 ∉ ω

In Proofgold the corresponding term root is c19a3b... and proposition id is 31d5c0...

Proof:
Proof not loaded.

Theorem. (rational_numbers_neq_omega)

rational_numbers ≠ ω

In Proofgold the corresponding term root is 4482ba... and proposition id is c8556a...

Proof:
Proof not loaded.

Theorem. (rational_numbers_neq_omega_nonzero)

rational_numbers ≠ (ω ∖ {0})

In Proofgold the corresponding term root is aafd44... and proposition id is 103b63...

Proof:
Proof not loaded.

Theorem. (Zplus_neq_omega)

Zplus ≠ ω

In Proofgold the corresponding term root is 13b7b4... and proposition id is b79364...

Proof:
Proof not loaded.

Theorem. (Zplus_neq_R)

Zplus ≠ R

In Proofgold the corresponding term root is 1bb3ff... and proposition id is 6a953e...

Proof:
Proof not loaded.

Theorem. (Zplus_neq_rational_numbers)

Zplus ≠ rational_numbers

In Proofgold the corresponding term root is 52fdb1... and proposition id is 71cc37...

Proof:
Proof not loaded.

Theorem. (not_TransSet_singleton_1)

¬ TransSet {1}

In Proofgold the corresponding term root is 8f2d67... and proposition id is be9a70...

Proof:
Proof not loaded.

Theorem. (Sing1_not_in_omega)

{1} ∉ ω

In Proofgold the corresponding term root is 42627d... and proposition id is 0905d5...

Proof:
Proof not loaded.

Theorem. (setprod_R_R_neq_omega)

setprod R R ≠ ω

In Proofgold the corresponding term root is aa42e1... and proposition id is 6bc216...

Proof:
Proof not loaded.

Theorem. (setprod_R_R_neq_omega_nonzero)

setprod R R ≠ (ω ∖ {0})

In Proofgold the corresponding term root is 16fb20... and proposition id is c22fec...

Proof:
Proof not loaded.

Theorem. (setprod_2_omega_neq_omega)

setprod 2 ω ≠ ω

In Proofgold the corresponding term root is f53822... and proposition id is 44d10b...

Proof:
Proof not loaded.

Theorem. (setprod_2_omega_neq_omega_nonzero)

setprod 2 ω ≠ (ω ∖ {0})

In Proofgold the corresponding term root is 6cfc28... and proposition id is 86b6ee...

Proof:
Proof not loaded.

Theorem. (setprod_2_omega_neq_setprod_R_R)

setprod 2 ω ≠ setprod R R

In Proofgold the corresponding term root is 27818d... and proposition id is cc09a9...

Proof:
Proof not loaded.

Theorem. (setprod_2_omega_neq_R)

setprod 2 ω ≠ R

In Proofgold the corresponding term root is ac8095... and proposition id is 3104fa...

Proof:
Proof not loaded.

Theorem. (setprod_2_omega_neq_rational_numbers)

setprod 2 ω ≠ rational_numbers

In Proofgold the corresponding term root is 361728... and proposition id is 0c67b7...

Proof:
Proof not loaded.

Theorem. (rational_numbers_neq_setprod_2_omega)

rational_numbers ≠ setprod 2 ω

In Proofgold the corresponding term root is a99dfa... and proposition id is 90879f...

Proof:
Proof not loaded.

Theorem. (rational_numbers_neq_setprod_R_R)

rational_numbers ≠ setprod R R

In Proofgold the corresponding term root is 87c24b... and proposition id is 2b8f8c...

Proof:
Proof not loaded.

Theorem. (rational_numbers_neq_R)

rational_numbers ≠ R

In Proofgold the corresponding term root is 873c44... and proposition id is 005852...

Proof:
Proof not loaded.

Theorem. (order_rel_Q_implies_Rlt)

∀a b : set, order_rel rational_numbers a b → Rlt a b

In Proofgold the corresponding term root is 7f5c17... and proposition id is dc6ad2...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_2_omega_unfold)

∀a b : set, order_rel (setprod 2 ω) a b → ∃i m j n : set, (i ∈ 2 ∧ m ∈ ω ∧ j ∈ 2 ∧ n ∈ ω ∧ a = (i,m) ∧ b = (j,n) ∧ (i ∈ j ∨ (i = j ∧ m ∈ n)))

In Proofgold the corresponding term root is 629858... and proposition id is 364aae...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_R_R_unfold)

∀a b : set, order_rel (setprod R R) a b → ∃a1 a2 b1 b2 : set, a = (a1,a2) ∧ b = (b1,b2) ∧ (Rlt a1 b1 ∨ (a1 = b1 ∧ Rlt a2 b2))

In Proofgold the corresponding term root is 3fb6e4... and proposition id is dcc911...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_R_R_intro)

∀a1 a2 b1 b2 : set, (Rlt a1 b1 ∨ (a1 = b1 ∧ Rlt a2 b2)) → order_rel (setprod R R) (a1,a2) (b1,b2)

In Proofgold the corresponding term root is 989413... and proposition id is ad6f17...

Proof:
Proof not loaded.

Theorem. (Zplus_neq_setprod_2_omega)

Zplus ≠ setprod 2 ω

In Proofgold the corresponding term root is fda20b... and proposition id is 05c673...

Proof:
Proof not loaded.

Theorem. (Zplus_neq_setprod_R_R)

Zplus ≠ setprod R R

In Proofgold the corresponding term root is 037894... and proposition id is 1d241d...

Proof:
Proof not loaded.

Theorem. (order_rel_Zplus_iff_mem)

∀a b : set, a ∈ Zplus → b ∈ Zplus → (order_rel Zplus a b ↔ a ∈ b)

In Proofgold the corresponding term root is bd9bc9... and proposition id is fb5971...

Proof:
Proof not loaded.

Theorem. (Zplus_mem_omega)

∀n : set, n ∈ Zplus → n ∈ ω

In Proofgold the corresponding term root is baeb05... and proposition id is b35176...

Proof:
Proof not loaded.

Theorem. (Zplus_mem_nonzero)

∀n : set, n ∈ Zplus → n ≠ 0

In Proofgold the corresponding term root is 100be8... and proposition id is 487ea8...

Proof:
Proof not loaded.

Theorem. (Zplus_Subq_omega)

Zplus ⊆ ω

In Proofgold the corresponding term root is 7f6b42... and proposition id is 3e29d2...

Proof:
Proof not loaded.

Theorem. (Zplus_ordsucc_closed)

∀n : set, n ∈ Zplus → ordsucc n ∈ Zplus

In Proofgold the corresponding term root is 1ef4b7... and proposition id is 8011b8...

Proof:
Proof not loaded.

Theorem. (nat_nonzero_in_Zplus)

∀n : set, nat_p n → n ≠ 0 → n ∈ Zplus

In Proofgold the corresponding term root is 09d720... and proposition id is bd0390...

Proof:
Proof not loaded.

Theorem. (singleton_ordsucc0_in_order_topology_basis_Zplus)

{ordsucc 0} ∈ order_topology_basis Zplus

In Proofgold the corresponding term root is aeb064... and proposition id is 0619aa...

Proof:
Proof not loaded.

Theorem. (singleton_ordsucc_in_order_topology_basis_Zplus)

∀m : set, m ∈ Zplus → {ordsucc m} ∈ order_topology_basis Zplus

In Proofgold the corresponding term root is 697c05... and proposition id is 321654...

Proof:
Proof not loaded.

Theorem. (singleton_in_order_topology_basis_Zplus)

∀n : set, n ∈ Zplus → {n} ∈ order_topology_basis Zplus

In Proofgold the corresponding term root is a28c0e... and proposition id is 960f5c...

Proof:
Proof not loaded.

Theorem. (Zplus_order_topology_is_discrete)

generated_topology Zplus (order_topology_basis Zplus) = 𝒫 Zplus

In Proofgold the corresponding term root is 5a676e... and proposition id is 72e0a8...

Proof:
Proof not loaded.

Theorem. (order_topology_on_Zplus_discrete)

order_topology Zplus = discrete_topology Zplus

In Proofgold the corresponding term root is 8a9eb0... and proposition id is f589fc...

Proof:
Proof not loaded.

Theorem. (omega_order_topology_is_discrete)

generated_topology ω (order_topology_basis ω) = 𝒫 ω

In Proofgold the corresponding term root is b19793... and proposition id is 891401...

Proof:
Proof not loaded.

Theorem. (order_topology_on_omega_discrete)

order_topology ω = discrete_topology ω

In Proofgold the corresponding term root is 5e4520... and proposition id is ca97fb...

Proof:
Proof not loaded.

Theorem. (and_assoc)

∀A B C : prop, (A ∧ B) ∧ C → A ∧ (B ∧ C)

In Proofgold the corresponding term root is 1ad119... and proposition id is 017793...

Proof:
Proof not loaded.

Theorem. (and_assoc_rev)

∀A B C : prop, A ∧ (B ∧ C) → (A ∧ B) ∧ C

In Proofgold the corresponding term root is 5c7924... and proposition id is 8afdbb...

Proof:
Proof not loaded.

Theorem. (and7E)

∀A B C D E F G : prop, A ∧ B ∧ C ∧ D ∧ E ∧ F ∧ G → ∀p : prop, (A → B → C → D → E → F → G → p) → p

In Proofgold the corresponding term root is 2ea6ee... and proposition id is 02714a...

Proof:
Proof not loaded.

Theorem. (and4E)

∀A B C D : prop, A ∧ B ∧ C ∧ D → ∀p : prop, (A → B → C → D → p) → p

In Proofgold the corresponding term root is 7f6e6c... and proposition id is b41cb7...

Proof:
Proof not loaded.

Theorem. (and5E)

∀A B C D E : prop, A ∧ B ∧ C ∧ D ∧ E → ∀p : prop, (A → B → C → D → E → p) → p

In Proofgold the corresponding term root is b9e36e... and proposition id is a02e0e...

Proof:
Proof not loaded.

Theorem. (and6E)

∀A B C D E F : prop, A ∧ B ∧ C ∧ D ∧ E ∧ F → ∀p : prop, (A → B → C → D → E → F → p) → p

In Proofgold the corresponding term root is 3668da... and proposition id is 1c4fc6...

Proof:
Proof not loaded.

Theorem. (mem_eqR)

∀x A B : set, A = B → x ∈ A → x ∈ B

In Proofgold the corresponding term root is 2b6b4c... and proposition id is b91a3c...

Proof:
Proof not loaded.

Theorem. (mem_eqL)

∀x A B : set, A = B → x ∈ B → x ∈ A

In Proofgold the corresponding term root is e6ff42... and proposition id is 612e14...

Proof:
Proof not loaded.

Theorem. (conj7_last_disjE)

∀i m j n a b : set, i ∈ 2 ∧ m ∈ ω ∧ j ∈ 2 ∧ n ∈ ω ∧ a = (i,m) ∧ b = (j,n) ∧ (i ∈ j ∨ (i = j ∧ m ∈ n)) → (i ∈ j ∨ (i = j ∧ m ∈ n))

In Proofgold the corresponding term root is f36328... and proposition id is 04d430...

Proof:
Proof not loaded.

Definition. We define two_by_nat to be setprod 2 ω of type set.

In Proofgold the corresponding term root is e4fb85... and object id is 0fd1b8...

Definition. We define two_by_nat_order_topology to be order_topology two_by_nat of type set.

In Proofgold the corresponding term root is 551d3e... and object id is b8f011...

Theorem. (two_by_nat_singleton_not_in_basis)

{(1,0)} ∉ order_topology_basis two_by_nat

In Proofgold the corresponding term root is 07530b... and proposition id is 9690ab...

Proof:
Proof not loaded.

Theorem. (two_by_nat_singleton_not_open)

¬ ({(1,0)} ∈ two_by_nat_order_topology)

In Proofgold the corresponding term root is 776c6f... and proposition id is cc90be...

Proof:
Proof not loaded.

Theorem. (two_by_nat_not_discrete)

¬ (two_by_nat_order_topology = discrete_topology two_by_nat)

In Proofgold the corresponding term root is 290be1... and proposition id is b42b2b...

Proof:
Proof not loaded.

Theorem. (simply_ordered_set_setprod_2_omega)

simply_ordered_set (setprod 2 ω)

In Proofgold the corresponding term root is f9d891... and proposition id is 8e43c4...

Proof:
Proof not loaded.

Theorem. (pair_eq_fst)

∀a b c d : set, (a,b) = (c,d) → a = c

In Proofgold the corresponding term root is a3f8c1... and proposition id is e20aa2...

Proof:
Proof not loaded.

Theorem. (pair_eq_snd)

∀a b c d : set, (a,b) = (c,d) → b = d

In Proofgold the corresponding term root is e14ece... and proposition id is fb67cf...

Proof:
Proof not loaded.

Theorem. (no_point_between_setprod_2_omega_0k_0succk)

∀k y : set, k ∈ ω → y ∈ setprod 2 ω → order_rel (setprod 2 ω) (0,k) y → order_rel (setprod 2 ω) y (0,ordsucc k) → False

In Proofgold the corresponding term root is 804f46... and proposition id is 59e370...

Proof:
Proof not loaded.

Theorem. (no_point_between_setprod_2_omega_1n_1succn)

∀n y : set, n ∈ ω → y ∈ setprod 2 ω → order_rel (setprod 2 ω) (1,n) y → order_rel (setprod 2 ω) y (1,ordsucc n) → False

In Proofgold the corresponding term root is 5b9312... and proposition id is 1d1e50...

Proof:
Proof not loaded.

Theorem. (open_ray_lower_setprod_2_omega_01_eq_Sing00)

open_ray_lower (setprod 2 ω) (0,1) = {(0,0)}

In Proofgold the corresponding term root is 817346... and proposition id is f8d24f...

Proof:
Proof not loaded.

Theorem. (singleton_setprod_2_omega_00_open)

{(0,0)} ∈ order_topology (setprod 2 ω)

In Proofgold the corresponding term root is e08482... and proposition id is 0f0cd9...

Proof:
Proof not loaded.

Theorem. (singleton_setprod_2_omega_0k_open)

∀k : set, k ∈ ω → {(0,k)} ∈ order_topology (setprod 2 ω)

In Proofgold the corresponding term root is 48fb9e... and proposition id is dac25a...

Proof:
Proof not loaded.

Theorem. (singleton_setprod_2_omega_1n_open_if_nonzero)

∀n : set, n ∈ ω → n ≠ 0 → {(1,n)} ∈ order_topology (setprod 2 ω)

In Proofgold the corresponding term root is dcbd6c... and proposition id is 3ed5ba...

Proof:
Proof not loaded.

Theorem. (setminus_open_ray_upper_setprod_2_omega_0k)

∀k : set, k ∈ ω → (setprod 2 ω) ∖ open_ray_upper (setprod 2 ω) (0,k) = open_ray_lower (setprod 2 ω) (0,ordsucc k)

In Proofgold the corresponding term root is af0613... and proposition id is e61c12...

Proof:
Proof not loaded.

Theorem. (setminus_open_ray_lower_setprod_2_omega_1succ)

∀n : set, n ∈ ω → (setprod 2 ω) ∖ open_ray_lower (setprod 2 ω) (1,ordsucc n) = open_ray_upper (setprod 2 ω) (1,n)

In Proofgold the corresponding term root is b2f45c... and proposition id is bd13a6...

Proof:
Proof not loaded.

Definition. We define rectangle_set to be λU V ⇒ setprod U V of type set → set → set.

In Proofgold the corresponding term root is fc0b60... and object id is 107b6b...

Theorem. (rectangle_set_def)

∀U V : set, rectangle_set U V = setprod U V

In Proofgold the corresponding term root is 39613e... and proposition id is 4d1d40...

Proof:
Proof not loaded.

Theorem. (tuple_2_rectangle_set)

∀U V : set, ∀x y : set, x ∈ U → y ∈ V → (x,y) ∈ rectangle_set U V

In Proofgold the corresponding term root is 7f845b... and proposition id is 8ae2b0...

Proof:
Proof not loaded.

Theorem. (setprod_eta)

∀X Y p : set, p ∈ setprod X Y → p = (p 0,p 1)

In Proofgold the corresponding term root is af978c... and proposition id is f6b455...

Proof:
Proof not loaded.

Theorem. (setprod_Subq)

∀U V X Y : set, U ⊆ X → V ⊆ Y → setprod U V ⊆ setprod X Y

In Proofgold the corresponding term root is 7c33ee... and proposition id is 513b9c...

Proof:
Proof not loaded.

Theorem. (setprod_elem_decompose)

∀X Y p : set, p ∈ setprod X Y → ∃x ∈ X, ∃y ∈ Y, p ∈ setprod {x} {y}

In Proofgold the corresponding term root is 80c0ef... and proposition id is 9033b8...

Proof:
Proof not loaded.

Theorem. (singleton_subset)

∀x U : set, x ∈ U → {x} ⊆ U

In Proofgold the corresponding term root is 821ced... and proposition id is d0ea13...

Proof:
Proof not loaded.

Theorem. (singleton_elem)

∀x y : set, x ∈ {y} → x = y

In Proofgold the corresponding term root is cd02ab... and proposition id is 4d60e6...

Proof:
Proof not loaded.

Theorem. (setprod_coords_in)

∀x y U V p : set, p ∈ setprod {x} {y} → p ∈ setprod U V → x ∈ U ∧ y ∈ V

In Proofgold the corresponding term root is f0e18d... and proposition id is b86901...

Proof:
Proof not loaded.

Theorem. (setprod_intersection)

∀U1 V1 U2 V2 : set, setprod U1 V1 ∩ setprod U2 V2 = setprod (U1 ∩ U2) (V1 ∩ V2)

In Proofgold the corresponding term root is 8610a3... and proposition id is c8f3bb...

Proof:
Proof not loaded.

Definition. We define product_subbasis to be λX Tx Y Ty ⇒ ⋃_{U ∈ Tx}{rectangle_set U V|V ∈ Ty} of type set → set → set → set → set.

In Proofgold the corresponding term root is 560b6b... and object id is cbbdda...

Definition. We define product_topology to be λX Tx Y Ty ⇒ generated_topology (setprod X Y) (product_subbasis X Tx Y Ty) of type set → set → set → set → set.

In Proofgold the corresponding term root is 4c7bad... and object id is d7760a...

Theorem. (product_subbasis_is_basis)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → basis_on (setprod X Y) (product_subbasis X Tx Y Ty)

In Proofgold the corresponding term root is 58aaac... and proposition id is c41e81...

Proof:
Proof not loaded.

Theorem. (product_topology_is_topology)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → topology_on (setprod X Y) (product_topology X Tx Y Ty)

In Proofgold the corresponding term root is 58a933... and proposition id is 3b8a65...

Proof:
Proof not loaded.

Definition. We define product_basis_from to be λBx By ⇒ ⋃_{U ∈ Bx}{setprod U V|V ∈ By} of type set → set → set.

In Proofgold the corresponding term root is 1b8da4... and object id is d60694...

Theorem. (product_basis_generates_product_topology)

∀X Y Bx By Tx Ty : set, basis_on X Bx → generated_topology X Bx = Tx → basis_on Y By → generated_topology Y By = Ty → generated_topology (setprod X Y) (product_basis_from Bx By) = product_topology X Tx Y Ty

In Proofgold the corresponding term root is 6ca0f7... and proposition id is 2cd7af...

Proof:
Proof not loaded.

Theorem. (product_basis_from_is_basis_on)

∀X Y Bx By Tx Ty : set, basis_on X Bx → generated_topology X Bx = Tx → basis_on Y By → generated_topology Y By = Ty → basis_on (setprod X Y) (product_basis_from Bx By)

In Proofgold the corresponding term root is 6fa6a3... and proposition id is ab5f6e...

Proof:
Proof not loaded.

Theorem. (product_basis_generates)

∀X Tx Y Ty Bx By : set, basis_on X Bx ∧ generated_topology X Bx = Tx → basis_on Y By ∧ generated_topology Y By = Ty → ∃B : set, basis_on (setprod X Y) B ∧ (∀U ∈ Bx, ∀V ∈ By, setprod U V ∈ B) ∧ generated_topology (setprod X Y) B = product_topology X Tx Y Ty

In Proofgold the corresponding term root is c3768b... and proposition id is 25f1da...

Proof:
Proof not loaded.

Definition. We define projection1 to be λX Y ⇒ {(p,p 0)|p ∈ setprod X Y} of type set → set → set.

In Proofgold the corresponding term root is 2a96db... and object id is 1d9cd6...

Definition. We define projection2 to be λX Y ⇒ {(p,p 1)|p ∈ setprod X Y} of type set → set → set.

In Proofgold the corresponding term root is f24403... and object id is 91b1f3...

Theorem. (product_subbasis_from_projections)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → ∃S : set, S = product_subbasis X Tx Y Ty ∧ generated_topology (setprod X Y) S = product_topology X Tx Y Ty

In Proofgold the corresponding term root is 7c4cc8... and proposition id is 5b2b9b...

Proof:
Proof not loaded.

Definition. We define apply_fun to be λf x ⇒ Eps_i (λy ⇒ (x,y) ∈ f) of type set → set → set.

In Proofgold the corresponding term root is 7f6e4a... and object id is 10d730...

Definition. We define function_on to be λf X Y ⇒ ∀x : set, x ∈ X → apply_fun f x ∈ Y of type set → set → set → prop.

In Proofgold the corresponding term root is f12768... and object id is b628b6...

Definition. We define function_space to be λX Y ⇒ {f ∈ 𝒫 (setprod X Y)|function_on f X Y} of type set → set → set.

In Proofgold the corresponding term root is 9c8fa0... and object id is db7aa5...

Theorem. (function_on_restrict_dom)

∀f X X' Y : set, function_on f X Y → X' ⊆ X → function_on f X' Y

In Proofgold the corresponding term root is d430b6... and proposition id is 218cb4...

Proof:
Proof not loaded.

Theorem. (function_on_of_function_space)

∀f X Y : set, f ∈ function_space X Y → function_on f X Y

In Proofgold the corresponding term root is f58059... and proposition id is 677aed...

Proof:
Proof not loaded.

Definition. We define functional_graph to be λf ⇒ ∀x y1 y2 : set, (x,y1) ∈ f → (x,y2) ∈ f → y1 = y2 of type set → prop.

In Proofgold the corresponding term root is d048d3... and object id is 718d03...

Definition. We define graph_domain_subset to be λf X ⇒ ∀x y : set, (x,y) ∈ f → x ∈ X of type set → set → prop.

In Proofgold the corresponding term root is be7d6d... and object id is b6b8b9...

Definition. We define graph_range_subset to be λf Y ⇒ ∀x y : set, (x,y) ∈ f → y ∈ Y of type set → set → prop.

In Proofgold the corresponding term root is 881387... and object id is de40aa...

Theorem. (graph_domain_subset_of_sub_setprod)

∀f X Y : set, f ⊆ setprod X Y → graph_domain_subset f X

In Proofgold the corresponding term root is cad45a... and proposition id is 8ef468...

Proof:
Proof not loaded.

Theorem. (graph_range_subset_of_sub_setprod)

∀f X Y : set, f ⊆ setprod X Y → graph_range_subset f Y

In Proofgold the corresponding term root is 0832f7... and proposition id is e47a51...

Proof:
Proof not loaded.

Theorem. (graph_domain_subset_graph)

∀A : set, ∀g : set → set, graph_domain_subset {(a,g a)|a ∈ A} A

In Proofgold the corresponding term root is f5017f... and proposition id is 22f6b7...

Proof:
Proof not loaded.

Theorem. (graph_domain_subset_const_fun)

∀A c : set, graph_domain_subset {(a,c)|a ∈ A} A

In Proofgold the corresponding term root is 5564cd... and proposition id is a3f99e...

Proof:
Proof not loaded.

Theorem. (graph_range_subset_graph)

∀A Y : set, ∀g : set → set, (∀a : set, a ∈ A → g a ∈ Y) → graph_range_subset {(a,g a)|a ∈ A} Y

In Proofgold the corresponding term root is 998270... and proposition id is a3f9ca...

Proof:
Proof not loaded.

Theorem. (graph_range_subset_const_fun)

∀A Y c : set, c ∈ Y → graph_range_subset {(a,c)|a ∈ A} Y

In Proofgold the corresponding term root is f5d4c5... and proposition id is 9f815a...

Proof:
Proof not loaded.

Theorem. (functional_graph_graph)

∀A : set, ∀g : set → set, functional_graph {(a,g a)|a ∈ A}

In Proofgold the corresponding term root is 81b7e2... and proposition id is 7b1c79...

Proof:
Proof not loaded.

Theorem. (functional_graph_const_fun)

∀A c : set, functional_graph {(a,c)|a ∈ A}

In Proofgold the corresponding term root is 9f9cf9... and proposition id is ec8f2f...

Proof:
Proof not loaded.

Theorem. (apply_fun_in_graph_of_ex)

∀f x : set, (∃y : set, (x,y) ∈ f) → (x,apply_fun f x) ∈ f

In Proofgold the corresponding term root is bf5870... and proposition id is 0850ca...

Proof:
Proof not loaded.

Theorem. (functional_graph_apply_fun_eq)

∀f x y : set, functional_graph f → (x,y) ∈ f → apply_fun f x = y

In Proofgold the corresponding term root is 082f59... and proposition id is 7e07f1...

Proof:
Proof not loaded.

Theorem. (graph_domain_subset_binunion)

∀A B f g : set, graph_domain_subset f A → graph_domain_subset g B → graph_domain_subset (f ∪ g) (A ∪ B)

In Proofgold the corresponding term root is 68f295... and proposition id is 85c94e...

Proof:
Proof not loaded.

Theorem. (graph_range_subset_binunion)

∀f g Y : set, graph_range_subset f Y → graph_range_subset g Y → graph_range_subset (f ∪ g) Y

In Proofgold the corresponding term root is 2eedde... and proposition id is f4cc74...

Proof:
Proof not loaded.

Theorem. (function_on_from_totality_and_range)

∀X Y f : set, (∀x : set, x ∈ X → ∃y : set, y ∈ Y ∧ (x,y) ∈ f) → graph_range_subset f Y → function_on f X Y

In Proofgold the corresponding term root is 5409d1... and proposition id is eba96c...

Proof:
Proof not loaded.

Definition. We define graph to be λA g ⇒ {(a,g a)|a ∈ A} of type set → (set → set) → set.

In Proofgold the corresponding term root is 8d2453... and object id is f069c2...

Theorem. (apply_fun_graph)

∀A : set, ∀g : set → set, ∀a : set, a ∈ A → apply_fun (graph A g) a = g a

In Proofgold the corresponding term root is 121b08... and proposition id is 23987d...

Proof:
Proof not loaded.

Definition. We define total_function_on to be λf X Y ⇒ function_on f X Y ∧ ∀x : set, x ∈ X → ∃y : set, y ∈ Y ∧ (x,y) ∈ f of type set → set → set → prop.

In Proofgold the corresponding term root is 57596b... and object id is 237936...

Theorem. (total_function_on_function_on)

∀f X Y : set, total_function_on f X Y → function_on f X Y

In Proofgold the corresponding term root is c8550d... and proposition id is bbd5e0...

Proof:
Proof not loaded.

Theorem. (total_function_on_totality)

∀f X Y : set, total_function_on f X Y → ∀x : set, x ∈ X → ∃y : set, y ∈ Y ∧ (x,y) ∈ f

In Proofgold the corresponding term root is 791304... and proposition id is 603038...

Proof:
Proof not loaded.

Theorem. (total_functional_graph_eq_graph_of_apply_fun)

∀f X Y : set, total_function_on f X Y → functional_graph f → f ⊆ setprod X Y → f = graph X (λx : set ⇒ apply_fun f x)

In Proofgold the corresponding term root is 39e617... and proposition id is 5f5c84...

Proof:
Proof not loaded.

Theorem. (total_function_on_domain_iff_exists_pair)

∀f X Y x : set, total_function_on f X Y → graph_domain_subset f X → (x ∈ X ↔ ∃y : set, (x,y) ∈ f)

In Proofgold the corresponding term root is a301c1... and proposition id is 82a698...

Proof:
Proof not loaded.

Theorem. (total_function_on_domain_unique)

∀f X X' Y : set, total_function_on f X Y → graph_domain_subset f X → total_function_on f X' Y → graph_domain_subset f X' → X = X'

In Proofgold the corresponding term root is 4e11b4... and proposition id is 457182...

Proof:
Proof not loaded.

Theorem. (total_function_on_apply_fun_in_graph)

∀f X Y x : set, total_function_on f X Y → x ∈ X → (x,apply_fun f x) ∈ f

In Proofgold the corresponding term root is 6e403d... and proposition id is 718002...

Proof:
Proof not loaded.

Theorem. (total_function_on_apply_fun_in_Y)

∀f X Y x : set, total_function_on f X Y → x ∈ X → apply_fun f x ∈ Y

In Proofgold the corresponding term root is 146efd... and proposition id is e0ac26...

Proof:
Proof not loaded.

Theorem. (graph_range_subset_from_total_functional)

∀A Y f : set, graph_domain_subset f A → total_function_on f A Y → functional_graph f → graph_range_subset f Y

In Proofgold the corresponding term root is 9d3bfd... and proposition id is 662a46...

Proof:
Proof not loaded.

Definition. We define const_fun to be λA x ⇒ {(a,x)|a ∈ A} of type set → set → set.

In Proofgold the corresponding term root is dd27ad... and object id is 9c5c68...

Theorem. (const_fun_apply)

∀A x a : set, a ∈ A → apply_fun (const_fun A x) a = x

In Proofgold the corresponding term root is 224122... and proposition id is 9ea7df...

Proof:
Proof not loaded.

Theorem. (const_fun_total_function_on)

∀A Y x : set, x ∈ Y → total_function_on (const_fun A x) A Y

In Proofgold the corresponding term root is 17c8be... and proposition id is e3793a...

Proof:
Proof not loaded.

Theorem. (identity_function_apply)

∀X x : set, x ∈ X → apply_fun {(y,y)|y ∈ X} x = x

In Proofgold the corresponding term root is 49ced2... and proposition id is 465cb3...

Proof:
Proof not loaded.

Theorem. (identity_total_function_on)

∀X : set, total_function_on {(y,y)|y ∈ X} X X

In Proofgold the corresponding term root is 19607d... and proposition id is 8312a9...

Proof:
Proof not loaded.

Definition. We define const_family to be λI X ⇒ {(i,X)|i ∈ I} of type set → set → set.

In Proofgold the corresponding term root is dd27ad... and object id is 9c5c68...

Theorem. (const_family_apply)

∀I X i : set, i ∈ I → apply_fun (const_family I X) i = X

In Proofgold the corresponding term root is 224122... and proposition id is 9ea7df...

Proof:
Proof not loaded.

Definition. We define product_component to be λXi i ⇒ (apply_fun Xi i) 0 of type set → set → set.

In Proofgold the corresponding term root is 2a79d3... and object id is 191ecd...

Definition. We define product_component_topology to be λXi i ⇒ (apply_fun Xi i) 1 of type set → set → set.

In Proofgold the corresponding term root is e541fe... and object id is a12553...

Theorem. (product_component_def)

∀Xi i : set, product_component Xi i = (apply_fun Xi i) 0

In Proofgold the corresponding term root is 1dfd96... and proposition id is 644065...

Proof:
Proof not loaded.

Theorem. (product_component_topology_def)

∀Xi i : set, product_component_topology Xi i = (apply_fun Xi i) 1

In Proofgold the corresponding term root is d4d82a... and proposition id is e81a6b...

Proof:
Proof not loaded.

Definition. We define const_space_family to be λI X Tx ⇒ {(i,(X,Tx))|i ∈ I} of type set → set → set → set.

In Proofgold the corresponding term root is da9a83... and object id is 97127c...

Theorem. (const_space_family_apply)

∀I X Tx i : set, i ∈ I → apply_fun (const_space_family I X Tx) i = (X,Tx)

In Proofgold the corresponding term root is ef5131... and proposition id is 2e95dc...

Proof:
Proof not loaded.

Definition. We define space_family_set to be λXi i ⇒ (apply_fun Xi i) 0 of type set → set → set.

In Proofgold the corresponding term root is 2a79d3... and object id is 191ecd...

Definition. We define space_family_topology to be λXi i ⇒ (apply_fun Xi i) 1 of type set → set → set.

In Proofgold the corresponding term root is e541fe... and object id is a12553...

Theorem. (space_family_set_def)

∀Xi i : set, space_family_set Xi i = (apply_fun Xi i) 0

In Proofgold the corresponding term root is 1dfd96... and proposition id is 644065...

Proof:
Proof not loaded.

Theorem. (space_family_topology_def)

∀Xi i : set, space_family_topology Xi i = (apply_fun Xi i) 1

In Proofgold the corresponding term root is d4d82a... and proposition id is e81a6b...

Proof:
Proof not loaded.

Theorem. (space_family_set_eq_product_component)

∀Xi i : set, space_family_set Xi i = product_component Xi i

In Proofgold the corresponding term root is e80147... and proposition id is 8f743c...

Proof:
Proof not loaded.

Theorem. (space_family_topology_eq_product_component_topology)

∀Xi i : set, space_family_topology Xi i = product_component_topology Xi i

In Proofgold the corresponding term root is ab7e88... and proposition id is d3434a...

Proof:
Proof not loaded.

Theorem. (space_family_set_const_space_family)

∀I X Tx i : set, i ∈ I → space_family_set (const_space_family I X Tx) i = X

In Proofgold the corresponding term root is 090dd8... and proposition id is dd2ecc...

Proof:
Proof not loaded.

Theorem. (space_family_topology_const_space_family)

∀I X Tx i : set, i ∈ I → space_family_topology (const_space_family I X Tx) i = Tx

In Proofgold the corresponding term root is 1dc2d4... and proposition id is d14453...

Proof:
Proof not loaded.

Definition. We define space_family_union to be λI Xi ⇒ ⋃ {space_family_set Xi i|i ∈ I} of type set → set → set.

In Proofgold the corresponding term root is 0b21af... and object id is 96e1d8...

Definition. We define topology_family_union to be λI Xi ⇒ ⋃ {space_family_topology Xi i|i ∈ I} of type set → set → set.

In Proofgold the corresponding term root is fd963d... and object id is 39ce72...

Theorem. (space_family_union_const_space_family)

∀I X Tx : set, I ≠ Empty → space_family_union I (const_space_family I X Tx) = X

In Proofgold the corresponding term root is f35652... and proposition id is 740394...

Proof:
Proof not loaded.

Theorem. (topology_family_union_const_space_family)

∀I X Tx : set, I ≠ Empty → topology_family_union I (const_space_family I X Tx) = Tx

In Proofgold the corresponding term root is e60840... and proposition id is c82dd7...

Proof:
Proof not loaded.

Definition. We define product_space to be λI Xi ⇒ {f ∈ 𝒫 (setprod I (space_family_union I Xi))|total_function_on f I (space_family_union I Xi) ∧ functional_graph f ∧ ∀i : set, i ∈ I → apply_fun f i ∈ space_family_set Xi i} of type set → set → set.

In Proofgold the corresponding term root is 02b365... and object id is 988d81...

Theorem. (product_space_graphI)

∀I Xi : set, ∀g : set → set, (∀i : set, i ∈ I → g i ∈ space_family_set Xi i) → graph I g ∈ product_space I Xi

In Proofgold the corresponding term root is a56ded... and proposition id is 3ef7cc...

Proof:
Proof not loaded.

Definition. We define product_cylinder to be λI Xi i U ⇒ {f ∈ product_space I Xi|i ∈ I ∧ U ∈ space_family_topology Xi i ∧ apply_fun f i ∈ U} of type set → set → set → set → set.

In Proofgold the corresponding term root is f8db2f... and object id is e5df89...

Definition. We define product_subbasis_full to be λI Xi ⇒ ⋃_{i ∈ I}{product_cylinder I Xi i U|U ∈ space_family_topology Xi i} of type set → set → set.

In Proofgold the corresponding term root is 499562... and object id is e1be62...

Definition. We define product_topology_full to be λI Xi ⇒ generated_topology_from_subbasis (product_space I Xi) (product_subbasis_full I Xi) of type set → set → set.

In Proofgold the corresponding term root is fe4ae5... and object id is ba39d5...

Definition. We define box_basis to be λI Xi ⇒ {B ∈ 𝒫 (product_space I Xi)|∃U : set, total_function_on U I (topology_family_union I Xi) ∧ functional_graph U ∧ (∀i : set, i ∈ I → apply_fun U i ∈ space_family_topology Xi i) ∧ B = {f ∈ product_space I Xi|∀i : set, i ∈ I → apply_fun f i ∈ apply_fun U i}} of type set → set → set.

In Proofgold the corresponding term root is c0dea2... and object id is ff9c97...

Definition. We define box_topology to be λI Xi ⇒ generated_topology (product_space I Xi) (box_basis I Xi) of type set → set → set.

In Proofgold the corresponding term root is b52874... and object id is bcb672...

Definition. We define countable_product_space to be λI Xi ⇒ product_space I Xi of type set → set → set.

In Proofgold the corresponding term root is 02b365... and object id is 988d81...

Definition. We define countable_product_topology to be λI Xi ⇒ product_topology_full I Xi of type set → set → set.

In Proofgold the corresponding term root is fe4ae5... and object id is ba39d5...

Definition. We define euclidean_space to be λn ⇒ product_space n (const_space_family n R R_standard_topology) of type set → set.

In Proofgold the corresponding term root is 215510... and object id is 69af4b...

Definition. We define euclidean_topology to be λn ⇒ product_topology_full n (const_space_family n R R_standard_topology) of type set → set.

In Proofgold the corresponding term root is 56dcfc... and object id is b1f952...

Definition. We define R2_standard_topology to be product_topology R R_standard_topology R R_standard_topology of type set.

In Proofgold the corresponding term root is 82b01e... and object id is a8843f...

Theorem. (R2_standard_equals_product)

R2_standard_topology = product_topology R R_standard_topology R R_standard_topology

In Proofgold the corresponding term root is 82f29c... and proposition id is 7e3ec9...

Proof:
Proof not loaded.

Theorem. (open_rectangle_set_eq_rectangle_set_intervals)

∀a b c d : set, {p ∈ EuclidPlane|∃x y : set, p = (x,y) ∧ Rlt a x ∧ Rlt x b ∧ Rlt c y ∧ Rlt y d} = rectangle_set (open_interval a b) (open_interval c d)

In Proofgold the corresponding term root is 6abc1a... and proposition id is 17f32c...

Proof:
Proof not loaded.

Theorem. (open_rectangle_in_R2_standard_topology)

∀a b c d : set, Rlt a b → Rlt c d → {p ∈ EuclidPlane|∃x y : set, p = (x,y) ∧ Rlt a x ∧ Rlt x b ∧ Rlt c y ∧ Rlt y d} ∈ R2_standard_topology

In Proofgold the corresponding term root is 2c62c5... and proposition id is 487ae4...

Proof:
Proof not loaded.

Theorem. (rectangular_regions_subset_R2_standard_topology)

∀U : set, U ∈ rectangular_regions → U ∈ R2_standard_topology

In Proofgold the corresponding term root is 1d8a0b... and proposition id is 909836...

Proof:
Proof not loaded.

Theorem. (generated_topology_rectangular_regions_sub_R2_standard_topology)

generated_topology EuclidPlane rectangular_regions ⊆ R2_standard_topology

In Proofgold the corresponding term root is 0be580... and proposition id is 89dd51...

Proof:
Proof not loaded.

Theorem. (generated_topology_rectangular_regions_eq_R2_standard_topology)

generated_topology EuclidPlane rectangular_regions = R2_standard_topology

In Proofgold the corresponding term root is abe239... and proposition id is 686810...

Proof:
Proof not loaded.

Theorem. (generated_topology_circular_regions_eq_R2_standard_topology)

generated_topology EuclidPlane circular_regions = R2_standard_topology

In Proofgold the corresponding term root is a358b7... and proposition id is ecc928...

Proof:
Proof not loaded.

Theorem. (rectangular_regions_generate_R2_standard_topology)

basis_generates EuclidPlane rectangular_regions R2_standard_topology

In Proofgold the corresponding term root is 1d7bfc... and proposition id is dda6e3...

Proof:
Proof not loaded.

Theorem. (circular_regions_generate_R2_standard_topology)

basis_generates EuclidPlane circular_regions R2_standard_topology

In Proofgold the corresponding term root is e08e25... and proposition id is a41dd7...

Proof:
Proof not loaded.

Theorem. (R2_standard_topology_eq_generated_rectangular_regions)

R2_standard_topology = generated_topology EuclidPlane rectangular_regions

In Proofgold the corresponding term root is 780586... and proposition id is 03901c...

Proof:
Proof not loaded.

Theorem. (R2_standard_topology_eq_generated_circular_regions)

R2_standard_topology = generated_topology EuclidPlane circular_regions

In Proofgold the corresponding term root is 810ee4... and proposition id is e3bf94...

Proof:
Proof not loaded.

Definition. We define subspace_topology to be λX Tx Y ⇒ {U ∈ 𝒫 Y|∃V ∈ Tx, U = V ∩ Y} of type set → set → set → set.

In Proofgold the corresponding term root is f2cf4f... and object id is 09e9a9...

Theorem. (subspace_topologyE)

∀X Tx Y U : set, U ∈ subspace_topology X Tx Y → ∃V ∈ Tx, U = V ∩ Y

In Proofgold the corresponding term root is 0db81f... and proposition id is 9250d3...

Proof:
Proof not loaded.

Theorem. (subspace_topologyI)

∀X Tx Y V : set, V ∈ Tx → (V ∩ Y) ∈ subspace_topology X Tx Y

In Proofgold the corresponding term root is 797204... and proposition id is 322ada...

Proof:
Proof not loaded.

Theorem. (subspace_topology_subset)

∀X Tx Y U : set, U ∈ subspace_topology X Tx Y → U ⊆ Y

In Proofgold the corresponding term root is effaa1... and proposition id is 30c65b...

Proof:
Proof not loaded.

Theorem. (subspace_topology_in_Power)

∀X Tx Y U : set, U ∈ subspace_topology X Tx Y → U ∈ 𝒫 Y

In Proofgold the corresponding term root is ce6bec... and proposition id is fdcf13...

Proof:
Proof not loaded.

Theorem. (subspace_topology_binintersect_witness)

∀X Tx Y U V VU VV : set, U = VU ∩ Y → V = VV ∩ Y → U ∩ V = (VU ∩ VV) ∩ Y

In Proofgold the corresponding term root is 793699... and proposition id is 0bda5c...

Proof:
Proof not loaded.

Theorem. (subspace_topology_binintersect)

∀X Tx Y U V : set, topology_on X Tx → Y ⊆ X → U ∈ subspace_topology X Tx Y → V ∈ subspace_topology X Tx Y → U ∩ V ∈ subspace_topology X Tx Y

In Proofgold the corresponding term root is 548ecf... and proposition id is 223b8b...

Proof:
Proof not loaded.

Theorem. (subspace_topology_mono)

∀X Tx1 Tx2 Y U : set, Tx1 ⊆ Tx2 → U ∈ subspace_topology X Tx1 Y → U ∈ subspace_topology X Tx2 Y

In Proofgold the corresponding term root is 000b6f... and proposition id is 727c68...

Proof:
Proof not loaded.

Theorem. (subspace_topology_union_closed)

∀X Tx Y UFam : set, topology_on X Tx → Y ⊆ X → UFam ⊆ subspace_topology X Tx Y → ⋃ UFam ∈ subspace_topology X Tx Y

In Proofgold the corresponding term root is 81e45a... and proposition id is a49995...

Proof:
Proof not loaded.

Theorem. (subspace_topology_whole)

∀X Tx : set, topology_on X Tx → subspace_topology X Tx X = Tx

In Proofgold the corresponding term root is 4d6d5e... and proposition id is 1d4a3d...

Proof:
Proof not loaded.

Theorem. (subspace_topology_is_topology)

∀X Tx Y : set, topology_on X Tx → Y ⊆ X → topology_on Y (subspace_topology X Tx Y)

In Proofgold the corresponding term root is ead4cf... and proposition id is 957991...

Proof:
Proof not loaded.

Theorem. (open_in_subspace_iff)

∀X Tx Y U : set, topology_on X Tx → Y ⊆ X → U ⊆ Y → (open_in Y (subspace_topology X Tx Y) U ↔ ∃V ∈ Tx, U = V ∩ Y)

In Proofgold the corresponding term root is bce457... and proposition id is c2f7fe...

Proof:
Proof not loaded.

Theorem. (subspace_basis)

∀X Tx Y B : set, topology_on X Tx → Y ⊆ X → basis_on X B ∧ generated_topology X B = Tx → basis_on Y {b ∩ Y|b ∈ B} ∧ generated_topology Y {b ∩ Y|b ∈ B} = subspace_topology X Tx Y

In Proofgold the corresponding term root is 36423a... and proposition id is bcb3a8...

Proof:
Proof not loaded.

Theorem. (open_in_subspace_if_ambient_open)

∀X Tx Y U : set, topology_on X Tx → Y ∈ Tx → U ⊆ Y → open_in Y (subspace_topology X Tx Y) U → U ∈ Tx

In Proofgold the corresponding term root is 9cf0dc... and proposition id is 941699...

Proof:
Proof not loaded.

Theorem. (product_subspace_topology)

∀X Tx Y Ty A B : set, topology_on X Tx → topology_on Y Ty → A ⊆ X → B ⊆ Y → product_topology A (subspace_topology X Tx A) B (subspace_topology Y Ty B) = subspace_topology (setprod X Y) (product_topology X Tx Y Ty) (setprod A B)

In Proofgold the corresponding term root is c5398b... and proposition id is 360e38...

Proof:
Proof not loaded.

Definition. We define unit_interval to be {x ∈ R|¬ (Rlt x 0) ∧ ¬ (Rlt 1 x)} of type set.

In Proofgold the corresponding term root is 1a7bfd... and object id is a53174...

Theorem. (unit_interval_sub_R)

unit_interval ⊆ R

In Proofgold the corresponding term root is 31e9de... and proposition id is 936042...

Proof:
Proof not loaded.

Theorem. (eps_1_in_R)

eps_ 1 ∈ R

In Proofgold the corresponding term root is 4f562d... and proposition id is 425d48...

Proof:
Proof not loaded.

Theorem. (eps_1_pos_R)

Rlt 0 (eps_ 1)

In Proofgold the corresponding term root is 7124d3... and proposition id is 3db897...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_R_R_x0_xeps1)

∀x : set, x ∈ R → order_rel (setprod R R) (x,0) (x,eps_ 1)

In Proofgold the corresponding term root is 115785... and proposition id is ab152c...

Proof:
Proof not loaded.

Theorem. (eps_1_lt1_R)

Rlt (eps_ 1) 1

In Proofgold the corresponding term root is 12013f... and proposition id is 8ef7ad...

Proof:
Proof not loaded.

Theorem. (zero_in_unit_interval)

0 ∈ unit_interval

In Proofgold the corresponding term root is 05626c... and proposition id is 6429f6...

Proof:
Proof not loaded.

Theorem. (one_in_unit_interval)

1 ∈ unit_interval

In Proofgold the corresponding term root is 69b723... and proposition id is 957c26...

Proof:
Proof not loaded.

Theorem. (eps_1_in_unit_interval)

eps_ 1 ∈ unit_interval

In Proofgold the corresponding term root is f1b1a9... and proposition id is fb1516...

Proof:
Proof not loaded.

Theorem. (unit_interval_Rle0)

∀t : set, t ∈ unit_interval → Rle 0 t

In Proofgold the corresponding term root is fa422a... and proposition id is e63d41...

Proof:
Proof not loaded.

Theorem. (unit_interval_Rle1)

∀t : set, t ∈ unit_interval → Rle t 1

In Proofgold the corresponding term root is 3935c2... and proposition id is bc475b...

Proof:
Proof not loaded.

Theorem. (unit_interval_mul_closed)

∀a b : set, a ∈ unit_interval → b ∈ unit_interval → mul_SNo a b ∈ unit_interval

In Proofgold the corresponding term root is 5f4be7... and proposition id is 593dbb...

Proof:
Proof not loaded.

Theorem. (two_not_in_unit_interval)

2 ∉ unit_interval

In Proofgold the corresponding term root is 555642... and proposition id is 04fdec...

Proof:
Proof not loaded.

Theorem. (two_in_R)

2 ∈ R

In Proofgold the corresponding term root is d2b6c1... and proposition id is 162da2...

Proof:
Proof not loaded.

Theorem. (unit_interval_neq_R)

unit_interval ≠ R

In Proofgold the corresponding term root is 2e36d3... and proposition id is 1c2217...

Proof:
Proof not loaded.

Definition. We define unit_interval_topology to be subspace_topology R R_standard_topology unit_interval of type set.

In Proofgold the corresponding term root is 59fcc5... and object id is aea1f7...

Theorem. (unit_interval_topology_on)

topology_on unit_interval unit_interval_topology

In Proofgold the corresponding term root is 14fbd8... and proposition id is a8fa0f...

Proof:
Proof not loaded.

Definition. We define flip_unit_interval to be {(t,add_SNo 1 (minus_SNo t))|t ∈ unit_interval} of type set.

In Proofgold the corresponding term root is 37e267... and object id is dc3815...

Theorem. (flip_unit_interval_apply)

∀t : set, t ∈ unit_interval → apply_fun flip_unit_interval t = add_SNo 1 (minus_SNo t)

In Proofgold the corresponding term root is e379a6... and proposition id is f48a51...

Proof:
Proof not loaded.

Theorem. (flip_unit_interval_in_R)

∀t : set, t ∈ unit_interval → apply_fun flip_unit_interval t ∈ R

In Proofgold the corresponding term root is 0ca7d1... and proposition id is bcfa2e...

Proof:
Proof not loaded.

Theorem. (flip_unit_interval_at_0)

apply_fun flip_unit_interval 0 = 1

In Proofgold the corresponding term root is 5c7681... and proposition id is 0a31e0...

Proof:
Proof not loaded.

Theorem. (flip_unit_interval_at_1)

apply_fun flip_unit_interval 1 = 0

In Proofgold the corresponding term root is 5d2560... and proposition id is 4df1df...

Proof:
Proof not loaded.

Theorem. (flip_unit_interval_function_on)

function_on flip_unit_interval unit_interval unit_interval

In Proofgold the corresponding term root is 43215b... and proposition id is ab3805...

Proof:
Proof not loaded.

Theorem. (unit_interval_open_neighborhood_has_other_point)

∀U0 : set, U0 ∈ unit_interval_topology → eps_ 1 ∈ U0 → ∃x : set, x ∈ U0 ∧ x ≠ eps_ 1

In Proofgold the corresponding term root is 6fb02f... and proposition id is 7314a3...

Proof:
Proof not loaded.

Definition. We define ordered_square to be setprod unit_interval unit_interval of type set.

In Proofgold the corresponding term root is 78d4ca... and object id is 50135f...

Definition. We define ordered_square_order_basis to be ({I ∈ 𝒫 ordered_square|∃a ∈ ordered_square, ∃b ∈ ordered_square, I = {x ∈ ordered_square|order_rel (setprod R R) a x ∧ order_rel (setprod R R) x b}} ∪ {I ∈ 𝒫 ordered_square|∃b ∈ ordered_square, I = {x ∈ ordered_square|order_rel (setprod R R) x b}} ∪ {I ∈ 𝒫 ordered_square|∃a ∈ ordered_square, I = {x ∈ ordered_square|order_rel (setprod R R) a x}}) of type set.

In Proofgold the corresponding term root is 13a289... and object id is 801487...

Definition. We define ordered_square_topology to be generated_topology ordered_square ordered_square_order_basis of type set.

In Proofgold the corresponding term root is ad35cd... and object id is 7fc3d0...

Definition. We define ordered_square_open_strip to be {p ∈ ordered_square|∃y : set, p = (eps_ 1,y) ∧ Rlt (eps_ 1) y ∧ ¬ (Rlt 1 y)} of type set.

In Proofgold the corresponding term root is 6ea097... and object id is 38b428...

Definition. We define ordered_square_subspace_topology to be subspace_topology (setprod R R) R2_dictionary_order_topology ordered_square of type set.

In Proofgold the corresponding term root is 62fdad... and object id is 9cc4ee...

Theorem. (ordered_square_neq_setprod_R_R)

ordered_square ≠ setprod R R

In Proofgold the corresponding term root is 02009d... and proposition id is e1de34...

Proof:
Proof not loaded.

Theorem. (ordered_square_standard_subspace_equals_product)

subspace_topology (setprod R R) R2_standard_topology ordered_square = product_topology unit_interval unit_interval_topology unit_interval unit_interval_topology

In Proofgold the corresponding term root is c72575... and proposition id is 6740d0...

Proof:
Proof not loaded.

Theorem. (ordered_square_not_subspace_dictionary)

ordered_square_topology ≠ subspace_topology (setprod R R) R2_dictionary_order_topology ordered_square

In Proofgold the corresponding term root is f579f3... and proposition id is adfca7...

Proof:
Proof not loaded.

Definition. We define order_interval to be λX a b ⇒ {x ∈ X|order_rel X a x ∧ order_rel X x b} of type set → set → set → set.

In Proofgold the corresponding term root is 9d609f... and object id is 756ccb...

Theorem. (order_interval_R_eq_open_interval)

∀a b : set, order_interval R a b = open_interval a b

In Proofgold the corresponding term root is 5bfcd0... and proposition id is 7e1f35...

Proof:
Proof not loaded.

Theorem. (order_interval_subset)

∀X a b : set, order_interval X a b ⊆ X

In Proofgold the corresponding term root is 772071... and proposition id is 6e66b2...

Proof:
Proof not loaded.

Theorem. (order_intervalE)

∀X a b x : set, x ∈ order_interval X a b → x ∈ X ∧ (order_rel X a x ∧ order_rel X x b)

In Proofgold the corresponding term root is d293c0... and proposition id is 0b69ee...

Proof:
Proof not loaded.

Theorem. (order_intervalI)

∀X a b x : set, x ∈ X → order_rel X a x → order_rel X x b → x ∈ order_interval X a b

In Proofgold the corresponding term root is 4df301... and proposition id is a75a6c...

Proof:
Proof not loaded.

Definition. We define convex_in to be λX Y ⇒ Y ⊆ X ∧ ∀a b : set, a ∈ Y → b ∈ Y → order_interval X a b ⊆ Y of type set → set → prop.

In Proofgold the corresponding term root is 80a12d... and object id is c64aaf...

Theorem. (convex_in_subset)

∀X Y : set, convex_in X Y → Y ⊆ X

In Proofgold the corresponding term root is 156c58... and proposition id is 7f5891...

Proof:
Proof not loaded.

Theorem. (convex_in_interval_property)

∀X Y : set, convex_in X Y → ∀a b : set, a ∈ Y → b ∈ Y → order_interval X a b ⊆ Y

In Proofgold the corresponding term root is cf6de9... and proposition id is 012c58...

Proof:
Proof not loaded.

Definition. We define order_topology_basis_inherited to be λX Y ⇒ (({I ∈ 𝒫 Y|∃a ∈ Y, ∃b ∈ Y, I = {x ∈ Y|order_rel X a x ∧ order_rel X x b}} ∪ {I ∈ 𝒫 Y|∃b ∈ Y, I = {x ∈ Y|order_rel X x b}} ∪ {I ∈ 𝒫 Y|∃a ∈ Y, I = {x ∈ Y|order_rel X a x}}) ∪ {Y}) of type set → set → set.

In Proofgold the corresponding term root is 4fc513... and object id is 0c1488...

Definition. We define order_topology_inherited to be λX Y ⇒ generated_topology Y (order_topology_basis_inherited X Y) of type set → set → set.

In Proofgold the corresponding term root is 4b2ec3... and object id is c3985e...

Theorem. (convex_subspace_order_topology)

∀X Y : set, simply_ordered_set X → convex_in X Y → order_topology_inherited X Y = subspace_topology X (order_topology X) Y

In Proofgold the corresponding term root is 30b60b... and proposition id is bb9036...

Proof:
Proof not loaded.

Theorem. (binintersect_right_absorb_subset)

∀W Y A : set, A ⊆ Y → (W ∩ Y) ∩ A = W ∩ A

In Proofgold the corresponding term root is 0d33fc... and proposition id is afcf75...

Proof:
Proof not loaded.

Theorem. (ex16_1_subspace_transitive)

∀X Tx Y A : set, topology_on X Tx → Y ⊆ X → A ⊆ Y → subspace_topology Y (subspace_topology X Tx Y) A = subspace_topology X Tx A

In Proofgold the corresponding term root is 8ebb42... and proposition id is de9f7c...

Proof:
Proof not loaded.

Theorem. (ex16_2_finer_subspaces)

∀X T T' Y : set, topology_on X T → topology_on X T' → T' ⊆ T → Y ⊆ X → subspace_topology X T' Y ⊆ subspace_topology X T Y

In Proofgold the corresponding term root is 4e66a7... and proposition id is 74d02b...

Proof:
Proof not loaded.

Theorem. (subspace_topology_eq_of_eq)

∀X T T' Y : set, T = T' → subspace_topology X T Y = subspace_topology X T' Y

In Proofgold the corresponding term root is c1dd92... and proposition id is 7c002f...

Proof:
Proof not loaded.

Definition. We define one_half to be inv_nat 2 of type set.

In Proofgold the corresponding term root is d13ab8... and object id is e07582...

Definition. We define interval_A to be {x ∈ R|one_half < abs_SNo x ∧ abs_SNo x < 1} of type set.

In Proofgold the corresponding term root is 3c5644... and object id is 158667...

Definition. We define interval_B to be {x ∈ R|one_half < abs_SNo x ∧ ¬ (1 < abs_SNo x)} of type set.

In Proofgold the corresponding term root is 227327... and object id is eddafb...

Definition. We define interval_C to be {x ∈ R|¬ (abs_SNo x < one_half) ∧ abs_SNo x < 1} of type set.

In Proofgold the corresponding term root is 74dfd4... and object id is f08d46...

Definition. We define interval_D to be {x ∈ R|¬ (abs_SNo x < one_half) ∧ ¬ (1 < abs_SNo x)} of type set.

In Proofgold the corresponding term root is 5304d1... and object id is bd91e1...

Definition. We define interval_E to be {x ∈ R|0 < abs_SNo x ∧ abs_SNo x < 1 ∧ ¬ (div_SNo 1 x ∈ Zplus)} of type set.

In Proofgold the corresponding term root is 35f361... and object id is c80962...

Theorem. (ex16_3_open_sets_subspace)

∀X Tx Y : set, topology_on X Tx → Y ⊆ X → ∀U : set, open_in Y (subspace_topology X Tx Y) U → ∃V : set, open_in X Tx V ∧ U = V ∩ Y

In Proofgold the corresponding term root is b6764f... and proposition id is 7d86a7...

Proof:
Proof not loaded.

Definition. We define projection_image1 to be λX Y U ⇒ {x ∈ X|∃y : set, (x,y) ∈ U} of type set → set → set → set.

In Proofgold the corresponding term root is 42178c... and object id is a6ee25...

Definition. We define projection_image2 to be λX Y U ⇒ {y ∈ Y|∃x : set, (x,y) ∈ U} of type set → set → set → set.

In Proofgold the corresponding term root is b18ef5... and object id is d31401...

Theorem. (nonempty_has_element)

∀V : set, V ≠ Empty → ∃y : set, y ∈ V

In Proofgold the corresponding term root is 21158b... and proposition id is b90275...

Proof:
Proof not loaded.

Theorem. (elem_implies_nonempty)

∀V y : set, y ∈ V → V ≠ Empty

In Proofgold the corresponding term root is a174eb... and proposition id is 1df70a...

Proof:
Proof not loaded.

Theorem. (projection_image1_rectangle_nonempty)

∀X Y U V : set, U ⊆ X → V ⊆ Y → V ≠ Empty → projection_image1 X Y (setprod U V) = U

In Proofgold the corresponding term root is 9f457a... and proposition id is 760f2f...

Proof:
Proof not loaded.

Theorem. (projection_image1_rectangle_empty)

∀X Y U : set, projection_image1 X Y (setprod U Empty) = Empty

In Proofgold the corresponding term root is 3506b6... and proposition id is c58817...

Proof:
Proof not loaded.

Theorem. (projection_image2_rectangle_nonempty)

∀X Y U V : set, U ⊆ X → V ⊆ Y → U ≠ Empty → projection_image2 X Y (setprod U V) = V

In Proofgold the corresponding term root is 9c32c7... and proposition id is 4a85a2...

Proof:
Proof not loaded.

Theorem. (projection_image2_rectangle_empty)

∀X Y V : set, projection_image2 X Y (setprod Empty V) = Empty

In Proofgold the corresponding term root is 767a2d... and proposition id is 323c73...

Proof:
Proof not loaded.

Theorem. (ex16_4_projections_open)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → ∀U : set, U ∈ product_topology X Tx Y Ty → open_in X Tx (projection_image1 X Y U) ∧ open_in Y Ty (projection_image2 X Y U)

In Proofgold the corresponding term root is ecfdca... and proposition id is 879ae1...

Proof:
Proof not loaded.

Theorem. (ex16_5a_product_monotone)

∀X T T' Y U U' : set, X ≠ Empty → Y ≠ Empty → topology_on X T → topology_on X T' → topology_on Y U → topology_on Y U' → T ⊆ T' ∧ U ⊆ U' → product_topology X T Y U ⊆ product_topology X T' Y U'

In Proofgold the corresponding term root is 9c59a5... and proposition id is 7a3280...

Proof:
Proof not loaded.

Theorem. (ex16_5b_product_converse)

∀X T T' Y U U' : set, X ≠ Empty → Y ≠ Empty → topology_on X T → topology_on X T' → topology_on Y U → topology_on Y U' → product_topology X T Y U ⊆ product_topology X T' Y U' → T ⊆ T' ∧ U ⊆ U'

In Proofgold the corresponding term root is e618f1... and proposition id is a89f9d...

Proof:
Proof not loaded.

Definition. We define rational_rectangle_basis to be {r ∈ 𝒫 (setprod R R)|∃a b c d : set, a ∈ rational_numbers ∧ b ∈ rational_numbers ∧ c ∈ rational_numbers ∧ d ∈ rational_numbers ∧ r = setprod (open_interval a b) (open_interval c d)} of type set.

In Proofgold the corresponding term root is 8e49e6... and object id is 90f4f8...

Theorem. (open_interval_in_R_standard_topology_endpoints)

∀a b : set, a ∈ R → b ∈ R → open_interval a b ∈ R_standard_topology

In Proofgold the corresponding term root is 99e1de... and proposition id is 5e93c1...

Proof:
Proof not loaded.

Theorem. (open_interval_in_rational_open_intervals_basis)

∀q1 q2 : set, q1 ∈ rational_numbers → q2 ∈ rational_numbers → open_interval q1 q2 ∈ rational_open_intervals_basis

In Proofgold the corresponding term root is 252f7c... and proposition id is d3381d...

Proof:
Proof not loaded.

Theorem. (rational_open_intervals_basisE)

∀b : set, b ∈ rational_open_intervals_basis → ∃q1, ∃q2, (q1 ∈ rational_numbers ∧ q2 ∈ rational_numbers ∧ b = open_interval q1 q2)

In Proofgold the corresponding term root is 9b9b9b... and proposition id is ca63c7...

Proof:
Proof not loaded.

Theorem. (rational_rectangle_basis_eq_product_basis_from)

rational_rectangle_basis = product_basis_from rational_open_intervals_basis rational_open_intervals_basis

In Proofgold the corresponding term root is 752136... and proposition id is e278a2...

Proof:
Proof not loaded.

Theorem. (ex16_6_rational_rectangles_basis)

basis_on (setprod R R) rational_rectangle_basis ∧ generated_topology (setprod R R) rational_rectangle_basis = R2_standard_topology

In Proofgold the corresponding term root is 43d035... and proposition id is 98318b...

Proof:
Proof not loaded.

Definition. We define closed_interval_in to be λX a b ⇒ {x ∈ X|(x = a ∨ order_rel X a x) ∧ (x = b ∨ order_rel X x b)} of type set → set → set → set.

In Proofgold the corresponding term root is 7dc572... and object id is 379964...

Definition. We define halfopen_interval_left_in to be λX a b ⇒ {x ∈ X|(x = a ∨ order_rel X a x) ∧ order_rel X x b} of type set → set → set → set.

In Proofgold the corresponding term root is cbf5a5... and object id is f78238...

Definition. We define halfopen_interval_right_in to be λX a b ⇒ {x ∈ X|order_rel X a x ∧ (x = b ∨ order_rel X x b)} of type set → set → set → set.

In Proofgold the corresponding term root is 4bbfc6... and object id is 21de64...

Definition. We define closed_ray_upper to be λX a ⇒ {x ∈ X|x = a ∨ order_rel X a x} of type set → set → set.

In Proofgold the corresponding term root is 44e73e... and object id is e8dc15...

Definition. We define closed_ray_lower to be λX a ⇒ {x ∈ X|x = a ∨ order_rel X x a} of type set → set → set.

In Proofgold the corresponding term root is dea699... and object id is e64469...

Theorem. (closed_interval_in_eq_ray_intersection)

∀X a b : set, closed_interval_in X a b = (closed_ray_upper X a) ∩ (closed_ray_lower X b)

In Proofgold the corresponding term root is 49ac95... and proposition id is fb1fda...

Proof:
Proof not loaded.

Definition. We define interval_in to be λX a b Y ⇒ Y = order_interval X a b ∨ Y = halfopen_interval_left_in X a b ∨ Y = halfopen_interval_right_in X a b ∨ Y = closed_interval_in X a b of type set → set → set → set → prop.

In Proofgold the corresponding term root is a1a622... and object id is c727e9...

Definition. We define ray_in to be λX a Y ⇒ Y = open_ray_upper X a ∨ Y = closed_ray_upper X a ∨ Y = open_ray_lower X a ∨ Y = closed_ray_lower X a of type set → set → set → prop.

In Proofgold the corresponding term root is b98697... and object id is 8d4a65...

Definition. We define interval_or_ray_in to be λX Y ⇒ (∃a b : set, a ∈ X ∧ b ∈ X ∧ interval_in X a b Y) ∨ (∃a : set, a ∈ X ∧ ray_in X a Y) of type set → set → prop.

In Proofgold the corresponding term root is b33546... and object id is b91173...

Definition. We define Q_sqrt2_cut to be {q ∈ rational_numbers|mul_SNo q q < 2} of type set.

In Proofgold the corresponding term root is 98cb60... and object id is 744a25...

Theorem. (two_in_rational_numbers)

2 ∈ rational_numbers

In Proofgold the corresponding term root is 45944d... and proposition id is 106a14...

Proof:
Proof not loaded.

Theorem. (two_not_in_Q_sqrt2_cut)

2 ∉ Q_sqrt2_cut

In Proofgold the corresponding term root is 296a9b... and proposition id is d1fffe...

Proof:
Proof not loaded.

Theorem. (minus_two_not_in_Q_sqrt2_cut)

minus_SNo 2 ∉ Q_sqrt2_cut

In Proofgold the corresponding term root is b7a2d7... and proposition id is 6ce4d5...

Proof:
Proof not loaded.

Theorem. (Q_sqrt2_cut_neq_Q)

Q_sqrt2_cut ≠ rational_numbers

In Proofgold the corresponding term root is 9b86a3... and proposition id is ccac93...

Proof:
Proof not loaded.

Theorem. (Q_sqrt2_cut_sub_Q)

Q_sqrt2_cut ⊆ rational_numbers

In Proofgold the corresponding term root is 5da8ff... and proposition id is 8cdde9...

Proof:
Proof not loaded.

Theorem. (zero_in_Q_sqrt2_cut)

0 ∈ Q_sqrt2_cut

In Proofgold the corresponding term root is a8e70a... and proposition id is dd5f8a...

Proof:
Proof not loaded.

Theorem. (Q_sqrt2_cut_between_square)

∀a b x : set, a ∈ Q_sqrt2_cut → b ∈ Q_sqrt2_cut → x ∈ rational_numbers → Rlt a x → Rlt x b → mul_SNo x x < 2

In Proofgold the corresponding term root is 48a843... and proposition id is cc0504...

Proof:
Proof not loaded.

Theorem. (Q_sqrt2_cut_convex)

convex_in rational_numbers Q_sqrt2_cut

In Proofgold the corresponding term root is e96a2c... and proposition id is 6079d7...

Proof:
Proof not loaded.

Theorem. (Q_sqrt2_cut_no_max)

∀q : set, q ∈ Q_sqrt2_cut → ∃r : set, r ∈ Q_sqrt2_cut ∧ Rlt q r

In Proofgold the corresponding term root is cf4b6d... and proposition id is 66dbd3...

Proof:
Proof not loaded.

Theorem. (Q_sqrt2_cut_neg_closed)

∀q : set, q ∈ Q_sqrt2_cut → minus_SNo q ∈ Q_sqrt2_cut

In Proofgold the corresponding term root is fabacf... and proposition id is a54f38...

Proof:
Proof not loaded.

Theorem. (Q_sqrt2_cut_no_min)

∀q : set, q ∈ Q_sqrt2_cut → ∃r : set, r ∈ Q_sqrt2_cut ∧ Rlt r q

In Proofgold the corresponding term root is 4d0ec7... and proposition id is fc7b6e...

Proof:
Proof not loaded.

Theorem. (Q_sqrt2_cut_not_interval_or_ray)

¬ interval_or_ray_in rational_numbers Q_sqrt2_cut

In Proofgold the corresponding term root is ff6793... and proposition id is bef2a6...

Proof:
Proof not loaded.

Theorem. (ex16_7_convex_interval_or_ray)

∃X Y : set, convex_in X Y ∧ Y ≠ X ∧ ¬ interval_or_ray_in X Y

In Proofgold the corresponding term root is 033708... and proposition id is f01f7f...

Proof:
Proof not loaded.

Theorem. (singleton_rectangle_in_dictionary)

∀a V : set, a ∈ R → V ∈ R_standard_topology → rectangle_set {a} V ∈ R2_dictionary_order_topology

In Proofgold the corresponding term root is dbb496... and proposition id is c06b30...

Proof:
Proof not loaded.

Theorem. (rectangle_in_dictionary)

∀U V : set, U ∈ discrete_topology R → V ∈ R_standard_topology → rectangle_set U V ∈ R2_dictionary_order_topology

In Proofgold the corresponding term root is 9f2bf6... and proposition id is 8cbd7d...

Proof:
Proof not loaded.

Theorem. (singleton_not_open_R_standard_topology)

∀a : set, a ∈ R → {a} ∉ R_standard_topology

In Proofgold the corresponding term root is aca2bd... and proposition id is 6b4f20...

Proof:
Proof not loaded.

Theorem. (ex16_9_dictionary_equals_product)

R2_dictionary_order_topology = product_topology R (discrete_topology R) R R_standard_topology ∧ R2_dictionary_order_topology ≠ R2_standard_topology

In Proofgold the corresponding term root is 5181e6... and proposition id is cac17a...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is dff310... and proposition id is 79adf3...

Proof:
Proof not loaded.

Definition. We define interior_of to be λX T A ⇒ {x ∈ X|∃U : set, U ∈ T ∧ x ∈ U ∧ U ⊆ A} of type set → set → set → set.

In Proofgold the corresponding term root is 3b8bd0... and object id is 0b3836...

Definition. We define closure_of to be λX T A ⇒ {x ∈ X|∀U : set, U ∈ T → x ∈ U → U ∩ A ≠ Empty} of type set → set → set → set.

In Proofgold the corresponding term root is 66761a... and object id is 355f66...

Theorem. (interior_of_contains_open_subset_point)

∀X Tx A U x : set, topology_on X Tx → U ∈ Tx → x ∈ U → U ⊆ A → x ∈ interior_of X Tx A

In Proofgold the corresponding term root is a30042... and proposition id is edc25e...

Proof:
Proof not loaded.

Theorem. (point_in_open_empty_interior_not_subset)

∀X Tx A U x : set, topology_on X Tx → interior_of X Tx A = Empty → U ∈ Tx → x ∈ U → ¬ (U ⊆ A)

In Proofgold the corresponding term root is dc57f3... and proposition id is 21d763...

Proof:
Proof not loaded.

Theorem. (closure_infinite_finite_complement)

∀X A : set, A ⊆ X → infinite A → closure_of X (finite_complement_topology X) A = X

In Proofgold the corresponding term root is 096688... and proposition id is d001e8...

Proof:
Proof not loaded.

Theorem. (subset_of_closure)

∀X Tx A : set, topology_on X Tx → A ⊆ X → A ⊆ closure_of X Tx A

In Proofgold the corresponding term root is 8bb62a... and proposition id is 34e2cf...

Proof:
Proof not loaded.

Theorem. (closure_monotone)

∀X Tx A B : set, topology_on X Tx → A ⊆ B → B ⊆ X → closure_of X Tx A ⊆ closure_of X Tx B

In Proofgold the corresponding term root is 7e4362... and proposition id is 96a594...

Proof:
Proof not loaded.

Theorem. (closure_finer_than_Subq)

∀X T1 T2 A : set, finer_than T1 T2 → closure_of X T1 A ⊆ closure_of X T2 A

In Proofgold the corresponding term root is 96e633... and proposition id is 53b782...

Proof:
Proof not loaded.

Theorem. (interior_subset)

∀X Tx A : set, topology_on X Tx → interior_of X Tx A ⊆ A

In Proofgold the corresponding term root is d5f872... and proposition id is 2bc9c5...

Proof:
Proof not loaded.

Theorem. (interior_monotone)

∀X Tx A B : set, topology_on X Tx → A ⊆ B → interior_of X Tx A ⊆ interior_of X Tx B

In Proofgold the corresponding term root is 9e824d... and proposition id is 7aee49...

Proof:
Proof not loaded.

Theorem. (open_interior_eq)

∀X Tx U : set, topology_on X Tx → U ∈ Tx → interior_of X Tx U = U

In Proofgold the corresponding term root is 2231ac... and proposition id is 9e1c33...

Proof:
Proof not loaded.

Theorem. (open_subset_interior)

∀X Tx U A : set, topology_on X Tx → U ∈ Tx → U ⊆ A → U ⊆ interior_of X Tx A

In Proofgold the corresponding term root is 48f813... and proposition id is 2ad571...

Proof:
Proof not loaded.

Theorem. (open_subset_empty_interior_implies_empty)

∀X Tx U A : set, topology_on X Tx → U ∈ Tx → U ⊆ A → interior_of X Tx A = Empty → U = Empty

In Proofgold the corresponding term root is 12c129... and proposition id is 3aa032...

Proof:
Proof not loaded.

Theorem. (interior_of_empty)

∀X Tx : set, topology_on X Tx → interior_of X Tx Empty = Empty

In Proofgold the corresponding term root is 744616... and proposition id is ca8b2d...

Proof:
Proof not loaded.

Theorem. (interior_of_space)

∀X Tx : set, topology_on X Tx → interior_of X Tx X = X

In Proofgold the corresponding term root is f0c83b... and proposition id is bfa24d...

Proof:
Proof not loaded.

Theorem. (interior_is_open)

∀X Tx A : set, topology_on X Tx → A ⊆ X → interior_of X Tx A ∈ Tx

In Proofgold the corresponding term root is 1e05a3... and proposition id is 420da6...

Proof:
Proof not loaded.

Theorem. (interior_union_contains_union_interiors)

∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → interior_of X Tx A ∪ interior_of X Tx B ⊆ interior_of X Tx (A ∪ B)

In Proofgold the corresponding term root is 2f3def... and proposition id is 873cfd...

Proof:
Proof not loaded.

Theorem. (interior_intersection_contains_intersection)

∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → interior_of X Tx (A ∩ B) ⊆ interior_of X Tx A ∩ interior_of X Tx B

In Proofgold the corresponding term root is 50a55e... and proposition id is e9f03f...

Proof:
Proof not loaded.

Theorem. (interior_intersection_of_opens)

∀X Tx U V : set, topology_on X Tx → U ∈ Tx → V ∈ Tx → interior_of X Tx (U ∩ V) = U ∩ V

In Proofgold the corresponding term root is c81097... and proposition id is 2b8186...

Proof:
Proof not loaded.

Theorem. (interior_idempotent)

∀X Tx A : set, topology_on X Tx → A ⊆ X → interior_of X Tx (interior_of X Tx A) = interior_of X Tx A

In Proofgold the corresponding term root is bf6e75... and proposition id is f33a2a...

Proof:
Proof not loaded.

Theorem. (not_in_closure_has_disjoint_open)

∀X Tx A x : set, topology_on X Tx → A ⊆ X → x ∈ X → x ∉ closure_of X Tx A → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ U ∩ A = Empty

In Proofgold the corresponding term root is 247e6c... and proposition id is d5aa32...

Proof:
Proof not loaded.

Theorem. (closure_is_closed)

∀X Tx A : set, topology_on X Tx → A ⊆ X → closed_in X Tx (closure_of X Tx A)

In Proofgold the corresponding term root is 0f5992... and proposition id is 7e5158...

Proof:
Proof not loaded.

Theorem. (interior_closure_complement_duality)

∀X Tx A : set, topology_on X Tx → A ⊆ X → interior_of X Tx A = X ∖ closure_of X Tx (X ∖ A)

In Proofgold the corresponding term root is 54ea57... and proposition id is f0f0aa...

Proof:
Proof not loaded.

Theorem. (closure_contains_set)

∀X Tx A : set, topology_on X Tx → A ⊆ X → A ⊆ closure_of X Tx A

In Proofgold the corresponding term root is 8bb62a... and proposition id is 34e2cf...

Proof:
Proof not loaded.

Theorem. (closure_in_space)

∀X Tx A : set, topology_on X Tx → closure_of X Tx A ⊆ X

In Proofgold the corresponding term root is f74310... and proposition id is 9643ef...

Proof:
Proof not loaded.

Theorem. (closure_subset_of_closed_superset)

∀X Tx A C : set, topology_on X Tx → A ⊆ C → closed_in X Tx C → closure_of X Tx A ⊆ C

In Proofgold the corresponding term root is 31eb90... and proposition id is 85eb2e...

Proof:
Proof not loaded.

Theorem. (closure_union_contains_union_closures)

∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → closure_of X Tx A ∪ closure_of X Tx B ⊆ closure_of X Tx (A ∪ B)

In Proofgold the corresponding term root is bc6216... and proposition id is a03032...

Proof:
Proof not loaded.

Theorem. (closure_of_empty)

∀X Tx : set, topology_on X Tx → closure_of X Tx Empty = Empty

In Proofgold the corresponding term root is 0431e4... and proposition id is 064815...

Proof:
Proof not loaded.

Theorem. (closure_of_space)

∀X Tx : set, topology_on X Tx → closure_of X Tx X = X

In Proofgold the corresponding term root is 34e1a3... and proposition id is 3f7d0c...

Proof:
Proof not loaded.

Theorem. (union_of_closed_is_closed)

∀X Tx C D : set, topology_on X Tx → closed_in X Tx C → closed_in X Tx D → closed_in X Tx (C ∪ D)

In Proofgold the corresponding term root is 8f9f1a... and proposition id is 5750f2...

Proof:
Proof not loaded.

Theorem. (empty_is_closed)

∀X Tx : set, topology_on X Tx → closed_in X Tx Empty

In Proofgold the corresponding term root is b914ec... and proposition id is e6cbc6...

Proof:
Proof not loaded.

Theorem. (space_is_closed)

∀X Tx : set, topology_on X Tx → closed_in X Tx X

In Proofgold the corresponding term root is 9ebffc... and proposition id is ea3315...

Proof:
Proof not loaded.

Theorem. (binunion_eq_Union_UPair)

∀U V : set, U ∪ V = ⋃ (UPair U V)

In Proofgold the corresponding term root is b94f4c... and proposition id is ed8d87...

Proof:
Proof not loaded.

Theorem. (Union_UPair_eq_binunion)

∀U V : set, ⋃ (UPair U V) = U ∪ V

In Proofgold the corresponding term root is 865e0d... and proposition id is 7e009e...

Proof:
Proof not loaded.

Theorem. (lemma_union_two_open)

∀X T U V : set, topology_on X T → U ∈ T → V ∈ T → U ∪ V ∈ T

In Proofgold the corresponding term root is 9163f4... and proposition id is 2f5926...

Proof:
Proof not loaded.

Theorem. (intersection_of_closed_is_closed)

∀X Tx C D : set, topology_on X Tx → closed_in X Tx C → closed_in X Tx D → closed_in X Tx (C ∩ D)

In Proofgold the corresponding term root is 918983... and proposition id is 7fc16b...

Proof:
Proof not loaded.

Theorem. (closed_closure_eq)

∀X Tx C : set, topology_on X Tx → closed_in X Tx C → closure_of X Tx C = C

In Proofgold the corresponding term root is 7a3894... and proposition id is 8ff2b2...

Proof:
Proof not loaded.

Theorem. (closure_intersection_of_closed)

∀X Tx C D : set, topology_on X Tx → closed_in X Tx C → closed_in X Tx D → closure_of X Tx (C ∩ D) = C ∩ D

In Proofgold the corresponding term root is 1fdfd7... and proposition id is 58dc2c...

Proof:
Proof not loaded.

Theorem. (closure_union_of_closed)

∀X Tx C D : set, topology_on X Tx → closed_in X Tx C → closed_in X Tx D → closure_of X Tx (C ∪ D) = C ∪ D

In Proofgold the corresponding term root is a62e83... and proposition id is 687ff5...

Proof:
Proof not loaded.

Theorem. (closure_idempotent)

∀X Tx A : set, topology_on X Tx → A ⊆ X → closure_of X Tx (closure_of X Tx A) = closure_of X Tx A

In Proofgold the corresponding term root is 11169d... and proposition id is 68da76...

Proof:
Proof not loaded.

Theorem. (closure_intersection_contained)

∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → closure_of X Tx (A ∩ B) ⊆ closure_of X Tx A ∩ closure_of X Tx B

In Proofgold the corresponding term root is 272d8b... and proposition id is 3106eb...

Proof:
Proof not loaded.

Theorem. (closed_sets_axioms)

∀X T : set, topology_on X T → let C ≔ {X ∖ U|U ∈ T} in X ∈ C ∧ Empty ∈ C ∧ (∀F : set, F ∈ 𝒫 C → intersection_of_family X F ∈ C) ∧ (∀A B : set, A ∈ C → B ∈ C → A ∪ B ∈ C)

In Proofgold the corresponding term root is ae7753... and proposition id is b63534...

Proof:
Proof not loaded.

Theorem. (closed_in_subspace_iff_intersection)

∀X Tx Y A : set, topology_on X Tx → Y ⊆ X → (closed_in Y (subspace_topology X Tx Y) A ↔ ∃C : set, closed_in X Tx C ∧ A = C ∩ Y)

In Proofgold the corresponding term root is be4297... and proposition id is f144d1...

Proof:
Proof not loaded.

Theorem. (closed_in_closed_subspace)

∀X Tx Y A : set, topology_on X Tx → closed_in X Tx Y → closed_in Y (subspace_topology X Tx Y) A → closed_in X Tx A

In Proofgold the corresponding term root is 793cf4... and proposition id is 20f112...

Proof:
Proof not loaded.

Theorem. (closure_in_subspace)

∀X Tx Y A : set, topology_on X Tx → Y ⊆ X → A ⊆ Y → closure_of Y (subspace_topology X Tx Y) A = (closure_of X Tx A) ∩ Y

In Proofgold the corresponding term root is 4f7055... and proposition id is eef100...

Proof:
Proof not loaded.

Theorem. (closure_characterization)

∀X Tx A x : set, topology_on X Tx → x ∈ X → (x ∈ closure_of X Tx A ↔ (∀U ∈ Tx, x ∈ U → U ∩ A ≠ Empty))

In Proofgold the corresponding term root is c7ff2d... and proposition id is 800c64...

Proof:
Proof not loaded.

Definition. We define limit_point_of to be λX Tx A x ⇒ topology_on X Tx ∧ x ∈ X ∧ ∀U : set, U ∈ Tx → x ∈ U → ∃y : set, y ∈ A ∧ y ≠ x ∧ y ∈ U of type set → set → set → set → prop.

In Proofgold the corresponding term root is 05a110... and object id is 456f0c...

Definition. We define limit_points_of to be λX Tx A ⇒ {x ∈ X|limit_point_of X Tx A x} of type set → set → set → set.

In Proofgold the corresponding term root is d2aeb4... and object id is 31b1d2...

Theorem. (closure_equals_set_plus_limit_points)

∀X Tx A : set, topology_on X Tx → A ⊆ X → closure_of X Tx A = A ∪ limit_points_of X Tx A

In Proofgold the corresponding term root is b6fe0c... and proposition id is 27ca37...

Proof:
Proof not loaded.

Theorem. (closed_iff_contains_limit_points)

∀X Tx A : set, topology_on X Tx → A ⊆ X → (closed_in X Tx A ↔ limit_points_of X Tx A ⊆ A)

In Proofgold the corresponding term root is be75f6... and proposition id is 7b997b...

Proof:
Proof not loaded.

Definition. We define Hausdorff_space to be λX Tx ⇒ topology_on X Tx ∧ ∀x1 x2 : set, x1 ∈ X → x2 ∈ X → x1 ≠ x2 → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ x1 ∈ U ∧ x2 ∈ V ∧ U ∩ V = Empty of type set → set → prop.

In Proofgold the corresponding term root is 16b62c... and object id is 85b16d...

Theorem. (Hausdorff_space_topology)

∀X Tx : set, Hausdorff_space X Tx → topology_on X Tx

In Proofgold the corresponding term root is d82fce... and proposition id is a11744...

Proof:
Proof not loaded.

Theorem. (Hausdorff_space_separation)

∀X Tx x1 x2 : set, Hausdorff_space X Tx → x1 ∈ X → x2 ∈ X → x1 ≠ x2 → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ x1 ∈ U ∧ x2 ∈ V ∧ U ∩ V = Empty

In Proofgold the corresponding term root is e6e98c... and proposition id is ed327c...

Proof:
Proof not loaded.

Definition. We define T1_space to be λX Tx ⇒ topology_on X Tx ∧ (∀F : set, F ⊆ X → finite F → closed_in X Tx F) of type set → set → prop.

In Proofgold the corresponding term root is 4967e1... and object id is fe3bd7...

Theorem. (T1_space_topology)

∀X Tx : set, T1_space X Tx → topology_on X Tx

In Proofgold the corresponding term root is 6662be... and proposition id is 368447...

Proof:
Proof not loaded.

Theorem. (T1_space_finite_closed)

∀X Tx F : set, T1_space X Tx → F ⊆ X → finite F → closed_in X Tx F

In Proofgold the corresponding term root is d95432... and proposition id is 05340b...

Proof:
Proof not loaded.

Theorem. (Hausdorff_singleton_complement_open)

∀X Tx x : set, Hausdorff_space X Tx → x ∈ X → X ∖ {x} ∈ Tx

In Proofgold the corresponding term root is a300c4... and proposition id is 28642f...

Proof:
Proof not loaded.

Theorem. (Hausdorff_singletons_closed)

∀X Tx x : set, Hausdorff_space X Tx → x ∈ X → closed_in X Tx {x}

In Proofgold the corresponding term root is 3f67f3... and proposition id is 941084...

Proof:
Proof not loaded.

Theorem. (finite_sets_closed_in_Hausdorff)

∀X Tx : set, Hausdorff_space X Tx → ∀F : set, F ⊆ X → finite F → closed_in X Tx F

In Proofgold the corresponding term root is 2c4b5d... and proposition id is 3f6f4a...

Proof:
Proof not loaded.

Theorem. (limit_points_infinite_neighborhoods)

∀X Tx A x : set, T1_space X Tx → x ∈ X → (limit_point_of X Tx A x ↔ (∀U ∈ Tx, x ∈ U → infinite (U ∩ A)))

In Proofgold the corresponding term root is b94fd7... and proposition id is 612e8c...

Proof:
Proof not loaded.

Theorem. (Hausdorff_unique_limits)

∀X Tx seq x y : set, Hausdorff_space X Tx → x ∈ X → y ∈ X → x ≠ y → function_on seq ω X → (∀U : set, U ∈ Tx → x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U) → (∀U : set, U ∈ Tx → y ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U) → False

In Proofgold the corresponding term root is b2ab24... and proposition id is c21a29...

Proof:
Proof not loaded.

Theorem. (Hausdorff_stability)

∀X Tx Y Ty : set, Hausdorff_space X Tx ∧ Hausdorff_space Y Ty → Hausdorff_space (setprod X Y) (product_topology X Tx Y Ty)

In Proofgold the corresponding term root is e4f560... and proposition id is 16ff0a...

Proof:
Proof not loaded.

Theorem. (ex17_1_topology_from_closed_sets)

∀X Tx : set, closed_in X Tx X → (∀A : set, closed_in X Tx A → closed_in X Tx (X ∖ A)) → topology_on X Tx

In Proofgold the corresponding term root is 380cd0... and proposition id is b87d64...

Proof:
Proof not loaded.

Theorem. (ex17_2_closed_in_closed_subspace)

∀X Tx Y A : set, closed_in X Tx Y → closed_in Y (subspace_topology X Tx Y) A → closed_in X Tx A

In Proofgold the corresponding term root is f25458... and proposition id is acc8de...

Proof:
Proof not loaded.

Theorem. (ex17_3_product_of_closed_sets_closed)

∀X Tx Y Ty A B : set, closed_in X Tx A → closed_in Y Ty B → closed_in (setprod X Y) (product_topology X Tx Y Ty) (setprod A B)

In Proofgold the corresponding term root is 3d3b09... and proposition id is 29000d...

Proof:
Proof not loaded.

Theorem. (ex17_4_open_minus_closed_and_closed_minus_open)

∀X Tx U A : set, topology_on X Tx → open_in X Tx U → closed_in X Tx A → open_in X Tx (U ∖ A) ∧ closed_in X Tx (A ∖ U)

In Proofgold the corresponding term root is 80566d... and proposition id is 091281...

Proof:
Proof not loaded.

Theorem. (ex17_5_closure_of_interval_in_order_topology)

∀X a b : set, simply_ordered_set X → a ∈ X → b ∈ X → closure_of X (order_topology X) (order_interval X a b) ⊆ closed_interval_in X a b

In Proofgold the corresponding term root is c55c2f... and proposition id is 2812f8...

Proof:
Proof not loaded.

Definition. We define no_immediate_successor to be λX a ⇒ ∀c : set, c ∈ X → order_rel X a c → ∃x : set, x ∈ X ∧ order_rel X a x ∧ order_rel X x c of type set → set → prop.

In Proofgold the corresponding term root is d86a96... and object id is 4771c6...

Definition. We define no_immediate_predecessor to be λX b ⇒ ∀c : set, c ∈ X → order_rel X c b → ∃x : set, x ∈ X ∧ order_rel X c x ∧ order_rel X x b of type set → set → prop.

In Proofgold the corresponding term root is 37cbdd... and object id is d7d7d1...

Theorem. (ex17_5_basis_elem_meets_interval)

∀X a b x b0 : set, simply_ordered_set X → a ∈ X → b ∈ X → order_rel X a b → no_immediate_successor X a → no_immediate_predecessor X b → x ∈ closed_interval_in X a b → b0 ∈ order_topology_basis X → x ∈ b0 → ∃y : set, y ∈ b0 ∧ y ∈ order_interval X a b

In Proofgold the corresponding term root is a8d77d... and proposition id is f53628...

Proof:
Proof not loaded.

Theorem. (ex17_5_closure_of_interval_eq_conditions)

∀X a b : set, simply_ordered_set X → a ∈ X → b ∈ X → order_rel X a b → no_immediate_successor X a → no_immediate_predecessor X b → closure_of X (order_topology X) (order_interval X a b) = closed_interval_in X a b

In Proofgold the corresponding term root is 6fd46c... and proposition id is 74786e...

Proof:
Proof not loaded.

Theorem. (closure_idempotent_and_closed)

∀X Tx A : set, topology_on X Tx → closure_of X Tx (closure_of X Tx A) = closure_of X Tx A ∧ closed_in X Tx (closure_of X Tx A)

In Proofgold the corresponding term root is 9b66b0... and proposition id is 967e99...

Proof:
Proof not loaded.

Theorem. (ex17_6a_closure_monotone)

∀X Tx A B : set, topology_on X Tx → A ⊆ B → B ⊆ X → closure_of X Tx A ⊆ closure_of X Tx B

In Proofgold the corresponding term root is 7e4362... and proposition id is 96a594...

Proof:
Proof not loaded.

Theorem. (ex17_6b_closure_binunion)

∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → closure_of X Tx (A ∪ B) = closure_of X Tx A ∪ closure_of X Tx B

In Proofgold the corresponding term root is d4c4a5... and proposition id is 14181e...

Proof:
Proof not loaded.

Theorem. (ex17_6c_closure_Union_contains_Union_closures)

∀X Tx Fam : set, topology_on X Tx → (∀A : set, A ∈ Fam → A ⊆ X) → ⋃ {closure_of X Tx A|A ∈ Fam} ⊆ closure_of X Tx (⋃ Fam)

In Proofgold the corresponding term root is 518a5a... and proposition id is a52096...

Proof:
Proof not loaded.

Theorem. (ex17_7_counterexample_union_closure)

∃X Tx Fam : set, topology_on X Tx ∧ (∀A : set, A ∈ Fam → A ⊆ X) ∧ ¬ (closure_of X Tx (⋃ Fam) ⊆ ⋃ {closure_of X Tx A|A ∈ Fam})

In Proofgold the corresponding term root is fa1f53... and proposition id is b6aef3...

Proof:
Proof not loaded.

Theorem. (ex17_8a_closure_intersection_Subq_intersection_closures)

∀X Tx A B : set, topology_on X Tx → closure_of X Tx (A ∩ B) ⊆ closure_of X Tx A ∩ closure_of X Tx B

In Proofgold the corresponding term root is c96078... and proposition id is a800db...

Proof:
Proof not loaded.

Theorem. (ex17_8b_closure_intersection_family_Subq_intersection_closures)

∀X Tx Fam : set, topology_on X Tx → (∀A : set, A ∈ Fam → A ⊆ X) → closure_of X Tx (intersection_of_family X Fam) ⊆ intersection_of_family X {closure_of X Tx A|A ∈ Fam}

In Proofgold the corresponding term root is 45f50f... and proposition id is a95345...

Proof:
Proof not loaded.

Theorem. (ex17_8c_closure_setminus_Subq_closure_left)

∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → closure_of X Tx (A ∖ B) ⊆ closure_of X Tx A

In Proofgold the corresponding term root is 5957e2... and proposition id is a1a1b6...

Proof:
Proof not loaded.

Theorem. (ex17_8c_counterexample_equality_fails)

∃X Tx A B : set, topology_on X Tx ∧ A ⊆ X ∧ B ⊆ X ∧ closure_of X Tx (A ∖ B) ≠ (closure_of X Tx A ∖ closure_of X Tx B)

In Proofgold the corresponding term root is ea2fd0... and proposition id is 887958...

Proof:
Proof not loaded.

Theorem. (ex17_9_closure_of_product_subset)

∀X Y Tx Ty A B : set, topology_on X Tx → topology_on Y Ty → closure_of (setprod X Y) (product_topology X Tx Y Ty) (setprod A B) = setprod (closure_of X Tx A) (closure_of Y Ty B)

In Proofgold the corresponding term root is 59643f... and proposition id is 6246c1...

Proof:
Proof not loaded.

Theorem. (ex17_10_order_topology_Hausdorff)

∀X : set, simply_ordered_set X → Hausdorff_space X (order_topology X)

In Proofgold the corresponding term root is 87495d... and proposition id is 239e98...

Proof:
Proof not loaded.

Theorem. (ex17_11_product_Hausdorff)

∀X Tx Y Ty : set, Hausdorff_space X Tx → Hausdorff_space Y Ty → Hausdorff_space (setprod X Y) (product_topology X Tx Y Ty)

In Proofgold the corresponding term root is 09d25c... and proposition id is 9c43ac...

Proof:
Proof not loaded.

Theorem. (ex17_12_subspace_Hausdorff)

∀X Tx Y : set, Hausdorff_space X Tx → Y ⊆ X → Hausdorff_space Y (subspace_topology X Tx Y)

In Proofgold the corresponding term root is 5f9b35... and proposition id is 87bcfe...

Proof:
Proof not loaded.

Theorem. (ex17_13_diagonal_closed_iff_Hausdorff)

∀X Tx : set, topology_on X Tx → (Hausdorff_space X Tx ↔ closed_in (setprod X X) (product_topology X Tx X Tx) {(x,x)|x ∈ X})

In Proofgold the corresponding term root is e1f58e... and proposition id is f4ea1e...

Proof:
Proof not loaded.

Definition. We define seq_one_over_n to be (λn ∈ ω ⇒ {inv_nat (ordsucc n)}) of type set.

In Proofgold the corresponding term root is faf3a7... and object id is 55e1dc...

Theorem. (seq_one_over_n_apply)

∀n : set, n ∈ ω → apply_fun seq_one_over_n n = inv_nat (ordsucc n)

In Proofgold the corresponding term root is 71ec05... and proposition id is e65c8e...

Proof:
Proof not loaded.

Theorem. (inv_nat_ordsucc_inj)

∀n m : set, n ∈ ω → m ∈ ω → inv_nat (ordsucc n) = inv_nat (ordsucc m) → n = m

In Proofgold the corresponding term root is 1857ca... and proposition id is 67ab46...

Proof:
Proof not loaded.

Theorem. (omega_binunion)

∀a b : set, a ∈ ω → b ∈ ω → a ∪ b ∈ ω

In Proofgold the corresponding term root is 79572b... and proposition id is e14a6f...

Proof:
Proof not loaded.

Theorem. (seq_one_over_n_inj)

∀n m : set, n ∈ ω → m ∈ ω → apply_fun seq_one_over_n n = apply_fun seq_one_over_n m → n = m

In Proofgold the corresponding term root is 2b18f2... and proposition id is ad8990...

Proof:
Proof not loaded.

Theorem. (K_set_infinite)

infinite K_set

In Proofgold the corresponding term root is b07104... and proposition id is f6d95c...

Proof:
Proof not loaded.

Theorem. (ex17_14_sequence_in_finite_complement_topology)

∀x : set, x ∈ R → ∀U : set, U ∈ finite_complement_topology R → x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq_one_over_n n ∈ U

In Proofgold the corresponding term root is 7c7e1c... and proposition id is 7ae57b...

Proof:
Proof not loaded.

Theorem. (lemma_T1_singletons_closed)

∀X Tx : set, topology_on X Tx → (T1_space X Tx ↔ (∀x : set, x ∈ X → closed_in X Tx {x}))

In Proofgold the corresponding term root is 1cdf66... and proposition id is eacb50...

Proof:
Proof not loaded.

Theorem. (T1_singleton_complement_open)

∀X Tx x : set, T1_space X Tx → x ∈ X → X ∖ {x} ∈ Tx

In Proofgold the corresponding term root is 21d299... and proposition id is db2948...

Proof:
Proof not loaded.

Theorem. (subspace_T1)

∀X Tx Y : set, topology_on X Tx → Y ⊆ X → T1_space X Tx → T1_space Y (subspace_topology X Tx Y)

In Proofgold the corresponding term root is ed10e9... and proposition id is 926a41...

Proof:
Proof not loaded.

Theorem. (ex17_15_T1_characterization)

∀X Tx : set, topology_on X Tx → (T1_space X Tx ↔ ∀x y : set, x ∈ X → y ∈ X → x ≠ y → (∃U : set, U ∈ Tx ∧ x ∈ U ∧ y ∉ U) ∧ (∃V : set, V ∈ Tx ∧ y ∈ V ∧ x ∉ V))

In Proofgold the corresponding term root is 2bc34f... and proposition id is 10250e...

Proof:
Proof not loaded.

Definition. We define R_nonneg_set to be {x ∈ R|0 ≤ x} of type set.

In Proofgold the corresponding term root is d38283... and object id is 125ff2...

Theorem. (closure_of_K_in_R_K_topology)

closure_of R R_K_topology K_set = K_set

In Proofgold the corresponding term root is 183b7e... and proposition id is 2a5a58...

Proof:
Proof not loaded.

Theorem. (closure_of_K_in_R_ray_topology)

closure_of R R_ray_topology K_set = R_nonneg_set

In Proofgold the corresponding term root is d91f0a... and proposition id is 0e6ea9...

Proof:
Proof not loaded.

Theorem. (K_set_above_positive_bound_finite)

∀b : set, b ∈ R → Rlt 0 b → finite (K_set ∩ {y ∈ R|Rlt b y})

In Proofgold the corresponding term root is f2316b... and proposition id is 878585...

Proof:
Proof not loaded.

Theorem. (standard_open_neighborhood_disjoint_from_K_set_pos)

∀x : set, x ∈ R → 0 < x → ¬ (x ∈ K_set) → ∃U : set, U ∈ R_standard_topology ∧ x ∈ U ∧ U ∩ K_set = Empty

In Proofgold the corresponding term root is 9d2e2a... and proposition id is d64311...

Proof:
Proof not loaded.

Theorem. (closure_of_K_in_R_upper_limit_topology)

closure_of R R_upper_limit_topology K_set = K_set

In Proofgold the corresponding term root is 8b3dca... and proposition id is b60123...

Proof:
Proof not loaded.

Theorem. (closure_of_K_in_R_standard_topology)

closure_of R R_standard_topology K_set = K_set ∪ {0}

In Proofgold the corresponding term root is 16caaf... and proposition id is 4fa7a3...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is 5d56ff... and proposition id is f72363...

Proof:
Proof not loaded.

Theorem. (ray_topology_contains_0_if_contains_1)

∀U : set, U ∈ R_ray_topology → 1 ∈ U → 0 ∈ U

In Proofgold the corresponding term root is 2e8e87... and proposition id is 7186b7...

Proof:
Proof not loaded.

Theorem. (ray_topology_not_Hausdorff)

¬ Hausdorff_space R R_ray_topology

In Proofgold the corresponding term root is 4ea54f... and proposition id is 7fe48e...

Proof:
Proof not loaded.

Theorem. (ray_topology_not_T1)

¬ T1_space R R_ray_topology

In Proofgold the corresponding term root is 8bbd09... and proposition id is 631ffd...

Proof:
Proof not loaded.

Theorem. (setminus_Empty_eq)

∀X : set, X ∖ Empty = X

In Proofgold the corresponding term root is 20e108... and proposition id is ebb8ec...

Proof:
Proof not loaded.

Theorem. (infinite_R)

infinite R

In Proofgold the corresponding term root is c9b5ed... and proposition id is 1b6993...

Proof:
Proof not loaded.

Theorem. (finite_complement_topology_T1)

∀X : set, T1_space X (finite_complement_topology X)

In Proofgold the corresponding term root is 595ad3... and proposition id is 149038...

Proof:
Proof not loaded.

Theorem. (R_finite_complement_not_Hausdorff)

¬ Hausdorff_space R R_finite_complement_topology

In Proofgold the corresponding term root is 131871... and proposition id is 6fb3a5...

Proof:
Proof not loaded.

Theorem. (R_standard_topology_Hausdorff)

Hausdorff_space R R_standard_topology

In Proofgold the corresponding term root is a359cb... and proposition id is 5d3c0e...

Proof:
Proof not loaded.

Theorem. (R_standard_topology_T1)

T1_space R R_standard_topology

In Proofgold the corresponding term root is ab1e25... and proposition id is 1d92d8...

Proof:
Proof not loaded.

Theorem. (finer_preserves_Hausdorff)

∀X Tx Ty : set, Hausdorff_space X Tx → topology_on X Ty → Tx ⊆ Ty → Hausdorff_space X Ty

In Proofgold the corresponding term root is 44a607... and proposition id is fd0010...

Proof:
Proof not loaded.

Theorem. (R_upper_limit_topology_Hausdorff)

Hausdorff_space R R_upper_limit_topology

In Proofgold the corresponding term root is 02306c... and proposition id is 88f4f3...

Proof:
Proof not loaded.

Theorem. (R_upper_limit_topology_T1)

T1_space R R_upper_limit_topology

In Proofgold the corresponding term root is 763ab7... and proposition id is 731981...

Proof:
Proof not loaded.

Theorem. (R_K_topology_Hausdorff)

Hausdorff_space R R_K_topology

In Proofgold the corresponding term root is 305af2... and proposition id is 2304cd...

Proof:
Proof not loaded.

Theorem. (R_K_topology_T1)

T1_space R R_K_topology

In Proofgold the corresponding term root is c78c5b... and proposition id is 425e17...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is 093dbd... and proposition id is 6b012f...

Proof:
Proof not loaded.

Theorem. (real_2)

2 ∈ R

In Proofgold the corresponding term root is d2b6c1... and proposition id is 162da2...

Proof:
Proof not loaded.

Theorem. (ordsucc_2_eq_3)

ordsucc 2 = 3

In Proofgold the corresponding term root is 8a0684... and proposition id is 0f7314...

Proof:
Proof not loaded.

Theorem. (add_SNo_2_1_eq_3)

add_SNo 2 1 = 3

In Proofgold the corresponding term root is 4a4763... and proposition id is ed457c...

Proof:
Proof not loaded.

Theorem. (real_3)

3 ∈ R

In Proofgold the corresponding term root is d2c03e... and proposition id is 65aaa4...

Proof:
Proof not loaded.

Definition. We define sqrt2 to be sqrt_SNo_nonneg 2 of type set.

In Proofgold the corresponding term root is e4d87f... and object id is 5d2c5f...

Theorem. (sqrt2_in_R)

sqrt2 ∈ R

In Proofgold the corresponding term root is 4f429d... and proposition id is 6d2b5b...

Proof:
Proof not loaded.

Theorem. (SNoLt_0_sqrt2)

0 < sqrt2

In Proofgold the corresponding term root is 026aa1... and proposition id is 0b6a26...

Proof:
Proof not loaded.

Theorem. (three_in_rational_numbers)

3 ∈ rational_numbers

In Proofgold the corresponding term root is d69548... and proposition id is fbbd17...

Proof:
Proof not loaded.

Theorem. (sqrt2_not_rational_numbers)

sqrt2 ∉ rational_numbers

In Proofgold the corresponding term root is aab579... and proposition id is 1a9e30...

Proof:
Proof not loaded.

Definition. We define R_C_topology to be generated_topology R rational_halfopen_intervals_basis of type set.

In Proofgold the corresponding term root is 8a14c0... and object id is 101b91...

Theorem. (R_lower_limit_finer_than_R_C)

finer_than R_lower_limit_topology R_C_topology

In Proofgold the corresponding term root is f36bc2... and proposition id is 6352a1...

Proof:
Proof not loaded.

Definition. We define ex17_17_interval_A to be open_interval 0 sqrt2 of type set.

In Proofgold the corresponding term root is 8ae440... and object id is 0887ca...

Definition. We define ex17_17_interval_B to be open_interval sqrt2 3 of type set.

In Proofgold the corresponding term root is 514f5f... and object id is bdca0e...

Definition. We define ex17_17_interval_A_closure_lower to be {x ∈ R|0 ≤ x ∧ x < sqrt2} of type set.

In Proofgold the corresponding term root is ac703e... and object id is 3cb65e...

Definition. We define ex17_17_interval_A_closure_C to be {x ∈ R|0 ≤ x ∧ x ≤ sqrt2} of type set.

In Proofgold the corresponding term root is d50c0b... and object id is 643504...

Definition. We define ex17_17_interval_B_closure_lower to be {x ∈ R|sqrt2 ≤ x ∧ x < 3} of type set.

In Proofgold the corresponding term root is 6a60cf... and object id is f01f05...

Theorem. (ex17_17_closure_A_lower_Subq)

closure_of R R_lower_limit_topology ex17_17_interval_A ⊆ ex17_17_interval_A_closure_lower

In Proofgold the corresponding term root is c902ea... and proposition id is d6f275...

Proof:
Proof not loaded.

Theorem. (ex17_17_closure_A_lower_Supq)

ex17_17_interval_A_closure_lower ⊆ closure_of R R_lower_limit_topology ex17_17_interval_A

In Proofgold the corresponding term root is 3ce800... and proposition id is 9eedd6...

Proof:
Proof not loaded.

Theorem. (ex17_17_closure_B_lower_Subq)

closure_of R R_lower_limit_topology ex17_17_interval_B ⊆ ex17_17_interval_B_closure_lower

In Proofgold the corresponding term root is 356671... and proposition id is 2e3abb...

Proof:
Proof not loaded.

Theorem. (ex17_17_closure_B_lower_Supq)

ex17_17_interval_B_closure_lower ⊆ closure_of R R_lower_limit_topology ex17_17_interval_B

In Proofgold the corresponding term root is 6c61ac... and proposition id is 55c809...

Proof:
Proof not loaded.

Theorem. (ex17_17_closure_A_C_Subq)

closure_of R R_C_topology ex17_17_interval_A ⊆ ex17_17_interval_A_closure_C

In Proofgold the corresponding term root is b85997... and proposition id is ed3394...

Proof:
Proof not loaded.

Theorem. (ex17_17_closure_A_C_Supq)

ex17_17_interval_A_closure_C ⊆ closure_of R R_C_topology ex17_17_interval_A

In Proofgold the corresponding term root is 854c16... and proposition id is beb37e...

Proof:
Proof not loaded.

Theorem. (ex17_17_closure_B_C_Subq)

closure_of R R_C_topology ex17_17_interval_B ⊆ ex17_17_interval_B_closure_lower

In Proofgold the corresponding term root is 4a718a... and proposition id is 87e52d...

Proof:
Proof not loaded.

Theorem. (ex17_17_closure_B_C_Supq)

ex17_17_interval_B_closure_lower ⊆ closure_of R R_C_topology ex17_17_interval_B

In Proofgold the corresponding term root is ee3a8f... and proposition id is b1d40d...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is cfbfc5... and proposition id is c18e61...

Proof:
Proof not loaded.

Definition. We define ordsq_A to be {(inv_nat n,0)|n ∈ ω ∖ {0}} of type set.

In Proofgold the corresponding term root is ee31ec... and object id is 74f256...

Definition. We define ordsq_B to be {(add_SNo 1 (minus_SNo (inv_nat n)),eps_ 1)|n ∈ ω ∖ {0}} of type set.

In Proofgold the corresponding term root is 0775d8... and object id is 5cb27e...

Definition. We define ordsq_C to be {p ∈ ordered_square|∃x : set, p = (x,0) ∧ Rlt 0 x ∧ Rlt x 1} of type set.

In Proofgold the corresponding term root is 1a1314... and object id is f14d96...

Definition. We define ordsq_D to be {p ∈ ordered_square|∃x : set, p = (x,eps_ 1) ∧ Rlt 0 x ∧ Rlt x 1} of type set.

In Proofgold the corresponding term root is 86e730... and object id is d7a1f8...

Definition. We define ordsq_E to be {p ∈ ordered_square|∃y : set, p = (eps_ 1,y) ∧ Rlt 0 y ∧ Rlt y 1} of type set.

In Proofgold the corresponding term root is 27a712... and object id is 66450d...

Definition. We define ordsq_p01 to be (0,1) of type set.

In Proofgold the corresponding term root is 4d20b0... and object id is 2a85c6...

Definition. We define ordsq_p10 to be (1,0) of type set.

In Proofgold the corresponding term root is 719ba8... and object id is 392713...

Definition. We define ordsq_E_closure to be ordsq_E ∪ {(eps_ 1,0)} ∪ {(eps_ 1,1)} of type set.

In Proofgold the corresponding term root is ffc2cc... and object id is be6dcd...

Definition. We define ordsq_top_edge_lt1 to be {p ∈ ordered_square|∃x : set, p = (x,1) ∧ x ∈ unit_interval ∧ Rlt x 1} of type set.

In Proofgold the corresponding term root is a72a0c... and object id is 08bb6e...

Definition. We define ordsq_C_closure to be ordsq_C ∪ ordsq_top_edge_lt1 ∪ {ordsq_p10} of type set.

In Proofgold the corresponding term root is 7056c1... and object id is 3c3d78...

Definition. We define ordsq_D_closure to be ordsq_D ∪ ordsq_C ∪ ordsq_top_edge_lt1 ∪ {ordsq_p10} of type set.

In Proofgold the corresponding term root is 613b43... and object id is c646fe...

Theorem. (simply_ordered_set_setprod_R_R)

simply_ordered_set (setprod R R)

In Proofgold the corresponding term root is b2d9ff... and proposition id is 89baec...

Proof:
Proof not loaded.

Theorem. (ordered_square_sub_setprod_R_R)

ordered_square ⊆ setprod R R

In Proofgold the corresponding term root is 2c34e1... and proposition id is cbd236...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_R_R_same_first)

∀x a2 b2 : set, Rlt a2 b2 → order_rel (setprod R R) (x,a2) (x,b2)

In Proofgold the corresponding term root is cbb9e1... and proposition id is b662d6...

Proof:
Proof not loaded.

Theorem. (order_rel_setprod_R_R_disj)

∀a1 a2 b1 b2 : set, order_rel (setprod R R) (a1,a2) (b1,b2) → Rlt a1 b1 ∨ (a1 = b1 ∧ Rlt a2 b2)

In Proofgold the corresponding term root is f29984... and proposition id is 4156ad...

Proof:
Proof not loaded.

Theorem. (ordsq_eps1_eps1_in_E)

(eps_ 1,eps_ 1) ∈ ordsq_E

In Proofgold the corresponding term root is 34bd8a... and proposition id is 316624...

Proof:
Proof not loaded.

Theorem. (ex17_18_closure_E)

closure_of ordered_square ordered_square_topology ordsq_E = ordsq_E_closure

In Proofgold the corresponding term root is abb7dc... and proposition id is 1e7621...

Proof:
Proof not loaded.

Theorem. (inv_nat_Rle_1_local)

∀n : set, n ∈ ω ∖ {0} → Rle (inv_nat n) 1

In Proofgold the corresponding term root is f96a98... and proposition id is 725836...

Proof:
Proof not loaded.

Theorem. (inv_nat_in_unit_interval)

∀n : set, n ∈ ω ∖ {0} → inv_nat n ∈ unit_interval

In Proofgold the corresponding term root is 3c77ed... and proposition id is 2782b7...

Proof:
Proof not loaded.

Theorem. (ordsq_B_sub_ordered_square)

ordsq_B ⊆ ordered_square

In Proofgold the corresponding term root is f06734... and proposition id is 537445...

Proof:
Proof not loaded.

Theorem. (div_SNo_1_eq_inv_nat_local)

∀n : set, SNo n → div_SNo 1 n = inv_nat n

In Proofgold the corresponding term root is 695f62... and proposition id is 9f4e83...

Proof:
Proof not loaded.

Theorem. (exists_inv_nat_ordsucc_lt_local)

∀r : set, r ∈ R → Rlt 0 r → ∃N : set, N ∈ ω ∧ Rlt (inv_nat (ordsucc N)) r

In Proofgold the corresponding term root is 93ca9a... and proposition id is 423ad2...

Proof:
Proof not loaded.

Theorem. (inv_nat_ordsucc_antitone_local2)

∀i j : set, i ∈ ω → j ∈ ω → i ∈ j → Rlt (inv_nat (ordsucc j)) (inv_nat (ordsucc i))

In Proofgold the corresponding term root is ce3bf0... and proposition id is 808d77...

Proof:
Proof not loaded.

Theorem. (ex17_18_closure_A_Supq)

ordsq_A ∪ {ordsq_p01} ⊆ closure_of ordered_square ordered_square_topology ordsq_A

In Proofgold the corresponding term root is 61a693... and proposition id is 27da31...

Proof:
Proof not loaded.

Theorem. (ex17_18_closure_A_Subq)

closure_of ordered_square ordered_square_topology ordsq_A ⊆ ordsq_A ∪ {ordsq_p01}

In Proofgold the corresponding term root is 98dd6a... and proposition id is 4533ff...

Proof:
Proof not loaded.

Theorem. (ex17_18_p10_interval_basis_meets_B)

∀a0 b0 : set, a0 ∈ ordered_square → b0 ∈ ordered_square → order_rel (setprod R R) a0 ordsq_p10 → order_rel (setprod R R) ordsq_p10 b0 → ∃w : set, w ∈ {x0 ∈ ordered_square|order_rel (setprod R R) a0 x0 ∧ order_rel (setprod R R) x0 b0} ∧ w ∈ ordsq_B

In Proofgold the corresponding term root is 6d9fea... and proposition id is fa48cb...

Proof:
Proof not loaded.

Theorem. (ex17_18_p10_lower_basis_meets_B)

∀b0 : set, b0 ∈ ordered_square → order_rel (setprod R R) ordsq_p10 b0 → ∃w : set, w ∈ {x0 ∈ ordered_square|order_rel (setprod R R) x0 b0} ∧ w ∈ ordsq_B

In Proofgold the corresponding term root is 68d1b9... and proposition id is 5da623...

Proof:
Proof not loaded.

Theorem. (ex17_18_p10_upper_basis_meets_B)

∀a0 : set, a0 ∈ ordered_square → order_rel (setprod R R) a0 ordsq_p10 → ∃w : set, w ∈ {x0 ∈ ordered_square|order_rel (setprod R R) a0 x0} ∧ w ∈ ordsq_B

In Proofgold the corresponding term root is 7be464... and proposition id is 0e42c1...

Proof:
Proof not loaded.

Theorem. (ordsq_B_lt_p10)

∀w : set, w ∈ ordsq_B → order_rel (setprod R R) w ordsq_p10

In Proofgold the corresponding term root is b1743b... and proposition id is 5d653c...

Proof:
Proof not loaded.

Theorem. (ordsq_B_ordsucc_index_increasing)

∀i j : set, i ∈ ω → j ∈ ω → i ∈ j → order_rel (setprod R R) (add_SNo 1 (minus_SNo (inv_nat (ordsucc i))),eps_ 1) (add_SNo 1 (minus_SNo (inv_nat (ordsucc j))),eps_ 1)

In Proofgold the corresponding term root is 7a784d... and proposition id is 9502ec...

Proof:
Proof not loaded.

Theorem. (ex17_18_closure_B_eq)

closure_of ordered_square ordered_square_topology ordsq_B = ordsq_B ∪ {ordsq_p10}

In Proofgold the corresponding term root is da585a... and proposition id is 2fed81...

Proof:
Proof not loaded.

Theorem. (ex17_18_top_edge_lt1_Sub_closure_C)

ordsq_top_edge_lt1 ⊆ closure_of ordered_square ordered_square_topology ordsq_C

In Proofgold the corresponding term root is 0d55ec... and proposition id is 3a0d00...

Proof:
Proof not loaded.

Theorem. (ex17_18_p10_in_closure_C)

ordsq_p10 ∈ closure_of ordered_square ordered_square_topology ordsq_C

In Proofgold the corresponding term root is 8c8a29... and proposition id is f6d59d...

Proof:
Proof not loaded.

Theorem. (ex17_18_closure_C_eq)

closure_of ordered_square ordered_square_topology ordsq_C = ordsq_C_closure

In Proofgold the corresponding term root is 616f08... and proposition id is 4efcbb...

Proof:
Proof not loaded.

Theorem. (ex17_18_p10_in_closure_D)

ordsq_p10 ∈ closure_of ordered_square ordered_square_topology ordsq_D

In Proofgold the corresponding term root is 148ca7... and proposition id is 50e5ae...

Proof:
Proof not loaded.

Theorem. (ex17_18_C_Sub_closure_D)

∀p : set, p ∈ ordsq_C → p ∈ closure_of ordered_square ordered_square_topology ordsq_D

In Proofgold the corresponding term root is bcdab9... and proposition id is 44e6a2...

Proof:
Proof not loaded.

Theorem. (ex17_18_top_edge_lt1_Sub_closure_D)

ordsq_top_edge_lt1 ⊆ closure_of ordered_square ordered_square_topology ordsq_D

In Proofgold the corresponding term root is 9f1e25... and proposition id is 4e0f8f...

Proof:
Proof not loaded.

Theorem. (ex17_18_closure_D_eq)

closure_of ordered_square ordered_square_topology ordsq_D = ordsq_D_closure

In Proofgold the corresponding term root is 511808... and proposition id is 28724e...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is 114b37... and proposition id is d188c1...

Proof:
Proof not loaded.

Definition. We define boundary_of to be λX Tx A ⇒ closure_of X Tx A ∩ closure_of X Tx (X ∖ A) of type set → set → set → set.

In Proofgold the corresponding term root is da6db2... and object id is 6a949d...

Theorem. (ex17_19_boundary_properties)

∀X Tx A : set, topology_on X Tx → boundary_of X Tx A ⊆ closure_of X Tx A ∧ boundary_of X Tx A ⊆ closure_of X Tx (X ∖ A)

In Proofgold the corresponding term root is 9cecb1... and proposition id is e14723...

Proof:
Proof not loaded.

Theorem. (ex17_20_boundaries_and_interiors_in_R2)

boundary_of (setprod R R) R2_standard_topology ordered_square_open_strip ≠ boundary_of (setprod R R) R2_dictionary_order_topology ordered_square_open_strip

In Proofgold the corresponding term root is 288de0... and proposition id is 9da9db...

Proof:
Proof not loaded.

Theorem. (ex17_21_Kuratowski_closure_complement_maximal)

∀X : set, closure_of X (discrete_topology X) (X ∖ Empty) = X

In Proofgold the corresponding term root is 9bcfc3... and proposition id is 87a00f...

Proof:
Proof not loaded.

Definition. We define preimage_of to be λX f V ⇒ {x ∈ X|apply_fun f x ∈ V} of type set → set → set → set.

In Proofgold the corresponding term root is 59c401... and object id is d2d45d...

Theorem. (preimage_of_Union)

∀X f Fam : set, preimage_of X f (⋃ Fam) = ⋃ {preimage_of X f V|V ∈ Fam}

In Proofgold the corresponding term root is ec775d... and proposition id is 0df6b5...

Proof:
Proof not loaded.

Theorem. (preimage_of_binintersect)

∀X f U V : set, preimage_of X f (U ∩ V) = (preimage_of X f U) ∩ (preimage_of X f V)

In Proofgold the corresponding term root is 57395f... and proposition id is a343b2...

Proof:
Proof not loaded.

Theorem. (preimage_of_mono)

∀X f V W : set, V ⊆ W → preimage_of X f V ⊆ preimage_of X f W

In Proofgold the corresponding term root is 4c2e95... and proposition id is bebc4c...

Proof:
Proof not loaded.

Theorem. (preimage_of_binunion)

∀X f U V : set, preimage_of X f (U ∪ V) = (preimage_of X f U) ∪ (preimage_of X f V)

In Proofgold the corresponding term root is 82f712... and proposition id is dcf1e3...

Proof:
Proof not loaded.

Theorem. (preimage_of_setminus)

∀X f U V : set, preimage_of X f (U ∖ V) = (preimage_of X f U) ∖ (preimage_of X f V)

In Proofgold the corresponding term root is aeb03d... and proposition id is f8cf3f...

Proof:
Proof not loaded.

Theorem. (preimage_of_Empty)

∀X f : set, preimage_of X f Empty = Empty

In Proofgold the corresponding term root is 90c82f... and proposition id is ec465a...

Proof:
Proof not loaded.

Theorem. (preimage_of_whole)

∀X Y f : set, function_on f X Y → preimage_of X f Y = X

In Proofgold the corresponding term root is 52d7b6... and proposition id is 1e1fcb...

Proof:
Proof not loaded.

Theorem. (preimage_of_complement)

∀X Y f V : set, function_on f X Y → preimage_of X f (Y ∖ V) = X ∖ preimage_of X f V

In Proofgold the corresponding term root is 2349d0... and proposition id is 3cfb27...

Proof:
Proof not loaded.

Theorem. (projection1_apply)

∀X Y p : set, p ∈ setprod X Y → apply_fun (projection1 X Y) p = p 0

In Proofgold the corresponding term root is 1c4abc... and proposition id is 6e6364...

Proof:
Proof not loaded.

Theorem. (projection2_apply)

∀X Y p : set, p ∈ setprod X Y → apply_fun (projection2 X Y) p = p 1

In Proofgold the corresponding term root is 5c50db... and proposition id is 63bd70...

Proof:
Proof not loaded.

Theorem. (preimage_projection1_rectangle)

∀X Y U : set, U ⊆ X → preimage_of (setprod X Y) (projection1 X Y) U = rectangle_set U Y

In Proofgold the corresponding term root is f958c2... and proposition id is 338bcc...

Proof:
Proof not loaded.

Theorem. (preimage_projection2_rectangle)

∀X Y V : set, V ⊆ Y → preimage_of (setprod X Y) (projection2 X Y) V = rectangle_set X V

In Proofgold the corresponding term root is 04aff6... and proposition id is 91440e...

Proof:
Proof not loaded.

Definition. We define continuous_map to be λX Tx Y Ty f ⇒ topology_on X Tx ∧ topology_on Y Ty ∧ function_on f X Y ∧ ∀V : set, V ∈ Ty → preimage_of X f V ∈ Tx of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is 872dec... and object id is 946fc1...

Theorem. (continuous_map_topology_dom)

∀X Tx Y Ty f : set, continuous_map X Tx Y Ty f → topology_on X Tx

In Proofgold the corresponding term root is fe7968... and proposition id is c62e02...

Proof:
Proof not loaded.

Theorem. (continuous_map_topology_cod)

∀X Tx Y Ty f : set, continuous_map X Tx Y Ty f → topology_on Y Ty

In Proofgold the corresponding term root is 3e3457... and proposition id is 541523...

Proof:
Proof not loaded.

Theorem. (continuous_map_function_on)

∀X Tx Y Ty f : set, continuous_map X Tx Y Ty f → function_on f X Y

In Proofgold the corresponding term root is 430212... and proposition id is 346c87...

Proof:
Proof not loaded.

Theorem. (continuous_map_preimage)

∀X Tx Y Ty f : set, continuous_map X Tx Y Ty f → ∀V : set, V ∈ Ty → preimage_of X f V ∈ Tx

In Proofgold the corresponding term root is 706824... and proposition id is 96f34c...

Proof:
Proof not loaded.

Theorem. (continuous_map_domain_finer)

∀X Tx Tx' Y Ty f : set, continuous_map X Tx Y Ty f → topology_on X Tx' → Tx ⊆ Tx' → continuous_map X Tx' Y Ty f

In Proofgold the corresponding term root is de9153... and proposition id is fab995...

Proof:
Proof not loaded.

Theorem. (continuous_map_codomain_coarser)

∀X Tx Y Ty Ty' f : set, continuous_map X Tx Y Ty f → topology_on Y Ty' → Ty' ⊆ Ty → continuous_map X Tx Y Ty' f

In Proofgold the corresponding term root is e9c83b... and proposition id is 92a245...

Proof:
Proof not loaded.

Theorem. (continuous_map_range_restrict)

∀X Tx Y Ty f Z0 : set, continuous_map X Tx Y Ty f → Z0 ⊆ Y → (∀x : set, x ∈ X → apply_fun f x ∈ Z0) → continuous_map X Tx Z0 (subspace_topology Y Ty Z0) f

In Proofgold the corresponding term root is efbb0f... and proposition id is 4e5733...

Proof:
Proof not loaded.

Theorem. (continuous_map_range_expand)

∀X Tx Y Ty Z0 Tz0 f : set, continuous_map X Tx Y Ty f → Y ⊆ Z0 → topology_on Z0 Tz0 → Ty = subspace_topology Z0 Tz0 Y → continuous_map X Tx Z0 Tz0 f

In Proofgold the corresponding term root is 019fc1... and proposition id is 4ce562...

Proof:
Proof not loaded.

Theorem. (continuous_map_from_subbasis)

∀X Tx Y S f : set, topology_on X Tx → function_on f X Y → subbasis_on Y S → (∀s : set, s ∈ S → preimage_of X f s ∈ Tx) → continuous_map X Tx Y (generated_topology_from_subbasis Y S) f

In Proofgold the corresponding term root is c838a3... and proposition id is 4f4535...

Proof:
Proof not loaded.

Definition. We define continuous_map_total to be λX Tx Y Ty f ⇒ topology_on X Tx ∧ topology_on Y Ty ∧ total_function_on f X Y ∧ ∀V : set, V ∈ Ty → preimage_of X f V ∈ Tx of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is ae524c... and object id is db349d...

Theorem. (continuous_map_total_topology_dom)

∀X Tx Y Ty f : set, continuous_map_total X Tx Y Ty f → topology_on X Tx

In Proofgold the corresponding term root is 060e4a... and proposition id is b9c681...

Proof:
Proof not loaded.

Theorem. (continuous_map_total_topology_cod)

∀X Tx Y Ty f : set, continuous_map_total X Tx Y Ty f → topology_on Y Ty

In Proofgold the corresponding term root is 6ad3f5... and proposition id is f44b69...

Proof:
Proof not loaded.

Theorem. (continuous_map_total_total_function_on)

∀X Tx Y Ty f : set, continuous_map_total X Tx Y Ty f → total_function_on f X Y

In Proofgold the corresponding term root is 3bfeae... and proposition id is 9f39e1...

Proof:
Proof not loaded.

Theorem. (continuous_map_total_preimage)

∀X Tx Y Ty f : set, continuous_map_total X Tx Y Ty f → ∀V : set, V ∈ Ty → preimage_of X f V ∈ Tx

In Proofgold the corresponding term root is a089c3... and proposition id is b7bf7e...

Proof:
Proof not loaded.

Theorem. (continuous_map_total_imp)

∀X Tx Y Ty f : set, continuous_map_total X Tx Y Ty f → continuous_map X Tx Y Ty f

In Proofgold the corresponding term root is 3c55cc... and proposition id is 0c77f2...

Proof:
Proof not loaded.

Theorem. (const_fun_continuous)

∀X Tx Y Ty x : set, topology_on X Tx → topology_on Y Ty → x ∈ Y → continuous_map X Tx Y Ty (const_fun X x)

In Proofgold the corresponding term root is a2349b... and proposition id is 3f0b0f...

Proof:
Proof not loaded.

Theorem. (const_fun_continuous_total)

∀X Tx Y Ty x : set, topology_on X Tx → topology_on Y Ty → x ∈ Y → continuous_map_total X Tx Y Ty (const_fun X x)

In Proofgold the corresponding term root is 2e5ac7... and proposition id is 40a04d...

Proof:
Proof not loaded.

Theorem. (continuous_preserves_closed)

∀X Tx Y Ty f : set, continuous_map X Tx Y Ty f → ∀C : set, closed_in Y Ty C → closed_in X Tx (preimage_of X f C)

In Proofgold the corresponding term root is 79b548... and proposition id is f7321e...

Proof:
Proof not loaded.

Theorem. (continuous_local_neighborhood)

∀X Tx Y Ty f : set, topology_on X Tx → topology_on Y Ty → function_on f X Y → (∀V : set, V ∈ Ty → preimage_of X f V ∈ Tx) → ∀x : set, x ∈ X → ∀V : set, V ∈ Ty → apply_fun f x ∈ V → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ ∀u : set, u ∈ U → apply_fun f u ∈ V

In Proofgold the corresponding term root is d1227f... and proposition id is 2c38e5...

Proof:
Proof not loaded.

Theorem. (topology_elem_of_local_neighborhoods)

∀X Tx U : set, topology_on X Tx → U ⊆ X → (∀x : set, x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ V ⊆ U) → U ∈ Tx

In Proofgold the corresponding term root is 813a61... and proposition id is 632e28...

Proof:
Proof not loaded.

Definition. We define continuous_at to be λf x ⇒ function_on f R R ∧ x ∈ R ∧ ∀V : set, V ∈ R_standard_topology → apply_fun f x ∈ V → ∃U : set, U ∈ R_standard_topology ∧ x ∈ U ∧ ∀u : set, u ∈ U → apply_fun f u ∈ V of type set → set → prop.

In Proofgold the corresponding term root is 2e4eaa... and object id is b36247...

Theorem. (continuity_equiv_forms)

∀X Tx Y Ty f : set, topology_on X Tx → topology_on Y Ty → (continuous_map X Tx Y Ty f ↔ function_on f X Y ∧ (∀V : set, V ∈ Ty → preimage_of X f V ∈ Tx) ∧ (∀C : set, closed_in Y Ty C → closed_in X Tx (preimage_of X f C)) ∧ (∀x : set, x ∈ X → ∀V : set, V ∈ Ty → apply_fun f x ∈ V → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ ∀u : set, u ∈ U → apply_fun f u ∈ V))

In Proofgold the corresponding term root is 49bf9b... and proposition id is 7c2746...

Proof:
Proof not loaded.

Theorem. (identity_continuous)

∀X Tx : set, topology_on X Tx → let id ≔ {(x,x)|x ∈ X} in continuous_map X Tx X Tx id

In Proofgold the corresponding term root is 5cdcf8... and proposition id is 9e9468...

Proof:
Proof not loaded.

Theorem. (identity_continuous_R_lower_to_standard)

continuous_map R R_lower_limit_topology R R_standard_topology {(x,x)|x ∈ R}

In Proofgold the corresponding term root is 3e33c9... and proposition id is 77f691...

Proof:
Proof not loaded.

Definition. We define compose_fun to be λX f g ⇒ {(x,apply_fun g (apply_fun f x))|x ∈ X} of type set → set → set → set.

In Proofgold the corresponding term root is 133736... and object id is e504e1...

Theorem. (graph_domain_subset_compose_fun)

∀X f g : set, graph_domain_subset (compose_fun X f g) X

In Proofgold the corresponding term root is 23821f... and proposition id is 154736...

Proof:
Proof not loaded.

Theorem. (functional_graph_compose_fun)

∀X f g : set, functional_graph (compose_fun X f g)

In Proofgold the corresponding term root is fbc1b2... and proposition id is b399cd...

Proof:
Proof not loaded.

Theorem. (function_on_compose_fun)

∀X Y Z f g : set, function_on f X Y → function_on g Y Z → function_on (compose_fun X f g) X Z

In Proofgold the corresponding term root is 797127... and proposition id is 6351c6...

Proof:
Proof not loaded.

Theorem. (total_function_on_compose_fun)

∀X Y Z f g : set, function_on f X Y → function_on g Y Z → total_function_on (compose_fun X f g) X Z

In Proofgold the corresponding term root is a27cbf... and proposition id is 4619a9...

Proof:
Proof not loaded.

Theorem. (compose_fun_in_function_space)

∀X Y Z f g : set, function_on f X Y → function_on g Y Z → compose_fun X f g ∈ function_space X Z

In Proofgold the corresponding term root is 5bf797... and proposition id is a285b6...

Proof:
Proof not loaded.

Theorem. (compose_fun_apply)

∀X f g x : set, x ∈ X → apply_fun (compose_fun X f g) x = apply_fun g (apply_fun f x)

In Proofgold the corresponding term root is f6c9af... and proposition id is ac28df...

Proof:
Proof not loaded.

Theorem. (preimage_compose_fun)

∀X Y f g W : set, function_on f X Y → preimage_of X (compose_fun X f g) W = preimage_of X f (preimage_of Y g W)

In Proofgold the corresponding term root is cd8eb9... and proposition id is a6683c...

Proof:
Proof not loaded.

Theorem. (composition_continuous)

∀X Tx Y Ty Z Tz f g : set, continuous_map X Tx Y Ty f → continuous_map Y Ty Z Tz g → continuous_map X Tx Z Tz (compose_fun X f g)

In Proofgold the corresponding term root is b17b80... and proposition id is 9d3a43...

Proof:
Proof not loaded.

Theorem. (continuous_construction_rules)

∀X Tx Y Ty Z Tz : set, topology_on X Tx → topology_on Y Ty → topology_on Z Tz → (∀y0 : set, y0 ∈ Y → continuous_map X Tx Y Ty (const_fun X y0)) ∧ (∀A : set, A ⊆ X → continuous_map A (subspace_topology X Tx A) X Tx {(y,y)|y ∈ A}) ∧ (∀f g : set, continuous_map X Tx Y Ty f → continuous_map Y Ty Z Tz g → continuous_map X Tx Z Tz (compose_fun X f g)) ∧ (∀f A : set, A ⊆ X → continuous_map X Tx Y Ty f → continuous_map A (subspace_topology X Tx A) Y Ty f) ∧ ((∀f Z0 : set, continuous_map X Tx Y Ty f → Z0 ⊆ Y → (∀x : set, x ∈ X → apply_fun f x ∈ Z0) → continuous_map X Tx Z0 (subspace_topology Y Ty Z0) f) ∧ (∀f Z0 Tz0 : set, continuous_map X Tx Y Ty f → Y ⊆ Z0 → topology_on Z0 Tz0 → Ty = subspace_topology Z0 Tz0 Y → continuous_map X Tx Z0 Tz0 f)) ∧ (∀f : set, (∃UFam : set, UFam ⊆ Tx ∧ ⋃ UFam = X ∧ (∀U : set, U ∈ UFam → continuous_map U (subspace_topology X Tx U) Y Ty f)) → continuous_map X Tx Y Ty f)

In Proofgold the corresponding term root is 3f7276... and proposition id is 101fc6...

Proof:
Proof not loaded.

Theorem. (continuous_map_local_cover)

∀X Tx Y Ty f : set, topology_on X Tx → topology_on Y Ty → (∃UFam : set, UFam ⊆ Tx ∧ ⋃ UFam = X ∧ (∀U : set, U ∈ UFam → continuous_map U (subspace_topology X Tx U) Y Ty f)) → continuous_map X Tx Y Ty f

In Proofgold the corresponding term root is e20eca... and proposition id is 8aad77...

Proof:
Proof not loaded.

Theorem. (continuous_on_subspace_rule)

∀X Tx Y Ty f A : set, topology_on X Tx → topology_on Y Ty → A ⊆ X → continuous_map X Tx Y Ty f → continuous_map A (subspace_topology X Tx A) Y Ty f

In Proofgold the corresponding term root is 92d27c... and proposition id is 31726e...

Proof:
Proof not loaded.

Theorem. (function_on_subdomain)

∀f X Y A : set, function_on f X Y → A ⊆ X → function_on f A Y

In Proofgold the corresponding term root is d0a4a6... and proposition id is 61765c...

Proof:
Proof not loaded.

Theorem. (function_on_codomain)

∀f X Y Z : set, function_on f X Y → Y ⊆ Z → function_on f X Z

In Proofgold the corresponding term root is 9f903b... and proposition id is bef3a0...

Proof:
Proof not loaded.

Theorem. (total_function_on_codomain)

∀f X Y Z : set, total_function_on f X Y → Y ⊆ Z → total_function_on f X Z

In Proofgold the corresponding term root is 18a881... and proposition id is 739fa6...

Proof:
Proof not loaded.

Theorem. (continuous_map_congr_on)

∀X Tx Y Ty f g : set, continuous_map X Tx Y Ty f → function_on g X Y → (∀x : set, x ∈ X → apply_fun f x = apply_fun g x) → continuous_map X Tx Y Ty g

In Proofgold the corresponding term root is bf32bb... and proposition id is 95755c...

Proof:
Proof not loaded.

Theorem. (flip_unit_interval_continuous)

continuous_map unit_interval unit_interval_topology unit_interval unit_interval_topology flip_unit_interval

In Proofgold the corresponding term root is 0b962c... and proposition id is baeb84...

Proof:
Proof not loaded.

Definition. We define homeomorphism to be λX Tx Y Ty f ⇒ continuous_map X Tx Y Ty f ∧ ∃g : set, continuous_map Y Ty X Tx g ∧ (∀x : set, x ∈ X → apply_fun g (apply_fun f x) = x) ∧ (∀y : set, y ∈ Y → apply_fun f (apply_fun g y) = y) of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is b745fd... and object id is 7e1d0c...

Theorem. (homeomorphism_continuous)

∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty f → continuous_map X Tx Y Ty f

In Proofgold the corresponding term root is cacfde... and proposition id is 5c0d8d...

Proof:
Proof not loaded.

Theorem. (homeomorphism_inverse_package)

∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty f → ∃g : set, continuous_map Y Ty X Tx g ∧ (∀x : set, x ∈ X → apply_fun g (apply_fun f x) = x) ∧ (∀y : set, y ∈ Y → apply_fun f (apply_fun g y) = y)

In Proofgold the corresponding term root is 9c4a1c... and proposition id is cf0161...

Proof:
Proof not loaded.

Theorem. (homeomorphism_topology_left)

∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty f → topology_on X Tx

In Proofgold the corresponding term root is 8238f7... and proposition id is 0ce950...

Proof:
Proof not loaded.

Theorem. (homeomorphism_topology_right)

∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty f → topology_on Y Ty

In Proofgold the corresponding term root is 83e52a... and proposition id is 817342...

Proof:
Proof not loaded.

Definition. We define affine_line_R2 to be λa b c ⇒ {p ∈ EuclidPlane|add_SNo (mul_SNo a (R2_xcoord p)) (mul_SNo b (R2_ycoord p)) = c} of type set → set → set → set.

In Proofgold the corresponding term root is 46756c... and object id is 9a215c...

Theorem. (affine_line_R2_subset)

∀a b c : set, affine_line_R2 a b c ⊆ EuclidPlane

In Proofgold the corresponding term root is aa4623... and proposition id is 13444b...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_in_Power)

∀a b c : set, affine_line_R2 a b c ∈ 𝒫 EuclidPlane

In Proofgold the corresponding term root is 47ea5a... and proposition id is 1f082e...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_subset_R2)

∀a b c : set, affine_line_R2 a b c ⊆ setprod R R

In Proofgold the corresponding term root is aa4623... and proposition id is 13444b...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_in_Power_R2)

∀a b c : set, affine_line_R2 a b c ∈ 𝒫 (setprod R R)

In Proofgold the corresponding term root is 47ea5a... and proposition id is 1f082e...

Proof:
Proof not loaded.

Definition. We define affine_line_R2_param_by_x to be λa b c ⇒ graph R (λx : set ⇒ (x,div_SNo (add_SNo c (minus_SNo (mul_SNo a x))) b)) of type set → set → set → set.

In Proofgold the corresponding term root is 757214... and object id is 919d92...

Definition. We define affine_line_R2_param_by_y to be λa b c ⇒ graph R (λy : set ⇒ (div_SNo c a,y)) of type set → set → set → set.

In Proofgold the corresponding term root is 6e605e... and object id is 2c0cdd...

Theorem. (affine_line_R2_param_by_x_apply)

∀a b c x : set, x ∈ R → apply_fun (affine_line_R2_param_by_x a b c) x = (x,div_SNo (add_SNo c (minus_SNo (mul_SNo a x))) b)

In Proofgold the corresponding term root is 0badb6... and proposition id is 01d371...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_param_by_y_apply)

∀a b c y : set, y ∈ R → apply_fun (affine_line_R2_param_by_y a b c) y = (div_SNo c a,y)

In Proofgold the corresponding term root is 277183... and proposition id is 40f53e...

Proof:
Proof not loaded.

Theorem. (projection2_after_affine_line_R2_param_by_y)

∀a b c y : set, a ∈ R → c ∈ R → y ∈ R → apply_fun (projection2 R R) (apply_fun (affine_line_R2_param_by_y a b c) y) = y

In Proofgold the corresponding term root is 4be9d5... and proposition id is f75a8a...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_param_by_y_after_projection2_on_slice)

∀a b c p : set, a ∈ R → c ∈ R → p ∈ setprod {div_SNo c a} R → apply_fun (affine_line_R2_param_by_y a b c) (apply_fun (projection2 R R) p) = p

In Proofgold the corresponding term root is dae393... and proposition id is 8c6eeb...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_param_by_x_function_on)

∀a b c : set, a ∈ R → b ∈ R → c ∈ R → function_on (affine_line_R2_param_by_x a b c) R EuclidPlane

In Proofgold the corresponding term root is 12c326... and proposition id is 6efeff...

Proof:
Proof not loaded.

Theorem. (projection1_after_affine_line_R2_param_by_x)

∀a b c x : set, a ∈ R → b ∈ R → c ∈ R → x ∈ R → apply_fun (projection1 R R) (apply_fun (affine_line_R2_param_by_x a b c) x) = x

In Proofgold the corresponding term root is 10cf50... and proposition id is 3d171a...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_param_by_y_function_on)

∀a b c : set, a ∈ R → b ∈ R → c ∈ R → function_on (affine_line_R2_param_by_y a b c) R EuclidPlane

In Proofgold the corresponding term root is 51e4aa... and proposition id is 728ca2...

Proof:
Proof not loaded.

Theorem. (projection1_continuous_in_product)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → continuous_map (setprod X Y) (product_topology X Tx Y Ty) X Tx (projection1 X Y)

In Proofgold the corresponding term root is 593051... and proposition id is 0cbcad...

Proof:
Proof not loaded.

Theorem. (projection2_continuous_in_product)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → continuous_map (setprod X Y) (product_topology X Tx Y Ty) Y Ty (projection2 X Y)

In Proofgold the corresponding term root is a274a9... and proposition id is de92f0...

Proof:
Proof not loaded.

Theorem. (projection2_continuous_on_vertical_slice)

∀x0 Tx Ty : set, topology_on R Tx → topology_on R Ty → x0 ∈ R → continuous_map (setprod {x0} R) (subspace_topology EuclidPlane (product_topology R Tx R Ty) (setprod {x0} R)) R Ty (projection2 R R)

In Proofgold the corresponding term root is 64fa69... and proposition id is b77228...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_param_by_y_continuous_in_product)

∀a b c Tx Ty : set, topology_on R Tx → topology_on R Ty → a ∈ R → c ∈ R → continuous_map R Ty EuclidPlane (product_topology R Tx R Ty) (affine_line_R2_param_by_y a b c)

In Proofgold the corresponding term root is 5d36a8... and proposition id is 7ccc5c...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_param_by_x_in_line)

∀a b c x : set, a ∈ R → b ∈ R → c ∈ R → x ∈ R → b ≠ 0 → apply_fun (affine_line_R2_param_by_x a b c) x ∈ affine_line_R2 a b c

In Proofgold the corresponding term root is a65133... and proposition id is 62b757...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_param_by_x_after_projection1_on_line)

∀a b c p : set, a ∈ R → b ∈ R → c ∈ R → b ≠ 0 → p ∈ affine_line_R2 a b c → apply_fun (affine_line_R2_param_by_x a b c) (apply_fun (projection1 R R) p) = p

In Proofgold the corresponding term root is a8c0c5... and proposition id is a4bbc3...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_param_by_y_in_line)

∀a b c y : set, a ∈ R → b ∈ R → c ∈ R → y ∈ R → b = 0 → a ≠ 0 → apply_fun (affine_line_R2_param_by_y a b c) y ∈ affine_line_R2 a b c

In Proofgold the corresponding term root is 64c35d... and proposition id is 76908f...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_b0_eq_slice)

∀a b c : set, a ∈ R → b ∈ R → c ∈ R → b = 0 → a ≠ 0 → affine_line_R2 a b c = setprod {div_SNo c a} R

In Proofgold the corresponding term root is 12b117... and proposition id is 37d8d9...

Proof:
Proof not loaded.

Definition. We define same_sign_nonzero_R to be λa b ⇒ (Rlt 0 a ∧ Rlt 0 b) ∨ (Rlt a 0 ∧ Rlt b 0) of type set → set → prop.

In Proofgold the corresponding term root is a9ae6e... and object id is 2ae719...

Theorem. (affine_line_R2_param_by_x_y_decreases_same_sign)

∀a b c x1 x2 : set, a ∈ R → b ∈ R → c ∈ R → b ≠ 0 → same_sign_nonzero_R a b → x1 ∈ R → x2 ∈ R → Rlt x1 x2 → Rlt (div_SNo (add_SNo c (minus_SNo (mul_SNo a x2))) b) (div_SNo (add_SNo c (minus_SNo (mul_SNo a x1))) b)

In Proofgold the corresponding term root is bc3577... and proposition id is 2f6da3...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_singleton_open_same_sign)

∀a b c p : set, a ∈ R → b ∈ R → c ∈ R → b ≠ 0 → same_sign_nonzero_R a b → p ∈ affine_line_R2 a b c → {p} ∈ subspace_topology (setprod R R) (product_topology R R_lower_limit_topology R R_lower_limit_topology) (affine_line_R2 a b c)

In Proofgold the corresponding term root is 0c785f... and proposition id is e31708...

Proof:
Proof not loaded.

Theorem. (projection1_continuous_on_affine_line_same_sign)

∀a b c : set, a ∈ R → b ∈ R → c ∈ R → b ≠ 0 → same_sign_nonzero_R a b → continuous_map (affine_line_R2 a b c) (subspace_topology (setprod R R) (product_topology R R_lower_limit_topology R R_lower_limit_topology) (affine_line_R2 a b c)) R (discrete_topology R) (projection1 R R)

In Proofgold the corresponding term root is 8d1820... and proposition id is 6fedf3...

Proof:
Proof not loaded.

Theorem. (affine_scalar_div_continuous_R_lower_to_standard)

∀a b c : set, a ∈ R → b ∈ R → c ∈ R → b ≠ 0 → continuous_map R R_lower_limit_topology R R_standard_topology (graph R (λx : set ⇒ div_SNo (add_SNo c (minus_SNo (mul_SNo a x))) b))

In Proofgold the corresponding term root is 06c4c6... and proposition id is c49fdc...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_param_by_x_continuous_to_ll_std_line)

∀a b c : set, a ∈ R → b ∈ R → c ∈ R → b ≠ 0 → continuous_map R R_lower_limit_topology (affine_line_R2 a b c) (subspace_topology (setprod R R) (product_topology R R_lower_limit_topology R R_standard_topology) (affine_line_R2 a b c)) (affine_line_R2_param_by_x a b c)

In Proofgold the corresponding term root is 966937... and proposition id is b2e9dc...

Proof:
Proof not loaded.

Theorem. (affine_line_R2_param_by_x_continuous_in_product_ll_ll)

∀a b c : set, a ∈ R → b ∈ R → c ∈ R → b ≠ 0 → ¬ same_sign_nonzero_R a b → continuous_map R R_lower_limit_topology (setprod R R) (product_topology R R_lower_limit_topology R R_lower_limit_topology) (affine_line_R2_param_by_x a b c)

In Proofgold the corresponding term root is 574353... and proposition id is 64f5ce...

Proof:
Proof not loaded.

Theorem. (ex16_8_lines_in_lower_limit_products)

In Proofgold the corresponding term root is f064e2... and proposition id is b1a62c...

Proof:
Proof not loaded.

Theorem. (homeomorphism_injective)

∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty f → ∀x1 x2 : set, x1 ∈ X → x2 ∈ X → apply_fun f x1 = apply_fun f x2 → x1 = x2

In Proofgold the corresponding term root is d62371... and proposition id is 2be3da...

Proof:
Proof not loaded.

Theorem. (continuous_on_subspace)

∀X Tx Y Ty f A : set, topology_on X Tx → A ⊆ X → continuous_map X Tx Y Ty f → continuous_map A (subspace_topology X Tx A) Y Ty f

In Proofgold the corresponding term root is ac0a80... and proposition id is 8e8379...

Proof:
Proof not loaded.

Theorem. (homeomorphism_inverse_continuous)

In Proofgold the corresponding term root is 9c4a1c... and proposition id is cf0161...

Proof:
Proof not loaded.

Theorem. (function_union_on_disjoint_total_functional)

∀A B Y f g : set, A ∩ B = Empty → graph_domain_subset f A → graph_domain_subset g B → total_function_on f A Y → total_function_on g B Y → functional_graph f → functional_graph g → function_on (f ∪ g) (A ∪ B) Y

In Proofgold the corresponding term root is b19168... and proposition id is 8c15b0...

Proof:
Proof not loaded.

Theorem. (total_function_union_on_disjoint_total_functional)

∀A B Y f g : set, A ∩ B = Empty → graph_domain_subset f A → graph_domain_subset g B → total_function_on f A Y → total_function_on g B Y → functional_graph f → functional_graph g → total_function_on (f ∪ g) (A ∪ B) Y

In Proofgold the corresponding term root is e8ea52... and proposition id is 74eb6d...

Proof:
Proof not loaded.

Theorem. (functional_graph_union_disjoint_domains)

∀A B f g : set, A ∩ B = Empty → graph_domain_subset f A → graph_domain_subset g B → functional_graph f → functional_graph g → functional_graph (f ∪ g)

In Proofgold the corresponding term root is d10de5... and proposition id is b6a23d...

Proof:
Proof not loaded.

Theorem. (apply_fun_union_left)

∀A B Y f g x : set, A ∩ B = Empty → graph_domain_subset f A → graph_domain_subset g B → total_function_on f A Y → total_function_on g B Y → functional_graph f → functional_graph g → x ∈ A → apply_fun (f ∪ g) x = apply_fun f x

In Proofgold the corresponding term root is ad32fa... and proposition id is 9340a7...

Proof:
Proof not loaded.

Theorem. (apply_fun_union_right)

∀A B Y f g x : set, A ∩ B = Empty → graph_domain_subset f A → graph_domain_subset g B → total_function_on f A Y → total_function_on g B Y → functional_graph f → functional_graph g → x ∈ B → apply_fun (f ∪ g) x = apply_fun g x

In Proofgold the corresponding term root is d452ec... and proposition id is d228ba...

Proof:
Proof not loaded.

Theorem. (preimage_of_union_functions_total)

∀A B Y f g V : set, A ∩ B = Empty → graph_domain_subset f A → graph_domain_subset g B → total_function_on f A Y → total_function_on g B Y → functional_graph f → functional_graph g → preimage_of (A ∪ B) (f ∪ g) V = (preimage_of A f V) ∪ (preimage_of B g V)

In Proofgold the corresponding term root is 93f3a1... and proposition id is 800ad7...

Proof:
Proof not loaded.

Theorem. (subspace_union_of_opens)

∀X Tx A B U V : set, topology_on X Tx → A ∈ Tx → B ∈ Tx → A ∩ B = Empty → U ∈ subspace_topology X Tx A → V ∈ subspace_topology X Tx B → (U ∪ V) ∈ subspace_topology X Tx (A ∪ B)

In Proofgold the corresponding term root is 4c2dcf... and proposition id is aa0a96...

Proof:
Proof not loaded.

Theorem. (pasting_lemma_total_functional)

∀X A B Y Tx Ty f g : set, topology_on X Tx → A ∈ Tx → B ∈ Tx → A ∩ B = Empty → graph_domain_subset f A → graph_domain_subset g B → functional_graph f → functional_graph g → continuous_map_total A (subspace_topology X Tx A) Y Ty f → continuous_map_total B (subspace_topology X Tx B) Y Ty g → continuous_map_total (A ∪ B) (subspace_topology X Tx (A ∪ B)) Y Ty (f ∪ g)

In Proofgold the corresponding term root is 3d2dec... and proposition id is 9d5559...

Proof:
Proof not loaded.

Theorem. (pasting_lemma_total_functional_imp)

∀X A B Y Tx Ty f g : set, topology_on X Tx → A ∈ Tx → B ∈ Tx → A ∩ B = Empty → graph_domain_subset f A → graph_domain_subset g B → functional_graph f → functional_graph g → continuous_map_total A (subspace_topology X Tx A) Y Ty f → continuous_map_total B (subspace_topology X Tx B) Y Ty g → continuous_map (A ∪ B) (subspace_topology X Tx (A ∪ B)) Y Ty (f ∪ g)

In Proofgold the corresponding term root is f2d5a4... and proposition id is d3e988...

Proof:
Proof not loaded.

Theorem. (pasting_lemma)

∀X A B Y Tx Ty f g : set, topology_on X Tx → closed_in X Tx A → closed_in X Tx B → A ∪ B = X → continuous_map A (subspace_topology X Tx A) Y Ty f → continuous_map B (subspace_topology X Tx B) Y Ty g → (∀x : set, x ∈ (A ∩ B) → apply_fun f x = apply_fun g x) → ∃h : set, continuous_map X Tx Y Ty h ∧ ((∀x : set, x ∈ A → apply_fun h x = apply_fun f x) ∧ (∀x : set, x ∈ B → apply_fun h x = apply_fun g x))

In Proofgold the corresponding term root is a3343d... and proposition id is a55db9...

Proof:
Proof not loaded.

Definition. We define pair_map to be λA f g ⇒ {(a,(apply_fun f a,apply_fun g a))|a ∈ A} of type set → set → set → set.

In Proofgold the corresponding term root is b587bb... and object id is 65096b...

Theorem. (pair_map_apply)

∀A X Y f g a : set, a ∈ A → apply_fun (pair_map A f g) a = (apply_fun f a,apply_fun g a)

In Proofgold the corresponding term root is e6ad3e... and proposition id is 2e1823...

Proof:
Proof not loaded.

Theorem. (pair_map_in_function_space)

∀A X Y f g : set, function_on f A X → function_on g A Y → pair_map A f g ∈ function_space A (setprod X Y)

In Proofgold the corresponding term root is 7c9ae8... and proposition id is f5269d...

Proof:
Proof not loaded.

Theorem. (preimage_pair_map_rectangle)

∀A X Y f g U V : set, preimage_of A (pair_map A f g) (rectangle_set U V) = (preimage_of A f U) ∩ (preimage_of A g V)

In Proofgold the corresponding term root is 1f297d... and proposition id is 2ae7ff...

Proof:
Proof not loaded.

Theorem. (maps_into_products_axiom)

∀A Ta X Tx Y Ty f g : set, continuous_map A Ta X Tx f → continuous_map A Ta Y Ty g → continuous_map A Ta (setprod X Y) (product_topology X Tx Y Ty) (pair_map A f g)

In Proofgold the corresponding term root is f10cc7... and proposition id is d0dbbd...

Proof:
Proof not loaded.

Theorem. (maps_into_products)

∀A Ta X Tx Y Ty f g : set, continuous_map A Ta X Tx f → continuous_map A Ta Y Ty g → continuous_map A Ta (setprod X Y) (product_topology X Tx Y Ty) (pair_map A f g)

In Proofgold the corresponding term root is f10cc7... and proposition id is d0dbbd...

Proof:
Proof not loaded.

Definition. We define projection_map1 to be λX Y ⇒ projection1 X Y of type set → set → set.

In Proofgold the corresponding term root is 2a96db... and object id is 1d9cd6...

Definition. We define projection_map2 to be λX Y ⇒ projection2 X Y of type set → set → set.

In Proofgold the corresponding term root is f24403... and object id is 91b1f3...

Theorem. (preimage_of_rectangle_via_projections)

∀A X Y h U V : set, function_on h A (setprod X Y) → U ⊆ X → V ⊆ Y → preimage_of A h (rectangle_set U V) = (preimage_of A (compose_fun A h (projection_map1 X Y)) U) ∩ (preimage_of A (compose_fun A h (projection_map2 X Y)) V)

In Proofgold the corresponding term root is 485f9b... and proposition id is 0c15fc...

Proof:
Proof not loaded.

Theorem. (projection_maps_continuous)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → continuous_map (setprod X Y) (product_topology X Tx Y Ty) X Tx (projection_map1 X Y) ∧ continuous_map (setprod X Y) (product_topology X Tx Y Ty) Y Ty (projection_map2 X Y)

In Proofgold the corresponding term root is 49af7a... and proposition id is 09dbf2...

Proof:
Proof not loaded.

Theorem. (maps_into_products_converse)

∀A Ta X Tx Y Ty h : set, topology_on X Tx → topology_on Y Ty → continuous_map A Ta (setprod X Y) (product_topology X Tx Y Ty) h → continuous_map A Ta X Tx (compose_fun A h (projection_map1 X Y)) ∧ continuous_map A Ta Y Ty (compose_fun A h (projection_map2 X Y))

In Proofgold the corresponding term root is 75a88d... and proposition id is 7c60b8...

Proof:
Proof not loaded.

Theorem. (maps_into_products_coords_imp)

∀A Ta X Tx Y Ty h : set, topology_on X Tx → topology_on Y Ty → function_on h A (setprod X Y) → continuous_map A Ta X Tx (compose_fun A h (projection_map1 X Y)) → continuous_map A Ta Y Ty (compose_fun A h (projection_map2 X Y)) → continuous_map A Ta (setprod X Y) (product_topology X Tx Y Ty) h

In Proofgold the corresponding term root is f4ca05... and proposition id is 240025...

Proof:
Proof not loaded.

Theorem. (maps_into_products_iff_coords)

∀A Ta X Tx Y Ty h : set, topology_on X Tx → topology_on Y Ty → function_on h A (setprod X Y) → (continuous_map A Ta (setprod X Y) (product_topology X Tx Y Ty) h ↔ (continuous_map A Ta X Tx (compose_fun A h (projection_map1 X Y)) ∧ continuous_map A Ta Y Ty (compose_fun A h (projection_map2 X Y))))

In Proofgold the corresponding term root is 49e6ad... and proposition id is 774543...

Proof:
Proof not loaded.

Theorem. (projections_are_continuous)

In Proofgold the corresponding term root is 49af7a... and proposition id is 09dbf2...

Proof:
Proof not loaded.

Theorem. (product_topology_universal)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → ∃Tprod : set, topology_on (setprod X Y) Tprod ∧ continuous_map (setprod X Y) Tprod X Tx (projection_map1 X Y) ∧ continuous_map (setprod X Y) Tprod Y Ty (projection_map2 X Y)

In Proofgold the corresponding term root is ac9271... and proposition id is 823473...

Proof:
Proof not loaded.

Theorem. (rectangle_set_as_intersection)

∀X Y U V : set, U ⊆ X → V ⊆ Y → rectangle_set U V = (rectangle_set U Y) ∩ (rectangle_set X V)

In Proofgold the corresponding term root is 95e2f5... and proposition id is 79c851...

Proof:
Proof not loaded.

Theorem. (product_topology_coarsest)

∀X Tx Y Ty Tprod : set, topology_on X Tx → topology_on Y Ty → topology_on (setprod X Y) Tprod → continuous_map (setprod X Y) Tprod X Tx (projection_map1 X Y) → continuous_map (setprod X Y) Tprod Y Ty (projection_map2 X Y) → coarser_than (product_topology X Tx Y Ty) Tprod

In Proofgold the corresponding term root is 602661... and proposition id is d0eba8...

Proof:
Proof not loaded.

Definition. We define metric_on to be λX d ⇒ function_on d (setprod X X) R ∧ (∀x y : set, x ∈ X → y ∈ X → apply_fun d (x,y) = apply_fun d (y,x)) ∧ (∀x : set, x ∈ X → apply_fun d (x,x) = 0) ∧ (∀x y : set, x ∈ X → y ∈ X → ¬ (Rlt (apply_fun d (x,y)) 0) ∧ (apply_fun d (x,y) = 0 → x = y)) ∧ (∀x y z : set, x ∈ X → y ∈ X → z ∈ X → ¬ (Rlt (add_SNo (apply_fun d (x,y)) (apply_fun d (y,z))) (apply_fun d (x,z)))) of type set → set → prop.

In Proofgold the corresponding term root is 143767... and object id is 29b73a...

Definition. We define metric_on_total to be λX d ⇒ metric_on X d ∧ total_function_on d (setprod X X) R of type set → set → prop.

In Proofgold the corresponding term root is eb232c... and object id is 88739a...

Theorem. (metric_on_function_on)

∀X d : set, metric_on X d → function_on d (setprod X X) R

In Proofgold the corresponding term root is 094617... and proposition id is b75cef...

Proof:
Proof not loaded.

Definition. We define EuclidPlane_metric to be graph (setprod EuclidPlane EuclidPlane) (λpq : set ⇒ distance_R2 (pq 0) (pq 1)) of type set.

In Proofgold the corresponding term root is aa1473... and object id is 82ecd8...

Theorem. (EuclidPlane_metric_apply)

∀p q : set, p ∈ EuclidPlane → q ∈ EuclidPlane → apply_fun EuclidPlane_metric (p,q) = distance_R2 p q

In Proofgold the corresponding term root is 794e1a... and proposition id is 0b244a...

Proof:
Proof not loaded.

Theorem. (EuclidPlane_metric_is_metric_on)

metric_on EuclidPlane EuclidPlane_metric

In Proofgold the corresponding term root is 6f1aa1... and proposition id is 8f2433...

Proof:
Proof not loaded.

Theorem. (EuclidPlane_metric_is_metric_on_total)

metric_on_total EuclidPlane EuclidPlane_metric

In Proofgold the corresponding term root is 9c8290... and proposition id is 6ece6c...

Proof:
Proof not loaded.

Theorem. (metric_on_symmetric)

∀X d x y : set, metric_on X d → x ∈ X → y ∈ X → apply_fun d (x,y) = apply_fun d (y,x)

In Proofgold the corresponding term root is 91a7e0... and proposition id is aea431...

Proof:
Proof not loaded.

Theorem. (metric_on_diag_zero)

∀X d x : set, metric_on X d → x ∈ X → apply_fun d (x,x) = 0

In Proofgold the corresponding term root is 0bccd1... and proposition id is c045a7...

Proof:
Proof not loaded.

Theorem. (metric_on_nonneg)

∀X d x y : set, metric_on X d → x ∈ X → y ∈ X → ¬ (Rlt (apply_fun d (x,y)) 0)

In Proofgold the corresponding term root is 97a663... and proposition id is afb093...

Proof:
Proof not loaded.

Theorem. (metric_on_zero_eq)

∀X d x y : set, metric_on X d → x ∈ X → y ∈ X → apply_fun d (x,y) = 0 → x = y

In Proofgold the corresponding term root is 3926d1... and proposition id is 1a58f5...

Proof:
Proof not loaded.

Theorem. (metric_on_pos_of_neq)

∀X d x y : set, metric_on X d → x ∈ X → y ∈ X → ¬ (x = y) → Rlt 0 (apply_fun d (x,y))

In Proofgold the corresponding term root is 57362a... and proposition id is 57d224...

Proof:
Proof not loaded.

Definition. We define metric_neg_dists to be λX d x A ⇒ {minus_SNo (apply_fun d (x,a))|a ∈ A} of type set → set → set → set → set.

In Proofgold the corresponding term root is 55f7d2... and object id is b25746...

Theorem. (metric_neg_dists_in_R)

∀X d x A : set, metric_on X d → x ∈ X → A ⊆ X → ∀t : set, t ∈ metric_neg_dists X d x A → t ∈ R

In Proofgold the corresponding term root is a138cb... and proposition id is a30925...

Proof:
Proof not loaded.

Theorem. (metric_neg_dists_le_0)

∀X d x A : set, metric_on X d → x ∈ X → A ⊆ X → ∀t : set, t ∈ metric_neg_dists X d x A → Rle t 0

In Proofgold the corresponding term root is 1bb9a2... and proposition id is e79fdc...

Proof:
Proof not loaded.

Theorem. (metric_on_triangle)

∀X d x y z : set, metric_on X d → x ∈ X → y ∈ X → z ∈ X → ¬ (Rlt (add_SNo (apply_fun d (x,y)) (apply_fun d (y,z))) (apply_fun d (x,z)))

In Proofgold the corresponding term root is 352bb7... and proposition id is 5861bc...

Proof:
Proof not loaded.

Theorem. (metric_on_total_imp_metric_on)

∀X d : set, metric_on_total X d → metric_on X d

In Proofgold the corresponding term root is d713ce... and proposition id is ce1c7a...

Proof:
Proof not loaded.

Theorem. (metric_on_total_total_function)

∀X d : set, metric_on_total X d → total_function_on d (setprod X X) R

In Proofgold the corresponding term root is d161a9... and proposition id is 8d9ab2...

Proof:
Proof not loaded.

Theorem. (metric_triangle_Rle)

∀X d x y z : set, metric_on X d → x ∈ X → y ∈ X → z ∈ X → Rle (apply_fun d (x,z)) (add_SNo (apply_fun d (x,y)) (apply_fun d (y,z)))

In Proofgold the corresponding term root is 01a413... and proposition id is 770c2f...

Proof:
Proof not loaded.

Definition. We define open_ball to be λX d x r ⇒ {y ∈ X|Rlt (apply_fun d (x,y)) r} of type set → set → set → set → set.

In Proofgold the corresponding term root is 837549... and object id is a8638a...

Theorem. (open_ballE1)

∀X d x r y : set, y ∈ open_ball X d x r → y ∈ X

In Proofgold the corresponding term root is edcdba... and proposition id is b1dc64...

Proof:
Proof not loaded.

Theorem. (open_ballE2)

∀X d x r y : set, y ∈ open_ball X d x r → Rlt (apply_fun d (x,y)) r

In Proofgold the corresponding term root is aaae45... and proposition id is c105e6...

Proof:
Proof not loaded.

Theorem. (open_ballI)

∀X d x r y : set, y ∈ X → Rlt (apply_fun d (x,y)) r → y ∈ open_ball X d x r

In Proofgold the corresponding term root is acaa39... and proposition id is d2d62a...

Proof:
Proof not loaded.

Theorem. (open_ball_EuclidPlane_metric_eq)

∀c r : set, c ∈ EuclidPlane → open_ball EuclidPlane EuclidPlane_metric c r = {p ∈ EuclidPlane|Rlt (distance_R2 p c) r}

In Proofgold the corresponding term root is 4b3bb0... and proposition id is 1b9c47...

Proof:
Proof not loaded.

Theorem. (open_ball_subset_X)

∀X d x r : set, open_ball X d x r ⊆ X

In Proofgold the corresponding term root is 180a58... and proposition id is 80a140...

Proof:
Proof not loaded.

Theorem. (open_ball_in_Power)

∀X d x r : set, open_ball X d x r ∈ 𝒫 X

In Proofgold the corresponding term root is 636d27... and proposition id is b6a638...

Proof:
Proof not loaded.

Theorem. (center_in_open_ball)

∀X d x r : set, metric_on X d → x ∈ X → Rlt 0 r → x ∈ open_ball X d x r

In Proofgold the corresponding term root is 43f8e2... and proposition id is adc4eb...

Proof:
Proof not loaded.

Theorem. (open_ball_nonempty)

∀X d x r : set, metric_on X d → x ∈ X → Rlt 0 r → open_ball X d x r ≠ Empty

In Proofgold the corresponding term root is 554db0... and proposition id is 130508...

Proof:
Proof not loaded.

Theorem. (exists_eps_lt_pos)

∀d : set, d ∈ R → Rlt 0 d → ∃N ∈ ω, eps_ N < d

In Proofgold the corresponding term root is 0307bb... and proposition id is 69e182...

Proof:
Proof not loaded.

Theorem. (exists_eps_lt_two_pos)

∀a b : set, a ∈ R → b ∈ R → Rlt 0 a → Rlt 0 b → ∃r3 : set, r3 ∈ R ∧ Rlt 0 r3 ∧ Rlt r3 a ∧ Rlt r3 b

In Proofgold the corresponding term root is 320e56... and proposition id is adb7af...

Proof:
Proof not loaded.

Theorem. (exists_eps_lt_four_pos)

∀a b c d : set, a ∈ R → b ∈ R → c ∈ R → d ∈ R → Rlt 0 a → Rlt 0 b → Rlt 0 c → Rlt 0 d → ∃r3 : set, r3 ∈ R ∧ Rlt 0 r3 ∧ Rlt r3 a ∧ Rlt r3 b ∧ Rlt r3 c ∧ Rlt r3 d

In Proofgold the corresponding term root is b9b5b6... and proposition id is 8e4f70...

Proof:
Proof not loaded.

Theorem. (Rle_Rlt_tra)

∀a b c : set, Rle a b → Rlt b c → Rlt a c

In Proofgold the corresponding term root is 5f7cc8... and proposition id is 5935d4...

Proof:
Proof not loaded.

Theorem. (Rlt_Rle_tra)

∀a b c : set, Rlt a b → Rle b c → Rlt a c

In Proofgold the corresponding term root is 7e5b32... and proposition id is 41af58...

Proof:
Proof not loaded.

Theorem. (Rlt_add_SNo)

∀a b c d : set, Rlt a b → Rlt c d → Rlt (add_SNo a c) (add_SNo b d)

In Proofgold the corresponding term root is 7065bf... and proposition id is eae9ce...

Proof:
Proof not loaded.

Theorem. (open_ball_pair_dist_lt_add_radius)

∀X d x r y z : set, metric_on X d → x ∈ X → y ∈ open_ball X d x r → z ∈ open_ball X d x r → Rlt (apply_fun d (y,z)) (add_SNo r r)

In Proofgold the corresponding term root is eede21... and proposition id is 975abc...

Proof:
Proof not loaded.

Theorem. (open_ball_pair_dist_lt_eps_pred)

∀X d x n y z : set, metric_on X d → x ∈ X → n ∈ ω → y ∈ open_ball X d x (eps_ (ordsucc n)) → z ∈ open_ball X d x (eps_ (ordsucc n)) → Rlt (apply_fun d (y,z)) (eps_ n)

In Proofgold the corresponding term root is 7ca5f4... and proposition id is d866f6...

Proof:
Proof not loaded.

Theorem. (metric_ball_intersect_centers_lt_add)

∀X d x y p r1 r2 : set, metric_on X d → x ∈ X → y ∈ X → p ∈ X → r1 ∈ R → r2 ∈ R → p ∈ open_ball X d x r1 → p ∈ open_ball X d y r2 → Rlt (apply_fun d (x,y)) (add_SNo r1 r2)

In Proofgold the corresponding term root is c29911... and proposition id is 0c846a...

Proof:
Proof not loaded.

Definition. We define well_ordered_set to be λX ⇒ ordinal X ∧ X ≠ R ∧ X ≠ rational_numbers ∧ X ≠ setprod 2 ω ∧ X ≠ setprod R R of type set → prop.

In Proofgold the corresponding term root is 110f16... and object id is 789976...

Axiom. (well_ordering_theorem_axiom) We take the following as an axiom:

∀A : set, ∃W : set, well_ordered_set W ∧ equip A W

In Proofgold the corresponding term root is 5c44c6... and proposition id is 98d5c0...

Theorem. (well_ordering_theorem_equip)

∀A : set, ∃W : set, well_ordered_set W ∧ equip A W

In Proofgold the corresponding term root is 5c44c6... and proposition id is 98d5c0...

Proof:
Proof not loaded.

Theorem. (well_ordered_set_is_ordinal)

∀X : set, well_ordered_set X → ordinal X

In Proofgold the corresponding term root is 81974b... and proposition id is 54d4bf...

Proof:
Proof not loaded.

Definition. We define eps_separated_set to be λX d S n ⇒ S ⊆ X ∧ ∀x y : set, x ∈ S → y ∈ S → ¬ (x = y) → ¬ (Rlt (apply_fun d (x,y)) (eps_ n)) of type set → set → set → set → prop.

In Proofgold the corresponding term root is e13f47... and object id is 8c2732...

Definition. We define maximal_eps_separated_set to be λX d S n ⇒ eps_separated_set X d S n ∧ ∀x : set, x ∈ X → (∀y : set, y ∈ S → ¬ (Rlt (apply_fun d (x,y)) (eps_ n))) → x ∈ S of type set → set → set → set → prop.

In Proofgold the corresponding term root is 76c467... and object id is 5604fa...

Theorem. (maximal_eps_separated_set_implies_cover)

∀X d S n : set, metric_on X d → n ∈ ω → maximal_eps_separated_set X d S n → X ⊆ (⋃_{c ∈ S}open_ball X d c (eps_ n))

In Proofgold the corresponding term root is ce7553... and proposition id is 398eac...

Proof:
Proof not loaded.

Definition. We define maximal_eps_separated_set_on to be λX d Y S n ⇒ (Y ⊆ X) ∧ (S ⊆ Y) ∧ (∀x y : set, x ∈ S → y ∈ S → ¬ (x = y) → ¬ (Rlt (apply_fun d (x,y)) (eps_ n))) ∧ (∀x : set, x ∈ Y → (∀y : set, y ∈ S → ¬ (Rlt (apply_fun d (x,y)) (eps_ n))) → x ∈ S) of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is 44373c... and object id is 3bd6aa...

Theorem. (maximal_eps_separated_set_on_exists)

∀X d Y n : set, metric_on X d → n ∈ ω → Y ⊆ X → ∃S : set, maximal_eps_separated_set_on X d Y S n

In Proofgold the corresponding term root is f5397e... and proposition id is 1acdf1...

Proof:
Proof not loaded.

Theorem. (maximal_eps_separated_set_on_implies_cover)

∀X d Y S n : set, metric_on X d → n ∈ ω → maximal_eps_separated_set_on X d Y S n → Y ⊆ (⋃_{c ∈ S}open_ball X d c (eps_ n))

In Proofgold the corresponding term root is d3f5e7... and proposition id is 0d02b6...

Proof:
Proof not loaded.

Theorem. (eps_separated_imp_eps_succ_ball_point_unique)

∀X d S n p x y : set, metric_on X d → n ∈ ω → eps_separated_set X d S n → p ∈ X → x ∈ S → y ∈ S → p ∈ open_ball X d x (eps_ (ordsucc n)) → p ∈ open_ball X d y (eps_ (ordsucc n)) → x = y

In Proofgold the corresponding term root is 37ca98... and proposition id is 34b36b...

Proof:
Proof not loaded.

Theorem. (open_ball_refine_intersection)

∀X d c1 c2 x r1 r2 : set, metric_on X d → c1 ∈ X → c2 ∈ X → x ∈ X → r1 ∈ R → r2 ∈ R → Rlt 0 r1 → Rlt 0 r2 → x ∈ open_ball X d c1 r1 → x ∈ open_ball X d c2 r2 → ∃r3 : set, r3 ∈ R ∧ Rlt 0 r3 ∧ open_ball X d x r3 ⊆ (open_ball X d c1 r1) ∩ (open_ball X d c2 r2)

In Proofgold the corresponding term root is 9ca246... and proposition id is b1e4f5...

Proof:
Proof not loaded.

Theorem. (open_ball_refine_center)

∀X d c x r : set, metric_on X d → c ∈ X → x ∈ X → r ∈ R → Rlt 0 r → x ∈ open_ball X d c r → ∃s : set, s ∈ R ∧ Rlt 0 s ∧ open_ball X d x s ⊆ open_ball X d c r

In Proofgold the corresponding term root is 1f3dbe... and proposition id is 07e31f...

Proof:
Proof not loaded.

Theorem. (open_ball_refine_center_eps)

∀X d c x r : set, metric_on X d → c ∈ X → x ∈ X → r ∈ R → Rlt 0 r → x ∈ open_ball X d c r → ∃N : set, N ∈ ω ∧ open_ball X d x (eps_ N) ⊆ open_ball X d c r

In Proofgold the corresponding term root is 8e4426... and proposition id is c1cad9...

Proof:
Proof not loaded.

Definition. We define metric_topology to be λX d ⇒ generated_topology X (famunion X (λx ⇒ {open_ball X d x r|r ∈ R, Rlt 0 r})) of type set → set → set.

In Proofgold the corresponding term root is 663151... and object id is e01782...

Theorem. (open_balls_form_basis)

∀X d : set, metric_on X d → basis_on X (famunion X (λx ⇒ {open_ball X d x r|r ∈ R, Rlt 0 r}))

In Proofgold the corresponding term root is 8fcc0d... and proposition id is 054fdd...

Proof:
Proof not loaded.

Theorem. (open_in_metric_topology_has_eps_ball_sub)

∀X d U x : set, metric_on X d → open_in X (metric_topology X d) U → x ∈ U → ∃N : set, N ∈ ω ∧ open_ball X d x (eps_ N) ⊆ U

In Proofgold the corresponding term root is 87ca1c... and proposition id is 06ec60...

Proof:
Proof not loaded.

Theorem. (metric_topology_is_topology)

∀X d : set, metric_on X d → topology_on X (metric_topology X d)

In Proofgold the corresponding term root is 902c19... and proposition id is 947da8...

Proof:
Proof not loaded.

Theorem. (metric_topology_generated_by_balls)

∀X d : set, metric_on X d → generated_topology X (famunion X (λx ⇒ {open_ball X d x r|r ∈ R, Rlt 0 r})) = metric_topology X d

In Proofgold the corresponding term root is 5add8f... and proposition id is e5ae39...

Proof:
Proof not loaded.

Theorem. (open_ball_EuclidPlane_metric_in_circular_regions)

∀c r : set, c ∈ EuclidPlane → r ∈ R → Rlt 0 r → open_ball EuclidPlane EuclidPlane_metric c r ∈ circular_regions

In Proofgold the corresponding term root is 5c34bb... and proposition id is 024602...

Proof:
Proof not loaded.

Theorem. (EuclidPlane_metric_open_balls_family_eq_circular_regions)

famunion EuclidPlane (λc : set ⇒ {open_ball EuclidPlane EuclidPlane_metric c r|r ∈ R, Rlt 0 r}) = circular_regions

In Proofgold the corresponding term root is b6a468... and proposition id is afcfa8...

Proof:
Proof not loaded.

Theorem. (metric_topology_EuclidPlane_metric_eq_generated_circular_regions)

metric_topology EuclidPlane EuclidPlane_metric = generated_topology EuclidPlane circular_regions

In Proofgold the corresponding term root is 578270... and proposition id is b1b8f2...

Proof:
Proof not loaded.

Theorem. (metric_topology_EuclidPlane_metric_eq_generated_rectangular_regions)

metric_topology EuclidPlane EuclidPlane_metric = generated_topology EuclidPlane rectangular_regions

In Proofgold the corresponding term root is 6ffd95... and proposition id is 6171ed...

Proof:
Proof not loaded.

Theorem. (metric_topology_EuclidPlane_metric_sub_R2_standard_topology)

metric_topology EuclidPlane EuclidPlane_metric ⊆ R2_standard_topology

In Proofgold the corresponding term root is a9d913... and proposition id is 8272af...

Proof:
Proof not loaded.

Theorem. (metric_topology_EuclidPlane_metric_eq_R2_standard_topology)

metric_topology EuclidPlane EuclidPlane_metric = R2_standard_topology

In Proofgold the corresponding term root is d2a8b6... and proposition id is 61e77f...

Proof:
Proof not loaded.

Theorem. (open_ball_open_in_metric_topology)

∀X d x r : set, metric_on X d → x ∈ X → Rlt 0 r → open_in X (metric_topology X d) (open_ball X d x r)

In Proofgold the corresponding term root is 38c698... and proposition id is 467311...

Proof:
Proof not loaded.

Theorem. (open_ball_in_metric_topology)

∀X d x r : set, metric_on X d → x ∈ X → Rlt 0 r → open_ball X d x r ∈ metric_topology X d

In Proofgold the corresponding term root is c37ea8... and proposition id is 15a677...

Proof:
Proof not loaded.

Theorem. (metric_topology_Hausdorff)

∀X d : set, metric_on X d → Hausdorff_space X (metric_topology X d)

In Proofgold the corresponding term root is 29f32d... and proposition id is 243f5d...

Proof:
Proof not loaded.

Theorem. (metric_epsilon_delta_continuity)

∀X dX Y dY f : set, metric_on X dX → metric_on Y dY → function_on f X Y → (continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f ↔ (∀x0 : set, x0 ∈ X → ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃delta : set, delta ∈ R ∧ Rlt 0 delta ∧ (∀x : set, x ∈ X → Rlt (apply_fun dX (x,x0)) delta → Rlt (apply_fun dY (apply_fun f x,apply_fun f x0)) eps)))

In Proofgold the corresponding term root is 1ca43e... and proposition id is 623a7e...

Proof:
Proof not loaded.

Definition. We define sequence_in to be λseq A ⇒ function_on seq ω A of type set → set → prop.

In Proofgold the corresponding term root is 0f5b10... and object id is 37a7ba...

Definition. We define sequence_on to be λseq A ⇒ function_on seq ω A of type set → set → prop.

In Proofgold the corresponding term root is 0f5b10... and object id is 37a7ba...

Theorem. (apply_fun_of_graph_eq)

∀f A : set, ∀g : set → set, ∀a : set, f = graph A g → a ∈ A → apply_fun f a = g a

In Proofgold the corresponding term root is 6b28ab... and proposition id is fd729c...

Proof:
Proof not loaded.

Definition. We define converges_to to be λX Tx seq x ⇒ topology_on X Tx ∧ sequence_on seq X ∧ x ∈ X ∧ ∀U : set, U ∈ Tx → x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U of type set → set → set → set → prop.

In Proofgold the corresponding term root is 9d454f... and object id is 124bc0...

Theorem. (converges_to_topology)

∀X Tx seq x : set, converges_to X Tx seq x → topology_on X Tx

In Proofgold the corresponding term root is 501674... and proposition id is 248b69...

Proof:
Proof not loaded.

Theorem. (converges_to_sequence_on)

∀X Tx seq x : set, converges_to X Tx seq x → sequence_on seq X

In Proofgold the corresponding term root is a8ef8b... and proposition id is 26a911...

Proof:
Proof not loaded.

Theorem. (converges_to_point_in_X)

∀X Tx seq x : set, converges_to X Tx seq x → x ∈ X

In Proofgold the corresponding term root is 760417... and proposition id is af3628...

Proof:
Proof not loaded.

Theorem. (converges_to_neighborhoods)

∀X Tx seq x : set, converges_to X Tx seq x → ∀U : set, U ∈ Tx → x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ U

In Proofgold the corresponding term root is f7fc5f... and proposition id is 1450fa...

Proof:
Proof not loaded.

Definition. We define map_sequence to be λf seq ⇒ compose_fun ω seq f of type set → set → set.

In Proofgold the corresponding term root is cfba6e... and object id is 1b215a...

Definition. We define image_of to be λf U ⇒ Repl U (λx ⇒ apply_fun f x) of type set → set → set.

In Proofgold the corresponding term root is d8bf37... and object id is d91390...

Definition. We define function_sequence_value to be λf_seq n x ⇒ apply_fun (apply_fun f_seq n) x of type set → set → set → set.

In Proofgold the corresponding term root is 379670... and object id is 44e7e2...

Theorem. (image_of_mono)

∀f U V : set, U ⊆ V → image_of f U ⊆ image_of f V

In Proofgold the corresponding term root is fc6483... and proposition id is 258ec9...

Proof:
Proof not loaded.

Theorem. (image_of_sub_codomain)

∀f X Y U : set, function_on f X Y → U ⊆ X → image_of f U ⊆ Y

In Proofgold the corresponding term root is ca65b2... and proposition id is dd5208...

Proof:
Proof not loaded.

Theorem. (image_of_Empty)

∀f : set, image_of f Empty = Empty

In Proofgold the corresponding term root is f05468... and proposition id is 1153b9...

Proof:
Proof not loaded.

Theorem. (image_of_binunion)

∀f U V : set, image_of f (U ∪ V) = (image_of f U) ∪ (image_of f V)

In Proofgold the corresponding term root is 81565b... and proposition id is f80de7...

Proof:
Proof not loaded.

Theorem. (image_of_Union)

∀f Fam : set, image_of f (⋃ Fam) = ⋃ {image_of f U|U ∈ Fam}

In Proofgold the corresponding term root is 03e648... and proposition id is 334666...

Proof:
Proof not loaded.

Theorem. (image_of_compose_fun)

∀X f g U : set, U ⊆ X → image_of (compose_fun X f g) U = image_of g (image_of f U)

In Proofgold the corresponding term root is 02ce76... and proposition id is 7a868b...

Proof:
Proof not loaded.

Definition. We define sequence_converges_metric to be λX d seq x ⇒ metric_on X d ∧ sequence_on seq X ∧ x ∈ X ∧ ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → Rlt (apply_fun d (apply_fun seq n,x)) eps of type set → set → set → set → prop.

In Proofgold the corresponding term root is 6d7281... and object id is 59bae7...

Theorem. (sequence_converges_metric_metric_on)

∀X d seq x : set, sequence_converges_metric X d seq x → metric_on X d

In Proofgold the corresponding term root is 9eee2a... and proposition id is d7b50d...

Proof:
Proof not loaded.

Theorem. (sequence_converges_metric_sequence_on)

∀X d seq x : set, sequence_converges_metric X d seq x → sequence_on seq X

In Proofgold the corresponding term root is 4af8cf... and proposition id is 15cc5e...

Proof:
Proof not loaded.

Theorem. (sequence_converges_metric_point_in_X)

∀X d seq x : set, sequence_converges_metric X d seq x → x ∈ X

In Proofgold the corresponding term root is cd906b... and proposition id is 46136a...

Proof:
Proof not loaded.

Theorem. (sequence_converges_metric_eps)

∀X d seq x : set, sequence_converges_metric X d seq x → ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → Rlt (apply_fun d (apply_fun seq n,x)) eps

In Proofgold the corresponding term root is 4b9cc1... and proposition id is 9f1317...

Proof:
Proof not loaded.

Theorem. (metric_limits_unique)

∀X d seq x y : set, metric_on X d → sequence_on seq X → sequence_converges_metric X d seq x → sequence_converges_metric X d seq y → x = y

In Proofgold the corresponding term root is 078c12... and proposition id is 3becee...

Proof:
Proof not loaded.

Definition. We define uniform_convergence_functions to be λX dX Y dY f_seq f ⇒ metric_on X dX ∧ metric_on Y dY ∧ function_on f_seq ω (function_space X Y) ∧ function_on f X Y ∧ (∀n : set, n ∈ ω → function_on (apply_fun f_seq n) X Y) ∧ ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun dY (apply_fun (apply_fun f_seq n) x,apply_fun f x)) eps of type set → set → set → set → set → set → prop.

In Proofgold the corresponding term root is 844dff... and object id is a60a87...

Theorem. (uniform_convergence_functions_metric_on_X)

∀X dX Y dY f_seq f : set, uniform_convergence_functions X dX Y dY f_seq f → metric_on X dX

In Proofgold the corresponding term root is 34a4da... and proposition id is 0033b3...

Proof:
Proof not loaded.

Theorem. (uniform_convergence_functions_metric_on_Y)

∀X dX Y dY f_seq f : set, uniform_convergence_functions X dX Y dY f_seq f → metric_on Y dY

In Proofgold the corresponding term root is 6eed6b... and proposition id is 6175b2...

Proof:
Proof not loaded.

Theorem. (uniform_convergence_functions_fseq)

∀X dX Y dY f_seq f : set, uniform_convergence_functions X dX Y dY f_seq f → function_on f_seq ω (function_space X Y)

In Proofgold the corresponding term root is 031917... and proposition id is a6210d...

Proof:
Proof not loaded.

Theorem. (uniform_convergence_functions_f)

∀X dX Y dY f_seq f : set, uniform_convergence_functions X dX Y dY f_seq f → function_on f X Y

In Proofgold the corresponding term root is ea2722... and proposition id is 92a2d9...

Proof:
Proof not loaded.

Theorem. (uniform_convergence_functions_pointwise)

∀X dX Y dY f_seq f : set, uniform_convergence_functions X dX Y dY f_seq f → ∀n : set, n ∈ ω → function_on (apply_fun f_seq n) X Y

In Proofgold the corresponding term root is a95332... and proposition id is 18f04c...

Proof:
Proof not loaded.

Theorem. (uniform_convergence_functions_eps)

∀X dX Y dY f_seq f : set, uniform_convergence_functions X dX Y dY f_seq f → ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun dY (apply_fun (apply_fun f_seq n) x,apply_fun f x)) eps

In Proofgold the corresponding term root is 60d18d... and proposition id is dc567b...

Proof:
Proof not loaded.

Theorem. (Rle_tra)

∀a b c : set, Rle a b → Rle b c → Rle a c

In Proofgold the corresponding term root is 7a5614... and proposition id is a55d62...

Proof:
Proof not loaded.

Theorem. (Rle_add_SNo_2)

∀x y z : set, x ∈ R → y ∈ R → z ∈ R → Rle y z → Rle (add_SNo x y) (add_SNo x z)

In Proofgold the corresponding term root is 3e10be... and proposition id is c063be...

Proof:
Proof not loaded.

Theorem. (uniform_limit_of_continuous_is_continuous)

∀X dX Y dY f_seq f : set, metric_on X dX → metric_on Y dY → function_on f_seq ω (function_space X Y) → (∀n : set, n ∈ ω → continuous_map X (metric_topology X dX) Y (metric_topology Y dY) (apply_fun f_seq n)) → uniform_convergence_functions X dX Y dY f_seq f → continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f

In Proofgold the corresponding term root is 9ac376... and proposition id is 9c1d24...

Proof:
Proof not loaded.

Theorem. (sequence_convergence_metric)

∀X d seq x : set, sequence_converges_metric X d seq x → sequence_converges_metric X d seq x

In Proofgold the corresponding term root is 551606... and proposition id is f8cab4...

Proof:
Proof not loaded.

Theorem. (continuity_via_sequences_metric)

∀X dX Y dY f : set, metric_on X dX → metric_on Y dY → function_on f X Y → (continuous_map X (metric_topology X dX) Y (metric_topology Y dY) f ↔ ∀seq x : set, sequence_converges_metric X dX seq x → sequence_converges_metric Y dY ({(n,apply_fun f (apply_fun seq n))|n ∈ ω}) (apply_fun f x))

In Proofgold the corresponding term root is d3b4de... and proposition id is 050869...

Proof:
Proof not loaded.

Definition. We define quotient_topology to be λX Tx Y f ⇒ {V ∈ 𝒫 Y|{x ∈ X|apply_fun f x ∈ V} ∈ Tx} of type set → set → set → set → set.

In Proofgold the corresponding term root is 95947d... and object id is 6eb794...

Definition. We define quotient_map to be λX Tx Y f ⇒ topology_on X Tx ∧ function_on f X Y ∧ (∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun f x = y) of type set → set → set → set → prop.

In Proofgold the corresponding term root is 109369... and object id is bb56a6...

Theorem. (quotient_topology_is_topology)

∀X Tx Y f : set, topology_on X Tx → quotient_map X Tx Y f → topology_on Y (quotient_topology X Tx Y f)

In Proofgold the corresponding term root is 48c532... and proposition id is 53fa8e...

Proof:
Proof not loaded.

Theorem. (quotient_universal_property)

∀X Tx Y Ty f : set, quotient_map X Tx Y f → topology_on Y Ty → Ty ⊆ quotient_topology X Tx Y f → continuous_map X Tx Y Ty f

In Proofgold the corresponding term root is ed7de9... and proposition id is aa9ca1...

Proof:
Proof not loaded.

Definition. We define separation_of to be λX U V ⇒ U ∈ 𝒫 X ∧ V ∈ 𝒫 X ∧ U ∩ V = Empty ∧ U ≠ Empty ∧ V ≠ Empty ∧ U ∪ V = X of type set → set → set → prop.

In Proofgold the corresponding term root is c836b6... and object id is 497542...

Theorem. (separation_of_complement)

∀X U : set, U ⊆ X → U ≠ Empty → U ≠ X → separation_of X U (X ∖ U)

In Proofgold the corresponding term root is 89772f... and proposition id is d5e425...

Proof:
Proof not loaded.

Definition. We define connected_space to be λX Tx ⇒ topology_on X Tx ∧ ¬ (∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ separation_of X U V) of type set → set → prop.

In Proofgold the corresponding term root is 4ec8c7... and object id is 60b46a...

Theorem. (connected_space_topology)

∀X Tx : set, connected_space X Tx → topology_on X Tx

In Proofgold the corresponding term root is 3b3c69... and proposition id is 74d4c3...

Proof:
Proof not loaded.

Theorem. (connected_space_no_separation)

∀X Tx : set, connected_space X Tx → ¬ (∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ separation_of X U V)

In Proofgold the corresponding term root is 021244... and proposition id is 9fa984...

Proof:
Proof not loaded.

Theorem. (homeomorphism_preserves_connected)

∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty f → connected_space X Tx → connected_space Y Ty

In Proofgold the corresponding term root is 3ea75f... and proposition id is 6c5296...

Proof:
Proof not loaded.

Theorem. (clopen_gives_separation)

∀X Tx A : set, topology_on X Tx → A ≠ Empty → A ≠ X → open_in X Tx A → closed_in X Tx A → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ separation_of X U V

In Proofgold the corresponding term root is f098f9... and proposition id is 5546d9...

Proof:
Proof not loaded.

Theorem. (separation_gives_clopen)

∀X Tx U V : set, topology_on X Tx → U ∈ Tx → V ∈ Tx → separation_of X U V → ∃A : set, A ≠ Empty ∧ A ≠ X ∧ open_in X Tx A ∧ closed_in X Tx A

In Proofgold the corresponding term root is 0e8ef8... and proposition id is 87e14d...

Proof:
Proof not loaded.

Theorem. (connected_iff_no_nontrivial_clopen)

∀X Tx : set, topology_on X Tx → (connected_space X Tx ↔ ¬ (∃A : set, A ≠ Empty ∧ A ≠ X ∧ open_in X Tx A ∧ closed_in X Tx A))

In Proofgold the corresponding term root is 94918b... and proposition id is 12287b...

Proof:
Proof not loaded.

Theorem. (separation_subspace_limit_points)

∀X Tx Y A B : set, topology_on X Tx → Y ⊆ X → (((A ∈ subspace_topology X Tx Y ∧ B ∈ subspace_topology X Tx Y) ∧ separation_of Y A B) ↔ (separation_of Y A B ∧ ¬ (∃b : set, b ∈ B ∧ limit_point_of X Tx A b) ∧ ¬ (∃a : set, a ∈ A ∧ limit_point_of X Tx B a)))

In Proofgold the corresponding term root is 8739f2... and proposition id is b292b0...

Proof:
Proof not loaded.

Theorem. (connected_subset_in_separation_side)

∀X Tx C D Y : set, topology_on X Tx → Y ⊆ X → connected_space Y (subspace_topology X Tx Y) → C ∈ Tx → D ∈ Tx → separation_of X C D → Y ⊆ C ∨ Y ⊆ D

In Proofgold the corresponding term root is 6b810a... and proposition id is 8a27e2...

Proof:
Proof not loaded.

Theorem. (union_connected_common_point)

∀X Tx F : set, topology_on X Tx → (∀C : set, C ∈ F → C ⊆ X) → (∀C : set, C ∈ F → connected_space C (subspace_topology X Tx C)) → (∃x : set, ∀C : set, C ∈ F → x ∈ C) → connected_space (⋃ F) (subspace_topology X Tx (⋃ F))

In Proofgold the corresponding term root is 6e31d2... and proposition id is 2503a5...

Proof:
Proof not loaded.

Theorem. (connected_with_limit_points)

∀X Tx A B : set, topology_on X Tx → A ⊆ X → B ⊆ X → connected_space A (subspace_topology X Tx A) → A ⊆ B → B ⊆ closure_of X Tx A → connected_space B (subspace_topology X Tx B)

In Proofgold the corresponding term root is 27359e... and proposition id is f1ddbc...

Proof:
Proof not loaded.

Theorem. (continuous_image_connected)

∀X Tx Y Ty f : set, connected_space X Tx → continuous_map X Tx Y Ty f → connected_space (image_of f X) (subspace_topology Y Ty (image_of f X))

In Proofgold the corresponding term root is d908de... and proposition id is be371f...

Proof:
Proof not loaded.

Definition. We define R_upper_bound to be λA u ⇒ u ∈ R ∧ ∀a : set, a ∈ A → a ∈ R → Rle a u of type set → set → prop.

In Proofgold the corresponding term root is a3e8f8... and object id is 817378...

Definition. We define R_lub to be λA l ⇒ l ∈ R ∧ (∀a : set, a ∈ A → a ∈ R → Rle a l) ∧ (∀u : set, u ∈ R → (∀a : set, a ∈ A → a ∈ R → Rle a u) → Rle l u) of type set → set → prop.

In Proofgold the corresponding term root is e7f414... and object id is 0b8e32...

Theorem. (R_lub_in_R)

∀A l : set, R_lub A l → l ∈ R

In Proofgold the corresponding term root is 1edde1... and proposition id is 4b3ba7...

Proof:
Proof not loaded.

Theorem. (R_lub_unique)

∀A l1 l2 : set, R_lub A l1 → R_lub A l2 → l1 = l2

In Proofgold the corresponding term root is 7a67fa... and proposition id is 2735d9...

Proof:
Proof not loaded.

Theorem. (R_lub_Sing0)

R_lub {0} 0

In Proofgold the corresponding term root is 55e07b... and proposition id is 23a9b2...

Proof:
Proof not loaded.

Theorem. (not_imp)

∀A B : prop, ¬ (A → B) → A ∧ ¬ B

In Proofgold the corresponding term root is 86d3b0... and proposition id is 128901...

Proof:
Proof not loaded.

Theorem. (R_lub_exists)

∀A : set, (∃a0 : set, a0 ∈ A) → (∀a : set, a ∈ A → a ∈ R) → (∃u : set, u ∈ R ∧ ∀a : set, a ∈ A → a ∈ R → Rle a u) → ∃l : set, R_lub A l

In Proofgold the corresponding term root is e1e45f... and proposition id is 1759e8...

Proof:
Proof not loaded.

Theorem. (R_lub_approx_from_below)

∀A l eps : set, R_lub A l → eps ∈ R → Rlt 0 eps → ∃a : set, a ∈ A ∧ a ∈ R ∧ Rlt (add_SNo l (minus_SNo eps)) a

In Proofgold the corresponding term root is a1bb59... and proposition id is 643479...

Proof:
Proof not loaded.

Theorem. (interval_connected)

connected_space R R_standard_topology

In Proofgold the corresponding term root is 690e5f... and proposition id is cba954...

Proof:
Proof not loaded.

Definition. We define between_in_order to be λY u r v ⇒ (order_rel Y u r ∧ order_rel Y r v) ∨ (order_rel Y v r ∧ order_rel Y r u) ∨ r = u ∨ r = v of type set → set → set → set → prop.

In Proofgold the corresponding term root is 468cf1... and object id is 36b712...

Theorem. (between_in_orderI_left)

∀Y u r v : set, order_rel Y u r → order_rel Y r v → between_in_order Y u r v

In Proofgold the corresponding term root is a56727... and proposition id is a5ac1f...

Proof:
Proof not loaded.

Theorem. (between_in_orderI_right)

∀Y u r v : set, order_rel Y v r → order_rel Y r u → between_in_order Y u r v

In Proofgold the corresponding term root is 66598f... and proposition id is c46701...

Proof:
Proof not loaded.

Theorem. (between_in_orderI_eq_left)

∀Y u v : set, between_in_order Y u u v

In Proofgold the corresponding term root is 870f90... and proposition id is ed251c...

Proof:
Proof not loaded.

Theorem. (between_in_orderI_eq_right)

∀Y u v : set, between_in_order Y u v v

In Proofgold the corresponding term root is f1eb2c... and proposition id is f41e3a...

Proof:
Proof not loaded.

Theorem. (intermediate_value_theorem)

∀X Tx Y f a b r : set, connected_space X Tx → simply_ordered_set Y → continuous_map X Tx Y (order_topology Y) f → a ∈ X → b ∈ X → r ∈ Y → between_in_order Y (apply_fun f a) r (apply_fun f b) → ∃c : set, c ∈ X ∧ apply_fun f c = r

In Proofgold the corresponding term root is dc514b... and proposition id is c84cf0...

Proof:
Proof not loaded.

Theorem. (connected_subsets_real_are_intervals)

∀A : set, A ⊆ R → connected_space A (subspace_topology R R_standard_topology A) → ∀x y z : set, x ∈ A → y ∈ A → z ∈ R → (Rlt x z ∧ Rlt z y ∨ Rlt y z ∧ Rlt z x) → z ∈ A

In Proofgold the corresponding term root is 20da2c... and proposition id is 3aa990...

Proof:
Proof not loaded.

Theorem. (image_of_id_const_is_slice)

∀X y0 : set, image_of (pair_map X {(x,x)|x ∈ X} (const_fun X y0)) X = setprod X {y0}

In Proofgold the corresponding term root is f8628f... and proposition id is 91997f...

Proof:
Proof not loaded.

Theorem. (slice_X_connected)

∀X Tx Y Ty y0 : set, connected_space X Tx → topology_on Y Ty → y0 ∈ Y → connected_space (setprod X {y0}) (subspace_topology (setprod X Y) (product_topology X Tx Y Ty) (setprod X {y0}))

In Proofgold the corresponding term root is 7e4a1a... and proposition id is e3c1a8...

Proof:
Proof not loaded.

Theorem. (image_of_const_id_is_slice)

∀Y x0 : set, image_of (pair_map Y (const_fun Y x0) {(y,y)|y ∈ Y}) Y = setprod {x0} Y

In Proofgold the corresponding term root is 8b72b4... and proposition id is 402f9e...

Proof:
Proof not loaded.

Theorem. (homeomorphism_const_id_slice)

∀X Tx Y Ty x0 : set, topology_on X Tx → topology_on Y Ty → x0 ∈ X → homeomorphism Y Ty (setprod {x0} Y) (subspace_topology (setprod X Y) (product_topology X Tx Y Ty) (setprod {x0} Y)) (pair_map Y (const_fun Y x0) {(y,y)|y ∈ Y})

In Proofgold the corresponding term root is 0ab0e3... and proposition id is a60ba7...

Proof:
Proof not loaded.

Theorem. (slice_Y_connected)

∀X Tx Y Ty x0 : set, connected_space Y Ty → topology_on X Tx → x0 ∈ X → connected_space (setprod {x0} Y) (subspace_topology (setprod X Y) (product_topology X Tx Y Ty) (setprod {x0} Y))

In Proofgold the corresponding term root is 23aac1... and proposition id is 7e7190...

Proof:
Proof not loaded.

Theorem. (finite_product_connected)

∀X Tx Y Ty : set, connected_space X Tx → connected_space Y Ty → connected_space (setprod X Y) (product_topology X Tx Y Ty)

In Proofgold the corresponding term root is 6659da... and proposition id is 98fadc...

Proof:
Proof not loaded.

Theorem. (total_function_on_graph)

∀A Y : set, ∀g : set → set, (∀a : set, a ∈ A → g a ∈ Y) → total_function_on (graph A g) A Y

In Proofgold the corresponding term root is aa4379... and proposition id is b84a57...

Proof:
Proof not loaded.

Theorem. (box_basis_is_basis_on)

∀I Xi : set, (∀i : set, i ∈ I → topology_on (space_family_set Xi i) (space_family_topology Xi i)) → basis_on (product_space I Xi) (box_basis I Xi)

In Proofgold the corresponding term root is 7c0976... and proposition id is a49ec4...

Proof:
Proof not loaded.

Theorem. (box_topology_is_topology_on)

∀I Xi : set, (∀i : set, i ∈ I → topology_on (space_family_set Xi i) (space_family_topology Xi i)) → topology_on (product_space I Xi) (box_topology I Xi)

In Proofgold the corresponding term root is 6a8bb0... and proposition id is c49916...

Proof:
Proof not loaded.

Definition. We define R_omega_space to be product_space ω (const_space_family ω R R_standard_topology) of type set.

In Proofgold the corresponding term root is 7e1e43... and object id is 37a532...

Definition. We define R_omega_box_topology to be box_topology ω (const_space_family ω R R_standard_topology) of type set.

In Proofgold the corresponding term root is eb2681... and object id is 7a1a75...

Definition. We define R_omega_product_topology to be product_topology_full ω (const_space_family ω R R_standard_topology) of type set.

In Proofgold the corresponding term root is fe3b20... and object id is e581cd...

Definition. We define bounded_sequence_Romega to be λf ⇒ ∃M : set, M ∈ R ∧ ∀n : set, n ∈ ω → apply_fun f n ∈ open_interval (minus_SNo M) M of type set → prop.

In Proofgold the corresponding term root is ded9b8... and object id is 7dc505...

Definition. We define bounded_sequences_Romega to be {f ∈ R_omega_space|bounded_sequence_Romega f} of type set.

In Proofgold the corresponding term root is 3d63fc... and object id is 5416b2...

Definition. We define unbounded_sequence_Romega to be λf ⇒ ∀M : set, M ∈ R → ∃n : set, n ∈ ω ∧ ¬ (apply_fun f n ∈ open_interval (minus_SNo M) M) of type set → prop.

In Proofgold the corresponding term root is 2e358e... and object id is efcb0a...

Definition. We define unbounded_sequences_Romega to be {f ∈ R_omega_space|unbounded_sequence_Romega f} of type set.

In Proofgold the corresponding term root is 09956e... and object id is 9013cd...

Theorem. (Romega_coord_in_R)

∀f i : set, f ∈ R_omega_space → i ∈ ω → apply_fun f i ∈ R

In Proofgold the corresponding term root is dcdf43... and proposition id is 5cfcaa...

Proof:
Proof not loaded.

Theorem. (bounded_sequences_Romega_in_Power)

bounded_sequences_Romega ∈ 𝒫 R_omega_space

In Proofgold the corresponding term root is e1275f... and proposition id is b9cff9...

Proof:
Proof not loaded.

Theorem. (unbounded_sequences_Romega_in_Power)

unbounded_sequences_Romega ∈ 𝒫 R_omega_space

In Proofgold the corresponding term root is 97be4c... and proposition id is 4012a8...

Proof:
Proof not loaded.

Theorem. (bounded_sequences_in_Romega_box_topology)

bounded_sequences_Romega ∈ R_omega_box_topology

In Proofgold the corresponding term root is bfbde5... and proposition id is c361f3...

Proof:
Proof not loaded.

Theorem. (unbounded_sequences_in_Romega_box_topology)

unbounded_sequences_Romega ∈ R_omega_box_topology

In Proofgold the corresponding term root is a4d927... and proposition id is ff98bd...

Proof:
Proof not loaded.

Theorem. (bounded_unbounded_disjoint_Romega)

bounded_sequences_Romega ∩ unbounded_sequences_Romega = Empty

In Proofgold the corresponding term root is 2a5cf8... and proposition id is 09319a...

Proof:
Proof not loaded.

Theorem. (bounded_union_unbounded_Romega)

bounded_sequences_Romega ∪ unbounded_sequences_Romega = R_omega_space

In Proofgold the corresponding term root is d2e074... and proposition id is 6e661e...

Proof:
Proof not loaded.

Theorem. (bounded_sequences_Romega_nonempty)

bounded_sequences_Romega ≠ Empty

In Proofgold the corresponding term root is ae880f... and proposition id is 673914...

Proof:
Proof not loaded.

Theorem. (unbounded_sequences_Romega_nonempty)

unbounded_sequences_Romega ≠ Empty

In Proofgold the corresponding term root is 0912c4... and proposition id is 5a91e8...

Proof:
Proof not loaded.

Theorem. (R_omega_box_not_connected)

¬ connected_space (product_space ω (const_space_family ω R R_standard_topology)) (box_topology ω (const_space_family ω R R_standard_topology))

In Proofgold the corresponding term root is 4127f9... and proposition id is d8b4b3...

Proof:
Proof not loaded.

Definition. We define Romega_tilde to be λn ⇒ {f ∈ R_omega_space|∀i : set, i ∈ ω → n ∈ i → apply_fun f i = 0} of type set → set.

In Proofgold the corresponding term root is 7c4679... and object id is b346f6...

Definition. We define Romega_infty to be ⋃ {Romega_tilde n|n ∈ ω} of type set.

In Proofgold the corresponding term root is 88c1c7... and object id is ec9c10...

Theorem. (Romega_tilde_sub_Romega)

∀n : set, Romega_tilde n ⊆ R_omega_space

In Proofgold the corresponding term root is 39e7b2... and proposition id is 7a4dbb...

Proof:
Proof not loaded.

Theorem. (Romega_infty_sub_Romega)

Romega_infty ⊆ R_omega_space

In Proofgold the corresponding term root is 7d07fe... and proposition id is 1babd5...

Proof:
Proof not loaded.

Definition. We define Romega_zero to be const_fun ω 0 of type set.

In Proofgold the corresponding term root is 6fc4e6... and object id is 992347...

Theorem. (Romega_zero_in_Romega_space)

Romega_zero ∈ R_omega_space

In Proofgold the corresponding term root is 213c8a... and proposition id is 3ada18...

Proof:
Proof not loaded.

Theorem. (Romega_zero_in_Romega_tilde)

∀n : set, n ∈ ω → Romega_zero ∈ Romega_tilde n

In Proofgold the corresponding term root is 70609d... and proposition id is d6257f...

Proof:
Proof not loaded.

Theorem. (Romega_tilde_nonempty)

∀n : set, n ∈ ω → Romega_tilde n ≠ Empty

In Proofgold the corresponding term root is 43ac69... and proposition id is f0e631...

Proof:
Proof not loaded.

Theorem. (Romega_product_topology_is_topology)

topology_on R_omega_space R_omega_product_topology

In Proofgold the corresponding term root is a5ae5c... and proposition id is 49a2f6...

Proof:
Proof not loaded.

Theorem. (Romega_product_topology_sub_Romega_box_topology)

R_omega_product_topology ⊆ R_omega_box_topology

In Proofgold the corresponding term root is b05ad0... and proposition id is 720bfd...

Proof:
Proof not loaded.

Theorem. (Romega_tilde_subspace_product_subspace_box)

∀n : set, n ∈ ω → subspace_topology R_omega_space R_omega_product_topology (Romega_tilde n) ⊆ subspace_topology R_omega_space R_omega_box_topology (Romega_tilde n)

In Proofgold the corresponding term root is 7c4ce5... and proposition id is 2333e0...

Proof:
Proof not loaded.

Theorem. (dense_in_meets_nonempty_open)

∀A X Tx U : set, topology_on X Tx → closure_of X Tx A = X → U ∈ Tx → U ≠ Empty → U ∩ A ≠ Empty

In Proofgold the corresponding term root is 5cf3c4... and proposition id is 3c9dff...

Proof:
Proof not loaded.

Theorem. (connected_space_if_dense_connected_subset)

∀X Tx A : set, topology_on X Tx → A ⊆ X → connected_space A (subspace_topology X Tx A) → closure_of X Tx A = X → connected_space X Tx

In Proofgold the corresponding term root is 9a78ad... and proposition id is 72c8d1...

Proof:
Proof not loaded.

Theorem. (graph_omega_in_Romega_space)

∀h : set → set, (∀i : set, i ∈ ω → h i ∈ R) → graph ω h ∈ R_omega_space

In Proofgold the corresponding term root is ca17ca... and proposition id is 07d8f0...

Proof:
Proof not loaded.

Definition. We define Romega_singleton_seq to be λr ⇒ graph ω (λi : set ⇒ If_i (0 ∈ i) 0 r) of type set → set.

In Proofgold the corresponding term root is cb97e8... and object id is 803f06...

Theorem. (Romega_singleton_seq_in_Romega_space)

∀r : set, r ∈ R → Romega_singleton_seq r ∈ R_omega_space

In Proofgold the corresponding term root is b34fc4... and proposition id is 3e6b40...

Proof:
Proof not loaded.

Theorem. (Romega_singleton_seq_apply)

∀r i : set, r ∈ R → i ∈ ω → apply_fun (Romega_singleton_seq r) i = If_i (0 ∈ i) 0 r

In Proofgold the corresponding term root is 25e4c7... and proposition id is 49ada9...

Proof:
Proof not loaded.

Theorem. (Romega_singleton_seq_in_Romega_tilde0)

∀r : set, r ∈ R → Romega_singleton_seq r ∈ Romega_tilde 0

In Proofgold the corresponding term root is 1d90e4... and proposition id is 305a43...

Proof:
Proof not loaded.

Definition. We define Romega_singleton_map to be graph R Romega_singleton_seq of type set.

In Proofgold the corresponding term root is dd89d8... and object id is e22bc4...

Theorem. (Romega_singleton_map_apply)

∀r : set, r ∈ R → apply_fun Romega_singleton_map r = Romega_singleton_seq r

In Proofgold the corresponding term root is 5f7f35... and proposition id is 9bcc04...

Proof:
Proof not loaded.

Theorem. (Romega_singleton_map_function_on)

function_on Romega_singleton_map R R_omega_space

In Proofgold the corresponding term root is 213993... and proposition id is de4232...

Proof:
Proof not loaded.

Theorem. (Romega_singleton_map_continuous_prod)

continuous_map R R_standard_topology R_omega_space R_omega_product_topology Romega_singleton_map

In Proofgold the corresponding term root is 2c4250... and proposition id is 5d0e8e...

Proof:
Proof not loaded.

Theorem. (image_of_Romega_singleton_map_sub_Romega_tilde0)

image_of Romega_singleton_map R ⊆ Romega_tilde 0

In Proofgold the corresponding term root is e56bcb... and proposition id is d25145...

Proof:
Proof not loaded.

Definition. We define Romega_extend_seq to be λk p ⇒ graph ω (λi : set ⇒ If_i (ordsucc k ∈ i) 0 (If_i (i = ordsucc k) (p 1) (apply_fun (p 0) i))) of type set → set → set.

In Proofgold the corresponding term root is 6cf88f... and object id is b2e39a...

Theorem. (Romega_extend_seq_apply)

∀k p i : set, i ∈ ω → apply_fun (Romega_extend_seq k p) i = If_i (ordsucc k ∈ i) 0 (If_i (i = ordsucc k) (p 1) (apply_fun (p 0) i))

In Proofgold the corresponding term root is 16faf1... and proposition id is fb6a1f...

Proof:
Proof not loaded.

Theorem. (Romega_extend_seq_in_Romega_tilde_succ)

∀k p : set, k ∈ ω → p ∈ setprod (Romega_tilde k) R → Romega_extend_seq k p ∈ Romega_tilde (ordsucc k)

In Proofgold the corresponding term root is 9e30d4... and proposition id is b6348f...

Proof:
Proof not loaded.

Definition. We define Romega_extend_map to be λk ⇒ graph (setprod (Romega_tilde k) R) (λp : set ⇒ Romega_extend_seq k p) of type set → set.

In Proofgold the corresponding term root is 8ee213... and object id is cf6744...

Theorem. (Romega_extend_map_apply)

∀k p : set, k ∈ ω → p ∈ setprod (Romega_tilde k) R → apply_fun (Romega_extend_map k) p = Romega_extend_seq k p

In Proofgold the corresponding term root is a36d6b... and proposition id is 56aa57...

Proof:
Proof not loaded.

Theorem. (Romega_extend_map_function_on)

∀k : set, k ∈ ω → function_on (Romega_extend_map k) (setprod (Romega_tilde k) R) R_omega_space

In Proofgold the corresponding term root is 8dee82... and proposition id is 58bc52...

Proof:
Proof not loaded.

Theorem. (Romega_extend_map_function_on_tilde_succ)

∀k : set, k ∈ ω → function_on (Romega_extend_map k) (setprod (Romega_tilde k) R) (Romega_tilde (ordsucc k))

In Proofgold the corresponding term root is 35eee6... and proposition id is 3f5296...

Proof:
Proof not loaded.

Theorem. (image_of_Romega_extend_map_sub_Romega_tilde_succ)

∀k : set, k ∈ ω → image_of (Romega_extend_map k) (setprod (Romega_tilde k) R) ⊆ Romega_tilde (ordsucc k)

In Proofgold the corresponding term root is 094b3b... and proposition id is 16d71a...

Proof:
Proof not loaded.

Theorem. (Romega_extend_map_continuous_in_Romega_product)

∀k : set, k ∈ ω → continuous_map (setprod (Romega_tilde k) R) (product_topology (Romega_tilde k) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde k)) R R_standard_topology) R_omega_space R_omega_product_topology (Romega_extend_map k)

In Proofgold the corresponding term root is ccce3a... and proposition id is 088601...

Proof:
Proof not loaded.

Theorem. (Romega_extend_map_continuous_to_Romega_tilde_succ)

∀k : set, k ∈ ω → continuous_map (setprod (Romega_tilde k) R) (product_topology (Romega_tilde k) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde k)) R R_standard_topology) (Romega_tilde (ordsucc k)) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde (ordsucc k))) (Romega_extend_map k)

In Proofgold the corresponding term root is 3c776f... and proposition id is 5824b2...

Proof:
Proof not loaded.

Definition. We define Romega_restrict_seq to be λk f ⇒ graph ω (λi : set ⇒ If_i (k ∈ i) 0 (apply_fun f i)) of type set → set → set.

In Proofgold the corresponding term root is b7db59... and object id is 9b8f6f...

Theorem. (Romega_restrict_seq_apply)

∀k f i : set, k ∈ ω → f ∈ R_omega_space → i ∈ ω → apply_fun (Romega_restrict_seq k f) i = If_i (k ∈ i) 0 (apply_fun f i)

In Proofgold the corresponding term root is 2c1537... and proposition id is b9fb0c...

Proof:
Proof not loaded.

Theorem. (Romega_restrict_seq_in_Romega_tilde)

∀k f : set, k ∈ ω → f ∈ R_omega_space → Romega_restrict_seq k f ∈ Romega_tilde k

In Proofgold the corresponding term root is 63eed2... and proposition id is 3d3ed3...

Proof:
Proof not loaded.

Definition. We define Romega_coord_map to be λi ⇒ graph R_omega_space (λf : set ⇒ apply_fun f i) of type set → set.

In Proofgold the corresponding term root is e278c6... and object id is 9521a2...

Theorem. (Romega_coord_map_apply)

∀i f : set, i ∈ ω → f ∈ R_omega_space → apply_fun (Romega_coord_map i) f = apply_fun f i

In Proofgold the corresponding term root is d1119f... and proposition id is b36393...

Proof:
Proof not loaded.

Theorem. (Romega_coord_map_function_on)

∀i : set, i ∈ ω → function_on (Romega_coord_map i) R_omega_space R

In Proofgold the corresponding term root is 2cb4b7... and proposition id is ce91ec...

Proof:
Proof not loaded.

Theorem. (Romega_product_subbasis_on)

subbasis_on R_omega_space (product_subbasis_full ω (const_space_family ω R R_standard_topology))

In Proofgold the corresponding term root is 45c0fb... and proposition id is 8a43a2...

Proof:
Proof not loaded.

Theorem. (Romega_coord_map_continuous_in_Romega_product)

∀i : set, i ∈ ω → continuous_map R_omega_space R_omega_product_topology R R_standard_topology (Romega_coord_map i)

In Proofgold the corresponding term root is fc6698... and proposition id is d9cb95...

Proof:
Proof not loaded.

Definition. We define Romega_restrict_map to be λk ⇒ graph R_omega_space (λf : set ⇒ Romega_restrict_seq k f) of type set → set.

In Proofgold the corresponding term root is 00d2c7... and object id is 3d59ab...

Theorem. (Romega_restrict_map_apply)

∀k f : set, k ∈ ω → f ∈ R_omega_space → apply_fun (Romega_restrict_map k) f = Romega_restrict_seq k f

In Proofgold the corresponding term root is 750d72... and proposition id is d12d4b...

Proof:
Proof not loaded.

Theorem. (Romega_restrict_map_function_on)

∀k : set, k ∈ ω → function_on (Romega_restrict_map k) R_omega_space R_omega_space

In Proofgold the corresponding term root is 87f662... and proposition id is 854954...

Proof:
Proof not loaded.

Theorem. (Romega_restrict_map_continuous_in_Romega_product)

∀k : set, k ∈ ω → continuous_map R_omega_space R_omega_product_topology R_omega_space R_omega_product_topology (Romega_restrict_map k)

In Proofgold the corresponding term root is 341341... and proposition id is 3b7c26...

Proof:
Proof not loaded.

Definition. We define Romega_split_map to be λk ⇒ graph (Romega_tilde (ordsucc k)) (λf : set ⇒ (Romega_restrict_seq k f,apply_fun f (ordsucc k))) of type set → set.

In Proofgold the corresponding term root is 1516c2... and object id is da96f4...

Theorem. (Romega_split_map_apply)

∀k f : set, k ∈ ω → f ∈ Romega_tilde (ordsucc k) → apply_fun (Romega_split_map k) f = (Romega_restrict_seq k f,apply_fun f (ordsucc k))

In Proofgold the corresponding term root is 4ac352... and proposition id is 617f6b...

Proof:
Proof not loaded.

Theorem. (Romega_split_map_function_on)

∀k : set, k ∈ ω → function_on (Romega_split_map k) (Romega_tilde (ordsucc k)) (setprod (Romega_tilde k) R)

In Proofgold the corresponding term root is 56089b... and proposition id is 4395a7...

Proof:
Proof not loaded.

Theorem. (Romega_split_map_continuous)

∀k : set, k ∈ ω → continuous_map (Romega_tilde (ordsucc k)) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde (ordsucc k))) (setprod (Romega_tilde k) R) (product_topology (Romega_tilde k) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde k)) R R_standard_topology) (Romega_split_map k)

In Proofgold the corresponding term root is f48844... and proposition id is 9da77e...

Proof:
Proof not loaded.

Theorem. (R_omega_space_ext)

∀f g : set, f ∈ R_omega_space → g ∈ R_omega_space → (∀i : set, i ∈ ω → apply_fun f i = apply_fun g i) → f = g

In Proofgold the corresponding term root is 815391... and proposition id is f033ed...

Proof:
Proof not loaded.

Theorem. (Romega_tilde_succ_homeomorphism)

∀k : set, k ∈ ω → homeomorphism (setprod (Romega_tilde k) R) (product_topology (Romega_tilde k) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde k)) R R_standard_topology) (Romega_tilde (ordsucc k)) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde (ordsucc k))) (Romega_extend_map k)

In Proofgold the corresponding term root is 790863... and proposition id is 552277...

Proof:
Proof not loaded.

Theorem. (finite_subset_of_omega_bounded)

∀F : set, F ⊆ ω → finite F → ∃n ∈ ω, ∀m ∈ F, m ∈ n

In Proofgold the corresponding term root is 34525f... and proposition id is d8d9d2...

Proof:
Proof not loaded.

Theorem. (Romega_infty_eq_finite_support)

Romega_infty = {f ∈ R_omega_space|∃F : set, finite F ∧ ∀i : set, i ∈ ω ∖ F → apply_fun f i = 0}

In Proofgold the corresponding term root is b102d2... and proposition id is 249f24...

Proof:
Proof not loaded.

Theorem. (Romega_product_cylinder_open)

∀i U : set, i ∈ ω → U ∈ space_family_topology (const_space_family ω R R_standard_topology) i → product_cylinder ω (const_space_family ω R R_standard_topology) i U ∈ R_omega_product_topology

In Proofgold the corresponding term root is a736cb... and proposition id is 5ed93e...

Proof:
Proof not loaded.

Theorem. (Romega_box_basis_cap_Romega_tilde_in_product_subspace)

∀n b : set, n ∈ ω → b ∈ box_basis ω (const_space_family ω R R_standard_topology) → (b ∩ Romega_tilde n) ∈ subspace_topology R_omega_space R_omega_product_topology (Romega_tilde n)

In Proofgold the corresponding term root is 98511d... and proposition id is 0c8ccd...

Proof:
Proof not loaded.

Theorem. (Romega_tilde_subspace_box_subspace_product)

∀n : set, n ∈ ω → subspace_topology R_omega_space R_omega_box_topology (Romega_tilde n) ⊆ subspace_topology R_omega_space R_omega_product_topology (Romega_tilde n)

In Proofgold the corresponding term root is a1da86... and proposition id is 7cf082...

Proof:
Proof not loaded.

Theorem. (Romega_tilde_subspace_box_eq_product)

∀n : set, n ∈ ω → subspace_topology R_omega_space R_omega_box_topology (Romega_tilde n) = subspace_topology R_omega_space R_omega_product_topology (Romega_tilde n)

In Proofgold the corresponding term root is 98b422... and proposition id is d1e444...

Proof:
Proof not loaded.

Theorem. (Romega_infty_closed_in_Romega_box_topology)

closed_in R_omega_space R_omega_box_topology Romega_infty

In Proofgold the corresponding term root is d27d06... and proposition id is 09d0ad...

Proof:
Proof not loaded.

Theorem. (Romega_infty_meets_product_basis)

∀b x : set, b ∈ basis_of_subbasis R_omega_space (product_subbasis_full ω (const_space_family ω R R_standard_topology)) → x ∈ b → b ∩ Romega_infty ≠ Empty

In Proofgold the corresponding term root is 575866... and proposition id is fa2680...

Proof:
Proof not loaded.

Theorem. (Romega_tilde0_meets_product_basis)

∀b x : set, b ∈ basis_of_subbasis R_omega_space (product_subbasis_full ω (const_space_family ω R R_standard_topology)) → x ∈ Romega_tilde 0 → x ∈ b → b ∩ image_of Romega_singleton_map R ≠ Empty

In Proofgold the corresponding term root is 94d32a... and proposition id is a45726...

Proof:
Proof not loaded.

Theorem. (Romega_tilde0_singletons_dense_in_subspace)

closure_of (Romega_tilde 0) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde 0)) (image_of Romega_singleton_map R) = Romega_tilde 0

In Proofgold the corresponding term root is 140897... and proposition id is bf8532...

Proof:
Proof not loaded.

Theorem. (Romega_tilde0_connected)

connected_space (Romega_tilde 0) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde 0))

In Proofgold the corresponding term root is a80d6b... and proposition id is b33628...

Proof:
Proof not loaded.

Theorem. (Romega_tilde_connected)

∀n : set, n ∈ ω → connected_space (Romega_tilde n) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde n))

In Proofgold the corresponding term root is b2dad9... and proposition id is 5dcb29...

Proof:
Proof not loaded.

Theorem. (Romega_infty_connected)

connected_space Romega_infty (subspace_topology R_omega_space R_omega_product_topology Romega_infty)

In Proofgold the corresponding term root is c7cb44... and proposition id is 5add09...

Proof:
Proof not loaded.

Theorem. (Romega_infty_dense)

closure_of R_omega_space R_omega_product_topology Romega_infty = R_omega_space

In Proofgold the corresponding term root is c248de... and proposition id is 4b000d...

Proof:
Proof not loaded.

Theorem. (R_omega_product_connected)

connected_space (product_space ω (const_space_family ω R R_standard_topology)) (product_topology_full ω (const_space_family ω R R_standard_topology))

In Proofgold the corresponding term root is cd713e... and proposition id is 94d7a8...

Proof:
Proof not loaded.

Theorem. (Romega_tilde_connected_box)

∀n : set, n ∈ ω → connected_space (Romega_tilde n) (subspace_topology R_omega_space R_omega_box_topology (Romega_tilde n))

In Proofgold the corresponding term root is 4ad09a... and proposition id is 9d259a...

Proof:
Proof not loaded.

Theorem. (Romega_infty_connected_box)

connected_space Romega_infty (subspace_topology R_omega_space R_omega_box_topology Romega_infty)

In Proofgold the corresponding term root is 7b5bd9... and proposition id is cf1e1b...

Proof:
Proof not loaded.

Definition. We define path_between to be λX x y p ⇒ function_on p unit_interval X ∧ apply_fun p 0 = x ∧ apply_fun p 1 = y of type set → set → set → set → prop.

In Proofgold the corresponding term root is a448fd... and object id is 527437...

Definition. We define path_connected_space to be λX Tx ⇒ topology_on X Tx ∧ ∀x y : set, x ∈ X → y ∈ X → ∃p : set, path_between X x y p ∧ continuous_map unit_interval unit_interval_topology X Tx p of type set → set → prop.

In Proofgold the corresponding term root is 1c23ff... and object id is 7703ca...

Theorem. (path_between_pair0)

∀X x y p : set, path_between X x y p → function_on p unit_interval X ∧ apply_fun p 0 = x

In Proofgold the corresponding term root is 9fadb3... and proposition id is 4e90c5...

Proof:
Proof not loaded.

Theorem. (path_between_function_on)

∀X x y p : set, path_between X x y p → function_on p unit_interval X

In Proofgold the corresponding term root is b560d3... and proposition id is 2dd264...

Proof:
Proof not loaded.

Theorem. (path_between_at_zero)

∀X x y p : set, path_between X x y p → apply_fun p 0 = x

In Proofgold the corresponding term root is 5b8f87... and proposition id is eba97a...

Proof:
Proof not loaded.

Theorem. (path_between_at_one)

∀X x y p : set, path_between X x y p → apply_fun p 1 = y

In Proofgold the corresponding term root is d089d1... and proposition id is 671dd2...

Proof:
Proof not loaded.

Theorem. (path_betweenI)

∀X x y p : set, function_on p unit_interval X → apply_fun p 0 = x → apply_fun p 1 = y → path_between X x y p

In Proofgold the corresponding term root is 4e12d4... and proposition id is 612aeb...

Proof:
Proof not loaded.

Theorem. (path_witness_between)

∀X Tx x y p : set, (path_between X x y p ∧ continuous_map unit_interval unit_interval_topology X Tx p) → path_between X x y p

In Proofgold the corresponding term root is b79e20... and proposition id is b7d68f...

Proof:
Proof not loaded.

Theorem. (path_witness_continuous)

∀X Tx x y p : set, (path_between X x y p ∧ continuous_map unit_interval unit_interval_topology X Tx p) → continuous_map unit_interval unit_interval_topology X Tx p

In Proofgold the corresponding term root is 5c2901... and proposition id is 0eec40...

Proof:
Proof not loaded.

Theorem. (path_connected_space_topology)

∀X Tx : set, path_connected_space X Tx → topology_on X Tx

In Proofgold the corresponding term root is a92c05... and proposition id is 1143f9...

Proof:
Proof not loaded.

Theorem. (path_connected_space_paths)

∀X Tx x y : set, path_connected_space X Tx → x ∈ X → y ∈ X → ∃p : set, path_between X x y p ∧ continuous_map unit_interval unit_interval_topology X Tx p

In Proofgold the corresponding term root is 5ea9da... and proposition id is 7d89b1...

Proof:
Proof not loaded.

Theorem. (unit_interval_connected)

connected_space unit_interval unit_interval_topology

In Proofgold the corresponding term root is 1a818a... and proposition id is fa72ad...

Proof:
Proof not loaded.

Theorem. (zero_one_in_unit_interval)

0 ∈ unit_interval ∧ 1 ∈ unit_interval

In Proofgold the corresponding term root is 063658... and proposition id is 7c7606...

Proof:
Proof not loaded.

Theorem. (path_between_exists)

∀X x y : set, x ∈ X → y ∈ X → ∃p : set, path_between X x y p

In Proofgold the corresponding term root is 37dfa1... and proposition id is 5670f7...

Proof:
Proof not loaded.

Theorem. (separation_has_elements)

∀X U V : set, separation_of X U V → (∃x : set, x ∈ U) ∧ (∃y : set, y ∈ V)

In Proofgold the corresponding term root is 046c81... and proposition id is a7df59...

Proof:
Proof not loaded.

Theorem. (separation_subsets)

∀X U V : set, separation_of X U V → U ⊆ X ∧ V ⊆ X

In Proofgold the corresponding term root is 2e3843... and proposition id is b1c520...

Proof:
Proof not loaded.

Theorem. (subset_elem)

∀A B x : set, A ⊆ B → x ∈ A → x ∈ B

In Proofgold the corresponding term root is a4373f... and proposition id is 02033b...

Proof:
Proof not loaded.

Theorem. (path_connected_implies_connected)

∀X Tx : set, path_connected_space X Tx → connected_space X Tx

In Proofgold the corresponding term root is 84494e... and proposition id is 88d4b0...

Proof:
Proof not loaded.

Theorem. (continuous_image_path_connected)

∀X Tx Y Ty f : set, path_connected_space X Tx → continuous_map X Tx Y Ty f → (∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun f x = y) → path_connected_space Y Ty

In Proofgold the corresponding term root is e76a99... and proposition id is d7a953...

Proof:
Proof not loaded.

Definition. We define path_component_of to be λX Tx x ⇒ {y ∈ X|∃p : set, function_on p unit_interval X ∧ continuous_map unit_interval unit_interval_topology X Tx p ∧ apply_fun p 0 = x ∧ apply_fun p 1 = y} of type set → set → set → set.

In Proofgold the corresponding term root is c6757c... and object id is 00ce56...

Theorem. (path_component_symmetric_axiom)

∀X Tx x y : set, topology_on X Tx → x ∈ X → y ∈ X → y ∈ path_component_of X Tx x → x ∈ path_component_of X Tx y

In Proofgold the corresponding term root is 4c013f... and proposition id is bf323a...

Proof:
Proof not loaded.

Definition. We define unit_interval_left_half to be {t ∈ unit_interval|¬ (Rlt (eps_ 1) t)} of type set.

In Proofgold the corresponding term root is 1c0cf6... and object id is effed8...

Definition. We define unit_interval_right_half to be {t ∈ unit_interval|¬ (Rlt t (eps_ 1))} of type set.

In Proofgold the corresponding term root is dddc5f... and object id is e6d097...

Theorem. (unit_interval_left_half_sub)

unit_interval_left_half ⊆ unit_interval

In Proofgold the corresponding term root is 122c62... and proposition id is b90f97...

Proof:
Proof not loaded.

Theorem. (unit_interval_right_half_sub)

unit_interval_right_half ⊆ unit_interval

In Proofgold the corresponding term root is 21b69f... and proposition id is c7ff6e...

Proof:
Proof not loaded.

Definition. We define double_map_left_half to be graph unit_interval_left_half (λt : set ⇒ mul_SNo 2 t) of type set.

In Proofgold the corresponding term root is 613617... and object id is 1e69dc...

Definition. We define double_minus_one_map_right_half to be graph unit_interval_right_half (λt : set ⇒ add_SNo (mul_SNo 2 t) (minus_SNo 1)) of type set.

In Proofgold the corresponding term root is 59228b... and object id is 5c3d6e...

Theorem. (double_map_apply)

∀t : set, t ∈ unit_interval_left_half → apply_fun double_map_left_half t = mul_SNo 2 t

In Proofgold the corresponding term root is 996c47... and proposition id is e9096e...

Proof:
Proof not loaded.

Theorem. (double_minus_one_map_apply)

∀t : set, t ∈ unit_interval_right_half → apply_fun double_minus_one_map_right_half t = add_SNo (mul_SNo 2 t) (minus_SNo 1)

In Proofgold the corresponding term root is 9434d0... and proposition id is ca235e...

Proof:
Proof not loaded.

Theorem. (Rlt_mul2_left_iff)

∀a t : set, a ∈ R → t ∈ R → (Rlt a (mul_SNo 2 t) ↔ Rlt (mul_SNo (eps_ 1) a) t)

In Proofgold the corresponding term root is 8ff8be... and proposition id is b519f8...

Proof:
Proof not loaded.

Theorem. (Rlt_mul2_right_iff)

∀t b : set, t ∈ R → b ∈ R → (Rlt (mul_SNo 2 t) b ↔ Rlt t (mul_SNo (eps_ 1) b))

In Proofgold the corresponding term root is 159113... and proposition id is 8c36c8...

Proof:
Proof not loaded.

Theorem. (zero_in_unit_interval_left_half)

0 ∈ unit_interval_left_half

In Proofgold the corresponding term root is d63c20... and proposition id is 5c4b4d...

Proof:
Proof not loaded.

Theorem. (eps_1_in_unit_interval_left_half)

eps_ 1 ∈ unit_interval_left_half

In Proofgold the corresponding term root is 9cbfcc... and proposition id is 1123f6...

Proof:
Proof not loaded.

Theorem. (eps_1_in_unit_interval_right_half)

eps_ 1 ∈ unit_interval_right_half

In Proofgold the corresponding term root is 4e5454... and proposition id is fd043e...

Proof:
Proof not loaded.

Theorem. (one_in_unit_interval_right_half)

1 ∈ unit_interval_right_half

In Proofgold the corresponding term root is 5e089f... and proposition id is e69e02...

Proof:
Proof not loaded.

Theorem. (double_map_at_0)

apply_fun double_map_left_half 0 = 0

In Proofgold the corresponding term root is 8d5ddf... and proposition id is 6508ab...

Proof:
Proof not loaded.

Theorem. (double_map_at_eps1)

apply_fun double_map_left_half (eps_ 1) = 1

In Proofgold the corresponding term root is 25dc24... and proposition id is 415c33...

Proof:
Proof not loaded.

Theorem. (double_minus_one_map_at_eps1)

apply_fun double_minus_one_map_right_half (eps_ 1) = 0

In Proofgold the corresponding term root is 6f77cb... and proposition id is dd5174...

Proof:
Proof not loaded.

Theorem. (double_minus_one_map_at_1)

apply_fun double_minus_one_map_right_half 1 = 1

In Proofgold the corresponding term root is abd0e4... and proposition id is ae09d0...

Proof:
Proof not loaded.

Theorem. (unit_interval_halves_cover)

unit_interval_left_half ∪ unit_interval_right_half = unit_interval

In Proofgold the corresponding term root is 00bf1e... and proposition id is 448e99...

Proof:
Proof not loaded.

Theorem. (unit_interval_halves_closed)

closed_in unit_interval unit_interval_topology unit_interval_left_half ∧ closed_in unit_interval unit_interval_topology unit_interval_right_half

In Proofgold the corresponding term root is 37bcd2... and proposition id is 4ff8cd...

Proof:
Proof not loaded.

Theorem. (unit_interval_halves_intersection)

(unit_interval_left_half ∩ unit_interval_right_half) = {eps_ 1}

In Proofgold the corresponding term root is 88a122... and proposition id is 469023...

Proof:
Proof not loaded.

Theorem. (double_map_function_on)

function_on double_map_left_half unit_interval_left_half unit_interval

In Proofgold the corresponding term root is fb7d95... and proposition id is 13ff2d...

Proof:
Proof not loaded.

Theorem. (double_minus_one_map_function_on)

function_on double_minus_one_map_right_half unit_interval_right_half unit_interval

In Proofgold the corresponding term root is fb5b6b... and proposition id is 76e2ae...

Proof:
Proof not loaded.

Theorem. (double_map_continuous)

continuous_map unit_interval_left_half (subspace_topology unit_interval unit_interval_topology unit_interval_left_half) unit_interval unit_interval_topology double_map_left_half

In Proofgold the corresponding term root is 6d7cc6... and proposition id is 3393c3...

Proof:
Proof not loaded.

Theorem. (double_minus_one_map_continuous)

continuous_map unit_interval_right_half (subspace_topology unit_interval unit_interval_topology unit_interval_right_half) unit_interval unit_interval_topology double_minus_one_map_right_half

In Proofgold the corresponding term root is 28cdc4... and proposition id is 7ebc86...

Proof:
Proof not loaded.

Theorem. (path_component_transitive_axiom)

∀X Tx x y z : set, topology_on X Tx → x ∈ X → y ∈ X → z ∈ X → y ∈ path_component_of X Tx x → z ∈ path_component_of X Tx y → z ∈ path_component_of X Tx x

In Proofgold the corresponding term root is 7cb0fa... and proposition id is b260ef...

Proof:
Proof not loaded.

Theorem. (unit_interval_inclusion_continuous)

continuous_map unit_interval unit_interval_topology R R_standard_topology {(t,t)|t ∈ unit_interval}

In Proofgold the corresponding term root is 9ae775... and proposition id is 750faf...

Proof:
Proof not loaded.

Theorem. (subspace_path_connected_implies_in_path_component)

∀X Tx V y z : set, topology_on X Tx → V ⊆ X → path_connected_space V (subspace_topology X Tx V) → y ∈ V → z ∈ V → z ∈ path_component_of X Tx y

In Proofgold the corresponding term root is 70da19... and proposition id is 86993e...

Proof:
Proof not loaded.

Theorem. (path_component_reflexive)

∀X Tx x : set, topology_on X Tx → x ∈ X → x ∈ path_component_of X Tx x

In Proofgold the corresponding term root is 5ce944... and proposition id is f90005...

Proof:
Proof not loaded.

Theorem. (path_components_equivalence_relation)

∀X Tx : set, topology_on X Tx → (∀x : set, x ∈ X → x ∈ path_component_of X Tx x) ∧ (∀x y : set, x ∈ X → y ∈ X → y ∈ path_component_of X Tx x → x ∈ path_component_of X Tx y) ∧ (∀x y z : set, x ∈ X → y ∈ X → z ∈ X → y ∈ path_component_of X Tx x → z ∈ path_component_of X Tx y → z ∈ path_component_of X Tx x)

In Proofgold the corresponding term root is 27c7ad... and proposition id is 518a0a...

Proof:
Proof not loaded.

Definition. We define component_of to be λX Tx x ⇒ {y ∈ X|∃C : set, connected_space C (subspace_topology X Tx C) ∧ x ∈ C ∧ y ∈ C} of type set → set → set → set.

In Proofgold the corresponding term root is 6486c3... and object id is 666165...

Definition. We define locally_connected to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∀U : set, U ∈ Tx → x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ V ⊆ U ∧ connected_space V (subspace_topology X Tx V) of type set → set → prop.

In Proofgold the corresponding term root is 1d4a5f... and object id is e09402...

Theorem. (locally_connected_topology)

∀X Tx : set, locally_connected X Tx → topology_on X Tx

In Proofgold the corresponding term root is c9188e... and proposition id is c0c692...

Proof:
Proof not loaded.

Theorem. (locally_connected_local)

∀X Tx x U : set, locally_connected X Tx → x ∈ X → U ∈ Tx → x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ V ⊆ U ∧ connected_space V (subspace_topology X Tx V)

In Proofgold the corresponding term root is ae5353... and proposition id is 1f1959...

Proof:
Proof not loaded.

Theorem. (open_subspace_locally_connected)

∀X Tx Y : set, locally_connected X Tx → Y ∈ Tx → locally_connected Y (subspace_topology X Tx Y)

In Proofgold the corresponding term root is ac11d1... and proposition id is 381bf0...

Proof:
Proof not loaded.

Definition. We define locally_path_connected to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∀U : set, U ∈ Tx → x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ V ⊆ U ∧ path_connected_space V (subspace_topology X Tx V) of type set → set → prop.

In Proofgold the corresponding term root is 216222... and object id is b36dc1...

Theorem. (locally_path_connected_topology)

∀X Tx : set, locally_path_connected X Tx → topology_on X Tx

In Proofgold the corresponding term root is 8f0194... and proposition id is ad2df2...

Proof:
Proof not loaded.

Theorem. (locally_path_connected_local)

∀X Tx x U : set, locally_path_connected X Tx → x ∈ X → U ∈ Tx → x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ V ⊆ U ∧ path_connected_space V (subspace_topology X Tx V)

In Proofgold the corresponding term root is e1f323... and proposition id is a16882...

Proof:
Proof not loaded.

Theorem. (singleton_subspace_connected)

∀X Tx x : set, topology_on X Tx → x ∈ X → connected_space {x} (subspace_topology X Tx {x})

In Proofgold the corresponding term root is c76463... and proposition id is 1374c3...

Proof:
Proof not loaded.

Theorem. (point_in_component)

∀X Tx x : set, topology_on X Tx → x ∈ X → x ∈ component_of X Tx x

In Proofgold the corresponding term root is 154f51... and proposition id is de6fd4...

Proof:
Proof not loaded.

Theorem. (connected_subspace_subset)

∀X Tx C : set, topology_on X Tx → connected_space C (subspace_topology X Tx C) → C ⊆ X

In Proofgold the corresponding term root is 035039... and proposition id is c63b27...

Proof:
Proof not loaded.

Theorem. (component_of_connected)

∀X Tx x : set, topology_on X Tx → x ∈ X → connected_space (component_of X Tx x) (subspace_topology X Tx (component_of X Tx x))

In Proofgold the corresponding term root is 60388d... and proposition id is 90cddf...

Proof:
Proof not loaded.

Theorem. (component_of_eq_if_in)

∀X Tx x y : set, topology_on X Tx → x ∈ X → y ∈ component_of X Tx x → component_of X Tx y = component_of X Tx x

In Proofgold the corresponding term root is 6ff84f... and proposition id is 273b4f...

Proof:
Proof not loaded.

Theorem. (component_of_whole)

∀X Tx x : set, connected_space X Tx → x ∈ X → component_of X Tx x = X

In Proofgold the corresponding term root is 6d2b7c... and proposition id is 2fe4cb...

Proof:
Proof not loaded.

Theorem. (path_component_of_whole)

∀X Tx x : set, path_connected_space X Tx → x ∈ X → path_component_of X Tx x = X

In Proofgold the corresponding term root is b8deec... and proposition id is 6e3b50...

Proof:
Proof not loaded.

Definition. We define pairwise_disjoint to be λFam ⇒ ∀U V : set, U ∈ Fam → V ∈ Fam → U ≠ V → U ∩ V = Empty of type set → prop.

In Proofgold the corresponding term root is e9cebe... and object id is 27c86b...

Definition. We define covers to be λX U ⇒ ∀x : set, x ∈ X → ∃u : set, u ∈ U ∧ x ∈ u of type set → set → prop.

In Proofgold the corresponding term root is 26937a... and object id is df2459...

Theorem. (Subq_Union_implies_covers)

∀X Fam : set, X ⊆ ⋃ Fam → covers X Fam

In Proofgold the corresponding term root is abe4b6... and proposition id is 0d08c4...

Proof:
Proof not loaded.

Theorem. (covers_implies_Subq_Union)

∀X Fam : set, covers X Fam → X ⊆ ⋃ Fam

In Proofgold the corresponding term root is 867cd8... and proposition id is 7eed64...

Proof:
Proof not loaded.

Theorem. (path_components_open)

∀X Tx : set, locally_path_connected X Tx → ∀x : set, x ∈ X → open_in X Tx (path_component_of X Tx x)

In Proofgold the corresponding term root is 426b74... and proposition id is d20113...

Proof:
Proof not loaded.

Theorem. (components_equal_path_components)

∀X Tx : set, locally_path_connected X Tx → ∀x : set, x ∈ X → path_component_of X Tx x = component_of X Tx x

In Proofgold the corresponding term root is e295e3... and proposition id is 5787de...

Proof:
Proof not loaded.

Theorem. (components_are_closed)

∀X Tx : set, topology_on X Tx → ∀x : set, x ∈ X → closed_in X Tx (component_of X Tx x)

In Proofgold the corresponding term root is d56ed4... and proposition id is 6b58cd...

Proof:
Proof not loaded.

Theorem. (components_are_open_in_locally_connected)

∀X Tx : set, locally_connected X Tx → ∀x : set, x ∈ X → open_in X Tx (component_of X Tx x)

In Proofgold the corresponding term root is eae32d... and proposition id is f42a15...

Proof:
Proof not loaded.

Theorem. (components_partition_space)

∀X Tx : set, topology_on X Tx → covers X {component_of X Tx x|x ∈ X} ∧ pairwise_disjoint {component_of X Tx x|x ∈ X}

In Proofgold the corresponding term root is 8c9141... and proposition id is a63ecd...

Proof:
Proof not loaded.

Theorem. (quotient_preserves_local_connectedness)

∀X Tx Y f : set, quotient_map X Tx Y f → locally_connected X Tx → locally_connected Y (quotient_topology X Tx Y f)

In Proofgold the corresponding term root is a1ad09... and proposition id is b5e0c8...

Proof:
Proof not loaded.

Definition. We define quasicomponent_of to be λX Tx x ⇒ {y ∈ X|∀U : set, open_in X Tx U → closed_in X Tx U → x ∈ U → y ∈ U} of type set → set → set → set.

In Proofgold the corresponding term root is b1bab5... and object id is 38153d...

Theorem. (components_vs_quasicomponents)

∀X Tx : set, topology_on X Tx → (∀x : set, component_of X Tx x ⊆ quasicomponent_of X Tx x) ∧ (locally_connected X Tx → ∀x : set, x ∈ X → component_of X Tx x = quasicomponent_of X Tx x)

In Proofgold the corresponding term root is 058462... and proposition id is c06ed1...

Proof:
Proof not loaded.

Theorem. (right_closed_ray_in_R_lower_limit_topology)

∀a : set, a ∈ R → {x ∈ R|¬ (Rlt x a)} ∈ R_lower_limit_topology

In Proofgold the corresponding term root is fde261... and proposition id is a52a0f...

Proof:
Proof not loaded.

Theorem. (halfopen_interval_left_complement_eq_rays)

∀a b : set, a ∈ R → b ∈ R → R ∖ (halfopen_interval_left a b) = {x ∈ R|Rlt x a} ∪ {x ∈ R|¬ (Rlt x b)}

In Proofgold the corresponding term root is 73f239... and proposition id is e47ab9...

Proof:
Proof not loaded.

Theorem. (complement_halfopen_interval_left_in_R_lower_limit_topology)

∀a b : set, a ∈ R → b ∈ R → (R ∖ halfopen_interval_left a b) ∈ R_lower_limit_topology

In Proofgold the corresponding term root is 9bc841... and proposition id is 8fbe01...

Proof:
Proof not loaded.

Theorem. (ex23_Rl_components)

component_of R R_lower_limit_topology 0 = {0} ∧ (∀x : set, x ∈ R → component_of R R_lower_limit_topology x = {x})

In Proofgold the corresponding term root is b589ed... and proposition id is 296af9...

Proof:
Proof not loaded.

Axiom. (component_of_Romega_box_zero_subset_Romega_infty) We take the following as an axiom:

∀f : set, f ∈ component_of (product_space ω (const_space_family ω R R_standard_topology)) (box_topology ω (const_space_family ω R R_standard_topology)) (const_family ω 0) → f ∈ Romega_infty

In Proofgold the corresponding term root is 5c1a67... and proposition id is a05eee...

Theorem. (component_of_Romega_box_zero_subset_Romega_infty_sketch)

In Proofgold the corresponding term root is 5c1a67... and proposition id is a05eee...

Proof:
Proof not loaded.

Theorem. (ex23_Romega_components)

component_of (product_space ω (const_space_family ω R R_standard_topology)) (product_topology_full ω (const_space_family ω R R_standard_topology)) (const_family ω 0) = product_space ω (const_space_family ω R R_standard_topology) ∧ component_of (product_space ω (const_space_family ω R R_standard_topology)) (box_topology ω (const_space_family ω R R_standard_topology)) (const_family ω 0) = {f ∈ product_space ω (const_space_family ω R R_standard_topology)|∃F : set, finite F ∧ ∀i : set, i ∈ ω ∖ F → apply_fun f i = 0}

In Proofgold the corresponding term root is 26c930... and proposition id is 0e919a...

Proof:
Proof not loaded.

Axiom. (ordered_square_locally_connected) We take the following as an axiom:

locally_connected ordered_square ordered_square_topology

In Proofgold the corresponding term root is f746b0... and proposition id is 1baad1...

Theorem. (ordered_square_locally_connected_sketch)

locally_connected ordered_square ordered_square_topology

In Proofgold the corresponding term root is f746b0... and proposition id is 1baad1...

Proof:
Proof not loaded.

Axiom. (ordered_square_not_locally_path_connected) We take the following as an axiom:

¬ locally_path_connected ordered_square ordered_square_topology

In Proofgold the corresponding term root is 4f1b1f... and proposition id is 998713...

Theorem. (ordered_square_not_locally_path_connected_sketch)

¬ locally_path_connected ordered_square ordered_square_topology

In Proofgold the corresponding term root is 4f1b1f... and proposition id is 998713...

Proof:
Proof not loaded.

Theorem. (ex23_ordered_square_locally_conn_not_pathconn)

locally_connected ordered_square ordered_square_topology ∧ ¬ locally_path_connected ordered_square ordered_square_topology

In Proofgold the corresponding term root is 327aa2... and proposition id is e3081f...

Proof:
Proof not loaded.

Theorem. (ex23_connected_open_sets_path_connected)

∀X Tx U : set, locally_path_connected X Tx → open_in X Tx U → connected_space U (subspace_topology X Tx U) → path_connected_space U (subspace_topology X Tx U)

In Proofgold the corresponding term root is 8990d7... and proposition id is 7b607f...

Proof:
Proof not loaded.

Axiom. (path_connected_not_locally_connected_subset_of_plane_exists) We take the following as an axiom:

∃A : set, A ⊆ EuclidPlane ∧ path_connected_space A (subspace_topology EuclidPlane R2_standard_topology A) ∧ ¬ locally_connected A (subspace_topology EuclidPlane R2_standard_topology A)

In Proofgold the corresponding term root is 234f9b... and proposition id is 1a2e29...

Theorem. (path_connected_not_locally_connected_subset_of_plane_exists_sketch)

∃A : set, A ⊆ EuclidPlane ∧ path_connected_space A (subspace_topology EuclidPlane R2_standard_topology A) ∧ ¬ locally_connected A (subspace_topology EuclidPlane R2_standard_topology A)

In Proofgold the corresponding term root is 234f9b... and proposition id is 1a2e29...

Proof:
Proof not loaded.

Theorem. (ex23_path_connected_not_locally_connected_examples)

∃A : set, A ⊆ EuclidPlane ∧ path_connected_space A (subspace_topology EuclidPlane R2_standard_topology A) ∧ ¬ locally_connected A (subspace_topology EuclidPlane R2_standard_topology A)

In Proofgold the corresponding term root is 234f9b... and proposition id is 1a2e29...

Proof:
Proof not loaded.

Definition. We define open_cover_of to be λX Tx Fam ⇒ topology_on X Tx ∧ Fam ⊆ 𝒫 X ∧ X ⊆ ⋃ Fam ∧ (∀U : set, U ∈ Fam → U ∈ Tx) of type set → set → set → prop.

In Proofgold the corresponding term root is b602bc... and object id is 45c1d0...

Theorem. (open_cover_ofI)

∀X Tx Fam : set, topology_on X Tx → Fam ⊆ 𝒫 X → X ⊆ ⋃ Fam → (∀U : set, U ∈ Fam → U ∈ Tx) → open_cover_of X Tx Fam

In Proofgold the corresponding term root is afbd15... and proposition id is 39d182...

Proof:
Proof not loaded.

Theorem. (open_cover_of_topology)

∀X Tx Fam : set, open_cover_of X Tx Fam → topology_on X Tx

In Proofgold the corresponding term root is 7034b2... and proposition id is cbf61b...

Proof:
Proof not loaded.

Theorem. (open_cover_of_family_sub)

∀X Tx Fam : set, open_cover_of X Tx Fam → Fam ⊆ 𝒫 X

In Proofgold the corresponding term root is fc1e87... and proposition id is 40dc17...

Proof:
Proof not loaded.

Theorem. (open_cover_of_covers)

∀X Tx Fam : set, open_cover_of X Tx Fam → X ⊆ ⋃ Fam

In Proofgold the corresponding term root is 188121... and proposition id is d3b152...

Proof:
Proof not loaded.

Theorem. (open_cover_of_implies_covers)

∀X Tx Fam : set, open_cover_of X Tx Fam → covers X Fam

In Proofgold the corresponding term root is 8f7652... and proposition id is de566a...

Proof:
Proof not loaded.

Theorem. (open_cover_of_members_open)

∀X Tx Fam U : set, open_cover_of X Tx Fam → U ∈ Fam → U ∈ Tx

In Proofgold the corresponding term root is 92dc60... and proposition id is b09f31...

Proof:
Proof not loaded.

Theorem. (open_cover_of_union_eq)

∀X Tx Fam : set, open_cover_of X Tx Fam → ⋃ Fam = X

In Proofgold the corresponding term root is da8007... and proposition id is 54d045...

Proof:
Proof not loaded.

Theorem. (open_cover_of_members_open_in)

∀X Tx Fam U : set, open_cover_of X Tx Fam → U ∈ Fam → open_in X Tx U

In Proofgold the corresponding term root is 75aa2c... and proposition id is a4b57f...

Proof:
Proof not loaded.

Definition. We define restrict_family_to_subspace to be λFam Y ⇒ {U ∩ Y|U ∈ Fam} of type set → set → set.

In Proofgold the corresponding term root is 6dccfe... and object id is afe8a1...

Theorem. (restrict_family_to_subspace_subset_Power)

∀Fam Y : set, restrict_family_to_subspace Fam Y ⊆ 𝒫 Y

In Proofgold the corresponding term root is 883732... and proposition id is 999537...

Proof:
Proof not loaded.

Theorem. (open_cover_of_restrict_to_subspace)

∀X Tx Fam Y : set, open_cover_of X Tx Fam → Y ⊆ X → open_cover_of Y (subspace_topology X Tx Y) (restrict_family_to_subspace Fam Y)

In Proofgold the corresponding term root is 55063e... and proposition id is 6e67ee...

Proof:
Proof not loaded.

Definition. We define has_finite_subcover to be λX Tx Fam ⇒ ∃G : set, G ⊆ Fam ∧ finite G ∧ X ⊆ ⋃ G of type set → set → set → prop.

In Proofgold the corresponding term root is 2ffe06... and object id is 54d485...

Theorem. (has_finite_subcoverI)

∀X Tx Fam G : set, G ⊆ Fam ∧ finite G ∧ X ⊆ ⋃ G → has_finite_subcover X Tx Fam

In Proofgold the corresponding term root is 044c2a... and proposition id is 8f2755...

Proof:
Proof not loaded.

Definition. We define compact_space to be λX Tx ⇒ topology_on X Tx ∧ ∀Fam : set, open_cover_of X Tx Fam → has_finite_subcover X Tx Fam of type set → set → prop.

In Proofgold the corresponding term root is c47023... and object id is 6c5f9a...

Theorem. (compact_space_topology)

∀X Tx : set, compact_space X Tx → topology_on X Tx

In Proofgold the corresponding term root is d2cde4... and proposition id is 38f352...

Proof:
Proof not loaded.

Theorem. (compact_space_subcover_property)

∀X Tx : set, compact_space X Tx → ∀Fam : set, open_cover_of X Tx Fam → has_finite_subcover X Tx Fam

In Proofgold the corresponding term root is 333ba8... and proposition id is bbb4c9...

Proof:
Proof not loaded.

Theorem. (compact_space_singleton)

∀X Tx a : set, X = {a} → topology_on X Tx → compact_space X Tx

In Proofgold the corresponding term root is b56bc9... and proposition id is 743dd2...

Proof:
Proof not loaded.

Theorem. (Heine_Borel_subcover)

∀X Tx Fam : set, compact_space X Tx → open_cover_of X Tx Fam → has_finite_subcover X Tx Fam

In Proofgold the corresponding term root is 5a6a6b... and proposition id is eb96fe...

Proof:
Proof not loaded.

Theorem. (compact_subspace_via_ambient_covers)

∀X Tx Y : set, topology_on X Tx → Y ⊆ X → (compact_space Y (subspace_topology X Tx Y) ↔ ∀Fam : set, (Fam ⊆ Tx ∧ Y ⊆ ⋃ Fam) → has_finite_subcover Y Tx Fam)

In Proofgold the corresponding term root is efebf9... and proposition id is aee242...

Proof:
Proof not loaded.

Theorem. (closed_subspace_compact)

∀X Tx Y : set, compact_space X Tx → closed_in X Tx Y → compact_space Y (subspace_topology X Tx Y)

In Proofgold the corresponding term root is 2f4cec... and proposition id is 49e639...

Proof:
Proof not loaded.

Theorem. (Hausdorff_separate_point_compact_set_aux)

∀X Tx Y x : set, Hausdorff_space X Tx → Y ⊆ X → compact_space Y (subspace_topology X Tx Y) → x ∈ X → x ∉ Y → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ x ∈ U ∧ Y ⊆ V ∧ U ∩ V = Empty

In Proofgold the corresponding term root is 2c37c8... and proposition id is d40d2a...

Proof:
Proof not loaded.

Theorem. (compact_subspace_in_Hausdorff_closed)

∀X Tx Y : set, Hausdorff_space X Tx → Y ⊆ X → compact_space Y (subspace_topology X Tx Y) → closed_in X Tx Y

In Proofgold the corresponding term root is cdd390... and proposition id is 7f52d0...

Proof:
Proof not loaded.

Theorem. (Hausdorff_separate_point_compact_set)

In Proofgold the corresponding term root is 2c37c8... and proposition id is d40d2a...

Proof:
Proof not loaded.

Definition. We define image_of_fun to be λf X ⇒ image_of f X of type set → set → set.

In Proofgold the corresponding term root is d8bf37... and object id is d91390...

Theorem. (continuous_image_compact)

∀X Tx Y Ty f : set, compact_space X Tx → continuous_map X Tx Y Ty f → compact_space (image_of_fun f X) (subspace_topology Y Ty (image_of_fun f X))

In Proofgold the corresponding term root is 6721ed... and proposition id is 398129...

Proof:
Proof not loaded.

Theorem. (tube_lemma)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → compact_space Y Ty → ∀x0 : set, x0 ∈ X → ∀N : set, N ∈ product_topology X Tx Y Ty ∧ setprod {x0} Y ⊆ N → ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ setprod U Y ⊆ N

In Proofgold the corresponding term root is 2ef321... and proposition id is 9a6943...

Proof:
Proof not loaded.

Definition. We define bijection to be λX Y f ⇒ function_on f X Y ∧ (∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun f x = y ∧ (∀x' : set, x' ∈ X → apply_fun f x' = y → x' = x)) of type set → set → set → prop.

In Proofgold the corresponding term root is 1789fc... and object id is d21a20...

Theorem. (equip_of_bijection)

∀X Y f : set, bijection X Y f → equip X Y

In Proofgold the corresponding term root is 01b8ef... and proposition id is cfb450...

Proof:
Proof not loaded.

Theorem. (bijection_graph_of_bij)

∀X Y : set, ∀g : set → set, bij X Y g → bijection X Y (graph X g)

In Proofgold the corresponding term root is 4e1e9b... and proposition id is 4bc882...

Proof:
Proof not loaded.

Theorem. (bijection_compose_fun)

∀X Y Z f g : set, bijection X Y f → bijection Y Z g → bijection X Z (compose_fun X f g)

In Proofgold the corresponding term root is 08f36e... and proposition id is c923bb...

Proof:
Proof not loaded.

Definition. We define inv_fun_graph to be λX f Y ⇒ {(y,inv X (λx : set ⇒ apply_fun f x) y)|y ∈ Y} of type set → set → set → set.

In Proofgold the corresponding term root is f3a67c... and object id is fa3ff0...

Theorem. (inv_fun_graph_apply)

∀X Y f y : set, y ∈ Y → apply_fun (inv_fun_graph X f Y) y = inv X (λx : set ⇒ apply_fun f x) y

In Proofgold the corresponding term root is f09286... and proposition id is 50727c...

Proof:
Proof not loaded.

Theorem. (bijection_surj)

∀X Y f y : set, bijection X Y f → y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun f x = y

In Proofgold the corresponding term root is 97ede1... and proposition id is 43de23...

Proof:
Proof not loaded.

Theorem. (bijection_inj)

∀X Y f u v : set, bijection X Y f → u ∈ X → v ∈ X → apply_fun f u = apply_fun f v → u = v

In Proofgold the corresponding term root is b76378... and proposition id is e7daa3...

Proof:
Proof not loaded.

Theorem. (bijection_drop_last)

∀m F e : set, m ∈ ω → bijection (ordsucc m) F e → bijection m (F ∖ {apply_fun e m}) e

In Proofgold the corresponding term root is 69f437... and proposition id is ace108...

Proof:
Proof not loaded.

Theorem. (inv_fun_graph_right_inverse)

∀X Y f y : set, bijection X Y f → y ∈ Y → apply_fun f (apply_fun (inv_fun_graph X f Y) y) = y

In Proofgold the corresponding term root is 8b50d8... and proposition id is c16d59...

Proof:
Proof not loaded.

Theorem. (inv_fun_graph_left_inverse)

∀X Y f x : set, bijection X Y f → x ∈ X → apply_fun (inv_fun_graph X f Y) (apply_fun f x) = x

In Proofgold the corresponding term root is c56214... and proposition id is b39107...

Proof:
Proof not loaded.

Theorem. (inv_fun_graph_preimage_eq_image)

∀X Y f A : set, bijection X Y f → A ⊆ X → preimage_of Y (inv_fun_graph X f Y) A = image_of_fun f A

In Proofgold the corresponding term root is ca0b17... and proposition id is 3bd208...

Proof:
Proof not loaded.

Definition. We define Abs to be abs_SNo of type set → set.

In Proofgold the corresponding term root is 34f6dc... and object id is 2990e5...

Theorem. (compact_to_Hausdorff_inverse_continuous)

∀X Tx Y Ty f : set, compact_space X Tx → Hausdorff_space Y Ty → continuous_map X Tx Y Ty f → bijection X Y f → continuous_map Y Ty X Tx (inv_fun_graph X f Y)

In Proofgold the corresponding term root is 654abc... and proposition id is a971fa...

Proof:
Proof not loaded.

Theorem. (compact_to_Hausdorff_bijection_homeomorphism)

∀X Tx Y Ty f : set, compact_space X Tx → Hausdorff_space Y Ty → continuous_map X Tx Y Ty f → bijection X Y f → homeomorphism X Tx Y Ty f

In Proofgold the corresponding term root is e2a416... and proposition id is 0f0fce...

Proof:
Proof not loaded.

Definition. We define bounded_subset_of_reals to be λA ⇒ ∃M : set, M ∈ R ∧ ∀x : set, x ∈ A → ¬ (Rlt M (Abs x)) of type set → prop.

In Proofgold the corresponding term root is fe3ccf... and object id is 6918ab...

Theorem. (omega_in_R)

∀n : set, n ∈ ω → n ∈ R

In Proofgold the corresponding term root is bfbb64... and proposition id is 79abc7...

Proof:
Proof not loaded.

Theorem. (ordsucc_in_R)

∀n : set, n ∈ ω → ordsucc n ∈ R

In Proofgold the corresponding term root is ddf6a0... and proposition id is 563aa4...

Proof:
Proof not loaded.

Theorem. (interval_bounds_Abs)

∀M x : set, M ∈ R → x ∈ R → Rlt (minus_SNo M) x → Rlt x M → ¬ (Rlt M (Abs x))

In Proofgold the corresponding term root is e6f53c... and proposition id is 605347...

Proof:
Proof not loaded.

Theorem. (finite_product_compact)

∀X Tx Y Ty : set, compact_space X Tx → compact_space Y Ty → compact_space (setprod X Y) (product_topology X Tx Y Ty)

In Proofgold the corresponding term root is 91ae71... and proposition id is 0f1fc3...

Proof:
Proof not loaded.

Theorem. (unit_interval_has_finite_subcover)

∀Fam : set, open_cover_of unit_interval unit_interval_topology Fam → has_finite_subcover unit_interval unit_interval_topology Fam

In Proofgold the corresponding term root is b05797... and proposition id is 47a320...

Proof:
Proof not loaded.

Theorem. (unit_interval_compact_axiom)

compact_space unit_interval unit_interval_topology

In Proofgold the corresponding term root is 6cd294... and proposition id is 597132...

Proof:
Proof not loaded.

Theorem. (ex26_compactness_exercises)

(closed_in R R_standard_topology unit_interval) ∧ (compact_space unit_interval unit_interval_topology)

In Proofgold the corresponding term root is 53c4c3... and proposition id is 3bd7ac...

Proof:
Proof not loaded.

Theorem. (Heine_Borel_closed_bounded)

∀A : set, A ⊆ R → compact_space A (subspace_topology R R_standard_topology A) → closed_in R R_standard_topology A ∧ bounded_subset_of_reals A

In Proofgold the corresponding term root is d79277... and proposition id is cf79f5...

Proof:
Proof not loaded.

Theorem. (compact_real_closed_bounded)

∀A : set, compact_space A (subspace_topology R R_standard_topology A) → closed_in R R_standard_topology A ∧ bounded_subset_of_reals A

In Proofgold the corresponding term root is 38b967... and proposition id is 969285...

Proof:
Proof not loaded.

Definition. We define limit_point_compact to be λX Tx ⇒ topology_on X Tx ∧ ∀A : set, A ⊆ X → infinite A → ∃x : set, limit_point_of X Tx A x of type set → set → prop.

In Proofgold the corresponding term root is 8e719a... and object id is bc5bd0...

Theorem. (limit_point_compact_topology)

∀X Tx : set, limit_point_compact X Tx → topology_on X Tx

In Proofgold the corresponding term root is b52e68... and proposition id is d8dc9d...

Proof:
Proof not loaded.

Theorem. (limit_point_compact_property)

∀X Tx A : set, limit_point_compact X Tx → A ⊆ X → infinite A → ∃x : set, limit_point_of X Tx A x

In Proofgold the corresponding term root is 335d90... and proposition id is 3b3faa...

Proof:
Proof not loaded.

Theorem. (compact_implies_limit_point_compact)

∀X Tx : set, compact_space X Tx → limit_point_compact X Tx

In Proofgold the corresponding term root is f5506e... and proposition id is 34f6b6...

Proof:
Proof not loaded.

Theorem. (limit_point_compact_not_necessarily_compact)

∃X Tx : set, limit_point_compact X Tx ∧ ¬ compact_space X Tx

In Proofgold the corresponding term root is 464ab7... and proposition id is 51bc23...

Proof:
Proof not loaded.

Definition. We define locally_compact to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ compact_space (closure_of X Tx U) (subspace_topology X Tx (closure_of X Tx U)) of type set → set → prop.

In Proofgold the corresponding term root is 47e875... and object id is b40691...

Theorem. (locally_compact_topology)

∀X Tx : set, locally_compact X Tx → topology_on X Tx

In Proofgold the corresponding term root is c2a462... and proposition id is 19a400...

Proof:
Proof not loaded.

Theorem. (locally_compact_local)

∀X Tx x : set, locally_compact X Tx → x ∈ X → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ compact_space (closure_of X Tx U) (subspace_topology X Tx (closure_of X Tx U))

In Proofgold the corresponding term root is 46f9b1... and proposition id is 6c91d8...

Proof:
Proof not loaded.

Theorem. (Hausdorff_compact_sets_closed)

∀X Tx A : set, Hausdorff_space X Tx → A ⊆ X → compact_space A (subspace_topology X Tx A) → closed_in X Tx A

In Proofgold the corresponding term root is cdd390... and proposition id is 7f52d0...

Proof:
Proof not loaded.

Theorem. (compact_empty_subspace)

∀X Tx : set, topology_on X Tx → compact_space Empty (subspace_topology X Tx Empty)

In Proofgold the corresponding term root is 2fd7d5... and proposition id is 0d04d8...

Proof:
Proof not loaded.

Definition. We define one_point_compactification to be λX Tx Y Ty ⇒ compact_space Y Ty ∧ Hausdorff_space Y Ty ∧ X ⊆ Y ∧ ∃p : set, p ∈ Y ∧ ¬ p ∈ X ∧ subspace_topology Y Ty X = Tx ∧ (∀y : set, y ∈ Y → y ∈ X ∨ y = p) of type set → set → set → set → prop.

In Proofgold the corresponding term root is e71ab6... and object id is 90a795...

Theorem. (one_point_compactification_exists)

∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → ∃Y Ty : set, one_point_compactification X Tx Y Ty

In Proofgold the corresponding term root is ac0732... and proposition id is 3e1a7f...

Proof:
Proof not loaded.

Theorem. (ex29_local_compactness_exercises)

∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → ∃Y Ty : set, one_point_compactification X Tx Y Ty

In Proofgold the corresponding term root is ac0732... and proposition id is 3e1a7f...

Proof:
Proof not loaded.

Definition. We define relation_on to be λle J ⇒ ∀a b : set, (a,b) ∈ le → a ∈ J ∧ b ∈ J of type set → set → prop.

In Proofgold the corresponding term root is 5fcabc... and object id is 958957...

Definition. We define partial_order_on to be λJ le ⇒ relation_on le J ∧ (∀a : set, a ∈ J → (a,a) ∈ le) ∧ (∀a b : set, a ∈ J → b ∈ J → (a,b) ∈ le → (b,a) ∈ le → a = b) ∧ (∀a b c : set, a ∈ J → b ∈ J → c ∈ J → (a,b) ∈ le → (b,c) ∈ le → (a,c) ∈ le) of type set → set → prop.

In Proofgold the corresponding term root is f2e415... and object id is a66b2f...

Theorem. (partial_order_on_refl)

∀J le : set, partial_order_on J le → ∀a : set, a ∈ J → (a,a) ∈ le

In Proofgold the corresponding term root is 346381... and proposition id is 8399d2...

Proof:
Proof not loaded.

Theorem. (partial_order_on_antisym)

∀J le : set, partial_order_on J le → ∀a b : set, a ∈ J → b ∈ J → (a,b) ∈ le → (b,a) ∈ le → a = b

In Proofgold the corresponding term root is 0d40fc... and proposition id is 89830b...

Proof:
Proof not loaded.

Theorem. (partial_order_on_trans)

∀J le : set, partial_order_on J le → ∀a b c : set, a ∈ J → b ∈ J → c ∈ J → (a,b) ∈ le → (b,c) ∈ le → (a,c) ∈ le

In Proofgold the corresponding term root is b1121d... and proposition id is 86c7a7...

Proof:
Proof not loaded.

Definition. We define directed_set to be λJ le ⇒ (J ≠ Empty ∧ partial_order_on J le) ∧ ∀a b : set, a ∈ J → b ∈ J → ∃c : set, c ∈ J ∧ (a,c) ∈ le ∧ (b,c) ∈ le of type set → set → prop.

In Proofgold the corresponding term root is 737b9a... and object id is 734ee2...

Theorem. (directed_set_partial_order)

∀J le : set, directed_set J le → partial_order_on J le

In Proofgold the corresponding term root is 33e9b0... and proposition id is e1884e...

Proof:
Proof not loaded.

Theorem. (directed_set_refl)

∀J le : set, directed_set J le → ∀a : set, a ∈ J → (a,a) ∈ le

In Proofgold the corresponding term root is 8877bf... and proposition id is 213b28...

Proof:
Proof not loaded.

Theorem. (directed_set_trans)

∀J le : set, directed_set J le → ∀a b c : set, a ∈ J → b ∈ J → c ∈ J → (a,b) ∈ le → (b,c) ∈ le → (a,c) ∈ le

In Proofgold the corresponding term root is 8f6842... and proposition id is bb6192...

Proof:
Proof not loaded.

Theorem. (directed_set_nonempty)

∀J le : set, directed_set J le → J ≠ Empty

In Proofgold the corresponding term root is a3414f... and proposition id is 50810b...

Proof:
Proof not loaded.

Theorem. (directed_set_upper_bound_property)

∀J le : set, directed_set J le → ∀i j : set, i ∈ J → j ∈ J → ∃k : set, k ∈ J ∧ (i,k) ∈ le ∧ (j,k) ∈ le

In Proofgold the corresponding term root is c8dadf... and proposition id is 182e74...

Proof:
Proof not loaded.

Theorem. (directed_set_upper_bound)

∀J le i : set, directed_set J le → i ∈ J → ∃k : set, k ∈ J ∧ (i,k) ∈ le

In Proofgold the corresponding term root is aaa083... and proposition id is ea705c...

Proof:
Proof not loaded.

Theorem. (directed_set_pair_upper_bound)

∀J le i j : set, directed_set J le → i ∈ J → j ∈ J → ∃k : set, k ∈ J ∧ (i,k) ∈ le ∧ (j,k) ∈ le

In Proofgold the corresponding term root is f71773... and proposition id is 2acdae...

Proof:
Proof not loaded.

Theorem. (examples_of_directed_sets)

∀J le : set, directed_set J le → directed_set J le

In Proofgold the corresponding term root is c299c6... and proposition id is 9ecdf6...

Proof:
Proof not loaded.

Definition. We define rel_restrict to be λle K ⇒ {p ∈ le|p 0 ∈ K ∧ p 1 ∈ K} of type set → set → set.

In Proofgold the corresponding term root is c45d53... and object id is a10cf4...

Theorem. (rel_restrict_def)

∀le K : set, rel_restrict le K = {p ∈ le|p 0 ∈ K ∧ p 1 ∈ K}

In Proofgold the corresponding term root is 071f84... and proposition id is 6f2ef1...

Proof:
Proof not loaded.

Theorem. (rel_restrictE)

∀le K a b : set, (a,b) ∈ rel_restrict le K → (a,b) ∈ le ∧ a ∈ K ∧ b ∈ K

In Proofgold the corresponding term root is 6098cf... and proposition id is a66cf9...

Proof:
Proof not loaded.

Theorem. (rel_restrictI)

∀le K a b : set, (a,b) ∈ le → a ∈ K → b ∈ K → (a,b) ∈ rel_restrict le K

In Proofgold the corresponding term root is 66021e... and proposition id is 51d439...

Proof:
Proof not loaded.

Theorem. (cofinal_subset_directed)

∀J le K : set, directed_set J le → K ⊆ J → (∀a : set, a ∈ J → ∃b : set, b ∈ K ∧ (a,b) ∈ le) → directed_set K (rel_restrict le K)

In Proofgold the corresponding term root is dfec40... and proposition id is 350a5f...

Proof:
Proof not loaded.

Definition. We define net_on to be λnet ⇒ ∃J le X : set, directed_set J le ∧ total_function_on net J X ∧ functional_graph net ∧ graph_domain_subset net J of type set → prop.

In Proofgold the corresponding term root is ecd7d4... and object id is a7510a...

Definition. We define net_in_space to be λX net ⇒ ∃J le : set, directed_set J le ∧ total_function_on net J X ∧ functional_graph net ∧ graph_domain_subset net J of type set → set → prop.

In Proofgold the corresponding term root is 752f34... and object id is 259fbe...

Theorem. (net_in_space_domain_unique)

∀X net J le J' le' : set, directed_set J le ∧ total_function_on net J X ∧ functional_graph net ∧ graph_domain_subset net J → directed_set J' le' ∧ total_function_on net J' X ∧ functional_graph net ∧ graph_domain_subset net J' → J = J'

In Proofgold the corresponding term root is 289fc2... and proposition id is 2b93cd...

Proof:
Proof not loaded.

Definition. We define net_pack to be λJ le net ⇒ (J,(le,net)) of type set → set → set → set.

In Proofgold the corresponding term root is 2bed4a... and object id is 85d78b...

Definition. We define net_pack_index to be λN ⇒ N 0 of type set → set.

In Proofgold the corresponding term root is f0bb69... and object id is 9211e4...

Definition. We define net_pack_le to be λN ⇒ (N 1) 0 of type set → set.

In Proofgold the corresponding term root is 870ff6... and object id is ce76b1...

Definition. We define net_pack_fun to be λN ⇒ (N 1) 1 of type set → set.

In Proofgold the corresponding term root is 9eae6d... and object id is fc8972...

Theorem. (net_pack_index_unfold)

∀N : set, net_pack_index N = N 0

In Proofgold the corresponding term root is bccc91... and proposition id is 6f4f3f...

Proof:
Proof not loaded.

Theorem. (net_pack_le_unfold)

∀N : set, net_pack_le N = (N 1) 0

In Proofgold the corresponding term root is 1fa277... and proposition id is 419113...

Proof:
Proof not loaded.

Theorem. (net_pack_fun_unfold)

∀N : set, net_pack_fun N = (N 1) 1

In Proofgold the corresponding term root is 0f518a... and proposition id is 00272d...

Proof:
Proof not loaded.

Theorem. (net_pack_def)

∀J le net : set, net_pack J le net = (J,(le,net))

In Proofgold the corresponding term root is 972065... and proposition id is bd542f...

Proof:
Proof not loaded.

Theorem. (net_pack_index_eq)

∀J le net : set, net_pack_index (net_pack J le net) = J

In Proofgold the corresponding term root is 189a6f... and proposition id is f9b0d8...

Proof:
Proof not loaded.

Theorem. (net_pack_le_eq)

∀J le net : set, net_pack_le (net_pack J le net) = le

In Proofgold the corresponding term root is b6efec... and proposition id is 9b6fdd...

Proof:
Proof not loaded.

Theorem. (net_pack_fun_eq)

∀J le net : set, net_pack_fun (net_pack J le net) = net

In Proofgold the corresponding term root is a55d24... and proposition id is 4427f6...

Proof:
Proof not loaded.

Definition. We define net_pack_in_space to be λX N ⇒ ∃J le net : set, N = net_pack J le net ∧ directed_set J le ∧ total_function_on net J X ∧ functional_graph net ∧ graph_domain_subset net J of type set → set → prop.

In Proofgold the corresponding term root is 460c95... and object id is 3ec152...

Theorem. (net_pack_in_space_unpack_eq)

∀X N J le net : set, net_pack_in_space X N → N = net_pack J le net → directed_set J le ∧ total_function_on net J X ∧ functional_graph net ∧ graph_domain_subset net J

In Proofgold the corresponding term root is 452d78... and proposition id is 4b8e8d...

Proof:
Proof not loaded.

Theorem. (net_pack_in_space_implies_net_in_space)

∀X N : set, net_pack_in_space X N → net_in_space X (net_pack_fun N)

In Proofgold the corresponding term root is 2b1812... and proposition id is 1bd250...

Proof:
Proof not loaded.

Theorem. (net_in_space_implies_exists_net_pack_in_space)

∀X net : set, net_in_space X net → ∃N : set, net_pack_in_space X N ∧ net_pack_fun N = net

In Proofgold the corresponding term root is 2ace82... and proposition id is 3d241b...

Proof:
Proof not loaded.

Theorem. (graph_subset_setprod)

∀A Y : set, ∀g : set → set, (∀a : set, a ∈ A → g a ∈ Y) → graph A g ⊆ setprod A Y

In Proofgold the corresponding term root is 6c9fba... and proposition id is 44744f...

Proof:
Proof not loaded.

Theorem. (graph_in_function_space)

∀A Y : set, ∀g : set → set, (∀a : set, a ∈ A → g a ∈ Y) → graph A g ∈ function_space A Y

In Proofgold the corresponding term root is bf4e14... and proposition id is 4e13b1...

Proof:
Proof not loaded.

Definition. We define subnet_of to be λnet sub ⇒ ∃J leJ K leK X phi : set, directed_set J leJ ∧ directed_set K leK ∧ total_function_on net J X ∧ functional_graph net ∧ total_function_on sub K X ∧ functional_graph sub ∧ total_function_on phi K J ∧ functional_graph phi ∧ graph_domain_subset net J ∧ graph_domain_subset sub K ∧ graph_domain_subset phi K ∧ (∀i j : set, i ∈ K → j ∈ K → (i,j) ∈ leK → (apply_fun phi i,apply_fun phi j) ∈ leJ) ∧ (∀j : set, j ∈ J → ∃k : set, k ∈ K ∧ (j,apply_fun phi k) ∈ leJ) ∧ (∀k : set, k ∈ K → apply_fun sub k = apply_fun net (apply_fun phi k)) of type set → set → prop.

In Proofgold the corresponding term root is fba523... and object id is 62f53b...

Theorem. (pair_order_pred)

∀K le i j : set, (i,j) ∈ {q ∈ setprod K K|(q 1) 0 ⊆ (q 0) 0 ∧ ((q 0) 1,(q 1) 1) ∈ le} → (j 0) ⊆ (i 0) ∧ ((i 1),(j 1)) ∈ le

In Proofgold the corresponding term root is bc4231... and proposition id is c5d38c...

Proof:
Proof not loaded.

Theorem. (subnet_implies_net_on)

∀net sub : set, subnet_of net sub → net_on sub

In Proofgold the corresponding term root is 11b18b... and proposition id is 6bb85c...

Proof:
Proof not loaded.

Theorem. (subnet_implies_net_on_source)

∀net sub : set, subnet_of net sub → net_on net

In Proofgold the corresponding term root is e5927f... and proposition id is dea528...

Proof:
Proof not loaded.

Beginning of Section PropN2

Variable P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 : prop

Theorem. (and8I)

P1 → P2 → P3 → P4 → P5 → P6 → P7 → P8 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7 ∧ P8

In Proofgold the corresponding term root is cea710... and proposition id is 0f9534...

Proof:
Proof not loaded.

Theorem. (and9I)

P1 → P2 → P3 → P4 → P5 → P6 → P7 → P8 → P9 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7 ∧ P8 ∧ P9

In Proofgold the corresponding term root is c64e51... and proposition id is fea103...

Proof:
Proof not loaded.

Theorem. (and10I)

P1 → P2 → P3 → P4 → P5 → P6 → P7 → P8 → P9 → P10 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7 ∧ P8 ∧ P9 ∧ P10

In Proofgold the corresponding term root is 5f06c8... and proposition id is dc2e04...

Proof:
Proof not loaded.

Theorem. (and11I)

P1 → P2 → P3 → P4 → P5 → P6 → P7 → P8 → P9 → P10 → P11 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7 ∧ P8 ∧ P9 ∧ P10 ∧ P11

In Proofgold the corresponding term root is 5eb069... and proposition id is 85832b...

Proof:
Proof not loaded.

Theorem. (and12I)

P1 → P2 → P3 → P4 → P5 → P6 → P7 → P8 → P9 → P10 → P11 → P12 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7 ∧ P8 ∧ P9 ∧ P10 ∧ P11 ∧ P12

In Proofgold the corresponding term root is 7a61be... and proposition id is 710a9f...

Proof:
Proof not loaded.

Theorem. (and13I)

P1 → P2 → P3 → P4 → P5 → P6 → P7 → P8 → P9 → P10 → P11 → P12 → P13 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7 ∧ P8 ∧ P9 ∧ P10 ∧ P11 ∧ P12 ∧ P13

In Proofgold the corresponding term root is 93d31d... and proposition id is cd5551...

Proof:
Proof not loaded.

Theorem. (and14I)

P1 → P2 → P3 → P4 → P5 → P6 → P7 → P8 → P9 → P10 → P11 → P12 → P13 → P14 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7 ∧ P8 ∧ P9 ∧ P10 ∧ P11 ∧ P12 ∧ P13 ∧ P14

In Proofgold the corresponding term root is 73ee62... and proposition id is 5a186e...

Proof:
Proof not loaded.

End of Section PropN2

Theorem. (subnet_of_refl_witnessed)

∀J le X net : set, directed_set J le → total_function_on net J X → functional_graph net → graph_domain_subset net J → subnet_of net net

In Proofgold the corresponding term root is 567c15... and proposition id is 78288d...

Proof:
Proof not loaded.

Theorem. (subnet_of_refl)

∀net : set, net_on net → subnet_of net net

In Proofgold the corresponding term root is 7ea3c9... and proposition id is 326a0d...

Proof:
Proof not loaded.

Definition. We define accumulation_point_of_net to be λX Tx net x ⇒ ∃J le : set, topology_on X Tx ∧ directed_set J le ∧ total_function_on net J X ∧ functional_graph net ∧ graph_domain_subset net J ∧ x ∈ X ∧ ∀U : set, U ∈ Tx → x ∈ U → ∀j0 : set, j0 ∈ J → ∃j : set, j ∈ J ∧ (j0,j) ∈ le ∧ apply_fun net j ∈ U of type set → set → set → set → prop.

In Proofgold the corresponding term root is ebbbc4... and object id is c90ccc...

Definition. We define accumulation_point_of_net_on to be λX Tx net J le x ⇒ topology_on X Tx ∧ directed_set J le ∧ total_function_on net J X ∧ functional_graph net ∧ graph_domain_subset net J ∧ x ∈ X ∧ ∀U : set, U ∈ Tx → x ∈ U → ∀j0 : set, j0 ∈ J → ∃j : set, j ∈ J ∧ (j0,j) ∈ le ∧ apply_fun net j ∈ U of type set → set → set → set → set → set → prop.

In Proofgold the corresponding term root is 54a3b4... and object id is c8f5cf...

Theorem. (accumulation_point_of_net_on_implies_accumulation_point_of_net)

∀X Tx net J le x : set, accumulation_point_of_net_on X Tx net J le x → accumulation_point_of_net X Tx net x

In Proofgold the corresponding term root is a31daf... and proposition id is dff7ab...

Proof:
Proof not loaded.

Definition. We define net_converges to be λX Tx net x ⇒ ∃J le : set, topology_on X Tx ∧ directed_set J le ∧ total_function_on net J X ∧ functional_graph net ∧ graph_domain_subset net J ∧ x ∈ X ∧ ∀U : set, U ∈ Tx → x ∈ U → ∃i0 : set, i0 ∈ J ∧ ∀i : set, i ∈ J → (i0,i) ∈ le → apply_fun net i ∈ U of type set → set → set → set → prop.

In Proofgold the corresponding term root is 4da009... and object id is dbc78b...

Definition. We define net_converges_on to be λX Tx net J le x ⇒ topology_on X Tx ∧ directed_set J le ∧ total_function_on net J X ∧ functional_graph net ∧ graph_domain_subset net J ∧ x ∈ X ∧ ∀U : set, U ∈ Tx → x ∈ U → ∃i0 : set, i0 ∈ J ∧ ∀i : set, i ∈ J → (i0,i) ∈ le → apply_fun net i ∈ U of type set → set → set → set → set → set → prop.

In Proofgold the corresponding term root is 2e97ad... and object id is d6aad1...

Theorem. (net_converges_on_implies_net_converges)

∀X Tx net J le x : set, net_converges_on X Tx net J le x → net_converges X Tx net x

In Proofgold the corresponding term root is b7bd5a... and proposition id is 22e4f9...

Proof:
Proof not loaded.

Theorem. (net_converges_on_implies_net_in_space)

∀X Tx net J le x : set, net_converges_on X Tx net J le x → net_in_space X net

In Proofgold the corresponding term root is 00bd80... and proposition id is 73e43c...

Proof:
Proof not loaded.

Theorem. (net_converges_implies_net_in_space)

∀X Tx net x : set, net_converges X Tx net x → net_in_space X net

In Proofgold the corresponding term root is dde273... and proposition id is 1537bc...

Proof:
Proof not loaded.

Theorem. (net_converges_on_in_subspace_topology)

∀X Tx A net J le x : set, topology_on X Tx → A ⊆ X → directed_set J le → total_function_on net J A → functional_graph net → graph_domain_subset net J → x ∈ A → net_converges_on X Tx net J le x → net_converges_on A (subspace_topology X Tx A) net J le x

In Proofgold the corresponding term root is f94b7d... and proposition id is 72db21...

Proof:
Proof not loaded.

Theorem. (net_converges_implies_exists_net_converges_on)

∀X Tx net x : set, net_converges X Tx net x → ∃J le : set, net_converges_on X Tx net J le x

In Proofgold the corresponding term root is b12947... and proposition id is 11caaf...

Proof:
Proof not loaded.

Theorem. (net_converges_on_domain_unique)

∀X Tx net J le J' le' x : set, net_converges_on X Tx net J le x → net_converges_on X Tx net J' le' x → J = J'

In Proofgold the corresponding term root is 282499... and proposition id is 201b6a...

Proof:
Proof not loaded.

Theorem. (net_converges_implies_accumulation_point)

∀X Tx net x : set, net_converges X Tx net x → accumulation_point_of_net X Tx net x

In Proofgold the corresponding term root is d9f2aa... and proposition id is 50fdc7...

Proof:
Proof not loaded.

Theorem. (net_converges_implies_net_on)

∀X Tx net x : set, net_converges X Tx net x → net_on net

In Proofgold the corresponding term root is 0c388e... and proposition id is ec0092...

Proof:
Proof not loaded.

Definition. We define subnet_of_witness to be λnet sub J leJ K leK X phi ⇒ directed_set J leJ ∧ directed_set K leK ∧ total_function_on net J X ∧ functional_graph net ∧ graph_domain_subset net J ∧ total_function_on sub K X ∧ functional_graph sub ∧ graph_domain_subset sub K ∧ total_function_on phi K J ∧ functional_graph phi ∧ graph_domain_subset phi K ∧ (∀i j : set, i ∈ K → j ∈ K → (i,j) ∈ leK → (apply_fun phi i,apply_fun phi j) ∈ leJ) ∧ (∀j : set, j ∈ J → ∃k : set, k ∈ K ∧ (j,apply_fun phi k) ∈ leJ) ∧ (∀k : set, k ∈ K → apply_fun sub k = apply_fun net (apply_fun phi k)) of type set → set → set → set → set → set → set → set → prop.

In Proofgold the corresponding term root is 75ada6... and object id is 9b9ebb...

Theorem. (subnet_of_witness_dirJ)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → directed_set J leJ

In Proofgold the corresponding term root is 7fa890... and proposition id is 23b849...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_dirK)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → directed_set K leK

In Proofgold the corresponding term root is 3f6900... and proposition id is 1146fc...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_compose_fun)

∀net J leJ K leK X phi : set, directed_set J leJ → directed_set K leK → total_function_on net J X → functional_graph net → graph_domain_subset net J → total_function_on phi K J → functional_graph phi → graph_domain_subset phi K → (∀i j : set, i ∈ K → j ∈ K → (i,j) ∈ leK → (apply_fun phi i,apply_fun phi j) ∈ leJ) → (∀j : set, j ∈ J → ∃k : set, k ∈ K ∧ (j,apply_fun phi k) ∈ leJ) → subnet_of_witness net (compose_fun K phi net) J leJ K leK X phi

In Proofgold the corresponding term root is 8a38f2... and proposition id is 95c651...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_map)

∀net sub J leJ K leK X phi Y f : set, subnet_of_witness net sub J leJ K leK X phi → function_on f X Y → subnet_of_witness (compose_fun J net f) (compose_fun K sub f) J leJ K leK Y phi

In Proofgold the corresponding term root is d29c09... and proposition id is 31ecb3...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_trans)

∀net sub sub2 J leJ K leK L leL X phi psi : set, subnet_of_witness net sub J leJ K leK X phi → subnet_of_witness sub sub2 K leK L leL X psi → subnet_of_witness net sub2 J leJ L leL X (compose_fun L psi phi)

In Proofgold the corresponding term root is 6a1ceb... and proposition id is 4994ab...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_totnet)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → total_function_on net J X

In Proofgold the corresponding term root is 459e41... and proposition id is 35cb70...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_graphnet)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → functional_graph net

In Proofgold the corresponding term root is a48682... and proposition id is 4276fb...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_domnet)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → graph_domain_subset net J

In Proofgold the corresponding term root is e34e85... and proposition id is e7f7e8...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_totsub)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → total_function_on sub K X

In Proofgold the corresponding term root is c775a6... and proposition id is ba03b4...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_graphsub)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → functional_graph sub

In Proofgold the corresponding term root is e90070... and proposition id is f227ac...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_domsub)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → graph_domain_subset sub K

In Proofgold the corresponding term root is e924bb... and proposition id is b0e58a...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_totphi)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → total_function_on phi K J

In Proofgold the corresponding term root is ab695a... and proposition id is 70d53e...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_graphphi)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → functional_graph phi

In Proofgold the corresponding term root is 4244db... and proposition id is bf98ac...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_domphi)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → graph_domain_subset phi K

In Proofgold the corresponding term root is dbd842... and proposition id is d90107...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_mono)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → (∀i j : set, i ∈ K → j ∈ K → (i,j) ∈ leK → (apply_fun phi i,apply_fun phi j) ∈ leJ)

In Proofgold the corresponding term root is 63f1de... and proposition id is 06647f...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_cofinal)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → (∀j : set, j ∈ J → ∃k : set, k ∈ K ∧ (j,apply_fun phi k) ∈ leJ)

In Proofgold the corresponding term root is dc32cb... and proposition id is d09922...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_tail_above)

∀net sub J leJ K leK X phi j0 : set, subnet_of_witness net sub J leJ K leK X phi → j0 ∈ J → ∃k0 : set, k0 ∈ K ∧ ∀k : set, k ∈ K → (k0,k) ∈ leK → (j0,apply_fun phi k) ∈ leJ

In Proofgold the corresponding term root is 5e4713... and proposition id is 916130...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_vals)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → (∀k : set, k ∈ K → apply_fun sub k = apply_fun net (apply_fun phi k))

In Proofgold the corresponding term root is a10593... and proposition id is 71751e...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_implies_subnet_of)

∀net sub J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi → subnet_of net sub

In Proofgold the corresponding term root is 51ba08... and proposition id is 16fe10...

Proof:
Proof not loaded.

Theorem. (subnet_of_implies_exists_subnet_of_witness)

∀net sub : set, subnet_of net sub → ∃J leJ K leK X phi : set, subnet_of_witness net sub J leJ K leK X phi

In Proofgold the corresponding term root is 7d82b9... and proposition id is 50bff3...

Proof:
Proof not loaded.

Theorem. (subnet_of_witness_congr)

∀net1 net2 sub1 sub2 J1 J2 leJ1 leJ2 K1 K2 leK1 leK2 X phi : set, net1 = net2 → sub1 = sub2 → J1 = J2 → leJ1 = leJ2 → K1 = K2 → leK1 = leK2 → subnet_of_witness net1 sub1 J1 leJ1 K1 leK1 X phi → subnet_of_witness net2 sub2 J2 leJ2 K2 leK2 X phi

In Proofgold the corresponding term root is 73552f... and proposition id is 434457...

Proof:
Proof not loaded.

Definition. We define net_pack_converges to be λX Tx N x ⇒ ∃J le net : set, N = net_pack J le net ∧ net_converges_on X Tx net J le x of type set → set → set → set → prop.

In Proofgold the corresponding term root is 59ce55... and object id is 26f675...

Theorem. (net_pack_converges_implies_x_in_space)

∀X Tx N x : set, net_pack_converges X Tx N x → x ∈ X

In Proofgold the corresponding term root is febe75... and proposition id is b04630...

Proof:
Proof not loaded.

Theorem. (net_pack_converges_implies_topology_on)

∀X Tx N x : set, net_pack_converges X Tx N x → topology_on X Tx

In Proofgold the corresponding term root is e70eb2... and proposition id is 8fc83c...

Proof:
Proof not loaded.

Theorem. (net_pack_converges_implies_net_converges)

∀X Tx N x : set, net_pack_converges X Tx N x → net_converges X Tx (net_pack_fun N) x

In Proofgold the corresponding term root is ef3dbb... and proposition id is de605e...

Proof:
Proof not loaded.

Definition. We define subnet_pack_of to be λN S ⇒ ∃X phi : set, subnet_of_witness (net_pack_fun N) (net_pack_fun S) (net_pack_index N) (net_pack_le N) (net_pack_index S) (net_pack_le S) X phi of type set → set → prop.

In Proofgold the corresponding term root is 62809f... and object id is 2fa02d...

Definition. We define subnet_pack_of_in to be λX N S ⇒ ∃phi : set, subnet_of_witness (net_pack_fun N) (net_pack_fun S) (net_pack_index N) (net_pack_le N) (net_pack_index S) (net_pack_le S) X phi of type set → set → set → prop.

In Proofgold the corresponding term root is 9c90c8... and object id is b9e564...

Theorem. (subnet_pack_of_in_trans)

∀X N S T : set, subnet_pack_of_in X N S → subnet_pack_of_in X S T → subnet_pack_of_in X N T

In Proofgold the corresponding term root is 117cc1... and proposition id is 03b4af...

Proof:
Proof not loaded.

Definition. We define setprod_le to be λK1 le1 K2 le2 ⇒ {q ∈ setprod (setprod K1 K2) (setprod K1 K2)|(((q 0) 0,(q 1) 0) ∈ le1) ∧ (((q 0) 1,(q 1) 1) ∈ le2)} of type set → set → set → set → set.

In Proofgold the corresponding term root is e79093... and object id is cf00c2...

Theorem. (setprod_le_pred)

∀K1 le1 K2 le2 a b : set, (a,b) ∈ setprod_le K1 le1 K2 le2 → ((a 0,b 0) ∈ le1) ∧ ((a 1,b 1) ∈ le2)

In Proofgold the corresponding term root is 19b968... and proposition id is 4ed9db...

Proof:
Proof not loaded.

Theorem. (directed_set_setprod)

∀K1 le1 K2 le2 : set, directed_set K1 le1 → directed_set K2 le2 → directed_set (setprod K1 K2) (setprod_le K1 le1 K2 le2)

In Proofgold the corresponding term root is 506eec... and proposition id is 8361ed...

Proof:
Proof not loaded.

Theorem. (subnet_pack_of_in_refl_from_subnet)

∀X N S : set, subnet_pack_of_in X N S → subnet_pack_of_in X N N

In Proofgold the corresponding term root is e8c6a1... and proposition id is ab7661...

Proof:
Proof not loaded.

Theorem. (subnet_pack_of_in_common_refinement)

∀X N S1 S2 : set, subnet_pack_of_in X N S1 → subnet_pack_of_in X N S2 → ∃R : set, subnet_pack_of_in X N R ∧ subnet_pack_of_in X R S1 ∧ subnet_pack_of_in X R S2

In Proofgold the corresponding term root is f38aed... and proposition id is 5f72e2...

Proof:
Proof not loaded.

Theorem. (subnet_pack_of_in_implies_subnet_of_fun)

∀X N S : set, subnet_pack_of_in X N S → subnet_of (net_pack_fun N) (net_pack_fun S)

In Proofgold the corresponding term root is 314d16... and proposition id is 9e17ef...

Proof:
Proof not loaded.

Theorem. (subnet_preserves_convergence_witnessed)

∀X Tx net sub x J leJ K leK phi : set, topology_on X Tx → directed_set J leJ → directed_set K leK → total_function_on net J X → functional_graph net → graph_domain_subset net J → total_function_on sub K X → functional_graph sub → graph_domain_subset sub K → total_function_on phi K J → functional_graph phi → graph_domain_subset phi K → (∀i j : set, i ∈ K → j ∈ K → (i,j) ∈ leK → (apply_fun phi i,apply_fun phi j) ∈ leJ) → (∀j : set, j ∈ J → ∃k : set, k ∈ K ∧ (j,apply_fun phi k) ∈ leJ) → (∀k : set, k ∈ K → apply_fun sub k = apply_fun net (apply_fun phi k)) → x ∈ X → (∀U : set, U ∈ Tx → x ∈ U → ∃j0 : set, j0 ∈ J ∧ ∀j : set, j ∈ J → (j0,j) ∈ leJ → apply_fun net j ∈ U) → net_converges X Tx sub x

In Proofgold the corresponding term root is bcd25b... and proposition id is 33f9b5...

Proof:
Proof not loaded.

Theorem. (subnet_preserves_convergence)

∀X Tx net sub x J leJ K leK phi : set, subnet_of_witness net sub J leJ K leK X phi → net_converges_on X Tx net J leJ x → net_converges_on X Tx sub K leK x

In Proofgold the corresponding term root is 3caf7b... and proposition id is 84ed8c...

Proof:
Proof not loaded.

Theorem. (subnet_pack_of_in_preserves_net_converges_on)

∀X Tx N S x : set, subnet_pack_of_in X N S → net_converges_on X Tx (net_pack_fun N) (net_pack_index N) (net_pack_le N) x → net_converges_on X Tx (net_pack_fun S) (net_pack_index S) (net_pack_le S) x

In Proofgold the corresponding term root is 1cd5b7... and proposition id is f212f9...

Proof:
Proof not loaded.

Theorem. (subnet_pack_of_in_preserves_net_pack_converges)

∀X Tx N S x : set, net_pack_in_space X N → net_pack_in_space X S → subnet_pack_of_in X N S → net_pack_converges X Tx N x → net_pack_converges X Tx S x

In Proofgold the corresponding term root is 25a271... and proposition id is ed92c4...

Proof:
Proof not loaded.

Definition. We define net_points_in to be λA net J ⇒ ∀j : set, j ∈ J → apply_fun net j ∈ A of type set → set → set → prop.

In Proofgold the corresponding term root is 36675c... and object id is e5c09d...

Definition. We define rev_inclusion_rel to be λJ ⇒ {p ∈ setprod J J|p 1 ⊆ p 0} of type set → set.

In Proofgold the corresponding term root is acb0cb... and object id is b11e32...

Theorem. (rev_inclusion_rel_def)

∀J : set, rev_inclusion_rel J = {p ∈ setprod J J|p 1 ⊆ p 0}

In Proofgold the corresponding term root is e341c3... and proposition id is ed6a78...

Proof:
Proof not loaded.

Theorem. (rev_inclusion_relE)

∀J a b : set, (a,b) ∈ rev_inclusion_rel J → (a,b) ∈ setprod J J ∧ b ⊆ a

In Proofgold the corresponding term root is d20ce6... and proposition id is b46f43...

Proof:
Proof not loaded.

Theorem. (rev_inclusion_relI)

∀J a b : set, (a,b) ∈ setprod J J → b ⊆ a → (a,b) ∈ rev_inclusion_rel J

In Proofgold the corresponding term root is a7f1d7... and proposition id is adaca4...

Proof:
Proof not loaded.

Definition. We define inclusion_rel to be λJ ⇒ {p ∈ setprod J J|(p 0) ⊆ (p 1)} of type set → set.

In Proofgold the corresponding term root is c60302... and object id is 18013d...

Theorem. (inclusion_rel_def)

∀J : set, inclusion_rel J = {p ∈ setprod J J|(p 0) ⊆ (p 1)}

In Proofgold the corresponding term root is f063e1... and proposition id is af34fe...

Proof:
Proof not loaded.

Theorem. (inclusion_relE)

∀J a b : set, (a,b) ∈ inclusion_rel J → (a,b) ∈ setprod J J ∧ a ⊆ b

In Proofgold the corresponding term root is ea269c... and proposition id is 5a7ba5...

Proof:
Proof not loaded.

Theorem. (inclusion_relI)

∀J a b : set, (a,b) ∈ setprod J J → a ⊆ b → (a,b) ∈ inclusion_rel J

In Proofgold the corresponding term root is c02860... and proposition id is 2a6da4...

Proof:
Proof not loaded.

Theorem. (omega_directed_by_inclusion_rel)

directed_set ω (inclusion_rel ω)

In Proofgold the corresponding term root is 4336fc... and proposition id is 010f36...

Proof:
Proof not loaded.

Theorem. (finite_subcollections_directed_by_inclusion_rel)

∀I : set, directed_set (finite_subcollections I) (inclusion_rel (finite_subcollections I))

In Proofgold the corresponding term root is d2fbfa... and proposition id is 82cb93...

Proof:
Proof not loaded.

Definition. We define sequence_as_net to be λseq ⇒ graph ω (λn : set ⇒ apply_fun seq n) of type set → set.

In Proofgold the corresponding term root is 7fcd24... and object id is 7ccbdf...

Theorem. (sequence_as_net_in_space)

∀X seq : set, sequence_on seq X → net_in_space X (sequence_as_net seq)

In Proofgold the corresponding term root is 777cb9... and proposition id is b0d252...

Proof:
Proof not loaded.

Theorem. (converges_to_iff_net_converges_on_sequence_as_net)

∀X Tx seq x : set, sequence_on seq X → (converges_to X Tx seq x ↔ net_converges_on X Tx (sequence_as_net seq) ω (inclusion_rel ω) x)

In Proofgold the corresponding term root is 06f042... and proposition id is 7d35ea...

Proof:
Proof not loaded.

Theorem. (converges_to_implies_net_converges_sequence_as_net)

∀X Tx seq x : set, converges_to X Tx seq x → net_converges X Tx (sequence_as_net seq) x

In Proofgold the corresponding term root is 1197aa... and proposition id is 0487a0...

Proof:
Proof not loaded.

Theorem. (neighborhoods_directed_by_reverse_inclusion)

∀X Tx x : set, topology_on X Tx → x ∈ X → directed_set {U ∈ Tx|x ∈ U} (rev_inclusion_rel {U ∈ Tx|x ∈ U})

In Proofgold the corresponding term root is 41c6ae... and proposition id is b5f242...

Proof:
Proof not loaded.

Theorem. (closure_via_nets)

∀X Tx A x : set, topology_on X Tx → (x ∈ closure_of X Tx A ↔ ∃net J le : set, net_converges_on X Tx net J le x ∧ net_points_in A net J)

In Proofgold the corresponding term root is 6e5e2c... and proposition id is 4fba7c...

Proof:
Proof not loaded.

Theorem. (continuity_via_nets)

∀X Tx Y Ty f : set, topology_on X Tx → topology_on Y Ty → (continuous_map X Tx Y Ty f ↔ ∀net J le x : set, net_converges_on X Tx net J le x → net_converges_on Y Ty (compose_fun J net f) J le (apply_fun f x))

In Proofgold the corresponding term root is 39a4ae... and proposition id is 10b8ee...

Proof:
Proof not loaded.

Theorem. (subnet_converges_to_accumulation)

∀X Tx net x : set, accumulation_point_of_net X Tx net x → ∃sub : set, subnet_of net sub ∧ net_converges X Tx sub x

In Proofgold the corresponding term root is 7734d8... and proposition id is 25ef66...

Proof:
Proof not loaded.

Theorem. (subnet_converges_to_accumulation_witnessed)

∀X Tx net x : set, accumulation_point_of_net X Tx net x → ∃J le K leK phi sub : set, subnet_of_witness net sub J le K leK X phi ∧ net_converges_on X Tx sub K leK x

In Proofgold the corresponding term root is 693b98... and proposition id is 7c5eb6...

Proof:
Proof not loaded.

Theorem. (subnet_converges_to_accumulation_witnessed_on)

∀X Tx net x J le : set, accumulation_point_of_net_on X Tx net J le x → ∃K leK phi sub : set, subnet_of_witness net sub J le K leK X phi ∧ net_converges_on X Tx sub K leK x

In Proofgold the corresponding term root is a87fd7... and proposition id is d0d5d1...

Proof:
Proof not loaded.

Theorem. (compact_space_net_has_accumulation_point)

∀X Tx net : set, compact_space X Tx → net_in_space X net → ∃x0 : set, accumulation_point_of_net X Tx net x0

In Proofgold the corresponding term root is 1c2edf... and proposition id is fcf312...

Proof:
Proof not loaded.

Theorem. (compact_space_net_has_accumulation_point_on)

∀X Tx net J le : set, compact_space X Tx → directed_set J le → total_function_on net J X → functional_graph net → graph_domain_subset net J → ∃x0 : set, accumulation_point_of_net_on X Tx net J le x0

In Proofgold the corresponding term root is 693174... and proposition id is e7fe45...

Proof:
Proof not loaded.

Theorem. (finite_subcollections_directed_by_subset)

∀S : set, directed_set (finite_subcollections S) (inclusion_rel (finite_subcollections S))

In Proofgold the corresponding term root is d2fbfa... and proposition id is 82cb93...

Proof:
Proof not loaded.

Theorem. (open_cover_no_finite_subcover_implies_net_counterexample)

∀X Tx Fam : set, topology_on X Tx → open_cover_of X Tx Fam → ¬ (has_finite_subcover X Tx Fam) → ∃J le net0 : set, directed_set J le ∧ total_function_on net0 J X ∧ functional_graph net0 ∧ graph_domain_subset net0 J ∧ (∀sub K leK phi x : set, subnet_of_witness net0 sub J le K leK X phi → ¬ (net_converges_on X Tx sub K leK x))

In Proofgold the corresponding term root is 8b8aba... and proposition id is 9b0182...

Proof:
Proof not loaded.

Theorem. (open_cover_no_finite_subcover_implies_net_pack_counterexample)

∀X Tx Fam : set, topology_on X Tx → open_cover_of X Tx Fam → ¬ (has_finite_subcover X Tx Fam) → ∃N : set, net_pack_in_space X N ∧ (∀S x : set, subnet_pack_of_in X N S → ¬ (net_pack_converges X Tx S x))

In Proofgold the corresponding term root is 7766dd... and proposition id is 0fd0e4...

Proof:
Proof not loaded.

Theorem. (compact_space_implies_every_net_has_convergent_subnet)

∀X Tx : set, compact_space X Tx → ∀net : set, net_in_space X net → ∃sub x : set, subnet_of net sub ∧ net_converges X Tx sub x

In Proofgold the corresponding term root is 3822dc... and proposition id is 553ef5...

Proof:
Proof not loaded.

Theorem. (compact_space_implies_every_net_pack_has_convergent_subnet_fun)

∀X Tx : set, compact_space X Tx → ∀N : set, net_pack_in_space X N → ∃sub x : set, subnet_of (net_pack_fun N) sub ∧ net_converges X Tx sub x

In Proofgold the corresponding term root is da1db5... and proposition id is bbd14d...

Proof:
Proof not loaded.

Theorem. (compact_space_implies_every_net_pack_has_convergent_subnet_pack)

∀X Tx : set, compact_space X Tx → ∀N : set, net_pack_in_space X N → ∃S x : set, subnet_pack_of_in X N S ∧ net_pack_converges X Tx S x

In Proofgold the corresponding term root is 47566b... and proposition id is dbc4e1...

Proof:
Proof not loaded.

Theorem. (compact_space_of_every_net_pack_has_convergent_subnet_pack)

∀X Tx : set, topology_on X Tx → (∀N : set, net_pack_in_space X N → ∃S x : set, subnet_pack_of_in X N S ∧ net_pack_converges X Tx S x) → compact_space X Tx

In Proofgold the corresponding term root is 89af4f... and proposition id is 1039c7...

Proof:
Proof not loaded.

Theorem. (compact_iff_every_net_pack_has_convergent_subnet_pack)

∀X Tx : set, topology_on X Tx → (compact_space X Tx ↔ ∀N : set, net_pack_in_space X N → ∃S x : set, subnet_pack_of_in X N S ∧ net_pack_converges X Tx S x)

In Proofgold the corresponding term root is 491284... and proposition id is 4d636b...

Proof:
Proof not loaded.

Definition. We define countable_set to be λA ⇒ countable A of type set → prop.

In Proofgold the corresponding term root is 9bb8ca... and object id is bdda90...

Theorem. (omega_countable_set)

countable_set ω

In Proofgold the corresponding term root is e84fd3... and proposition id is 3e633e...

Proof:
Proof not loaded.

Theorem. (equip_countable_set_left)

∀X Y : set, equip X Y → countable_set X → countable_set Y

In Proofgold the corresponding term root is f97ef9... and proposition id is c6c5ae...

Proof:
Proof not loaded.

Theorem. (equip_countable_set_right)

∀X Y : set, equip X Y → countable_set Y → countable_set X

In Proofgold the corresponding term root is d13b3e... and proposition id is 145707...

Proof:
Proof not loaded.

Theorem. (rational_numbers_countable)

countable_set rational_numbers

In Proofgold the corresponding term root is 52d69f... and proposition id is 79df81...

Proof:
Proof not loaded.

Definition. We define countable_subcollection to be λV U ⇒ V ⊆ U ∧ countable_set V of type set → set → prop.

In Proofgold the corresponding term root is 57a9fa... and object id is 374194...

Definition. We define countable_index_set to be λI ⇒ countable_set I of type set → prop.

In Proofgold the corresponding term root is 9bb8ca... and object id is bdda90...

Definition. We define countable_product_topology_subbasis to be λI Xi ⇒ generated_topology_from_subbasis (product_space I Xi) (product_subbasis_full I Xi) of type set → set → set.

In Proofgold the corresponding term root is fe4ae5... and object id is ba39d5...

Theorem. (product_subbasis_full_subbasis_on)

∀I Xi : set, I ≠ Empty → (∀i : set, i ∈ I → topology_on (space_family_set Xi i) (space_family_topology Xi i)) → subbasis_on (product_space I Xi) (product_subbasis_full I Xi)

In Proofgold the corresponding term root is 1c47f1... and proposition id is ae01c1...

Proof:
Proof not loaded.

Theorem. (setprod_Empty_left)

∀Y : set, setprod Empty Y = Empty

In Proofgold the corresponding term root is b78d50... and proposition id is a53031...

Proof:
Proof not loaded.

Theorem. (basis_on_singleton)

∀X : set, basis_on X {X}

In Proofgold the corresponding term root is e1acfd... and proposition id is dc6fc9...

Proof:
Proof not loaded.

Theorem. (product_space_empty_index)

∀Xi : set, product_space Empty Xi = {Empty}

In Proofgold the corresponding term root is bb11cd... and proposition id is 71813f...

Proof:
Proof not loaded.

Theorem. (basis_of_subbasis_empty_eq)

∀X : set, X ≠ Empty → basis_of_subbasis X Empty = {X}

In Proofgold the corresponding term root is 026283... and proposition id is 7bf951...

Proof:
Proof not loaded.

Theorem. (countable_product_topology_subbasis_empty_is_topology)

∀Xi : set, topology_on (product_space Empty Xi) (countable_product_topology_subbasis Empty Xi)

In Proofgold the corresponding term root is 93d0a0... and proposition id is 76368f...

Proof:
Proof not loaded.

Theorem. (product_topology_full_is_topology)

∀I Xi : set, (∀i : set, i ∈ I → topology_on (space_family_set Xi i) (space_family_topology Xi i)) → topology_on (product_space I Xi) (product_topology_full I Xi)

In Proofgold the corresponding term root is 8926bb... and proposition id is 3544b9...

Proof:
Proof not loaded.

Theorem. (product_topology_full_empty_is_topology)

∀Xi : set, topology_on (product_space Empty Xi) (product_topology_full Empty Xi)

In Proofgold the corresponding term root is 93d0a0... and proposition id is 76368f...

Proof:
Proof not loaded.

Theorem. (product_topology_full_empty_is_compact)

∀Xi : set, compact_space (product_space Empty Xi) (product_topology_full Empty Xi)

In Proofgold the corresponding term root is 2da33e... and proposition id is 542e42...

Proof:
Proof not loaded.

Theorem. (equip_Power_nat)

∀n : set, nat_p n → equip (𝒫 n) (exp_nat 2 n)

In Proofgold the corresponding term root is f5822e... and proposition id is 293d3d...

Proof:
Proof not loaded.

Theorem. (countable_image)

∀X : set, countable_set X → ∀F : set → set, countable_set {F x|x ∈ X}

In Proofgold the corresponding term root is 85bdf5... and proposition id is a70b93...

Proof:
Proof not loaded.

Theorem. (rational_open_intervals_basis_countable)

countable_set rational_open_intervals_basis

In Proofgold the corresponding term root is 9830f5... and proposition id is 1f0496...

Proof:
Proof not loaded.

Theorem. (finite_subcollections_omega_countable)

countable (finite_subcollections ω)

In Proofgold the corresponding term root is ff8084... and proposition id is ed218b...

Proof:
Proof not loaded.

Theorem. (finite_subcollections_countable)

∀S : set, countable_set S → countable_set (finite_subcollections S)

In Proofgold the corresponding term root is d255f8... and proposition id is d53e79...

Proof:
Proof not loaded.

Theorem. (finite_intersections_of_countable)

∀X S : set, countable_set S → countable_set (finite_intersections_of X S)

In Proofgold the corresponding term root is be9275... and proposition id is 4595bf...

Proof:
Proof not loaded.

Theorem. (basis_of_subbasis_countable)

∀X S : set, countable_set S → countable_set (basis_of_subbasis X S)

In Proofgold the corresponding term root is 89d624... and proposition id is b9d5d3...

Proof:
Proof not loaded.

Definition. We define real_sequences to be {f ∈ 𝒫 (setprod ω R)|total_function_on f ω R ∧ functional_graph f} of type set.

In Proofgold the corresponding term root is 161258... and object id is 029ab2...

Theorem. (real_sequences_ext)

∀f g : set, f ∈ real_sequences → g ∈ real_sequences → (∀n : set, n ∈ ω → apply_fun f n = apply_fun g n) → f = g

In Proofgold the corresponding term root is 8b084b... and proposition id is 36858a...

Proof:
Proof not loaded.

Theorem. (exists_metric_on)

∀X : set, ∃d : set, metric_on X d

In Proofgold the corresponding term root is 0de768... and proposition id is a4924f...

Proof:
Proof not loaded.

Theorem. (space_family_set_const_Romega)

∀i : set, i ∈ ω → space_family_set (const_space_family ω R R_standard_topology) i = R

In Proofgold the corresponding term root is 3f341c... and proposition id is e6b536...

Proof:
Proof not loaded.

Theorem. (space_family_union_const_Romega)

space_family_union ω (const_space_family ω R R_standard_topology) = R

In Proofgold the corresponding term root is 3c962a... and proposition id is fcd58a...

Proof:
Proof not loaded.

Theorem. (real_sequences_eq_Romega_space)

real_sequences = R_omega_space

In Proofgold the corresponding term root is bbc84d... and proposition id is 5267d0...

Proof:
Proof not loaded.

Definition. We define Romega_coord_abs_diff to be λf g n ⇒ abs_SNo (add_SNo (apply_fun f n) (minus_SNo (apply_fun g n))) of type set → set → set → set.

In Proofgold the corresponding term root is c1a5ae... and object id is 7900b4...

Definition. We define Romega_coord_clipped_diff to be λf g n ⇒ If_i (Rlt (Romega_coord_abs_diff f g n) 1) (Romega_coord_abs_diff f g n) 1 of type set → set → set → set.

In Proofgold the corresponding term root is 3ea59f... and object id is dd5dac...

Definition. We define Romega_clipped_diffs to be λf g ⇒ Repl ω (λn : set ⇒ Romega_coord_clipped_diff f g n) of type set → set → set.

In Proofgold the corresponding term root is 820e6e... and object id is aa45ef...

Theorem. (Romega_clipped_diffs_sym)

∀f g : set, f ∈ real_sequences → g ∈ real_sequences → Romega_clipped_diffs f g = Romega_clipped_diffs g f

In Proofgold the corresponding term root is 1bcbf5... and proposition id is d8baeb...

Proof:
Proof not loaded.

Theorem. (Romega_coord_clipped_diff_self_zero)

∀f n : set, f ∈ real_sequences → n ∈ ω → Romega_coord_clipped_diff f f n = 0

In Proofgold the corresponding term root is a56545... and proposition id is fa85fc...

Proof:
Proof not loaded.

Theorem. (Romega_clipped_diffs_diag_eq_Sing0)

∀f : set, f ∈ real_sequences → Romega_clipped_diffs f f = {0}

In Proofgold the corresponding term root is 810d8c... and proposition id is bea24d...

Proof:
Proof not loaded.

Definition. We define Romega_uniform_metric_value to be λf g ⇒ Eps_i (λr : set ⇒ R_lub (Romega_clipped_diffs f g) r) of type set → set → set.

In Proofgold the corresponding term root is a91886... and object id is dafcc8...

Theorem. (Romega_clipped_diffs_in_R)

∀f g : set, f ∈ real_sequences → g ∈ real_sequences → ∀a : set, a ∈ Romega_clipped_diffs f g → a ∈ R

In Proofgold the corresponding term root is b53997... and proposition id is 9fa0af...

Proof:
Proof not loaded.

Theorem. (Romega_clipped_diffs_bounded)

∀f g : set, f ∈ real_sequences → g ∈ real_sequences → ∃u : set, u ∈ R ∧ ∀a : set, a ∈ Romega_clipped_diffs f g → a ∈ R → Rle a u

In Proofgold the corresponding term root is 65b349... and proposition id is 35e99c...

Proof:
Proof not loaded.

Theorem. (Romega_uniform_metric_value_is_lub)

∀f g : set, f ∈ real_sequences → g ∈ real_sequences → R_lub (Romega_clipped_diffs f g) (Romega_uniform_metric_value f g)

In Proofgold the corresponding term root is 092fd4... and proposition id is a4a3d0...

Proof:
Proof not loaded.

Theorem. (Romega_coord_clipped_diff_le_uniform)

∀f g i : set, f ∈ real_sequences → g ∈ real_sequences → i ∈ ω → Rle (Romega_coord_clipped_diff f g i) (Romega_uniform_metric_value f g)

In Proofgold the corresponding term root is 9b1794... and proposition id is 29c3d0...

Proof:
Proof not loaded.

Theorem. (Romega_coord_clipped_diff_lt_of_uniform_lt)

∀f g i r : set, f ∈ real_sequences → g ∈ real_sequences → i ∈ ω → Rlt (Romega_uniform_metric_value f g) r → Rlt (Romega_coord_clipped_diff f g i) r

In Proofgold the corresponding term root is 0bc515... and proposition id is ee153a...

Proof:
Proof not loaded.

Theorem. (Romega_uniform_metric_value_sym)

∀f g : set, f ∈ real_sequences → g ∈ real_sequences → Romega_uniform_metric_value f g = Romega_uniform_metric_value g f

In Proofgold the corresponding term root is 7f135e... and proposition id is 0956b0...

Proof:
Proof not loaded.

Theorem. (Romega_uniform_metric_value_self_zero)

∀f : set, f ∈ real_sequences → Romega_uniform_metric_value f f = 0

In Proofgold the corresponding term root is 914420... and proposition id is a361a5...

Proof:
Proof not loaded.

Theorem. (Romega_uniform_metric_value_in_R)

∀f g : set, f ∈ real_sequences → g ∈ real_sequences → Romega_uniform_metric_value f g ∈ R

In Proofgold the corresponding term root is 8c2ef1... and proposition id is 043ef4...

Proof:
Proof not loaded.

Theorem. (Romega_uniform_metric_value_nonneg)

∀f g : set, f ∈ real_sequences → g ∈ real_sequences → ¬ (Rlt (Romega_uniform_metric_value f g) 0)

In Proofgold the corresponding term root is ce46c1... and proposition id is b7474e...

Proof:
Proof not loaded.

Theorem. (Romega_coord_clipped_diff_nonneg)

∀f g n : set, f ∈ real_sequences → g ∈ real_sequences → n ∈ ω → 0 ≤ Romega_coord_clipped_diff f g n

In Proofgold the corresponding term root is 2fb16c... and proposition id is c13d73...

Proof:
Proof not loaded.

Theorem. (Romega_uniform_metric_value_eq0_coord_eq)

∀f g : set, f ∈ real_sequences → g ∈ real_sequences → Romega_uniform_metric_value f g = 0 → ∀n : set, n ∈ ω → apply_fun f n = apply_fun g n

In Proofgold the corresponding term root is 924ea6... and proposition id is fcd4b9...

Proof:
Proof not loaded.

Definition. We define uniform_metric_Romega to be graph (setprod real_sequences real_sequences) (λp : set ⇒ Romega_uniform_metric_value (p 0) (p 1)) of type set.

In Proofgold the corresponding term root is b282a9... and object id is 620ef4...

Theorem. (uniform_metric_Romega_function_on)

function_on uniform_metric_Romega (setprod real_sequences real_sequences) R

In Proofgold the corresponding term root is 126dc1... and proposition id is 810bb1...

Proof:
Proof not loaded.

Definition. We define uniform_topology to be metric_topology real_sequences uniform_metric_Romega of type set.

In Proofgold the corresponding term root is 79d2f7... and object id is 88883e...

Theorem. (Rle_of_SNoLe)

∀a b : set, a ∈ R → b ∈ R → a ≤ b → Rle a b

In Proofgold the corresponding term root is ef86d3... and proposition id is d8f852...

Proof:
Proof not loaded.

Theorem. (recip_SNo_pos_le1_of_ge1)

∀x : set, SNo x → 0 < x → 1 ≤ x → recip_SNo_pos x ≤ 1

In Proofgold the corresponding term root is c63292... and proposition id is 4b4278...

Proof:
Proof not loaded.

Theorem. (recip_SNo_pos_in_unit_interval_of_ge1)

∀t : set, t ∈ R → Rle 1 t → Rlt 0 t → recip_SNo_pos t ∈ unit_interval

In Proofgold the corresponding term root is 1d7be2... and proposition id is d8f594...

Proof:
Proof not loaded.

Theorem. (abs_SNo_upper_bound)

∀x : set, SNo x → x ≤ abs_SNo x

In Proofgold the corresponding term root is 22a2fe... and proposition id is b14bb3...

Proof:
Proof not loaded.

Theorem. (abs_SNo_lower_bound)

∀x : set, SNo x → minus_SNo (abs_SNo x) ≤ x

In Proofgold the corresponding term root is 9bb980... and proposition id is e17561...

Proof:
Proof not loaded.

Theorem. (abs_SNo_Le_of_bounds)

∀t u : set, SNo t → SNo u → minus_SNo u ≤ t → t ≤ u → abs_SNo t ≤ u

In Proofgold the corresponding term root is 81033e... and proposition id is 7dbadf...

Proof:
Proof not loaded.

Theorem. (abs_SNo_subadd)

∀x y : set, SNo x → SNo y → abs_SNo (add_SNo x y) ≤ add_SNo (abs_SNo x) (abs_SNo y)

In Proofgold the corresponding term root is f7ed3b... and proposition id is db0357...

Proof:
Proof not loaded.

Theorem. (abs_SNo_triangle)

∀a b c : set, SNo a → SNo b → SNo c → abs_SNo (add_SNo a (minus_SNo c)) ≤ add_SNo (abs_SNo (add_SNo a (minus_SNo b))) (abs_SNo (add_SNo b (minus_SNo c)))

In Proofgold the corresponding term root is d54ddd... and proposition id is 8a76af...

Proof:
Proof not loaded.

Theorem. (SNo_If_i)

∀P : prop, ∀x y : set, SNo x → SNo y → SNo (If_i P x y)

In Proofgold the corresponding term root is e8dc93... and proposition id is 2d3bd7...

Proof:
Proof not loaded.

Theorem. (Romega_coord_abs_diff_in_R)

∀f g n : set, f ∈ real_sequences → g ∈ real_sequences → n ∈ ω → Romega_coord_abs_diff f g n ∈ R

In Proofgold the corresponding term root is 8c6cd1... and proposition id is dc0729...

Proof:
Proof not loaded.

Theorem. (Rclip_mono)

∀t s : set, t ∈ R → s ∈ R → t ≤ s → If_i (Rlt t 1) t 1 ≤ If_i (Rlt s 1) s 1

In Proofgold the corresponding term root is 05b7bb... and proposition id is 8679ae...

Proof:
Proof not loaded.

Theorem. (Rclip_subadd_nonneg)

∀p q : set, p ∈ R → q ∈ R → 0 ≤ p → 0 ≤ q → If_i (Rlt (add_SNo p q) 1) (add_SNo p q) 1 ≤ add_SNo (If_i (Rlt p 1) p 1) (If_i (Rlt q 1) q 1)

In Proofgold the corresponding term root is 944c6b... and proposition id is ba9492...

Proof:
Proof not loaded.

Theorem. (Romega_coord_abs_diff_triangle)

∀f g h n : set, f ∈ real_sequences → g ∈ real_sequences → h ∈ real_sequences → n ∈ ω → Romega_coord_abs_diff f h n ≤ add_SNo (Romega_coord_abs_diff f g n) (Romega_coord_abs_diff g h n)

In Proofgold the corresponding term root is 758d4d... and proposition id is e91791...

Proof:
Proof not loaded.

Theorem. (Romega_coord_clipped_diff_triangle)

∀f g h n : set, f ∈ real_sequences → g ∈ real_sequences → h ∈ real_sequences → n ∈ ω → Romega_coord_clipped_diff f h n ≤ add_SNo (Romega_coord_clipped_diff f g n) (Romega_coord_clipped_diff g h n)

In Proofgold the corresponding term root is f928ee... and proposition id is 1842d5...

Proof:
Proof not loaded.

Theorem. (Romega_uniform_metric_value_triangle)

∀x y z : set, x ∈ real_sequences → y ∈ real_sequences → z ∈ real_sequences → ¬ (Rlt (add_SNo (Romega_uniform_metric_value x y) (Romega_uniform_metric_value y z)) (Romega_uniform_metric_value x z))

In Proofgold the corresponding term root is d3ee2c... and proposition id is 0b0052...

Proof:
Proof not loaded.

Theorem. (uniform_metric_Romega_is_metric)

metric_on real_sequences uniform_metric_Romega

In Proofgold the corresponding term root is b95b60... and proposition id is 3c7eb1...

Proof:
Proof not loaded.

Theorem. (uniform_topology_is_topology)

topology_on real_sequences uniform_topology

In Proofgold the corresponding term root is 5167c8... and proposition id is 8ae96b...

Proof:
Proof not loaded.

Theorem. (uniform_topology_is_topology_on_Romega_space)

topology_on R_omega_space uniform_topology

In Proofgold the corresponding term root is 7a8b02... and proposition id is b54eb3...

Proof:
Proof not loaded.

Definition. We define Romega_product_topology_on_real_sequences to be subspace_topology R_omega_space (product_topology_full ω (const_space_family ω R R_standard_topology)) real_sequences of type set.

In Proofgold the corresponding term root is 619862... and object id is d46f80...

Definition. We define Romega_box_topology_on_real_sequences to be subspace_topology R_omega_space (box_topology ω (const_space_family ω R R_standard_topology)) real_sequences of type set.

In Proofgold the corresponding term root is daf946... and object id is 977a44...

Theorem. (Romega_product_topology_on_real_sequences_eq)

Romega_product_topology_on_real_sequences = R_omega_product_topology

In Proofgold the corresponding term root is 50ac73... and proposition id is 784578...

Proof:
Proof not loaded.

Theorem. (uniform_topology_finer_than_product_and_coarser_than_box)

finer_than uniform_topology Romega_product_topology_on_real_sequences ∧ coarser_than uniform_topology Romega_box_topology_on_real_sequences

In Proofgold the corresponding term root is fa1bbf... and proposition id is f07df9...

Proof:
Proof not loaded.

Definition. We define R_bounded_distance to be λa b ⇒ If_i (Rlt (abs_SNo (add_SNo a (minus_SNo b))) 1) (abs_SNo (add_SNo a (minus_SNo b))) 1 of type set → set → set.

In Proofgold the corresponding term root is 103b14... and object id is 366dca...

Definition. We define R_bounded_metric to be graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) of type set.

In Proofgold the corresponding term root is 3ec682... and object id is c6b86a...

Theorem. (R_bounded_metric_apply_early)

∀x y : set, x ∈ R → y ∈ R → apply_fun R_bounded_metric (x,y) = R_bounded_distance x y

In Proofgold the corresponding term root is beb60d... and proposition id is bc356a...

Proof:
Proof not loaded.

Theorem. (R_bounded_distance_sym)

∀a b : set, a ∈ R → b ∈ R → R_bounded_distance a b = R_bounded_distance b a

In Proofgold the corresponding term root is 998ca7... and proposition id is a6d5ed...

Proof:
Proof not loaded.

Theorem. (R_bounded_distance_self_zero)

∀a : set, a ∈ R → R_bounded_distance a a = 0

In Proofgold the corresponding term root is eead96... and proposition id is a8b11e...

Proof:
Proof not loaded.

Theorem. (R_bounded_distance_nonneg)

∀a b : set, a ∈ R → b ∈ R → 0 ≤ R_bounded_distance a b

In Proofgold the corresponding term root is 6962ed... and proposition id is 04e88d...

Proof:
Proof not loaded.

Theorem. (SNo_pos_ne0)

∀x : set, SNo x → 0 < x → x ≠ 0

In Proofgold the corresponding term root is 1fcd1d... and proposition id is 7aaa6a...

Proof:
Proof not loaded.

Theorem. (abs_SNo_eq0)

∀x : set, SNo x → abs_SNo x = 0 → x = 0

In Proofgold the corresponding term root is 37ddf1... and proposition id is d60c7c...

Proof:
Proof not loaded.

Theorem. (R_bounded_distance_eq0)

∀a b : set, a ∈ R → b ∈ R → R_bounded_distance a b = 0 → a = b

In Proofgold the corresponding term root is a67ef6... and proposition id is 4dfe1d...

Proof:
Proof not loaded.

Theorem. (R_bounded_distance_le_1)

∀a b : set, a ∈ R → b ∈ R → Rle (R_bounded_distance a b) 1

In Proofgold the corresponding term root is 4871bf... and proposition id is 33c9b7...

Proof:
Proof not loaded.

Theorem. (abs_SNo_in_R)

∀x : set, x ∈ R → abs_SNo x ∈ R

In Proofgold the corresponding term root is 6d9068... and proposition id is cd8fd9...

Proof:
Proof not loaded.

Theorem. (R_bounded_distance_in_R)

∀a b : set, a ∈ R → b ∈ R → R_bounded_distance a b ∈ R

In Proofgold the corresponding term root is f18276... and proposition id is 5b0328...

Proof:
Proof not loaded.

Theorem. (R_bounded_distance_triangle_Le)

∀a b c : set, a ∈ R → b ∈ R → c ∈ R → R_bounded_distance a c ≤ add_SNo (R_bounded_distance a b) (R_bounded_distance b c)

In Proofgold the corresponding term root is 5d3e09... and proposition id is 905fb9...

Proof:
Proof not loaded.

Definition. We define Romega_D_scaled_diffs to be λx y ⇒ Repl ω (λi : set ⇒ mul_SNo (R_bounded_distance (apply_fun x i) (apply_fun y i)) (inv_nat (ordsucc i))) of type set → set → set.

In Proofgold the corresponding term root is 60c086... and object id is c0cabc...

Theorem. (Romega_D_scaled_diffs_sym)

∀x y : set, x ∈ R_omega_space → y ∈ R_omega_space → Romega_D_scaled_diffs x y = Romega_D_scaled_diffs y x

In Proofgold the corresponding term root is 25c334... and proposition id is 90fe09...

Proof:
Proof not loaded.

Theorem. (Romega_D_scaled_diffs_diag_subset0)

∀x : set, x ∈ R_omega_space → Romega_D_scaled_diffs x x ⊆ {0}

In Proofgold the corresponding term root is c46b7c... and proposition id is 9d66ab...

Proof:
Proof not loaded.

Theorem. (Romega_D_scaled_diffs_diag_eq_Sing0)

∀x : set, x ∈ R_omega_space → Romega_D_scaled_diffs x x = {0}

In Proofgold the corresponding term root is 329780... and proposition id is b44a53...

Proof:
Proof not loaded.

Definition. We define Romega_D_metric_value to be λx y ⇒ Eps_i (λr : set ⇒ R_lub (Romega_D_scaled_diffs x y) r) of type set → set → set.

In Proofgold the corresponding term root is a1c424... and object id is 1a77eb...

Theorem. (inv_nat_Rle_1)

∀n : set, n ∈ ω ∖ {0} → Rle (inv_nat n) 1

In Proofgold the corresponding term root is f96a98... and proposition id is 725836...

Proof:
Proof not loaded.

Theorem. (Romega_D_scaled_diffs_in_R)

∀x y : set, x ∈ R_omega_space → y ∈ R_omega_space → ∀a : set, a ∈ Romega_D_scaled_diffs x y → a ∈ R

In Proofgold the corresponding term root is de87cf... and proposition id is 319f33...

Proof:
Proof not loaded.

Theorem. (Romega_D_scaled_diffs_bounded)

∀x y : set, x ∈ R_omega_space → y ∈ R_omega_space → ∃u : set, u ∈ R ∧ ∀a : set, a ∈ Romega_D_scaled_diffs x y → a ∈ R → Rle a u

In Proofgold the corresponding term root is 025ca8... and proposition id is eb25d1...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_value_is_lub)

∀x y : set, x ∈ R_omega_space → y ∈ R_omega_space → R_lub (Romega_D_scaled_diffs x y) (Romega_D_metric_value x y)

In Proofgold the corresponding term root is 41889f... and proposition id is 505999...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_value_sym)

∀x y : set, x ∈ R_omega_space → y ∈ R_omega_space → Romega_D_metric_value x y = Romega_D_metric_value y x

In Proofgold the corresponding term root is eaa05f... and proposition id is d4e490...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_value_self_zero)

∀x : set, x ∈ R_omega_space → Romega_D_metric_value x x = 0

In Proofgold the corresponding term root is 155dcf... and proposition id is 89b0ff...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_value_in_R)

∀x y : set, x ∈ R_omega_space → y ∈ R_omega_space → Romega_D_metric_value x y ∈ R

In Proofgold the corresponding term root is c1e3cb... and proposition id is e1ca82...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_value_nonneg)

∀x y : set, x ∈ R_omega_space → y ∈ R_omega_space → ¬ (Rlt (Romega_D_metric_value x y) 0)

In Proofgold the corresponding term root is a02b2c... and proposition id is 386a30...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_value_eq0_coord_eq)

∀x y : set, x ∈ R_omega_space → y ∈ R_omega_space → Romega_D_metric_value x y = 0 → ∀i : set, i ∈ ω → apply_fun x i = apply_fun y i

In Proofgold the corresponding term root is c70d5c... and proposition id is 689a7b...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_value_ub_scaled)

∀x y i : set, x ∈ R_omega_space → y ∈ R_omega_space → i ∈ ω → Rle (mul_SNo (R_bounded_distance (apply_fun x i) (apply_fun y i)) (inv_nat (ordsucc i))) (Romega_D_metric_value x y)

In Proofgold the corresponding term root is d53a98... and proposition id is ed722e...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_value_lt_implies_scaled_lt)

∀x y i r : set, x ∈ R_omega_space → y ∈ R_omega_space → i ∈ ω → Rlt (Romega_D_metric_value x y) r → Rlt (mul_SNo (R_bounded_distance (apply_fun x i) (apply_fun y i)) (inv_nat (ordsucc i))) r

In Proofgold the corresponding term root is 9ef2c8... and proposition id is b776a8...

Proof:
Proof not loaded.

Theorem. (R_bounded_distance_lt_lt1_imp_abs_lt)

∀a b r : set, a ∈ R → b ∈ R → r ∈ R → Rlt r 1 → Rlt (R_bounded_distance a b) r → abs_SNo (add_SNo a (minus_SNo b)) < r

In Proofgold the corresponding term root is b3549d... and proposition id is b53e70...

Proof:
Proof not loaded.

Theorem. (abs_lt_lt1_imp_R_bounded_distance_lt)

∀a b r : set, a ∈ R → b ∈ R → r ∈ R → Rlt r 1 → abs_SNo (add_SNo a (minus_SNo b)) < r → Rlt (R_bounded_distance a b) r

In Proofgold the corresponding term root is 6b5e3b... and proposition id is 85f527...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_coord_abs_lt)

∀x y i delta : set, x ∈ R_omega_space → y ∈ R_omega_space → i ∈ ω → delta ∈ R → Rlt 0 delta → Rlt delta 1 → Rlt (Romega_D_metric_value x y) (mul_SNo delta (inv_nat (ordsucc i))) → abs_SNo (add_SNo (apply_fun x i) (minus_SNo (apply_fun y i))) < delta

In Proofgold the corresponding term root is b72f38... and proposition id is 068dcb...

Proof:
Proof not loaded.

Theorem. (div_SNo_1_eq_inv_nat)

∀n : set, SNo n → div_SNo 1 n = inv_nat n

In Proofgold the corresponding term root is 695f62... and proposition id is 9f4e83...

Proof:
Proof not loaded.

Theorem. (inv_nat_1_eq_1)

inv_nat 1 = 1

In Proofgold the corresponding term root is cb7d70... and proposition id is 75c888...

Proof:
Proof not loaded.

Theorem. (inv_nat_ordsucc_antitone)

∀i j : set, i ∈ ω → j ∈ ω → i ∈ j → Rlt (inv_nat (ordsucc j)) (inv_nat (ordsucc i))

In Proofgold the corresponding term root is ce3bf0... and proposition id is 808d77...

Proof:
Proof not loaded.

Theorem. (inv_nat_2_lt_1)

Rlt (inv_nat 2) 1

In Proofgold the corresponding term root is 84c908... and proposition id is 160e15...

Proof:
Proof not loaded.

Theorem. (exists_inv_nat_ordsucc_lt)

∀r : set, r ∈ R → Rlt 0 r → ∃N : set, N ∈ ω ∧ Rlt (inv_nat (ordsucc N)) r

In Proofgold the corresponding term root is 93ca9a... and proposition id is 423ad2...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_value_triangle)

∀x y z : set, x ∈ R_omega_space → y ∈ R_omega_space → z ∈ R_omega_space → ¬ (Rlt (add_SNo (Romega_D_metric_value x y) (Romega_D_metric_value y z)) (Romega_D_metric_value x z))

In Proofgold the corresponding term root is 018d0a... and proposition id is 27d1cc...

Proof:
Proof not loaded.

Definition. We define Romega_D_metric to be graph (setprod R_omega_space R_omega_space) (λp : set ⇒ Romega_D_metric_value (p 0) (p 1)) of type set.

In Proofgold the corresponding term root is 43d5aa... and object id is 9bd8fe...

Theorem. (Romega_D_metric_function_on)

function_on Romega_D_metric (setprod R_omega_space R_omega_space) R

In Proofgold the corresponding term root is ae78fd... and proposition id is 375b1f...

Proof:
Proof not loaded.

Definition. We define Romega_D_metric_topology to be metric_topology R_omega_space Romega_D_metric of type set.

In Proofgold the corresponding term root is 8312fe... and object id is c375f9...

Theorem. (Romega_D_metric_open_ball_in_product_topology)

∀x r : set, metric_on R_omega_space Romega_D_metric → x ∈ R_omega_space → r ∈ R → Rlt 0 r → open_ball R_omega_space Romega_D_metric x r ∈ R_omega_product_topology

In Proofgold the corresponding term root is 2c81e7... and proposition id is 02c1e4...

Proof:
Proof not loaded.

Theorem. (Romega_product_cylinder_in_D_metric_topology)

∀i U : set, metric_on R_omega_space Romega_D_metric → i ∈ ω → U ∈ R_standard_topology → product_cylinder ω (const_space_family ω R R_standard_topology) i U ∈ Romega_D_metric_topology

In Proofgold the corresponding term root is 8cad46... and proposition id is e83cd8...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_induces_product_topology)

metric_on R_omega_space Romega_D_metric ∧ Romega_D_metric_topology = R_omega_product_topology

In Proofgold the corresponding term root is 8e3d4b... and proposition id is 14c05d...

Proof:
Proof not loaded.

Definition. We define open_cover to be λX Tx U ⇒ (∀u : set, u ∈ U → u ∈ Tx) ∧ covers X U of type set → set → set → prop.

In Proofgold the corresponding term root is 2d2289... and object id is be938f...

Theorem. (open_cover_members_open)

∀X Tx U u : set, open_cover X Tx U → u ∈ U → u ∈ Tx

In Proofgold the corresponding term root is 0afc9e... and proposition id is d6ff88...

Proof:
Proof not loaded.

Theorem. (open_cover_implies_covers)

∀X Tx U : set, open_cover X Tx U → covers X U

In Proofgold the corresponding term root is f6ae4d... and proposition id is f3f85d...

Proof:
Proof not loaded.

Theorem. (open_cover_of_implies_open_cover)

∀X Tx Fam : set, open_cover_of X Tx Fam → open_cover X Tx Fam

In Proofgold the corresponding term root is 204c49... and proposition id is 481324...

Proof:
Proof not loaded.

Theorem. (open_cover_family_sub)

∀X Tx U : set, topology_on X Tx → open_cover X Tx U → U ⊆ 𝒫 X

In Proofgold the corresponding term root is 137434... and proposition id is 5f24de...

Proof:
Proof not loaded.

Theorem. (open_cover_implies_open_cover_of)

∀X Tx Fam : set, topology_on X Tx → open_cover X Tx Fam → open_cover_of X Tx Fam

In Proofgold the corresponding term root is e3abd5... and proposition id is 49ced2...

Proof:
Proof not loaded.

Definition. We define Lindelof_space to be λX Tx ⇒ topology_on X Tx ∧ ∀U : set, open_cover X Tx U → ∃V : set, countable_subcollection V U ∧ covers X V of type set → set → prop.

In Proofgold the corresponding term root is 63ab86... and object id is 21f553...

Theorem. (countable_space_implies_Lindelof_space)

∀X Tx : set, topology_on X Tx → countable X → Lindelof_space X Tx

In Proofgold the corresponding term root is 677374... and proposition id is 8157b5...

Proof:
Proof not loaded.

Theorem. (compact_space_implies_Lindelof_space)

∀X Tx : set, compact_space X Tx → Lindelof_space X Tx

In Proofgold the corresponding term root is 6ed8ee... and proposition id is 74d965...

Proof:
Proof not loaded.

Definition. We define Sorgenfrey_line to be R of type set.

In Proofgold the corresponding term root is e26ffa... and object id is 5277ef...

Definition. We define Sorgenfrey_topology to be R_lower_limit_topology of type set.

In Proofgold the corresponding term root is ce1277... and object id is fb0317...

Theorem. (R_lower_limit_topology_Hausdorff)

Hausdorff_space R R_lower_limit_topology

In Proofgold the corresponding term root is 3666ea... and proposition id is eb0a13...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_line_Hausdorff)

Hausdorff_space Sorgenfrey_line Sorgenfrey_topology

In Proofgold the corresponding term root is 3666ea... and proposition id is eb0a13...

Proof:
Proof not loaded.

Definition. We define countable_basis_at to be λX Tx x ⇒ topology_on X Tx ∧ x ∈ X ∧ ∃B : set, B ⊆ Tx ∧ countable_set B ∧ (∀b : set, b ∈ B → x ∈ b) ∧ (∀U : set, U ∈ Tx → x ∈ U → ∃b : set, b ∈ B ∧ b ⊆ U) of type set → set → set → prop.

In Proofgold the corresponding term root is a51b9d... and object id is e9f4d1...

Definition. We define first_countable_space to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → countable_basis_at X Tx x of type set → set → prop.

In Proofgold the corresponding term root is 635722... and object id is ce10a4...

Theorem. (open_ball_radius_mono)

∀X d x r1 r2 : set, Rlt r1 r2 → open_ball X d x r1 ⊆ open_ball X d x r2

In Proofgold the corresponding term root is f2e366... and proposition id is 87e6e4...

Proof:
Proof not loaded.

Theorem. (metric_topology_first_countable)

∀X d : set, metric_on X d → first_countable_space X (metric_topology X d)

In Proofgold the corresponding term root is bff518... and proposition id is dc2a5b...

Proof:
Proof not loaded.

Theorem. (uniform_topology_first_countable)

first_countable_space real_sequences uniform_topology

In Proofgold the corresponding term root is 6bf16d... and proposition id is a8e5be...

Proof:
Proof not loaded.

Theorem. (product_countable_basis_at_point_if_components_first_countable)

∀I Xi f : set, countable_index_set I → I ≠ Empty → (∀i : set, i ∈ I → first_countable_space (space_family_set Xi i) (space_family_topology Xi i)) → f ∈ product_space I Xi → countable_basis_at (product_space I Xi) (countable_product_topology_subbasis I Xi) f

In Proofgold the corresponding term root is 5f3536... and proposition id is c0a0e9...

Proof:
Proof not loaded.

Theorem. (convergent_sequence_implies_closure)

∀X Tx A x : set, topology_on X Tx → A ⊆ X → (∃seq : set, sequence_in seq A ∧ converges_to X Tx seq x) → x ∈ closure_of X Tx A

In Proofgold the corresponding term root is d57b53... and proposition id is 9be52b...

Proof:
Proof not loaded.

Theorem. (equip_finite_transfer)

∀X Y : set, equip X Y → finite Y → finite X

In Proofgold the corresponding term root is b5a792... and proposition id is 9bcd25...

Proof:
Proof not loaded.

Theorem. (inj_equip_image)

∀X Y : set, ∀f : set → set, inj X Y f → equip X {f x|x ∈ X}

In Proofgold the corresponding term root is c42456... and proposition id is b898e7...

Proof:
Proof not loaded.

Theorem. (inj_into_finite_nat)

∀X n : set, ∀f : set → set, nat_p n → inj X n f → finite X

In Proofgold the corresponding term root is 6696bb... and proposition id is 1bcef6...

Proof:
Proof not loaded.

Theorem. (first_countable_sequences_detect_closure)

∀X Tx A x : set, first_countable_space X Tx → A ⊆ X → (x ∈ closure_of X Tx A ↔ ∃seq : set, sequence_in seq A ∧ converges_to X Tx seq x)

In Proofgold the corresponding term root is d424c6... and proposition id is 93c374...

Proof:
Proof not loaded.

Theorem. (first_countable_sequences_detect_continuity)

∀X Tx Y Ty f : set, topology_on X Tx → topology_on Y Ty → (continuous_map X Tx Y Ty f → ∀x seq : set, sequence_on seq X → converges_to X Tx seq x → converges_to Y Ty (map_sequence f seq) (apply_fun f x))

In Proofgold the corresponding term root is 34a038... and proposition id is 17038a...

Proof:
Proof not loaded.

Definition. We define second_countable_space to be λX Tx ⇒ topology_on X Tx ∧ ∃B : set, basis_on X B ∧ countable_set B ∧ basis_generates X B Tx of type set → set → prop.

In Proofgold the corresponding term root is a1a1ca... and object id is a641c8...

Theorem. (second_countable_implies_first_countable)

∀X Tx : set, second_countable_space X Tx → first_countable_space X Tx

In Proofgold the corresponding term root is 198bca... and proposition id is af0ded...

Proof:
Proof not loaded.

Theorem. (R_standard_topology_second_countable)

second_countable_space R R_standard_topology

In Proofgold the corresponding term root is c61305... and proposition id is d95a30...

Proof:
Proof not loaded.

Theorem. (euclidean_spaces_second_countable)

∀n : set, n ∈ ω → second_countable_space (euclidean_space n) (euclidean_topology n)

In Proofgold the corresponding term root is f82715... and proposition id is 03e78c...

Proof:
Proof not loaded.

Definition. We define discrete_subspace to be λX Tx A ⇒ A ⊆ X ∧ (∀a : set, a ∈ A → ∃U : set, U ∈ Tx ∧ U ∩ A = {a}) of type set → set → set → prop.

In Proofgold the corresponding term root is 22568f... and object id is 0c2854...

Theorem. (second_countable_discrete_subspace_countable)

∀X Tx A : set, second_countable_space X Tx → discrete_subspace X Tx A → countable_set A

In Proofgold the corresponding term root is 71732e... and proposition id is 0783e3...

Proof:
Proof not loaded.

Definition. We define binary_sequences_Romega to be {f ∈ real_sequences|∀n : set, n ∈ ω → apply_fun f n ∈ {0,1}} of type set.

In Proofgold the corresponding term root is d03358... and object id is 6b4eea...

Theorem. (binary_sequences_Romega_uncountable)

¬ countable_set binary_sequences_Romega

In Proofgold the corresponding term root is b1ed54... and proposition id is 9e7ad5...

Proof:
Proof not loaded.

Theorem. (real_sequences_in_Power_setprod)

∀f : set, f ∈ real_sequences → f ∈ 𝒫 (setprod ω R)

In Proofgold the corresponding term root is ce1615... and proposition id is e2c375...

Proof:
Proof not loaded.

Theorem. (real_sequences_total)

∀f : set, f ∈ real_sequences → total_function_on f ω R

In Proofgold the corresponding term root is 34f0ff... and proposition id is 50c676...

Proof:
Proof not loaded.

Theorem. (real_sequences_functional)

∀f : set, f ∈ real_sequences → functional_graph f

In Proofgold the corresponding term root is 9a2ad1... and proposition id is 8b9e40...

Proof:
Proof not loaded.

Theorem. (real_sequences_neq_exists_coord)

∀f g : set, f ∈ real_sequences → g ∈ real_sequences → f ≠ g → ∃n : set, n ∈ ω ∧ apply_fun f n ≠ apply_fun g n

In Proofgold the corresponding term root is ce4a78... and proposition id is 6b2437...

Proof:
Proof not loaded.

Theorem. (uniform_metric_Romega_binary_dist_1)

∀f g : set, f ∈ binary_sequences_Romega → g ∈ binary_sequences_Romega → f ≠ g → apply_fun uniform_metric_Romega (f,g) = 1

In Proofgold the corresponding term root is b5d0ee... and proposition id is ebfb66...

Proof:
Proof not loaded.

Theorem. (binary_sequences_Romega_discrete_in_uniform_topology)

discrete_subspace real_sequences uniform_topology binary_sequences_Romega

In Proofgold the corresponding term root is ebf1d3... and proposition id is d9c8a9...

Proof:
Proof not loaded.

Theorem. (Romega_uniform_first_not_second_countable)

first_countable_space real_sequences uniform_topology ∧ ¬ second_countable_space real_sequences uniform_topology

In Proofgold the corresponding term root is c91fed... and proposition id is 1be633...

Proof:
Proof not loaded.

Theorem. (countability_axioms_subspace_product)

∀X Tx : set, topology_on X Tx → (∀A : set, A ⊆ X → first_countable_space X Tx → first_countable_space A (subspace_topology X Tx A)) ∧ (∀A : set, A ⊆ X → second_countable_space X Tx → second_countable_space A (subspace_topology X Tx A)) ∧ (∀I Xi : set, countable_index_set I → (∀i : set, i ∈ I → first_countable_space (space_family_set Xi i) (space_family_topology Xi i)) → first_countable_space (countable_product_space I Xi) (countable_product_topology_subbasis I Xi)) ∧ (∀I Xi : set, countable_index_set I → (∀i : set, i ∈ I → second_countable_space (space_family_set Xi i) (space_family_topology Xi i)) → second_countable_space (countable_product_space I Xi) (countable_product_topology_subbasis I Xi))

In Proofgold the corresponding term root is da019d... and proposition id is b33328...

Proof:
Proof not loaded.

Definition. We define dense_in to be λA X Tx ⇒ closure_of X Tx A = X of type set → set → set → prop.

In Proofgold the corresponding term root is d0b42c... and object id is eb2621...

Theorem. (dense_in_superset)

∀A B X Tx : set, topology_on X Tx → A ⊆ B → B ⊆ X → dense_in A X Tx → dense_in B X Tx

In Proofgold the corresponding term root is d48728... and proposition id is 4fff39...

Proof:
Proof not loaded.

Theorem. (countable_basis_implies_Lindelof)

∀X Tx : set, topology_on X Tx → second_countable_space X Tx → ∀U : set, open_cover X Tx U → ∃V : set, countable_subcollection V U ∧ covers X V

In Proofgold the corresponding term root is 4468dd... and proposition id is c84b5d...

Proof:
Proof not loaded.

Theorem. (second_countable_implies_Lindelof_space)

∀X Tx : set, second_countable_space X Tx → Lindelof_space X Tx

In Proofgold the corresponding term root is 79552f... and proposition id is 767bec...

Proof:
Proof not loaded.

Theorem. (countable_basis_implies_separable)

∀X Tx : set, topology_on X Tx → second_countable_space X Tx → ∃D : set, countable_set D ∧ dense_in D X Tx

In Proofgold the corresponding term root is 6b921b... and proposition id is 8fa04c...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_line_first_countable)

first_countable_space Sorgenfrey_line Sorgenfrey_topology

In Proofgold the corresponding term root is 12b7ee... and proposition id is c93ecb...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_line_rationals_dense)

dense_in rational_numbers Sorgenfrey_line Sorgenfrey_topology

In Proofgold the corresponding term root is 88d0da... and proposition id is 157de8...

Proof:
Proof not loaded.

Theorem. (disjoint_open_family_countable_of_dense)

∀X Tx D Fam : set, topology_on X Tx → dense_in D X Tx → countable_set D → Fam ⊆ Tx → (∀U : set, U ∈ Fam → U ≠ Empty) → pairwise_disjoint Fam → countable_set Fam

In Proofgold the corresponding term root is bb8a5c... and proposition id is 3be5c4...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_line_not_second_countable)

¬ second_countable_space Sorgenfrey_line Sorgenfrey_topology

In Proofgold the corresponding term root is dc4a4e... and proposition id is 394287...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_line_Lindelof)

Lindelof_space Sorgenfrey_line Sorgenfrey_topology

In Proofgold the corresponding term root is f4b99d... and proposition id is faa96c...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_line_countability)

first_countable_space Sorgenfrey_line Sorgenfrey_topology ∧ dense_in rational_numbers Sorgenfrey_line Sorgenfrey_topology ∧ Lindelof_space Sorgenfrey_line Sorgenfrey_topology ∧ ¬ second_countable_space Sorgenfrey_line Sorgenfrey_topology

In Proofgold the corresponding term root is f3a7e9... and proposition id is 66cb9f...

Proof:
Proof not loaded.

Definition. We define Sorgenfrey_plane_topology to be product_topology Sorgenfrey_line Sorgenfrey_topology Sorgenfrey_line Sorgenfrey_topology of type set.

In Proofgold the corresponding term root is 52db81... and object id is a38ce0...

Definition. We define Sorgenfrey_plane_rational_points to be setprod rational_numbers rational_numbers of type set.

In Proofgold the corresponding term root is b9c5e4... and object id is 7f467a...

Theorem. (Sorgenfrey_plane_rational_points_countable)

countable_set Sorgenfrey_plane_rational_points

In Proofgold the corresponding term root is d27a86... and proposition id is 87df3a...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_rational_points_dense)

dense_in Sorgenfrey_plane_rational_points (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology

In Proofgold the corresponding term root is 8e7be2... and proposition id is d68f0a...

Proof:
Proof not loaded.

Definition. We define one_point_sets_closed to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → closed_in X Tx {x} of type set → set → prop.

In Proofgold the corresponding term root is 21cfa4... and object id is bd123f...

Theorem. (metric_topology_one_point_sets_closed)

∀X d : set, metric_on X d → one_point_sets_closed X (metric_topology X d)

In Proofgold the corresponding term root is 04f544... and proposition id is 5bc359...

Proof:
Proof not loaded.

Theorem. (Hausdorff_one_point_sets_closed)

∀X Tx : set, Hausdorff_space X Tx → one_point_sets_closed X Tx

In Proofgold the corresponding term root is 77b8b3... and proposition id is d3e283...

Proof:
Proof not loaded.

Definition. We define regular_space to be λX Tx ⇒ one_point_sets_closed X Tx ∧ ∀x : set, x ∈ X → ∀F : set, closed_in X Tx F → x ∉ F → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ x ∈ U ∧ F ⊆ V ∧ U ∩ V = Empty of type set → set → prop.

In Proofgold the corresponding term root is 66fb85... and object id is 342685...

Definition. We define normal_space to be λX Tx ⇒ one_point_sets_closed X Tx ∧ ∀A B : set, closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty of type set → set → prop.

In Proofgold the corresponding term root is 5c660f... and object id is 861332...

Theorem. (normal_space_topology_on)

∀X Tx : set, normal_space X Tx → topology_on X Tx

In Proofgold the corresponding term root is 4b796f... and proposition id is ae0f1d...

Proof:
Proof not loaded.

Theorem. (regular_space_topology_on)

∀X Tx : set, regular_space X Tx → topology_on X Tx

In Proofgold the corresponding term root is ffb997... and proposition id is 0a73f5...

Proof:
Proof not loaded.

Theorem. (regular_space_open_nbhd_closure_sub)

∀X Tx U x : set, regular_space X Tx → U ∈ Tx → x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ closure_of X Tx V ⊆ U

In Proofgold the corresponding term root is ccdf7c... and proposition id is a62254...

Proof:
Proof not loaded.

Theorem. (discrete_normal_space)

∀X : set, normal_space X (discrete_topology X)

In Proofgold the corresponding term root is 7e4195... and proposition id is ccc66f...

Proof:
Proof not loaded.

Theorem. (regular_space_implies_Hausdorff)

∀X Tx : set, regular_space X Tx → Hausdorff_space X Tx

In Proofgold the corresponding term root is d4c913... and proposition id is 7c5914...

Proof:
Proof not loaded.

Theorem. (normal_space_implies_regular)

∀X Tx : set, normal_space X Tx → regular_space X Tx

In Proofgold the corresponding term root is dfb1a4... and proposition id is b0dc88...

Proof:
Proof not loaded.

Definition. We define Hausdorff_spaces_family to be λI Xi ⇒ ∀i : set, i ∈ I → Hausdorff_space (product_component Xi i) (product_component_topology Xi i) of type set → set → prop.

In Proofgold the corresponding term root is d4f889... and object id is f23ae4...

Definition. We define regular_spaces_family to be λI Xi ⇒ ∀i : set, i ∈ I → regular_space (product_component Xi i) (product_component_topology Xi i) of type set → set → prop.

In Proofgold the corresponding term root is 52c86d... and object id is 20e0f3...

Definition. We define uncountable_set to be λX ⇒ ¬ countable_set X of type set → prop.

In Proofgold the corresponding term root is 2e1531... and object id is f60f73...

Theorem. (uncountable_set_ne_Empty)

∀X : set, uncountable_set X → X ≠ Empty

In Proofgold the corresponding term root is 32de79... and proposition id is 4aa5aa...

Proof:
Proof not loaded.

Definition. We define countable_subsets_bounded to be λX ⇒ ∀A : set, A ⊆ X → countable_set A → ∃b : set, b ∈ X ∧ ∀a : set, a ∈ A → a ∈ b of type set → prop.

In Proofgold the corresponding term root is 00b24e... and object id is e2e017...

Theorem. (mem_implies_order_rel_well_ordered)

∀X a b : set, well_ordered_set X → a ∈ b → order_rel X a b

In Proofgold the corresponding term root is 351435... and proposition id is 944f80...

Proof:
Proof not loaded.

Theorem. (order_rel_well_ordered_implies_mem)

∀X a b : set, well_ordered_set X → order_rel X a b → a ∈ b

In Proofgold the corresponding term root is b59494... and proposition id is db7a12...

Proof:
Proof not loaded.

Theorem. (open_ray_lower_well_ordered_1_eq_singleton0)

∀X : set, well_ordered_set X → 1 ∈ X → open_ray_lower X 1 = {0}

In Proofgold the corresponding term root is d53fab... and proposition id is a30c62...

Proof:
Proof not loaded.

Theorem. (open_ray_lower_in_order_topology)

∀X b : set, b ∈ X → open_ray_lower X b ∈ order_topology X

In Proofgold the corresponding term root is c41abd... and proposition id is b08daa...

Proof:
Proof not loaded.

Theorem. (open_ray_upper_in_order_topology)

∀X a : set, a ∈ X → open_ray_upper X a ∈ order_topology X

In Proofgold the corresponding term root is 90d4fc... and proposition id is 8075ba...

Proof:
Proof not loaded.

Theorem. (open_ray_upper_in_R_standard_topology)

∀a : set, a ∈ R → open_ray_upper R a ∈ R_standard_topology

In Proofgold the corresponding term root is 6daef3... and proposition id is 1fa3ab...

Proof:
Proof not loaded.

Theorem. (open_ray_lower_in_R_standard_topology)

∀b : set, b ∈ R → open_ray_lower R b ∈ R_standard_topology

In Proofgold the corresponding term root is e7cba3... and proposition id is 49b50f...

Proof:
Proof not loaded.

Theorem. (open_ray_rectangle_in_R2_standard_topology)

∀q : set, q ∈ R → rectangle_set (open_ray_lower R q) (open_ray_upper R q) ∈ R2_standard_topology

In Proofgold the corresponding term root is 9f6ccf... and proposition id is 136f67...

Proof:
Proof not loaded.

Theorem. (R2_strict_lt_open)

{p ∈ EuclidPlane|Rlt (p 0) (p 1)} ∈ R2_standard_topology

In Proofgold the corresponding term root is 5d245f... and proposition id is b47e89...

Proof:
Proof not loaded.

Theorem. (open_ray_rectangle_swapped_in_R2_standard_topology)

∀q : set, q ∈ R → rectangle_set (open_ray_upper R q) (open_ray_lower R q) ∈ R2_standard_topology

In Proofgold the corresponding term root is 44ec8c... and proposition id is 33d271...

Proof:
Proof not loaded.

Theorem. (R2_strict_gt_open)

{p ∈ EuclidPlane|Rlt (p 1) (p 0)} ∈ R2_standard_topology

In Proofgold the corresponding term root is 9fda27... and proposition id is 925785...

Proof:
Proof not loaded.

Theorem. (continuous_Rlt_preimage_open)

∀X Tx g h : set, continuous_map X Tx R R_standard_topology g → continuous_map X Tx R R_standard_topology h → {x ∈ X|Rlt (apply_fun g x) (apply_fun h x)} ∈ Tx

In Proofgold the corresponding term root is 3f54e3... and proposition id is 1c3429...

Proof:
Proof not loaded.

Theorem. (continuous_Rlt_preimage_open_swapped)

∀X Tx g h : set, continuous_map X Tx R R_standard_topology g → continuous_map X Tx R R_standard_topology h → {x ∈ X|Rlt (apply_fun h x) (apply_fun g x)} ∈ Tx

In Proofgold the corresponding term root is f0dfbb... and proposition id is 578d9d...

Proof:
Proof not loaded.

Theorem. (continuous_Rle_preimage_closed)

∀X Tx g h : set, continuous_map X Tx R R_standard_topology g → continuous_map X Tx R R_standard_topology h → closed_in X Tx {x ∈ X|Rle (apply_fun g x) (apply_fun h x)}

In Proofgold the corresponding term root is fef0dd... and proposition id is cc316b...

Proof:
Proof not loaded.

Theorem. (singleton0_open_in_order_topology_well_ordered)

∀X : set, well_ordered_set X → 1 ∈ X → {0} ∈ order_topology X

In Proofgold the corresponding term root is 13119b... and proposition id is cce1fe...

Proof:
Proof not loaded.

Theorem. (singleton0_open_in_order_topology_well_ordered_nonempty)

∀X : set, well_ordered_set X → simply_ordered_set X → 0 ∈ X → {0} ∈ order_topology X

In Proofgold the corresponding term root is 4da951... and proposition id is 8a4c8b...

Proof:
Proof not loaded.

Definition. We define separating_family_of_functions to be λX Tx F J ⇒ topology_on X Tx ∧ total_function_on F J (function_space X R) ∧ (∀j : set, j ∈ J → continuous_map X Tx R R_standard_topology (apply_fun F j)) ∧ (∀x0 : set, x0 ∈ X → ∀U : set, U ∈ Tx → x0 ∈ U → ∃j : set, j ∈ J ∧ Rlt 0 (apply_fun (apply_fun F j) x0) ∧ ∀x : set, x ∈ X ∖ U → apply_fun (apply_fun F j) x = 0) of type set → set → set → set → prop.

In Proofgold the corresponding term root is 9d4123... and object id is 0cc95d...

Theorem. (separating_family_of_functions_separates_points)

∀X Tx F J x y : set, topology_on X Tx → one_point_sets_closed X Tx → separating_family_of_functions X Tx F J → x ∈ X → y ∈ X → x ≠ y → ∃j : set, j ∈ J ∧ apply_fun (apply_fun F j) x ≠ apply_fun (apply_fun F j) y

In Proofgold the corresponding term root is 5f4fb2... and proposition id is 6d970f...

Proof:
Proof not loaded.

Definition. We define diagonal_map to be λX F J ⇒ graph X (λx : set ⇒ graph J (λj : set ⇒ apply_fun (apply_fun F j) x)) of type set → set → set → set.

In Proofgold the corresponding term root is 0f99b3... and object id is fc2e4c...

Theorem. (diagonal_map_coord_apply)

∀X F J x i : set, x ∈ X → i ∈ J → apply_fun (apply_fun (diagonal_map X F J) x) i = apply_fun (apply_fun F i) x

In Proofgold the corresponding term root is 639879... and proposition id is 0a86f8...

Proof:
Proof not loaded.

Definition. We define embedding_of to be λX Tx Y Ty f ⇒ continuous_map X Tx Y Ty f ∧ homeomorphism X Tx (image_of f X) (subspace_topology Y Ty (image_of f X)) f of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is fdfd9f... and object id is 6b1b9f...

Theorem. (embedding_of_continuous)

∀X Tx Y Ty f : set, embedding_of X Tx Y Ty f → continuous_map X Tx Y Ty f

In Proofgold the corresponding term root is 78d6bc... and proposition id is a8b65a...

Proof:
Proof not loaded.

Theorem. (embedding_of_homeomorphism)

∀X Tx Y Ty f : set, embedding_of X Tx Y Ty f → homeomorphism X Tx (image_of f X) (subspace_topology Y Ty (image_of f X)) f

In Proofgold the corresponding term root is fb291e... and proposition id is 546ac6...

Proof:
Proof not loaded.

Theorem. (embedding_of_from_local_refinement)

∀X Tx Y Ty f : set, continuous_map X Tx Y Ty f → (∀x U : set, x ∈ X → U ∈ Tx → x ∈ U → ∃V : set, V ∈ Ty ∧ apply_fun f x ∈ V ∧ preimage_of X f V ⊆ U) → (∀x y : set, x ∈ X → y ∈ X → apply_fun f x = apply_fun f y → x = y) → embedding_of X Tx Y Ty f

In Proofgold the corresponding term root is 163a03... and proposition id is 875fc5...

Proof:
Proof not loaded.

Definition. We define power_real to be λJ ⇒ product_space J (const_space_family J R R_standard_topology) of type set → set.

In Proofgold the corresponding term root is 215510... and object id is 69af4b...

Definition. We define unit_interval_power to be λJ ⇒ product_space J (const_space_family J unit_interval unit_interval_topology) of type set → set.

In Proofgold the corresponding term root is caf31a... and object id is 6b242b...

Definition. We define power_real_coord_abs_diff to be λf g j ⇒ abs_SNo (add_SNo (apply_fun f j) (minus_SNo (apply_fun g j))) of type set → set → set → set.

In Proofgold the corresponding term root is c1a5ae... and object id is 7900b4...

Definition. We define power_real_coord_clipped_diff to be λf g j ⇒ If_i (Rlt (power_real_coord_abs_diff f g j) 1) (power_real_coord_abs_diff f g j) 1 of type set → set → set → set.

In Proofgold the corresponding term root is 3ea59f... and object id is dd5dac...

Definition. We define power_real_clipped_diffs to be λJ f g ⇒ Repl J (λj : set ⇒ power_real_coord_clipped_diff f g j) of type set → set → set → set.

In Proofgold the corresponding term root is 404f3a... and object id is fa7f95...

Definition. We define power_real_uniform_metric_value to be λJ f g ⇒ If_i (J = Empty) 0 (Eps_i (λr : set ⇒ R_lub (power_real_clipped_diffs J f g) r)) of type set → set → set → set.

In Proofgold the corresponding term root is 02a3f4... and object id is 894c49...

Definition. We define uniform_metric_power_real to be λJ ⇒ graph (setprod (power_real J) (power_real J)) (λp : set ⇒ power_real_uniform_metric_value J (p 0) (p 1)) of type set → set.

In Proofgold the corresponding term root is ad1bbc... and object id is dcf460...

Definition. We define uniform_topology_power_real to be λJ ⇒ metric_topology (power_real J) (uniform_metric_power_real J) of type set → set.

In Proofgold the corresponding term root is c70f92... and object id is c85e4d...

Theorem. (power_real_coord_clipped_diff_eq_R_bounded_distance)

∀J f g j : set, j ∈ J → f ∈ power_real J → g ∈ power_real J → power_real_coord_clipped_diff f g j = R_bounded_distance (apply_fun f j) (apply_fun g j)

In Proofgold the corresponding term root is d0b625... and proposition id is 7185e7...

Proof:
Proof not loaded.

Theorem. (power_real_coord_in_R)

∀J f j : set, j ∈ J → f ∈ power_real J → apply_fun f j ∈ R

In Proofgold the corresponding term root is 638042... and proposition id is 3b1716...

Proof:
Proof not loaded.

Theorem. (power_real_ext)

∀J f g : set, f ∈ power_real J → g ∈ power_real J → (∀j : set, j ∈ J → apply_fun f j = apply_fun g j) → f = g

In Proofgold the corresponding term root is a60472... and proposition id is bc4d77...

Proof:
Proof not loaded.

Theorem. (power_real_clipped_diffs_in_R)

∀J f g a : set, f ∈ power_real J → g ∈ power_real J → a ∈ power_real_clipped_diffs J f g → a ∈ R

In Proofgold the corresponding term root is b0ea1c... and proposition id is 2f5d4c...

Proof:
Proof not loaded.

Theorem. (power_real_coord_clipped_diff_le_1)

∀J f g j : set, j ∈ J → f ∈ power_real J → g ∈ power_real J → Rle (power_real_coord_clipped_diff f g j) 1

In Proofgold the corresponding term root is 9c3119... and proposition id is da5644...

Proof:
Proof not loaded.

Theorem. (power_real_clipped_diffs_nonempty)

∀J f g : set, J ≠ Empty → f ∈ power_real J → g ∈ power_real J → ∃a0 : set, a0 ∈ power_real_clipped_diffs J f g

In Proofgold the corresponding term root is 0472aa... and proposition id is bc27f8...

Proof:
Proof not loaded.

Theorem. (power_real_uniform_metric_value_is_lub)

∀J f g : set, J ≠ Empty → f ∈ power_real J → g ∈ power_real J → R_lub (power_real_clipped_diffs J f g) (power_real_uniform_metric_value J f g)

In Proofgold the corresponding term root is ad1288... and proposition id is 691408...

Proof:
Proof not loaded.

Theorem. (power_real_uniform_metric_value_in_R)

∀J f g : set, f ∈ power_real J → g ∈ power_real J → power_real_uniform_metric_value J f g ∈ R

In Proofgold the corresponding term root is f5b20d... and proposition id is 6c1443...

Proof:
Proof not loaded.

Theorem. (uniform_metric_power_real_apply)

∀J x y : set, x ∈ power_real J → y ∈ power_real J → apply_fun (uniform_metric_power_real J) (x,y) = power_real_uniform_metric_value J x y

In Proofgold the corresponding term root is 824f16... and proposition id is 3bdb70...

Proof:
Proof not loaded.

Theorem. (power_real_clipped_diffs_sym)

∀J f g : set, f ∈ power_real J → g ∈ power_real J → power_real_clipped_diffs J f g = power_real_clipped_diffs J g f

In Proofgold the corresponding term root is b02fdb... and proposition id is c808c7...

Proof:
Proof not loaded.

Theorem. (power_real_clipped_diffs_diag_eq_Sing0)

∀J f : set, J ≠ Empty → f ∈ power_real J → power_real_clipped_diffs J f f = {0}

In Proofgold the corresponding term root is 8e687f... and proposition id is 55cef5...

Proof:
Proof not loaded.

Theorem. (power_real_uniform_metric_value_sym)

∀J f g : set, f ∈ power_real J → g ∈ power_real J → power_real_uniform_metric_value J f g = power_real_uniform_metric_value J g f

In Proofgold the corresponding term root is 544fd6... and proposition id is c794ea...

Proof:
Proof not loaded.

Theorem. (power_real_uniform_metric_value_self_zero)

∀J f : set, f ∈ power_real J → power_real_uniform_metric_value J f f = 0

In Proofgold the corresponding term root is c6911d... and proposition id is 51bd2e...

Proof:
Proof not loaded.

Theorem. (uniform_metric_power_real_function_on)

∀J : set, function_on (uniform_metric_power_real J) (setprod (power_real J) (power_real J)) R

In Proofgold the corresponding term root is 69564b... and proposition id is 0635a9...

Proof:
Proof not loaded.

Theorem. (power_real_uniform_metric_value_nonneg)

∀J f g : set, f ∈ power_real J → g ∈ power_real J → ¬ (Rlt (power_real_uniform_metric_value J f g) 0)

In Proofgold the corresponding term root is 8a5587... and proposition id is b50eec...

Proof:
Proof not loaded.

Theorem. (power_real_uniform_metric_value_eq0_coord_eq)

∀J x y j : set, J ≠ Empty → x ∈ power_real J → y ∈ power_real J → j ∈ J → power_real_uniform_metric_value J x y = 0 → apply_fun x j = apply_fun y j

In Proofgold the corresponding term root is d59d78... and proposition id is 641ba6...

Proof:
Proof not loaded.

Theorem. (power_real_uniform_metric_value_triangle)

∀J x y z : set, x ∈ power_real J → y ∈ power_real J → z ∈ power_real J → ¬ (Rlt (add_SNo (power_real_uniform_metric_value J x y) (power_real_uniform_metric_value J y z)) (power_real_uniform_metric_value J x z))

In Proofgold the corresponding term root is bec6fc... and proposition id is 441b77...

Proof:
Proof not loaded.

Theorem. (uniform_metric_power_real_is_metric_on)

∀J : set, metric_on (power_real J) (uniform_metric_power_real J)

In Proofgold the corresponding term root is 3202e3... and proposition id is 1bb7a2...

Proof:
Proof not loaded.

Definition. We define one_minus_fun to be graph R (λt : set ⇒ add_SNo 1 (minus_SNo t)) of type set.

In Proofgold the corresponding term root is 6a9f1f... and object id is 5151ec...

Theorem. (one_minus_fun_value_in_R)

∀t : set, t ∈ R → apply_fun one_minus_fun t ∈ R

In Proofgold the corresponding term root is 85a9c8... and proposition id is 413144...

Proof:
Proof not loaded.

Theorem. (one_minus_fun_0)

apply_fun one_minus_fun 0 = 1

In Proofgold the corresponding term root is d39437... and proposition id is bebbd8...

Proof:
Proof not loaded.

Theorem. (one_minus_fun_1)

apply_fun one_minus_fun 1 = 0

In Proofgold the corresponding term root is 291298... and proposition id is 5728d3...

Proof:
Proof not loaded.

Theorem. (one_minus_fun_continuous)

continuous_map R R_standard_topology R R_standard_topology one_minus_fun

In Proofgold the corresponding term root is a07289... and proposition id is d0918d...

Proof:
Proof not loaded.

Definition. We define neg_fun to be graph R (λt : set ⇒ minus_SNo t) of type set.

In Proofgold the corresponding term root is 3cec3a... and object id is 87c90a...

Theorem. (neg_fun_value_in_R)

∀t : set, t ∈ R → apply_fun neg_fun t ∈ R

In Proofgold the corresponding term root is 4fd54d... and proposition id is 1d4b0b...

Proof:
Proof not loaded.

Theorem. (neg_fun_in_function_space)

neg_fun ∈ function_space R R

In Proofgold the corresponding term root is 43c9f2... and proposition id is 50abb9...

Proof:
Proof not loaded.

Theorem. (neg_fun_apply)

∀t : set, t ∈ R → apply_fun neg_fun t = minus_SNo t

In Proofgold the corresponding term root is e39366... and proposition id is 6d8443...

Proof:
Proof not loaded.

Theorem. (minus_SNo_minus_SNo_R)

∀t : set, t ∈ R → minus_SNo (minus_SNo t) = t

In Proofgold the corresponding term root is 4ee422... and proposition id is 138661...

Proof:
Proof not loaded.

Theorem. (neg_fun_flip_upper)

∀a t : set, a ∈ R → t ∈ R → (Rlt a (minus_SNo t) ↔ Rlt t (minus_SNo a))

In Proofgold the corresponding term root is 4a9941... and proposition id is 54c7fd...

Proof:
Proof not loaded.

Theorem. (neg_fun_flip_lower)

∀t b : set, t ∈ R → b ∈ R → (Rlt (minus_SNo t) b ↔ Rlt (minus_SNo b) t)

In Proofgold the corresponding term root is 7ac59b... and proposition id is 43bfb3...

Proof:
Proof not loaded.

Theorem. (preimage_neg_fun_open_ray_upper)

∀a : set, a ∈ R → preimage_of R neg_fun (open_ray_upper R a) = open_ray_lower R (minus_SNo a)

In Proofgold the corresponding term root is 06eccd... and proposition id is 491a4a...

Proof:
Proof not loaded.

Theorem. (preimage_neg_fun_open_ray_lower)

∀b : set, b ∈ R → preimage_of R neg_fun (open_ray_lower R b) = open_ray_upper R (minus_SNo b)

In Proofgold the corresponding term root is 287715... and proposition id is 7d59e7...

Proof:
Proof not loaded.

Theorem. (neg_fun_continuous)

continuous_map R R_standard_topology R R_standard_topology neg_fun

In Proofgold the corresponding term root is 55eec7... and proposition id is 3a8e0b...

Proof:
Proof not loaded.

Definition. We define add_fun_R to be graph (setprod R R) (λp : set ⇒ add_SNo (p 0) (p 1)) of type set.

In Proofgold the corresponding term root is c4f491... and object id is 75013c...

Definition. We define mul_fun_R to be graph (setprod R R) (λp : set ⇒ mul_SNo (p 0) (p 1)) of type set.

In Proofgold the corresponding term root is df8564... and object id is d941ad...

Definition. We define mul_const_fun to be λc ⇒ graph R (λt : set ⇒ mul_SNo t c) of type set → set.

In Proofgold the corresponding term root is baafb2... and object id is 3b20b9...

Theorem. (add_fun_R_value_in_R)

∀p : set, p ∈ setprod R R → apply_fun add_fun_R p ∈ R

In Proofgold the corresponding term root is a32620... and proposition id is 25fa51...

Proof:
Proof not loaded.

Theorem. (add_fun_R_in_function_space)

add_fun_R ∈ function_space (setprod R R) R

In Proofgold the corresponding term root is b43880... and proposition id is 6c5e73...

Proof:
Proof not loaded.

Theorem. (mul_fun_R_in_function_space)

mul_fun_R ∈ function_space (setprod R R) R

In Proofgold the corresponding term root is f53a65... and proposition id is 896b3e...

Proof:
Proof not loaded.

Theorem. (add_fun_R_apply)

∀p : set, p ∈ setprod R R → apply_fun add_fun_R p = add_SNo (p 0) (p 1)

In Proofgold the corresponding term root is 734bc0... and proposition id is ad6272...

Proof:
Proof not loaded.

Theorem. (preimage_add_fun_R_open_ray_upper_in_product_topology)

∀a : set, a ∈ R → preimage_of (setprod R R) add_fun_R (open_ray_upper R a) ∈ (product_topology R R_standard_topology R R_standard_topology)

In Proofgold the corresponding term root is 0137ab... and proposition id is c8346b...

Proof:
Proof not loaded.

Theorem. (preimage_add_fun_R_open_ray_lower_in_product_topology)

∀b : set, b ∈ R → preimage_of (setprod R R) add_fun_R (open_ray_lower R b) ∈ (product_topology R R_standard_topology R R_standard_topology)

In Proofgold the corresponding term root is 652ed1... and proposition id is 75bf59...

Proof:
Proof not loaded.

Theorem. (add_fun_R_continuous)

continuous_map (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology add_fun_R

In Proofgold the corresponding term root is ba59cb... and proposition id is 4d3ddd...

Proof:
Proof not loaded.

Definition. We define normalize01_fun to be graph R (λt : set ⇒ div_SNo (mul_SNo t t) (add_SNo (mul_SNo t t) (mul_SNo (add_SNo t (minus_SNo 1)) (add_SNo t (minus_SNo 1))))) of type set.

In Proofgold the corresponding term root is 6941d4... and object id is a604fc...

Theorem. (normalize01_fun_value_in_R)

∀t : set, t ∈ R → apply_fun normalize01_fun t ∈ R

In Proofgold the corresponding term root is 711ae2... and proposition id is 09ea04...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_range_in_unit_interval)

∀t : set, t ∈ R → apply_fun normalize01_fun t ∈ unit_interval

In Proofgold the corresponding term root is bd7b97... and proposition id is 5ab4a1...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_img_in_open_ray_upper_iff)

∀a t : set, a ∈ R → t ∈ R → (apply_fun normalize01_fun t ∈ open_ray_upper R a ↔ Rlt a (apply_fun normalize01_fun t))

In Proofgold the corresponding term root is 0c0f47... and proposition id is 333d5f...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_img_in_open_ray_lower_iff)

∀b t : set, b ∈ R → t ∈ R → (apply_fun normalize01_fun t ∈ open_ray_lower R b ↔ Rlt (apply_fun normalize01_fun t) b)

In Proofgold the corresponding term root is b5b722... and proposition id is e87ad2...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_den_pos)

∀t : set, t ∈ R → 0 < add_SNo (mul_SNo t t) (mul_SNo (add_SNo t (minus_SNo 1)) (add_SNo t (minus_SNo 1)))

In Proofgold the corresponding term root is 3c1bab... and proposition id is 7307c0...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_pos_of_neq0)

∀t : set, t ∈ R → t ≠ 0 → Rlt 0 (apply_fun normalize01_fun t)

In Proofgold the corresponding term root is 03400f... and proposition id is 289e50...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_lt1_of_neq1)

∀t : set, t ∈ R → t ≠ 1 → Rlt (apply_fun normalize01_fun t) 1

In Proofgold the corresponding term root is 931d92... and proposition id is cf4690...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_0)

apply_fun normalize01_fun 0 = 0

In Proofgold the corresponding term root is 7c3133... and proposition id is 9f89b2...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_1)

apply_fun normalize01_fun 1 = 1

In Proofgold the corresponding term root is 5497d2... and proposition id is 6511d7...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_gt_iff_mul_den_lt_num)

∀a t : set, a ∈ R → t ∈ R → 0 < a → a < 1 → (a < apply_fun normalize01_fun t ↔ mul_SNo a (add_SNo (mul_SNo t t) (mul_SNo (add_SNo t (minus_SNo 1)) (add_SNo t (minus_SNo 1)))) < (mul_SNo t t))

In Proofgold the corresponding term root is fd34d1... and proposition id is 47766e...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_lt_iff_num_lt_mul_den)

∀b t : set, b ∈ R → t ∈ R → 0 < b → (apply_fun normalize01_fun t < b ↔ (mul_SNo t t) < mul_SNo b (add_SNo (mul_SNo t t) (mul_SNo (add_SNo t (minus_SNo 1)) (add_SNo t (minus_SNo 1)))))

In Proofgold the corresponding term root is 0a876d... and proposition id is b7620f...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_abs_early)

∀x r y : set, x ∈ R → r ∈ R → Rlt 0 r → Rlt r 1 → (y ∈ open_ball R R_bounded_metric x r ↔ (y ∈ R ∧ Rlt (R_bounded_distance x y) r))

In Proofgold the corresponding term root is 443e09... and proposition id is 58c3fb...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_absdiff_lt)

∀x r y : set, x ∈ R → r ∈ R → Rlt 0 r → Rlt r 1 → y ∈ open_ball R R_bounded_metric x r → abs_SNo (add_SNo x (minus_SNo y)) < r

In Proofgold the corresponding term root is 9b535d... and proposition id is 6b72ea...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_eq_R_if_1_lt_early)

∀c r : set, c ∈ R → r ∈ R → Rlt 1 r → open_ball R R_bounded_metric c r = R

In Proofgold the corresponding term root is 5c9a2e... and proposition id is be31ff...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_eq_open_interval_early)

∀c r : set, c ∈ R → r ∈ R → Rlt 0 r → Rlt r 1 → open_ball R R_bounded_metric c r = open_interval (add_SNo c (minus_SNo r)) (add_SNo c r)

In Proofgold the corresponding term root is 309694... and proposition id is 237a0b...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_r1_eq_open_interval_early)

∀c : set, c ∈ R → open_ball R R_bounded_metric c 1 = open_interval (add_SNo c (minus_SNo 1)) (add_SNo c 1)

In Proofgold the corresponding term root is f7ed43... and proposition id is 7f0efc...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_in_R_standard_topology_early)

∀c r : set, c ∈ R → r ∈ R → Rlt 0 r → open_ball R R_bounded_metric c r ∈ R_standard_topology

In Proofgold the corresponding term root is e3e485... and proposition id is 03dbbc...

Proof:
Proof not loaded.

Theorem. (R_bounded_metric_is_metric_on_early)

metric_on R R_bounded_metric

In Proofgold the corresponding term root is 6a523d... and proposition id is 469f1d...

Proof:
Proof not loaded.

Theorem. (mul_const_fun_apply)

∀c t : set, c ∈ R → t ∈ R → apply_fun (mul_const_fun c) t = mul_SNo t c

In Proofgold the corresponding term root is 3dfa84... and proposition id is c31924...

Proof:
Proof not loaded.

Theorem. (mul_const_fun_in_function_space)

∀c : set, c ∈ R → mul_const_fun c ∈ function_space R R

In Proofgold the corresponding term root is 426be0... and proposition id is 066cd7...

Proof:
Proof not loaded.

Theorem. (mul_const_fun_value_in_R)

∀c t : set, c ∈ R → t ∈ R → apply_fun (mul_const_fun c) t ∈ R

In Proofgold the corresponding term root is a9d3d7... and proposition id is 3f8c61...

Proof:
Proof not loaded.

Theorem. (preimage_mul_const_open_ray_upper)

∀c a : set, c ∈ R → 0 < c → a ∈ R → preimage_of R (mul_const_fun c) (open_ray_upper R a) = open_ray_upper R (div_SNo a c)

In Proofgold the corresponding term root is 69cd9d... and proposition id is 892b97...

Proof:
Proof not loaded.

Theorem. (preimage_mul_const_open_ray_lower)

∀c b : set, c ∈ R → 0 < c → b ∈ R → preimage_of R (mul_const_fun c) (open_ray_lower R b) = open_ray_lower R (div_SNo b c)

In Proofgold the corresponding term root is a1672a... and proposition id is 6e8037...

Proof:
Proof not loaded.

Theorem. (mul_const_fun_continuous_pos)

∀c : set, c ∈ R → 0 < c → continuous_map R R_standard_topology R R_standard_topology (mul_const_fun c)

In Proofgold the corresponding term root is 97e1b4... and proposition id is de32e4...

Proof:
Proof not loaded.

Theorem. (mul_const_fun_continuous)

∀c : set, c ∈ R → continuous_map R R_standard_topology R R_standard_topology (mul_const_fun c)

In Proofgold the corresponding term root is 8f2faa... and proposition id is 430809...

Proof:
Proof not loaded.

Theorem. (add_two_continuous_R)

∀X Tx g h : set, topology_on X Tx → continuous_map X Tx R R_standard_topology g → continuous_map X Tx R R_standard_topology h → continuous_map X Tx R R_standard_topology (compose_fun X (pair_map X g h) add_fun_R)

In Proofgold the corresponding term root is 285758... and proposition id is f59b58...

Proof:
Proof not loaded.

Theorem. (mul_fun_R_value_in_R)

∀p : set, p ∈ setprod R R → apply_fun mul_fun_R p ∈ R

In Proofgold the corresponding term root is 06d56a... and proposition id is 127776...

Proof:
Proof not loaded.

Theorem. (mul_fun_R_apply)

∀p : set, p ∈ setprod R R → apply_fun mul_fun_R p = mul_SNo (p 0) (p 1)

In Proofgold the corresponding term root is c432b7... and proposition id is 287fb7...

Proof:
Proof not loaded.

Theorem. (abs_SNo_mul_nonneg_left)

∀x y : set, SNo x → SNo y → 0 ≤ x → abs_SNo (mul_SNo x y) = mul_SNo x (abs_SNo y)

In Proofgold the corresponding term root is d54ec9... and proposition id is ef8f0a...

Proof:
Proof not loaded.

Theorem. (abs_SNo_mul_eq)

∀x y : set, SNo x → SNo y → abs_SNo (mul_SNo x y) = mul_SNo (abs_SNo x) (abs_SNo y)

In Proofgold the corresponding term root is eef86f... and proposition id is 7bcb73...

Proof:
Proof not loaded.

Theorem. (preimage_mul_fun_R_open_ray_upper_in_product_topology)

∀a : set, a ∈ R → preimage_of (setprod R R) mul_fun_R (open_ray_upper R a) ∈ (product_topology R R_standard_topology R R_standard_topology)

In Proofgold the corresponding term root is 3f2508... and proposition id is eaf896...

Proof:
Proof not loaded.

Theorem. (preimage_mul_fun_R_open_ray_lower_in_product_topology)

∀b : set, b ∈ R → preimage_of (setprod R R) mul_fun_R (open_ray_lower R b) ∈ (product_topology R R_standard_topology R R_standard_topology)

In Proofgold the corresponding term root is 6f11d7... and proposition id is 225137...

Proof:
Proof not loaded.

Theorem. (mul_fun_R_continuous)

continuous_map (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology mul_fun_R

In Proofgold the corresponding term root is fbcbc3... and proposition id is a0b580...

Proof:
Proof not loaded.

Theorem. (mul_two_continuous_R)

In Proofgold the corresponding term root is e90f54... and proposition id is 7cd797...

Proof:
Proof not loaded.

Theorem. (add_of_pair_map_apply)

∀A f g a : set, a ∈ A → apply_fun f a ∈ R → apply_fun g a ∈ R → apply_fun (compose_fun A (pair_map A f g) add_fun_R) a = add_SNo (apply_fun f a) (apply_fun g a)

In Proofgold the corresponding term root is ddf3c1... and proposition id is c781e4...

Proof:
Proof not loaded.

Theorem. (mul_of_pair_map_apply)

∀A f g a : set, a ∈ A → apply_fun f a ∈ R → apply_fun g a ∈ R → apply_fun (compose_fun A (pair_map A f g) mul_fun_R) a = mul_SNo (apply_fun f a) (apply_fun g a)

In Proofgold the corresponding term root is dcf03e... and proposition id is c242ae...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_preimage_open_ray_upper_mid)

∀a : set, a ∈ R → ¬ (Rlt a 0) → Rlt a 1 → preimage_of R normalize01_fun (open_ray_upper R a) ∈ R_standard_topology

In Proofgold the corresponding term root is 3af16b... and proposition id is 1399f7...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_preimage_open_ray_upper)

∀a : set, a ∈ R → preimage_of R normalize01_fun (open_ray_upper R a) ∈ R_standard_topology

In Proofgold the corresponding term root is 6c20e9... and proposition id is 1e24dd...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_preimage_open_ray_lower_mid)

∀b : set, b ∈ R → ¬ (Rlt 1 b) → Rlt 0 b → preimage_of R normalize01_fun (open_ray_lower R b) ∈ R_standard_topology

In Proofgold the corresponding term root is 7ac436... and proposition id is c10296...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_preimage_open_ray_lower)

∀b : set, b ∈ R → preimage_of R normalize01_fun (open_ray_lower R b) ∈ R_standard_topology

In Proofgold the corresponding term root is 6e453f... and proposition id is 3496cf...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_continuous)

continuous_map R R_standard_topology R R_standard_topology normalize01_fun

In Proofgold the corresponding term root is 26b120... and proposition id is dac8ce...

Proof:
Proof not loaded.

Theorem. (normalize01_fun_continuous_to_unit_interval)

continuous_map R R_standard_topology unit_interval unit_interval_topology normalize01_fun

In Proofgold the corresponding term root is 6879b2... and proposition id is f69ec7...

Proof:
Proof not loaded.

Theorem. (graph_to_R_in_power_real)

∀J : set, ∀g : set → set, (∀j : set, j ∈ J → g j ∈ R) → graph J g ∈ power_real J

In Proofgold the corresponding term root is 5b74b5... and proposition id is 1d3932...

Proof:
Proof not loaded.

Theorem. (diagonal_map_function_on_power_real)

∀X F J : set, total_function_on F J (function_space X R) → function_on (diagonal_map X F J) X (power_real J)

In Proofgold the corresponding term root is 768a0b... and proposition id is c49dbb...

Proof:
Proof not loaded.

Theorem. (graph_to_unit_interval_in_unit_interval_power)

∀J : set, ∀g : set → set, (∀j : set, j ∈ J → g j ∈ unit_interval) → graph J g ∈ unit_interval_power J

In Proofgold the corresponding term root is 8a5d28... and proposition id is 0f4f7f...

Proof:
Proof not loaded.

Theorem. (diagonal_map_function_on_unit_interval_power)

∀X F J : set, total_function_on F J (function_space X unit_interval) → function_on (diagonal_map X F J) X (unit_interval_power J)

In Proofgold the corresponding term root is fa7bbe... and proposition id is ad8184...

Proof:
Proof not loaded.

Theorem. (diagonal_map_continuous_unit_interval_power)

∀X Tx F J : set, topology_on X Tx → J ≠ Empty → total_function_on F J (function_space X unit_interval) → (∀j : set, j ∈ J → continuous_map X Tx unit_interval unit_interval_topology (apply_fun F j)) → continuous_map X Tx (unit_interval_power J) (product_topology_full J (const_space_family J unit_interval unit_interval_topology)) (diagonal_map X F J)

In Proofgold the corresponding term root is 0057a9... and proposition id is 9981a3...

Proof:
Proof not loaded.

Theorem. (power_real_omega_eq_Romega_space)

power_real ω = R_omega_space

In Proofgold the corresponding term root is c6f698... and proposition id is d2461d...

Proof:
Proof not loaded.

Theorem. (unit_interval_power_omega_def)

unit_interval_power ω = product_space ω (const_space_family ω unit_interval unit_interval_topology)

In Proofgold the corresponding term root is eff2ad... and proposition id is 6a6ec6...

Proof:
Proof not loaded.

Definition. We define metrizable to be λX Tx ⇒ ∃d : set, metric_on X d ∧ metric_topology X d = Tx of type set → set → prop.

In Proofgold the corresponding term root is 07f964... and object id is 81d439...

Theorem. (Romega_product_topology_metrizable)

metrizable R_omega_space R_omega_product_topology

In Proofgold the corresponding term root is 158fff... and proposition id is 01e3b7...

Proof:
Proof not loaded.

Definition. We define Sorgenfrey_plane_L to be {(x,minus_SNo x)|x ∈ Sorgenfrey_line} of type set.

In Proofgold the corresponding term root is 024c3f... and object id is 7f162d...

Definition. We define Sorgenfrey_plane_special_rectangle to be λa b d ⇒ rectangle_set (halfopen_interval_left a b) (halfopen_interval_left (minus_SNo a) d) of type set → set → set → set.

In Proofgold the corresponding term root is a83317... and object id is 0233df...

Theorem. (Sorgenfrey_plane_special_rectangle_open)

∀a b d : set, a ∈ R → b ∈ R → d ∈ R → Sorgenfrey_plane_special_rectangle a b d ∈ Sorgenfrey_plane_topology

In Proofgold the corresponding term root is f5646c... and proposition id is 5695fb...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_L_uncountable)

¬ countable_set Sorgenfrey_plane_L

In Proofgold the corresponding term root is 50f680... and proposition id is 3309fe...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_special_rectangle_L_point)

∀a b d x : set, a ∈ R → b ∈ R → d ∈ R → x ∈ R → (x,minus_SNo x) ∈ Sorgenfrey_plane_special_rectangle a b d → x = a

In Proofgold the corresponding term root is 4bb5df... and proposition id is c9a067...

Proof:
Proof not loaded.

Theorem. (tuple_2_ext)

∀a b c d : set, a = c → b = d → (a,b) = (c,d)

In Proofgold the corresponding term root is a77749... and proposition id is cbdc18...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_L_closed)

closed_in (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology Sorgenfrey_plane_L

In Proofgold the corresponding term root is d04e06... and proposition id is 876186...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_not_Lindelof)

¬ Lindelof_space (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology

In Proofgold the corresponding term root is 6c4cac... and proposition id is df0ee9...

Proof:
Proof not loaded.

Axiom. (ordered_square_Lindelof) We take the following as an axiom:

Lindelof_space ordered_square ordered_square_topology

In Proofgold the corresponding term root is 740c6d... and proposition id is c2fa12...

Theorem. (ordered_square_Lindelof_sketch)

Lindelof_space ordered_square ordered_square_topology

In Proofgold the corresponding term root is 740c6d... and proposition id is c2fa12...

Proof:
Proof not loaded.

Axiom. (ordered_square_open_strip_not_Lindelof) We take the following as an axiom:

¬ Lindelof_space ordered_square_open_strip ordered_square_subspace_topology

In Proofgold the corresponding term root is 1cbb79... and proposition id is c2bd43...

Theorem. (ordered_square_open_strip_not_Lindelof_sketch)

¬ Lindelof_space ordered_square_open_strip ordered_square_subspace_topology

In Proofgold the corresponding term root is 1cbb79... and proposition id is c2bd43...

Proof:
Proof not loaded.

Theorem. (ordered_square_subspace_not_Lindelof)

Lindelof_space ordered_square ordered_square_topology ∧ ¬ Lindelof_space ordered_square_open_strip ordered_square_subspace_topology

In Proofgold the corresponding term root is 747124... and proposition id is 47359e...

Proof:
Proof not loaded.

Theorem. (regular_normal_via_closure)

∀X Tx : set, topology_on X Tx → (one_point_sets_closed X Tx → (regular_space X Tx ↔ ∀x U : set, x ∈ X → U ∈ Tx → x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ closure_of X Tx V ⊆ U)) ∧ (one_point_sets_closed X Tx → (normal_space X Tx ↔ ∀A U : set, closed_in X Tx A → U ∈ Tx → A ⊆ U → ∃V : set, V ∈ Tx ∧ A ⊆ V ∧ closure_of X Tx V ⊆ U))

In Proofgold the corresponding term root is a0e2db... and proposition id is 944093...

Proof:
Proof not loaded.

Theorem. (regular_space_shrink_neighborhood)

∀X Tx x U : set, regular_space X Tx → x ∈ X → U ∈ Tx → x ∈ U → ∃V : set, V ∈ Tx ∧ x ∈ V ∧ closure_of X Tx V ⊆ U

In Proofgold the corresponding term root is e43f2e... and proposition id is 982199...

Proof:
Proof not loaded.

Theorem. (normal_space_shrink_closed)

∀X Tx A U : set, normal_space X Tx → closed_in X Tx A → U ∈ Tx → A ⊆ U → ∃V : set, V ∈ Tx ∧ A ⊆ V ∧ closure_of X Tx V ⊆ U

In Proofgold the corresponding term root is 322e1b... and proposition id is 875023...

Proof:
Proof not loaded.

Theorem. (normal_space_shrink_closed_cover2)

∀X Tx A U V : set, normal_space X Tx → closed_in X Tx A → U ∈ Tx → V ∈ Tx → A ⊆ U ∪ V → ∃U0 : set, U0 ∈ Tx ∧ (A ∖ V) ⊆ U0 ∧ closure_of X Tx U0 ⊆ U ∧ (A ∖ U0) ⊆ V

In Proofgold the corresponding term root is bc899b... and proposition id is 712799...

Proof:
Proof not loaded.

Theorem. (product_topology_regular)

∀X Tx Y Ty : set, regular_space X Tx → regular_space Y Ty → regular_space (setprod X Y) (product_topology X Tx Y Ty)

In Proofgold the corresponding term root is e4b9bd... and proposition id is ba770f...

Proof:
Proof not loaded.

Theorem. (product_space_points_differ_coord)

∀I Xi x1 x2 : set, x1 ∈ product_space I Xi → x2 ∈ product_space I Xi → x1 ≠ x2 → ∃i : set, i ∈ I ∧ apply_fun x1 i ≠ apply_fun x2 i

In Proofgold the corresponding term root is d27c38... and proposition id is 9ce2c3...

Proof:
Proof not loaded.

Theorem. (closure_of_product_cylinder_sub)

∀I Xi i V U : set, I ≠ Empty → (∀j : set, j ∈ I → topology_on (space_family_set Xi j) (space_family_topology Xi j)) → i ∈ I → V ∈ space_family_topology Xi i → U ∈ space_family_topology Xi i → closure_of (space_family_set Xi i) (space_family_topology Xi i) V ⊆ U → closure_of (product_space I Xi) (product_topology_full I Xi) (product_cylinder I Xi i V) ⊆ product_cylinder I Xi i U

In Proofgold the corresponding term root is d3626d... and proposition id is ceea7b...

Proof:
Proof not loaded.

Theorem. (product_topology_full_Hausdorff_axiom)

∀I Xi : set, Hausdorff_spaces_family I Xi → Hausdorff_space (product_space I Xi) (product_topology_full I Xi)

In Proofgold the corresponding term root is 96111e... and proposition id is bfde03...

Proof:
Proof not loaded.

Theorem. (product_topology_full_regular_axiom)

∀I Xi : set, regular_spaces_family I Xi → regular_space (product_space I Xi) (product_topology_full I Xi)

In Proofgold the corresponding term root is 614f77... and proposition id is 76ad74...

Proof:
Proof not loaded.

Theorem. (separation_axioms_subspace_product)

∀X Tx : set, topology_on X Tx → (∀Y : set, Y ⊆ X → Hausdorff_space X Tx → Hausdorff_space Y (subspace_topology X Tx Y)) ∧ (∀I Xi : set, Hausdorff_spaces_family I Xi → Hausdorff_space (product_space I Xi) (product_topology_full I Xi)) ∧ (∀Y : set, Y ⊆ X → regular_space X Tx → regular_space Y (subspace_topology X Tx Y)) ∧ (∀I Xi : set, regular_spaces_family I Xi → regular_space (product_space I Xi) (product_topology_full I Xi))

In Proofgold the corresponding term root is 4a69e7... and proposition id is 9b769e...

Proof:
Proof not loaded.

Definition. We define R_K to be R of type set.

In Proofgold the corresponding term root is e26ffa... and object id is 5277ef...

Theorem. (K_set_point_has_nonK_neighbor_in_intersection)

∀a b c d y : set, a ∈ R → b ∈ R → c ∈ R → d ∈ R → y ∈ K_set → y ∈ open_interval a b → y ∈ open_interval c d → ∃z : set, z ∈ (open_interval a b ∩ open_interval c d) ∧ z ∉ K_set

In Proofgold the corresponding term root is e54ce4... and proposition id is 2b6ddd...

Proof:
Proof not loaded.

Theorem. (RK_not_regular_axiom)

¬ regular_space R_K R_K_topology

In Proofgold the corresponding term root is 9a73c1... and proposition id is 9fadbb...

Proof:
Proof not loaded.

Theorem. (RK_Hausdorff_not_regular)

Hausdorff_space R_K R_K_topology ∧ ¬ regular_space R_K R_K_topology

In Proofgold the corresponding term root is deaf51... and proposition id is 02b3de...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_line_regular)

regular_space Sorgenfrey_line Sorgenfrey_topology

In Proofgold the corresponding term root is 3c8cd8... and proposition id is efab23...

Proof:
Proof not loaded.

Definition. We define Sorgenfrey_plane_diag_rational to be {(x,minus_SNo x)|x ∈ rational_numbers} of type set.

In Proofgold the corresponding term root is e1cd43... and object id is 236856...

Definition. We define Sorgenfrey_plane_diag_irrational to be {(x,minus_SNo x)|x ∈ (Sorgenfrey_line ∖ rational_numbers)} of type set.

In Proofgold the corresponding term root is 5dae80... and object id is 293d70...

Theorem. (Sorgenfrey_plane_diag_disjoint)

Sorgenfrey_plane_diag_rational ∩ Sorgenfrey_plane_diag_irrational = Empty

In Proofgold the corresponding term root is 25e547... and proposition id is 9663fd...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_diag_rational_closed)

closed_in (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology Sorgenfrey_plane_diag_rational

In Proofgold the corresponding term root is 22db7d... and proposition id is d7d93f...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_diag_irrational_closed)

closed_in (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology Sorgenfrey_plane_diag_irrational

In Proofgold the corresponding term root is f7c592... and proposition id is bb0759...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_L_subspace_discrete)

subspace_topology (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology Sorgenfrey_plane_L = discrete_topology Sorgenfrey_plane_L

In Proofgold the corresponding term root is 54afff... and proposition id is 87f362...

Proof:
Proof not loaded.

Theorem. (inj_Power_image)

∀A B : set, ∀f : set → set, inj A B f → inj (𝒫 A) (𝒫 B) (λS : set ⇒ {f x|x ∈ S})

In Proofgold the corresponding term root is e7b91e... and proposition id is 18a1db...

Proof:
Proof not loaded.

Theorem. (ex31_9_Sorgenfrey_plane_no_separation)

In Proofgold the corresponding term root is 8204de... and proposition id is 02a2df...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_not_normal)

regular_space (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology ∧ ¬ normal_space (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology

In Proofgold the corresponding term root is 718517... and proposition id is 1a9d5f...

Proof:
Proof not loaded.

Theorem. (inj_omega_preimage_ordsucc_finite)

∀A n : set, ∀f : set → set, nat_p n → inj A ω f → finite {a ∈ A|f a ∈ ordsucc n}

In Proofgold the corresponding term root is c8ffb5... and proposition id is 9821a0...

Proof:
Proof not loaded.

Theorem. (Union_Empty_eq)

⋃ Empty = Empty

In Proofgold the corresponding term root is 2c511e... and proposition id is 72b1c4...

Proof:
Proof not loaded.

Theorem. (Union_binunion_singleton_eq)

∀F y : set, ⋃ (F ∪ {y}) = (⋃ F) ∪ y

In Proofgold the corresponding term root is a29cf3... and proposition id is f6c12e...

Proof:
Proof not loaded.

Theorem. (finite_union_closed_in)

∀X Tx F : set, topology_on X Tx → finite F → (∀C : set, C ∈ F → closed_in X Tx C) → closed_in X Tx (⋃ F)

In Proofgold the corresponding term root is 04fd30... and proposition id is 8a9beb...

Proof:
Proof not loaded.

Theorem. (regular_countable_basis_normal)

∀X Tx : set, regular_space X Tx → second_countable_space X Tx → normal_space X Tx

In Proofgold the corresponding term root is 7dd405... and proposition id is a9147c...

Proof:
Proof not loaded.

Definition. We define order_interval_right_closed to be λX x y ⇒ {z ∈ X|order_rel X x z ∧ (z = y ∨ order_rel X z y)} of type set → set → set → set.

In Proofgold the corresponding term root is 4bbfc6... and object id is 21de64...

Theorem. (order_interval_right_closed_open)

∀X x y : set, well_ordered_set X → simply_ordered_set X → x ∈ X → y ∈ X → order_rel X x y → order_interval_right_closed X x y ∈ order_topology X

In Proofgold the corresponding term root is 38a13a... and proposition id is e107ac...

Proof:
Proof not loaded.

Theorem. (order_topology_basis_contains_right_closed_interval)

∀X b a : set, well_ordered_set X → simply_ordered_set X → a ∈ X → ¬ (a = 0) → b ∈ order_topology_basis X → a ∈ b → ∃x ∈ X, order_rel X x a ∧ order_interval_right_closed X x a ⊆ b

In Proofgold the corresponding term root is 459fb1... and proposition id is 2e5ffe...

Proof:
Proof not loaded.

Theorem. (order_topology_local_basis_elem)

∀X U y : set, simply_ordered_set X → U ∈ order_topology X → y ∈ U → ∃b ∈ order_topology_basis X, y ∈ b ∧ b ⊆ U

In Proofgold the corresponding term root is 884c6f... and proposition id is 023b45...

Proof:
Proof not loaded.

Theorem. (well_ordered_sets_normal_case_nonzero)

∀X A B UB VA : set, well_ordered_set X → simply_ordered_set X → topology_on X (order_topology X) → closed_in X (order_topology X) A → closed_in X (order_topology X) B → A ∩ B = Empty → A ≠ Empty → B ≠ Empty → UB ∈ order_topology X → B = X ∖ UB → VA ∈ order_topology X → A = X ∖ VA → ¬ (0 ∈ A) → ¬ (0 ∈ B) → ∃U V : set, U ∈ order_topology X ∧ V ∈ order_topology X ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty

In Proofgold the corresponding term root is 0ecaff... and proposition id is dc4b05...

Proof:
Proof not loaded.

Theorem. (well_ordered_sets_normal_case_0_in_A)

∀X A B : set, well_ordered_set X → simply_ordered_set X → closed_in X (order_topology X) A → closed_in X (order_topology X) B → A ∩ B = Empty → A ≠ Empty → B ≠ Empty → 0 ∈ A → ∃U V : set, U ∈ order_topology X ∧ V ∈ order_topology X ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty

In Proofgold the corresponding term root is 8e5bc2... and proposition id is bdbf0f...

Proof:
Proof not loaded.

Theorem. (well_ordered_sets_normal_case_0_in_B)

∀X A B : set, well_ordered_set X → simply_ordered_set X → closed_in X (order_topology X) A → closed_in X (order_topology X) B → A ∩ B = Empty → A ≠ Empty → B ≠ Empty → 0 ∈ B → ∃U V : set, U ∈ order_topology X ∧ V ∈ order_topology X ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty

In Proofgold the corresponding term root is 1f8024... and proposition id is f9e254...

Proof:
Proof not loaded.

Theorem. (well_ordered_sets_normal)

∀X : set, well_ordered_set X → simply_ordered_set X → normal_space X (order_topology X)

In Proofgold the corresponding term root is b61a85... and proposition id is a7fbca...

Proof:
Proof not loaded.

Definition. We define metric_sup_neg_dists to be λX d x A ⇒ Eps_i (λl : set ⇒ R_lub (metric_neg_dists X d x A) l) of type set → set → set → set → set.

In Proofgold the corresponding term root is ed2421... and object id is feb2bb...

Theorem. (metric_sup_neg_dists_is_lub)

∀X d x A : set, metric_on X d → x ∈ X → A ⊆ X → A ≠ Empty → R_lub (metric_neg_dists X d x A) (metric_sup_neg_dists X d x A)

In Proofgold the corresponding term root is 323172... and proposition id is 2b6bf0...

Proof:
Proof not loaded.

Definition. We define metric_dist_to_set to be λX d x A ⇒ minus_SNo (metric_sup_neg_dists X d x A) of type set → set → set → set → set.

In Proofgold the corresponding term root is cd6923... and object id is 5d7a99...

Theorem. (metric_dist_to_set_in_R)

∀X d x A : set, metric_on X d → x ∈ X → A ⊆ X → A ≠ Empty → metric_dist_to_set X d x A ∈ R

In Proofgold the corresponding term root is 5bde43... and proposition id is 559d2c...

Proof:
Proof not loaded.

Theorem. (metric_dist_to_set_eq_0_if_mem)

∀X d x A : set, metric_on X d → x ∈ X → A ⊆ X → x ∈ A → metric_dist_to_set X d x A = 0

In Proofgold the corresponding term root is 14dadc... and proposition id is 8ef4fd...

Proof:
Proof not loaded.

Theorem. (minus_SNo_minus_SNo_real)

∀t : set, t ∈ R → minus_SNo (minus_SNo t) = t

In Proofgold the corresponding term root is 4ee422... and proposition id is 138661...

Proof:
Proof not loaded.

Theorem. (metric_dist_to_set_le_dist)

∀X d x A a : set, metric_on X d → x ∈ X → A ⊆ X → A ≠ Empty → a ∈ A → Rle (metric_dist_to_set X d x A) (apply_fun d (x,a))

In Proofgold the corresponding term root is 6213d4... and proposition id is d230f3...

Proof:
Proof not loaded.

Theorem. (metric_dist_to_set_approx)

∀X d x A eps : set, metric_on X d → x ∈ X → A ⊆ X → A ≠ Empty → eps ∈ R → Rlt 0 eps → ∃a : set, a ∈ A ∧ a ∈ X ∧ Rlt (apply_fun d (x,a)) (add_SNo (metric_dist_to_set X d x A) eps)

In Proofgold the corresponding term root is 227d09... and proposition id is cff8f0...

Proof:
Proof not loaded.

Theorem. (metric_dist_to_set_Lipschitz)

∀X d x y A : set, metric_on X d → x ∈ X → y ∈ X → A ⊆ X → A ≠ Empty → Rle (metric_dist_to_set X d y A) (add_SNo (metric_dist_to_set X d x A) (apply_fun d (x,y)))

In Proofgold the corresponding term root is 1d9bfc... and proposition id is 89f6d1...

Proof:
Proof not loaded.

Theorem. (metric_dist_to_closed_set_pos_if_not_mem)

∀X d B x : set, metric_on X d → closed_in X (metric_topology X d) B → x ∈ X → x ∉ B → B ≠ Empty → Rlt 0 (metric_dist_to_set X d x B)

In Proofgold the corresponding term root is af2277... and proposition id is bc307b...

Proof:
Proof not loaded.

Theorem. (metrizable_spaces_normal)

∀X d : set, metric_on X d → normal_space X (metric_topology X d)

In Proofgold the corresponding term root is 88ae61... and proposition id is 4cdd05...

Proof:
Proof not loaded.

Theorem. (compact_Hausdorff_normal)

∀X Tx : set, compact_space X Tx → Hausdorff_space X Tx → normal_space X Tx

In Proofgold the corresponding term root is 4a1e19... and proposition id is 347551...

Proof:
Proof not loaded.

Axiom. (uncountable_product_R_not_normal) We take the following as an axiom:

∀J : set, uncountable_set J → ¬ normal_space (product_space J (const_space_family J R R_standard_topology)) (product_topology_full J (const_space_family J R R_standard_topology))

In Proofgold the corresponding term root is 4413ae... and proposition id is ad513d...

Axiom. (exists_uncountable_well_ordered_set) We take the following as an axiom:

∃X : set, well_ordered_set X ∧ uncountable_set X ∧ countable_subsets_bounded X

In Proofgold the corresponding term root is 75ed7a... and proposition id is 204cf9...

Theorem. (exists_uncountable_ordinal)

∃alpha : set, ordinal alpha ∧ uncountable_set alpha

In Proofgold the corresponding term root is 2208d9... and proposition id is fb93d9...

Proof:
Proof not loaded.

Definition. We define S_Omega to be Eps_i (λX : set ⇒ well_ordered_set X ∧ uncountable_set X ∧ countable_subsets_bounded X) of type set.

In Proofgold the corresponding term root is 07f34f... and object id is fa3159...

Theorem. (S_Omega_well_ordered_uncountable)

well_ordered_set S_Omega ∧ uncountable_set S_Omega

In Proofgold the corresponding term root is fd73da... and proposition id is 31908b...

Proof:
Proof not loaded.

Theorem. (S_Omega_notin_R)

S_Omega ∉ R

In Proofgold the corresponding term root is 4b39ce... and proposition id is ca6c4a...

Proof:
Proof not loaded.

Theorem. (S_Omega_countable_subsets_bounded)

countable_subsets_bounded S_Omega

In Proofgold the corresponding term root is 346d89... and proposition id is 9446a8...

Proof:
Proof not loaded.

Theorem. (S_Omega_omega_family_has_upper_bound)

∀seq : set, function_on seq ω S_Omega → ∃b : set, b ∈ S_Omega ∧ ∀n : set, n ∈ ω → apply_fun seq n ∈ b

In Proofgold the corresponding term root is e6e8e1... and proposition id is b94d0f...

Proof:
Proof not loaded.

Theorem. (S_Omega_ordinal_uncountable)

ordinal S_Omega ∧ uncountable_set S_Omega

In Proofgold the corresponding term root is 2c5695... and proposition id is 4c99cc...

Proof:
Proof not loaded.

Definition. We define Sbar_Omega to be SetAdjoin S_Omega S_Omega of type set.

In Proofgold the corresponding term root is 9b6699... and object id is 8418cc...

Theorem. (Sbar_Omega_eq_ordsucc)

Sbar_Omega = ordsucc S_Omega

In Proofgold the corresponding term root is c07529... and proposition id is 0a0e62...

Proof:
Proof not loaded.

Theorem. (Sbar_Omega_ordinal)

ordinal Sbar_Omega

In Proofgold the corresponding term root is 2c3be7... and proposition id is 885b87...

Proof:
Proof not loaded.

Theorem. (simply_ordered_set_Sbar_Omega)

simply_ordered_set Sbar_Omega

In Proofgold the corresponding term root is 8fdbb3... and proposition id is 72bc93...

Proof:
Proof not loaded.

Theorem. (S_Omega_Subq_Sbar_Omega)

S_Omega ⊆ Sbar_Omega

In Proofgold the corresponding term root is 6cd8a4... and proposition id is 20990e...

Proof:
Proof not loaded.

Theorem. (S_Omega_in_Sbar_Omega)

S_Omega ∈ Sbar_Omega

In Proofgold the corresponding term root is 8e839f... and proposition id is 9934eb...

Proof:
Proof not loaded.

Theorem. (Sbar_Omega_uncountable)

uncountable_set Sbar_Omega

In Proofgold the corresponding term root is b892e9... and proposition id is 5b8ccd...

Proof:
Proof not loaded.

Theorem. (Sbar_Omega_neq_R)

Sbar_Omega ≠ R

In Proofgold the corresponding term root is fd1443... and proposition id is d90ee8...

Proof:
Proof not loaded.

Theorem. (Sbar_Omega_neq_rational_numbers)

Sbar_Omega ≠ rational_numbers

In Proofgold the corresponding term root is 21cb1f... and proposition id is 4c8fe3...

Proof:
Proof not loaded.

Theorem. (Sbar_Omega_neq_setprod_2_omega)

Sbar_Omega ≠ setprod 2 ω

In Proofgold the corresponding term root is 4c1bd6... and proposition id is 18a2a8...

Proof:
Proof not loaded.

Theorem. (Sing1_in_setprod_R_R)

{1} ∈ setprod R R

In Proofgold the corresponding term root is 15b64b... and proposition id is f39271...

Proof:
Proof not loaded.

Theorem. (not_ordinal_setprod_R_R)

¬ ordinal (setprod R R)

In Proofgold the corresponding term root is fe7769... and proposition id is 0ce697...

Proof:
Proof not loaded.

Theorem. (Sbar_Omega_neq_setprod_R_R)

Sbar_Omega ≠ setprod R R

In Proofgold the corresponding term root is d10c9a... and proposition id is ed82ab...

Proof:
Proof not loaded.

Theorem. (Sbar_Omega_well_ordered_set)

well_ordered_set Sbar_Omega

In Proofgold the corresponding term root is 2d6e44... and proposition id is e1787e...

Proof:
Proof not loaded.

Theorem. (S_Omega_well_ordered_set)

well_ordered_set S_Omega

In Proofgold the corresponding term root is 4cf8ae... and proposition id is 421a26...

Proof:
Proof not loaded.

Theorem. (simply_ordered_set_S_Omega)

simply_ordered_set S_Omega

In Proofgold the corresponding term root is 912078... and proposition id is 0feda4...

Proof:
Proof not loaded.

Theorem. (S_Omega_TransSet)

TransSet S_Omega

In Proofgold the corresponding term root is fbc74a... and proposition id is 558605...

Proof:
Proof not loaded.

Theorem. (convex_in_Sbar_Omega_S_Omega)

convex_in Sbar_Omega S_Omega

In Proofgold the corresponding term root is 8d35fb... and proposition id is c968cc...

Proof:
Proof not loaded.

Theorem. (SOmega_interval_eq_inherited)

∀a b : set, a ∈ S_Omega → b ∈ S_Omega → {x ∈ S_Omega|order_rel S_Omega a x ∧ order_rel S_Omega x b} = {x ∈ S_Omega|order_rel Sbar_Omega a x ∧ order_rel Sbar_Omega x b}

In Proofgold the corresponding term root is 8025a5... and proposition id is b36f42...

Proof:
Proof not loaded.

Theorem. (SOmega_left_ray_eq_inherited)

∀b : set, b ∈ S_Omega → {x ∈ S_Omega|order_rel S_Omega x b} = {x ∈ S_Omega|order_rel Sbar_Omega x b}

In Proofgold the corresponding term root is f558d2... and proposition id is 248829...

Proof:
Proof not loaded.

Theorem. (SOmega_right_ray_eq_inherited)

∀a : set, a ∈ S_Omega → {x ∈ S_Omega|order_rel S_Omega a x} = {x ∈ S_Omega|order_rel Sbar_Omega a x}

In Proofgold the corresponding term root is 5ceae1... and proposition id is 5dcf45...

Proof:
Proof not loaded.

Theorem. (SOmega_order_topology_basis_inherited_eq)

order_topology_basis_inherited Sbar_Omega S_Omega = order_topology_basis S_Omega

In Proofgold the corresponding term root is a90994... and proposition id is 7de506...

Proof:
Proof not loaded.

Definition. We define SOmega_topology to be order_topology S_Omega of type set.

In Proofgold the corresponding term root is 1a9d4a... and object id is 74f270...

Definition. We define SbarOmega_topology to be order_topology Sbar_Omega of type set.

In Proofgold the corresponding term root is 07575a... and object id is 865ee1...

Theorem. (SOmega_topology_eq_subspace_SbarOmega)

SOmega_topology = subspace_topology Sbar_Omega SbarOmega_topology S_Omega

In Proofgold the corresponding term root is f4b60e... and proposition id is 111c39...

Proof:
Proof not loaded.

Theorem. (SOmega_SbarOmega_factors_normal)

normal_space S_Omega SOmega_topology ∧ normal_space Sbar_Omega SbarOmega_topology

In Proofgold the corresponding term root is 1cee25... and proposition id is d473d2...

Proof:
Proof not loaded.

Theorem. (SOmega_SbarOmega_product_not_normal)

¬ normal_space (setprod S_Omega Sbar_Omega) (product_topology S_Omega SOmega_topology Sbar_Omega SbarOmega_topology)

In Proofgold the corresponding term root is 2d110b... and proposition id is 43fdfe...

Proof:
Proof not loaded.

Theorem. (SOmega_SbarOmega_not_normal)

normal_space S_Omega SOmega_topology ∧ normal_space Sbar_Omega SbarOmega_topology ∧ ¬ normal_space (setprod S_Omega Sbar_Omega) (product_topology S_Omega SOmega_topology Sbar_Omega SbarOmega_topology)

In Proofgold the corresponding term root is 349b25... and proposition id is c1ba9d...

Proof:
Proof not loaded.

Definition. We define closed_interval to be λa b ⇒ {x ∈ R|¬ (Rlt x a) ∧ ¬ (Rlt b x)} of type set → set → set.

In Proofgold the corresponding term root is 1149df... and object id is a7c019...

Definition. We define closed_interval_topology to be λa b ⇒ subspace_topology R R_standard_topology (closed_interval a b) of type set → set → set.

In Proofgold the corresponding term root is 3d1a5d... and object id is 9a4557...

Theorem. (closed_interval_sub_R)

∀a b : set, closed_interval a b ⊆ R

In Proofgold the corresponding term root is d00d97... and proposition id is 502363...

Proof:
Proof not loaded.

Theorem. (closed_interval_bounds)

∀a b x : set, a ∈ R → b ∈ R → x ∈ closed_interval a b → Rle a x ∧ Rle x b

In Proofgold the corresponding term root is 9e779a... and proposition id is 602b6c...

Proof:
Proof not loaded.

Theorem. (closed_intervalI_of_Rle)

∀a b x : set, a ∈ R → b ∈ R → x ∈ R → Rle a x → Rle x b → x ∈ closed_interval a b

In Proofgold the corresponding term root is 53c929... and proposition id is 4834cd...

Proof:
Proof not loaded.

Theorem. (closed_interval_not_mem_cases)

∀a b x : set, a ∈ R → b ∈ R → x ∈ R → x ∉ closed_interval a b → Rlt x a ∨ Rlt b x

In Proofgold the corresponding term root is fc5997... and proposition id is 25b9c6...

Proof:
Proof not loaded.

Theorem. (closed_interval_topology_on)

∀a b : set, topology_on (closed_interval a b) (closed_interval_topology a b)

In Proofgold the corresponding term root is 68b34a... and proposition id is 4ec3bd...

Proof:
Proof not loaded.

Theorem. (left_endpoint_in_closed_interval)

∀a b : set, Rle a b → a ∈ closed_interval a b

In Proofgold the corresponding term root is 868735... and proposition id is 7b3cc0...

Proof:
Proof not loaded.

Theorem. (right_endpoint_in_closed_interval)

∀a b : set, Rle a b → b ∈ closed_interval a b

In Proofgold the corresponding term root is e35cab... and proposition id is 0ec925...

Proof:
Proof not loaded.

Theorem. (closed_interval_degenerate)

∀a y : set, a ∈ R → y ∈ closed_interval a a → y = a

In Proofgold the corresponding term root is 46302f... and proposition id is 76d95b...

Proof:
Proof not loaded.

Theorem. (closed_interval_closed_in_R_standard_topology)

∀a b : set, a ∈ R → b ∈ R → closed_in R R_standard_topology (closed_interval a b)

In Proofgold the corresponding term root is 8f0a8e... and proposition id is 9ccdf0...

Proof:
Proof not loaded.

Theorem. (closed_interval_closed_in_closed_interval_topology)

∀a b c d : set, a ∈ R → b ∈ R → c ∈ R → d ∈ R → closed_in (closed_interval a b) (closed_interval_topology a b) ((closed_interval c d) ∩ (closed_interval a b))

In Proofgold the corresponding term root is 6c11e1... and proposition id is 670f48...

Proof:
Proof not loaded.

Theorem. (normal_extend_nested_open_family_by_point)

∀X Tx : set, normal_space X Tx → ∀F : set, finite F → F ⊆ R → ∀Uof : set → set, (∀p : set, p ∈ F → Uof p ∈ Tx) → (∀p q : set, p ∈ F → q ∈ F → Rlt p q → closure_of X Tx (Uof p) ⊆ Uof q) → ∀r : set, r ∈ R → ∃Ur : set, Ur ∈ Tx ∧ (∀p : set, p ∈ F → Rlt p r → closure_of X Tx (Uof p) ⊆ Ur) ∧ (∀q : set, q ∈ F → Rlt r q → closure_of X Tx Ur ⊆ Uof q)

In Proofgold the corresponding term root is cad6fb... and proposition id is 32ad19...

Proof:
Proof not loaded.

Theorem. (Urysohn_lemma)

∀X Tx A B a b : set, Rle a b → normal_space X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → ∃f : set, continuous_map X Tx (closed_interval a b) (closed_interval_topology a b) f ∧ (∀x : set, x ∈ A → apply_fun f x = a) ∧ (∀x : set, x ∈ B → apply_fun f x = b)

In Proofgold the corresponding term root is 7c7ed2... and proposition id is f6db54...

Proof:
Proof not loaded.

Definition. We define completely_regular_space to be λX Tx ⇒ topology_on X Tx ∧ (one_point_sets_closed X Tx ∧ ∀x : set, x ∈ X → ∀F : set, closed_in X Tx F → x ∉ F → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ apply_fun f x = 0 ∧ ∀y : set, y ∈ F → apply_fun f y = 1) of type set → set → prop.

In Proofgold the corresponding term root is 994502... and object id is 1dc7c3...

Theorem. (completely_regular_space_open_separator)

∀X Tx x U : set, completely_regular_space X Tx → x ∈ X → U ∈ Tx → x ∈ U → ∃g : set, continuous_map X Tx R R_standard_topology g ∧ apply_fun g x = 1 ∧ ∀y : set, y ∈ X ∖ U → apply_fun g y = 0

In Proofgold the corresponding term root is a0e213... and proposition id is 04a41c...

Proof:
Proof not loaded.

Theorem. (completely_regular_space_open_separator_unit_interval)

∀X Tx x U : set, completely_regular_space X Tx → x ∈ X → U ∈ Tx → x ∈ U → ∃h : set, continuous_map X Tx unit_interval unit_interval_topology h ∧ apply_fun h x = 1 ∧ ∀y : set, y ∈ X ∖ U → apply_fun h y = 0

In Proofgold the corresponding term root is 75aac4... and proposition id is 7991e9...

Proof:
Proof not loaded.

Theorem. (completely_regular_space_open_separator_unit_interval_in_function_space)

∀X Tx x U : set, completely_regular_space X Tx → x ∈ X → U ∈ Tx → x ∈ U → ∃h : set, h ∈ function_space X unit_interval ∧ continuous_map X Tx unit_interval unit_interval_topology h ∧ apply_fun h x = 1 ∧ ∀y : set, y ∈ X ∖ U → apply_fun h y = 0

In Proofgold the corresponding term root is 975816... and proposition id is 630692...

Proof:
Proof not loaded.

Theorem. (completely_regular_space_separates_points)

∀X Tx x y : set, completely_regular_space X Tx → x ∈ X → y ∈ X → x ≠ y → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ apply_fun f x = 0 ∧ apply_fun f y = 1

In Proofgold the corresponding term root is 7a3316... and proposition id is 4e7e19...

Proof:
Proof not loaded.

Theorem. (completely_regular_space_topology_on)

∀X Tx : set, completely_regular_space X Tx → topology_on X Tx

In Proofgold the corresponding term root is cd3e94... and proposition id is de97a5...

Proof:
Proof not loaded.

Theorem. (completely_regular_space_one_point_sets_closed)

∀X Tx : set, completely_regular_space X Tx → one_point_sets_closed X Tx

In Proofgold the corresponding term root is 868cff... and proposition id is 877f4a...

Proof:
Proof not loaded.

Theorem. (completely_regular_space_separation)

∀X Tx x F : set, completely_regular_space X Tx → x ∈ X → closed_in X Tx F → x ∉ F → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ apply_fun f x = 0 ∧ ∀y : set, y ∈ F → apply_fun f y = 1

In Proofgold the corresponding term root is 6e35a4... and proposition id is 097f8c...

Proof:
Proof not loaded.

Theorem. (completely_regular_space_implies_regular)

∀X Tx : set, completely_regular_space X Tx → regular_space X Tx

In Proofgold the corresponding term root is 96746d... and proposition id is 0562da...

Proof:
Proof not loaded.

Definition. We define completely_regular_spaces_family to be λI Xi ⇒ ∀i : set, i ∈ I → completely_regular_space (product_component Xi i) (product_component_topology Xi i) of type set → set → prop.

In Proofgold the corresponding term root is ab668b... and object id is 8a9bbe...

Theorem. (continuous_from_discrete)

∀X Y Ty f : set, topology_on Y Ty → function_on f X Y → continuous_map X (discrete_topology X) Y Ty f

In Proofgold the corresponding term root is f1b959... and proposition id is 7f4ff5...

Proof:
Proof not loaded.

Theorem. (tuple_2_0_congr)

∀a b c d : set, (a,b) = (c,d) → a = c

In Proofgold the corresponding term root is a3f8c1... and proposition id is e20aa2...

Proof:
Proof not loaded.

Theorem. (tuple_2_1_congr)

∀a b c d : set, (a,b) = (c,d) → b = d

In Proofgold the corresponding term root is e14ece... and proposition id is fb67cf...

Proof:
Proof not loaded.

Theorem. (const_fun_pair_first)

∀A x a y : set, (a,y) ∈ const_fun A x → a ∈ A

In Proofgold the corresponding term root is 2d3c96... and proposition id is fd1ad6...

Proof:
Proof not loaded.

Theorem. (const_fun_pair_second)

∀A x a y : set, (a,y) ∈ const_fun A x → y = x

In Proofgold the corresponding term root is 03ef3b... and proposition id is 346ecf...

Proof:
Proof not loaded.

Theorem. (discrete_completely_regular_space)

∀X : set, completely_regular_space X (discrete_topology X)

In Proofgold the corresponding term root is 95a0c5... and proposition id is b59858...

Proof:
Proof not loaded.

Theorem. (discrete_Hausdorff_space)

∀X : set, Hausdorff_space X (discrete_topology X)

In Proofgold the corresponding term root is cf8256... and proposition id is 19deb1...

Proof:
Proof not loaded.

Definition. We define Tychonoff_space to be λX Tx ⇒ completely_regular_space X Tx ∧ Hausdorff_space X Tx of type set → set → prop.

In Proofgold the corresponding term root is 2e5f59... and object id is b27312...

Theorem. (generated_topology_from_subbasis_local_basis)

∀X S U x : set, U ∈ generated_topology_from_subbasis X S → x ∈ U → ∃b : set, b ∈ basis_of_subbasis X S ∧ x ∈ b ∧ b ⊆ U

In Proofgold the corresponding term root is e00972... and proposition id is 413537...

Proof:
Proof not loaded.

Theorem. (basis_of_subbasisE)

∀X S b : set, b ∈ basis_of_subbasis X S → ∃F : set, F ∈ finite_subcollections S ∧ b = intersection_of_family X F ∧ b ≠ Empty

In Proofgold the corresponding term root is fa884c... and proposition id is cc97b7...

Proof:
Proof not loaded.

Theorem. (generated_topology_from_subbasis_contains_subbasis_elem)

∀X S s : set, subbasis_on X S → s ∈ S → s ∈ generated_topology_from_subbasis X S

In Proofgold the corresponding term root is 6af98e... and proposition id is 185cf4...

Proof:
Proof not loaded.

Definition. We define product_eval_map to be λI Xi i ⇒ graph (product_space I Xi) (λf : set ⇒ apply_fun f i) of type set → set → set → set.

In Proofgold the corresponding term root is 7bec29... and object id is a6cd80...

Theorem. (product_eval_map_continuous)

∀I Xi i : set, (∀j : set, j ∈ I → topology_on (space_family_set Xi j) (space_family_topology Xi j)) → i ∈ I → continuous_map (product_space I Xi) (product_topology_full I Xi) (space_family_set Xi i) (space_family_topology Xi i) (product_eval_map I Xi i)

In Proofgold the corresponding term root is 1a3a9e... and proposition id is 7870c6...

Proof:
Proof not loaded.

Theorem. (net_converges_on_product_topology_full_implies_coord_converges)

∀I Xi net J le x i : set, (∀j : set, j ∈ I → topology_on (space_family_set Xi j) (space_family_topology Xi j)) → i ∈ I → net_converges_on (product_space I Xi) (product_topology_full I Xi) net J le x → net_converges_on (space_family_set Xi i) (space_family_topology Xi i) (compose_fun J net (product_eval_map I Xi i)) J le (apply_fun x i)

In Proofgold the corresponding term root is 2945fa... and proposition id is a56aa9...

Proof:
Proof not loaded.

Theorem. (generated_topology_from_subbasis_finite_neighborhood)

∀X S U x : set, subbasis_on X S → U ∈ generated_topology_from_subbasis X S → x ∈ U → ∃F : set, F ∈ finite_subcollections S ∧ x ∈ intersection_of_family X F ∧ intersection_of_family X F ⊆ U

In Proofgold the corresponding term root is 0ef8d7... and proposition id is 8f8b22...

Proof:
Proof not loaded.

Theorem. (net_converges_on_generated_topology_from_subbasis_of_subeventual)

∀X S net J le x : set, subbasis_on X S → directed_set J le → total_function_on net J X → functional_graph net → graph_domain_subset net J → x ∈ X → (∀s : set, s ∈ S → x ∈ s → ∃i0 : set, i0 ∈ J ∧ ∀i : set, i ∈ J → (i0,i) ∈ le → apply_fun net i ∈ s) → net_converges_on X (generated_topology_from_subbasis X S) net J le x

In Proofgold the corresponding term root is 125d4f... and proposition id is f9f50c...

Proof:
Proof not loaded.

Theorem. (product_subbasis_fullE)

∀I Xi s : set, s ∈ product_subbasis_full I Xi → ∃i : set, i ∈ I ∧ ∃U : set, U ∈ space_family_topology Xi i ∧ s = product_cylinder I Xi i U

In Proofgold the corresponding term root is 90b8fd... and proposition id is 42cca7...

Proof:
Proof not loaded.

Theorem. (net_converges_on_product_topology_full_of_all_coord_converge)

∀I Xi net J le x : set, I ≠ Empty → (∀i : set, i ∈ I → topology_on (space_family_set Xi i) (space_family_topology Xi i)) → directed_set J le → total_function_on net J (product_space I Xi) → functional_graph net → graph_domain_subset net J → x ∈ product_space I Xi → (∀i : set, i ∈ I → net_converges_on (space_family_set Xi i) (space_family_topology Xi i) (compose_fun J net (product_eval_map I Xi i)) J le (apply_fun x i)) → net_converges_on (product_space I Xi) (product_topology_full I Xi) net J le x

In Proofgold the corresponding term root is 207bc0... and proposition id is a5a7f6...

Proof:
Proof not loaded.

Theorem. (net_converges_on_product_topology_full_iff_all_coord_converge)

∀I Xi net J le x : set, I ≠ Empty → (∀i : set, i ∈ I → topology_on (space_family_set Xi i) (space_family_topology Xi i)) → directed_set J le → total_function_on net J (product_space I Xi) → functional_graph net → graph_domain_subset net J → x ∈ product_space I Xi → (net_converges_on (product_space I Xi) (product_topology_full I Xi) net J le x ↔ ∀i : set, i ∈ I → net_converges_on (space_family_set Xi i) (space_family_topology Xi i) (compose_fun J net (product_eval_map I Xi i)) J le (apply_fun x i))

In Proofgold the corresponding term root is b03c4f... and proposition id is e6ca7e...

Proof:
Proof not loaded.

Theorem. (continuous_combine_or_01)

∀X Tx f g x0 : set, topology_on X Tx → continuous_map X Tx R R_standard_topology f → continuous_map X Tx R R_standard_topology g → x0 ∈ X → apply_fun f x0 = 0 → apply_fun g x0 = 0 → ∃h : set, continuous_map X Tx R R_standard_topology h ∧ apply_fun h x0 = 0 ∧ ∀y : set, y ∈ X → (apply_fun f y = 1 ∨ apply_fun g y = 1) → apply_fun h y = 1

In Proofgold the corresponding term root is 14e5f9... and proposition id is b22465...

Proof:
Proof not loaded.

Theorem. (product_finite_subbasis_intersection_separator)

∀I Xi F x : set, I ≠ Empty → completely_regular_spaces_family I Xi → F ∈ finite_subcollections (product_subbasis_full I Xi) → x ∈ intersection_of_family (product_space I Xi) F → ∃f : set, continuous_map (product_space I Xi) (product_topology_full I Xi) R R_standard_topology f ∧ apply_fun f x = 0 ∧ ∀y : set, y ∈ product_space I Xi → (∃s : set, s ∈ F ∧ y ∉ s) → apply_fun f y = 1

In Proofgold the corresponding term root is 8c6b04... and proposition id is c71a4a...

Proof:
Proof not loaded.

Theorem. (completely_regular_subspace_product)

∀X Tx : set, topology_on X Tx → (∀Y : set, Y ⊆ X → completely_regular_space X Tx → completely_regular_space Y (subspace_topology X Tx Y)) ∧ (∀I Xi : set, completely_regular_spaces_family I Xi → completely_regular_space (product_space I Xi) (product_topology_full I Xi))

In Proofgold the corresponding term root is 4e5416... and proposition id is 112563...

Proof:
Proof not loaded.

Theorem. (continuous_map_preimage_closed)

∀X Tx Y Ty f C : set, continuous_map X Tx Y Ty f → closed_in Y Ty C → closed_in X Tx (preimage_of X f C)

In Proofgold the corresponding term root is 6e7fa2... and proposition id is a9c3fb...

Proof:
Proof not loaded.

Theorem. (homeomorphism_preserves_completely_regular)

∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty f → completely_regular_space Y Ty → completely_regular_space X Tx

In Proofgold the corresponding term root is d79dd6... and proposition id is 0d1cd5...

Proof:
Proof not loaded.

Theorem. (metric_spaces_completely_regular)

∀X d : set, metric_on X d → completely_regular_space X (metric_topology X d)

In Proofgold the corresponding term root is 2afb48... and proposition id is bb8a5a...

Proof:
Proof not loaded.

Theorem. (unit_interval_completely_regular)

completely_regular_space unit_interval unit_interval_topology

In Proofgold the corresponding term root is 3b0846... and proposition id is 70ce4d...

Proof:
Proof not loaded.

Theorem. (normal_space_implies_completely_regular)

∀X Tx : set, normal_space X Tx → completely_regular_space X Tx

In Proofgold the corresponding term root is c17d99... and proposition id is 0df18e...

Proof:
Proof not loaded.

Theorem. (continuous_map_to_generated_topology)

∀X Tx Y B f : set, topology_on X Tx → basis_on Y B → function_on f X Y → (∀b : set, b ∈ B → preimage_of X f b ∈ Tx) → continuous_map X Tx Y (generated_topology Y B) f

In Proofgold the corresponding term root is 3a7f0a... and proposition id is b90dd2...

Proof:
Proof not loaded.

Theorem. (completely_regular_product_topology)

∀X Tx Y Ty : set, completely_regular_space X Tx → completely_regular_space Y Ty → completely_regular_space (setprod X Y) (product_topology X Tx Y Ty)

In Proofgold the corresponding term root is 9dfeba... and proposition id is a4387b...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_line_completely_regular)

completely_regular_space Sorgenfrey_line Sorgenfrey_topology

In Proofgold the corresponding term root is 72915f... and proposition id is a6a368...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_completely_regular_not_normal)

completely_regular_space (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology ∧ ¬ normal_space (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology

In Proofgold the corresponding term root is 0f0052... and proposition id is 32f9c7...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is 63a560... and proposition id is ae4e46...

Proof:
Proof not loaded.

Theorem. (Urysohn_metrization_theorem)

∀X Tx : set, regular_space X Tx → second_countable_space X Tx → ∃d : set, metric_on X d ∧ metric_topology X d = Tx

In Proofgold the corresponding term root is 66094a... and proposition id is 49ee3c...

Proof:
Proof not loaded.

Theorem. (embedding_via_functions)

∀X Tx : set, topology_on X Tx → one_point_sets_closed X Tx → ∀F J : set, separating_family_of_functions X Tx F J → ∃Fmap : set, embedding_of X Tx (power_real J) (product_topology_full J (const_space_family J R R_standard_topology)) Fmap

In Proofgold the corresponding term root is 276d6e... and proposition id is f25317...

Proof:
Proof not loaded.

Theorem. (completely_regular_iff_embeds_in_cube)

∀X Tx : set, (completely_regular_space X Tx ↔ ∃J : set, ∃Fmap : set, embedding_of X Tx (unit_interval_power J) (product_topology_full J (const_space_family J unit_interval (subspace_topology R R_standard_topology unit_interval))) Fmap)

In Proofgold the corresponding term root is 4af436... and proposition id is 51bd1b...

Proof:
Proof not loaded.

Definition. We define cauchy_sequence to be λX d seq ⇒ metric_on X d ∧ sequence_on seq X ∧ ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → Rlt (apply_fun d (apply_fun seq m,apply_fun seq n)) eps of type set → set → set → prop.

In Proofgold the corresponding term root is 406860... and object id is 4f35ba...

Definition. We define cauchy_sequence_total to be λX d seq ⇒ metric_on_total X d ∧ sequence_on seq X ∧ ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → Rlt (apply_fun d (apply_fun seq m,apply_fun seq n)) eps of type set → set → set → prop.

In Proofgold the corresponding term root is a68f0b... and object id is ce843e...

Theorem. (cauchy_sequence_total_imp)

∀X d seq : set, cauchy_sequence_total X d seq → cauchy_sequence X d seq

In Proofgold the corresponding term root is 162a2a... and proposition id is fe48e2...

Proof:
Proof not loaded.

Definition. We define complete_metric_space to be λX d ⇒ metric_on X d ∧ ∀seq : set, sequence_on seq X → cauchy_sequence X d seq → ∃x : set, converges_to X (metric_topology X d) seq x of type set → set → prop.

In Proofgold the corresponding term root is 795586... and object id is 229bc3...

Definition. We define complete_metric_space_total to be λX d ⇒ metric_on_total X d ∧ ∀seq : set, sequence_on seq X → cauchy_sequence_total X d seq → ∃x : set, converges_to X (metric_topology X d) seq x of type set → set → prop.

In Proofgold the corresponding term root is c1d1c0... and object id is dcd473...

Definition. We define continuous_at_map to be λX Tx Y Ty f x ⇒ function_on f X Y ∧ x ∈ X ∧ ∀V : set, V ∈ Ty → apply_fun f x ∈ V → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ ∀u : set, u ∈ U → apply_fun f u ∈ V of type set → set → set → set → set → set → prop.

In Proofgold the corresponding term root is ce39c0... and object id is a7f586...

Definition. We define pointwise_limit_metric to be λX Y d fn f ⇒ ∀x : set, x ∈ X → ∀eps : set, eps ∈ R → Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → Rlt (apply_fun d (apply_fun (apply_fun fn n) x,apply_fun f x)) eps of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is a49e85... and object id is 413a0a...

Definition. We define uniform_limit_metric to be λX Y d fn f ⇒ ∀eps : set, eps ∈ R → Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun d (apply_fun (apply_fun fn n) x,apply_fun f x)) eps of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is a02f5c... and object id is 331779...

Theorem. (uniform_limit_metric_imp_pointwise_limit_metric)

∀X Y d fn f : set, uniform_limit_metric X Y d fn f → pointwise_limit_metric X Y d fn f

In Proofgold the corresponding term root is ccf0fa... and proposition id is be965a...

Proof:
Proof not loaded.

Theorem. (converges_to_imp_in_closure_of_terms)

∀X Tx A seq x : set, converges_to X Tx seq x → (∀n : set, n ∈ ω → apply_fun seq n ∈ A) → x ∈ closure_of X Tx A

In Proofgold the corresponding term root is 7d9525... and proposition id is 2ff200...

Proof:
Proof not loaded.

Theorem. (converges_to_imp_in_closure_of_eventually_terms)

∀X Tx A seq x N : set, converges_to X Tx seq x → N ∈ ω → (∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ A) → x ∈ closure_of X Tx A

In Proofgold the corresponding term root is e96835... and proposition id is ad52d4...

Proof:
Proof not loaded.

Theorem. (converges_to_closed_in_contains_limit)

∀X Tx A seq x : set, topology_on X Tx → A ⊆ X → closed_in X Tx A → converges_to X Tx seq x → (∀n : set, n ∈ ω → apply_fun seq n ∈ A) → x ∈ A

In Proofgold the corresponding term root is 418fb8... and proposition id is c4e7ef...

Proof:
Proof not loaded.

Theorem. (uniform_limit_metric_imp_sequence_converges_metric_at_point)

∀X Y d fn f x : set, metric_on Y d → function_on fn ω (function_space X Y) → function_on f X Y → x ∈ X → uniform_limit_metric X Y d fn f → sequence_converges_metric Y d (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)

In Proofgold the corresponding term root is efab88... and proposition id is 73f521...

Proof:
Proof not loaded.

Theorem. (metric_topology_neighborhood_contains_ball)

∀X d x U : set, metric_on X d → x ∈ X → U ∈ metric_topology X d → x ∈ U → ∃r : set, r ∈ R ∧ (Rlt 0 r ∧ open_ball X d x r ⊆ U)

In Proofgold the corresponding term root is b03e26... and proposition id is 93033c...

Proof:
Proof not loaded.

Theorem. (metric_topology_neighborhood_contains_ball_bounded)

∀X d x U a : set, metric_on X d → x ∈ X → U ∈ metric_topology X d → x ∈ U → a ∈ R → Rlt 0 a → ∃r : set, r ∈ R ∧ Rlt 0 r ∧ open_ball X d x r ⊆ U ∧ Rlt r a

In Proofgold the corresponding term root is 3e35ae... and proposition id is dfd348...

Proof:
Proof not loaded.

Theorem. (sequence_converges_metric_imp_converges_to_metric_topology)

∀X d seq x : set, sequence_converges_metric X d seq x → converges_to X (metric_topology X d) seq x

In Proofgold the corresponding term root is 29708b... and proposition id is 887af2...

Proof:
Proof not loaded.

Theorem. (converges_to_metric_topology_imp_sequence_converges_metric)

∀X d seq x : set, metric_on X d → converges_to X (metric_topology X d) seq x → sequence_converges_metric X d seq x

In Proofgold the corresponding term root is 303354... and proposition id is 0c5325...

Proof:
Proof not loaded.

Theorem. (uniform_limit_metric_imp_converges_to_metric_topology_at_point)

∀X Y d fn f x : set, metric_on Y d → function_on fn ω (function_space X Y) → function_on f X Y → x ∈ X → uniform_limit_metric X Y d fn f → converges_to Y (metric_topology Y d) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)

In Proofgold the corresponding term root is 254879... and proposition id is adcf9b...

Proof:
Proof not loaded.

Definition. We define uniform_cauchy_metric to be λX Y d fn ⇒ ∀eps : set, eps ∈ R → Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun d (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps of type set → set → set → set → prop.

In Proofgold the corresponding term root is 3e53ed... and object id is b18661...

Theorem. (uniform_cauchy_metric_complete_imp_uniform_limit_stub)

∀X Y d fn : set, complete_metric_space Y d → function_on fn ω (function_space X Y) → uniform_cauchy_metric X Y d fn → ∃f : set, function_on f X Y ∧ uniform_limit_metric X Y d fn f

In Proofgold the corresponding term root is e44ca4... and proposition id is 116664...

Proof:
Proof not loaded.

Theorem. (uniform_limit_of_continuous_to_metric_is_continuous)

∀X Tx Y d fn f : set, topology_on X Tx → metric_on Y d → function_on fn ω (function_space X Y) → function_on f X Y → (∀n : set, n ∈ ω → continuous_map X Tx Y (metric_topology Y d) (apply_fun fn n)) → uniform_limit_metric X Y d fn f → continuous_map X Tx Y (metric_topology Y d) f

In Proofgold the corresponding term root is 2b3a20... and proposition id is 36a169...

Proof:
Proof not loaded.

Theorem. (open_interval_in_metric_topology_R_bounded_metric_early)

∀a b : set, a ∈ R → b ∈ R → open_interval a b ∈ metric_topology R R_bounded_metric

In Proofgold the corresponding term root is bbb749... and proposition id is d8990a...

Proof:
Proof not loaded.

Theorem. (metric_topology_R_bounded_metric_eq_R_standard_topology_early)

metric_topology R R_bounded_metric = R_standard_topology

In Proofgold the corresponding term root is 3c2e15... and proposition id is d30c60...

Proof:
Proof not loaded.

Theorem. (cauchy_R_bounded_metric_abs_tail_early)

∀seq eps : set, cauchy_sequence R R_bounded_metric seq → eps ∈ R → Rlt 0 eps → Rlt eps 1 → ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → abs_SNo (add_SNo (apply_fun seq m) (minus_SNo (apply_fun seq n))) < eps

In Proofgold the corresponding term root is 1435c6... and proposition id is 87a936...

Proof:
Proof not loaded.

Theorem. (abs_Cauchy_sequence_converges_R_standard_topology_early)

∀seq : set, sequence_on seq R → (∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → abs_SNo (add_SNo (apply_fun seq m) (minus_SNo (apply_fun seq n))) < eps) → ∃x : set, converges_to R R_standard_topology seq x

In Proofgold the corresponding term root is c9fabe... and proposition id is 59126e...

Proof:
Proof not loaded.

Theorem. (R_bounded_metric_complete_early)

complete_metric_space R R_bounded_metric

In Proofgold the corresponding term root is 360bbb... and proposition id is f0ae42...

Proof:
Proof not loaded.

Theorem. (uniform_limit_of_continuous_to_R_standard_topology)

∀X Tx fn f : set, topology_on X Tx → function_on fn ω (function_space X R) → function_on f X R → (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)) → uniform_limit_metric X R R_bounded_metric fn f → continuous_map X Tx R R_standard_topology f

In Proofgold the corresponding term root is 420ad6... and proposition id is 080104...

Proof:
Proof not loaded.

Theorem. (uniform_cauchy_continuous_to_R_has_continuous_limit)

∀X Tx fn : set, topology_on X Tx → function_on fn ω (function_space X R) → (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)) → uniform_cauchy_metric X R R_bounded_metric fn → ∃f : set, function_on f X R ∧ uniform_limit_metric X R R_bounded_metric fn f ∧ continuous_map X Tx R R_standard_topology f

In Proofgold the corresponding term root is 3775b6... and proposition id is bf2104...

Proof:
Proof not loaded.

Definition. We define one_third to be inv_nat 3 of type set.

In Proofgold the corresponding term root is d14bda... and object id is f6f291...

Definition. We define two_thirds to be add_SNo one_third one_third of type set.

In Proofgold the corresponding term root is 064d8a... and object id is 750507...

Theorem. (one_third_in_R)

one_third ∈ R

In Proofgold the corresponding term root is 29f38d... and proposition id is 247017...

Proof:
Proof not loaded.

Theorem. (two_thirds_in_R)

two_thirds ∈ R

In Proofgold the corresponding term root is 8e2652... and proposition id is 510c48...

Proof:
Proof not loaded.

Theorem. (one_third_pos)

0 < one_third

In Proofgold the corresponding term root is 60ce7d... and proposition id is 2d4104...

Proof:
Proof not loaded.

Theorem. (two_thirds_pos)

0 < two_thirds

In Proofgold the corresponding term root is 0f81fe... and proposition id is 0fb392...

Proof:
Proof not loaded.

Theorem. (two_thirds_ne0)

two_thirds ≠ 0

In Proofgold the corresponding term root is 3fe756... and proposition id is 7e7595...

Proof:
Proof not loaded.

Definition. We define div_const_fun to be λc ⇒ graph R (λt : set ⇒ div_SNo t c) of type set → set.

In Proofgold the corresponding term root is 9b1806... and object id is a0b30d...

Theorem. (div_const_fun_apply)

∀c t : set, c ∈ R → t ∈ R → apply_fun (div_const_fun c) t = div_SNo t c

In Proofgold the corresponding term root is 579730... and proposition id is 270cd6...

Proof:
Proof not loaded.

Theorem. (compose_div_const_fun_apply)

∀A f c x : set, x ∈ A → c ∈ R → apply_fun f x ∈ R → apply_fun (compose_fun A f (div_const_fun c)) x = div_SNo (apply_fun f x) c

In Proofgold the corresponding term root is 88dcb2... and proposition id is 7e8be8...

Proof:
Proof not loaded.

Theorem. (div_const_fun_in_function_space)

∀c : set, c ∈ R → div_const_fun c ∈ function_space R R

In Proofgold the corresponding term root is 3c806d... and proposition id is 7fcb1a...

Proof:
Proof not loaded.

Theorem. (div_const_fun_value_in_R)

∀c t : set, c ∈ R → t ∈ R → apply_fun (div_const_fun c) t ∈ R

In Proofgold the corresponding term root is b87503... and proposition id is 6f2f6d...

Proof:
Proof not loaded.

Theorem. (preimage_div_const_open_ray_upper)

∀c a : set, c ∈ R → 0 < c → a ∈ R → preimage_of R (div_const_fun c) (open_ray_upper R a) = open_ray_upper R (mul_SNo a c)

In Proofgold the corresponding term root is 3e0ab6... and proposition id is 06ce24...

Proof:
Proof not loaded.

Theorem. (preimage_div_const_open_ray_lower)

∀c b : set, c ∈ R → 0 < c → b ∈ R → preimage_of R (div_const_fun c) (open_ray_lower R b) = open_ray_lower R (mul_SNo b c)

In Proofgold the corresponding term root is d3b8dd... and proposition id is cc9c3a...

Proof:
Proof not loaded.

Theorem. (div_const_fun_continuous_pos)

∀c : set, c ∈ R → 0 < c → continuous_map R R_standard_topology R R_standard_topology (div_const_fun c)

In Proofgold the corresponding term root is 575108... and proposition id is 01b2a5...

Proof:
Proof not loaded.

Theorem. (div_SNo_minus_den)

∀x y : set, x ∈ R → y ∈ R → y ≠ 0 → div_SNo x (minus_SNo y) = minus_SNo (div_SNo x y)

In Proofgold the corresponding term root is f04d30... and proposition id is 67be89...

Proof:
Proof not loaded.

Theorem. (div_const_fun_continuous)

∀c : set, c ∈ R → c ≠ 0 → continuous_map R R_standard_topology R R_standard_topology (div_const_fun c)

In Proofgold the corresponding term root is 689981... and proposition id is b15afa...

Proof:
Proof not loaded.

Theorem. (div_SNo_closed_interval_scale)

∀den y : set, den ∈ R → 0 < den → y ∈ closed_interval (minus_SNo den) den → div_SNo y den ∈ closed_interval (minus_SNo 1) 1

In Proofgold the corresponding term root is 9153c0... and proposition id is fa9421...

Proof:
Proof not loaded.

Theorem. (scale_continuous_mul_const_pos)

∀X Tx g c : set, topology_on X Tx → continuous_map X Tx R R_standard_topology g → c ∈ R → 0 < c → continuous_map X Tx R R_standard_topology (compose_fun X g (mul_const_fun c))

In Proofgold the corresponding term root is 554694... and proposition id is 04e29c...

Proof:
Proof not loaded.

Theorem. (punctured_horizontal_slice_path_component)

∀x0 x1 y0 : set, y0 ∈ R → y0 ≠ 0 → x0 ∈ R → x1 ∈ R → (x0,y0) ∈ path_component_of (EuclidPlane ∖ {(0,0)}) (subspace_topology EuclidPlane R2_standard_topology (EuclidPlane ∖ {(0,0)})) (x1,y0)

In Proofgold the corresponding term root is aa1daf... and proposition id is 09a401...

Proof:
Proof not loaded.

Theorem. (punctured_vertical_slice_path_component)

∀x0 y0 y1 : set, x0 ∈ R → x0 ≠ 0 → y0 ∈ R → y1 ∈ R → (x0,y0) ∈ path_component_of (EuclidPlane ∖ {(0,0)}) (subspace_topology EuclidPlane R2_standard_topology (EuclidPlane ∖ {(0,0)})) (x0,y1)

In Proofgold the corresponding term root is f76009... and proposition id is 6611ac...

Proof:
Proof not loaded.

Theorem. (punctured_space_path_connected)

path_connected_space (EuclidPlane ∖ {(0,0)}) (subspace_topology EuclidPlane R2_standard_topology (EuclidPlane ∖ {(0,0)}))

In Proofgold the corresponding term root is 102fb8... and proposition id is af25da...

Proof:
Proof not loaded.

Theorem. (minus_two_thirds_eq)

minus_SNo two_thirds = add_SNo (minus_SNo one_third) (minus_SNo one_third)

In Proofgold the corresponding term root is 902551... and proposition id is 741d6d...

Proof:
Proof not loaded.

Theorem. (one_third_eq_div_1_3)

one_third = div_SNo 1 3

In Proofgold the corresponding term root is 50b8a7... and proposition id is 61a575...

Proof:
Proof not loaded.

Theorem. (two_thirds_eq_div_2_3)

two_thirds = div_SNo 2 3

In Proofgold the corresponding term root is e519e1... and proposition id is 50c015...

Proof:
Proof not loaded.

Theorem. (div_SNo_1_2_eq_eps_1)

div_SNo 1 2 = eps_ 1

In Proofgold the corresponding term root is aa96d7... and proposition id is caf54e...

Proof:
Proof not loaded.

Theorem. (add_nat_4_4_eq_8)

add_nat 4 4 = 8

In Proofgold the corresponding term root is 58ff2b... and proposition id is 36602f...

Proof:
Proof not loaded.

Theorem. (mul_nat_4_2_eq_8)

mul_nat 4 2 = 8

In Proofgold the corresponding term root is e8b84d... and proposition id is efc978...

Proof:
Proof not loaded.

Theorem. (mul_SNo_4_2_lt_9)

mul_SNo 4 2 < 9

In Proofgold the corresponding term root is 5aa75b... and proposition id is 222704...

Proof:
Proof not loaded.

Theorem. (add_nat_3_3_eq_6)

add_nat 3 3 = 6

In Proofgold the corresponding term root is 008165... and proposition id is 008c0b...

Proof:
Proof not loaded.

Theorem. (add_nat_3_6_eq_9)

add_nat 3 6 = 9

In Proofgold the corresponding term root is c74c0b... and proposition id is fef9b3...

Proof:
Proof not loaded.

Theorem. (mul_nat_2_2_eq_4)

mul_nat 2 2 = 4

In Proofgold the corresponding term root is 74ac8a... and proposition id is c3da70...

Proof:
Proof not loaded.

Theorem. (mul_nat_3_3_eq_9)

mul_nat 3 3 = 9

In Proofgold the corresponding term root is 72f4ae... and proposition id is 86a725...

Proof:
Proof not loaded.

Theorem. (mul_SNo_2_2_eq_4)

mul_SNo 2 2 = 4

In Proofgold the corresponding term root is 44c5ad... and proposition id is ecc462...

Proof:
Proof not loaded.

Theorem. (mul_SNo_3_3_eq_9)

mul_SNo 3 3 = 9

In Proofgold the corresponding term root is d643b6... and proposition id is 073e8e...

Proof:
Proof not loaded.

Theorem. (two_thirds_sq_lt_eps_1)

mul_SNo two_thirds two_thirds < eps_ 1

In Proofgold the corresponding term root is 7b4165... and proposition id is 22452c...

Proof:
Proof not loaded.

Theorem. (add_one_third_two_thirds_eq_1)

add_SNo one_third two_thirds = 1

In Proofgold the corresponding term root is c815dc... and proposition id is 562412...

Proof:
Proof not loaded.

Theorem. (Rlt_two_thirds_1)

Rlt two_thirds 1

In Proofgold the corresponding term root is 26673f... and proposition id is 6a8e18...

Proof:
Proof not loaded.

Theorem. (one_third_lt_two_thirds)

one_third < two_thirds

In Proofgold the corresponding term root is e42b3a... and proposition id is 21346d...

Proof:
Proof not loaded.

Theorem. (one_third_lt_1_SNo)

one_third < 1

In Proofgold the corresponding term root is 1093ca... and proposition id is 3bb343...

Proof:
Proof not loaded.

Theorem. (Rlt_one_third_1)

Rlt one_third 1

In Proofgold the corresponding term root is dc97ed... and proposition id is d8fd40...

Proof:
Proof not loaded.

Theorem. (minus_1_lt_minus_one_third_SNo)

(minus_SNo 1) < (minus_SNo one_third)

In Proofgold the corresponding term root is 46f9ad... and proposition id is fb397a...

Proof:
Proof not loaded.

Theorem. (Rlt_minus1_minus_one_third)

Rlt (minus_SNo 1) (minus_SNo one_third)

In Proofgold the corresponding term root is e70a41... and proposition id is 629913...

Proof:
Proof not loaded.

Theorem. (closed_interval_minus_one_third_one_third_sub_closed_interval_minus1_1)

closed_interval (minus_SNo one_third) one_third ⊆ closed_interval (minus_SNo 1) 1

In Proofgold the corresponding term root is 879ebd... and proposition id is 256889...

Proof:
Proof not loaded.

Theorem. (two_thirds_eq_1_minus_one_third)

two_thirds = add_SNo 1 (minus_SNo one_third)

In Proofgold the corresponding term root is cc0fe3... and proposition id is 403c54...

Proof:
Proof not loaded.

Theorem. (minus_two_thirds_eq_minus1_plus_one_third)

minus_SNo two_thirds = add_SNo (minus_SNo 1) one_third

In Proofgold the corresponding term root is 84b42d... and proposition id is d19e06...

Proof:
Proof not loaded.

Theorem. (add_minus1_two_thirds_eq_minus_one_third)

add_SNo (minus_SNo 1) two_thirds = minus_SNo one_third

In Proofgold the corresponding term root is 534d47... and proposition id is 803b40...

Proof:
Proof not loaded.

Theorem. (add_1_minus_two_thirds_eq_one_third)

add_SNo 1 (minus_SNo two_thirds) = one_third

In Proofgold the corresponding term root is 63b541... and proposition id is 830f48...

Proof:
Proof not loaded.

Theorem. (Rle_0_two_thirds)

Rle 0 two_thirds

In Proofgold the corresponding term root is 396de5... and proposition id is 8adcad...

Proof:
Proof not loaded.

Theorem. (Rle_minus_two_thirds_0)

Rle (minus_SNo two_thirds) 0

In Proofgold the corresponding term root is 9eb979... and proposition id is abfed0...

Proof:
Proof not loaded.

Theorem. (Rlt_minus_one_third_one_third)

Rlt (minus_SNo one_third) one_third

In Proofgold the corresponding term root is 5033b7... and proposition id is fafdc6...

Proof:
Proof not loaded.

Theorem. (Rle_minus_one_third_one_third)

Rle (minus_SNo one_third) one_third

In Proofgold the corresponding term root is 69748a... and proposition id is ae69bc...

Proof:
Proof not loaded.

Theorem. (Rle_add_SNo_1)

∀x y z : set, x ∈ R → y ∈ R → z ∈ R → Rle x y → Rle (add_SNo x z) (add_SNo y z)

In Proofgold the corresponding term root is bb8e63... and proposition id is 8bcc84...

Proof:
Proof not loaded.

Theorem. (add_SNo_interval_expand_by_third_stub)

∀c y z : set, c ∈ R → 0 < c → y ∈ closed_interval (add_SNo (minus_SNo 1) c) (add_SNo 1 (minus_SNo c)) → z ∈ closed_interval (minus_SNo (mul_SNo c one_third)) (mul_SNo c one_third) → add_SNo y z ∈ closed_interval (add_SNo (minus_SNo 1) (mul_SNo c two_thirds)) (add_SNo 1 (minus_SNo (mul_SNo c two_thirds)))

In Proofgold the corresponding term root is 073e6d... and proposition id is 205818...

Proof:
Proof not loaded.

Theorem. (Rle_minus_contra)

∀x y : set, Rle x y → Rle (minus_SNo y) (minus_SNo x)

In Proofgold the corresponding term root is ba2cd2... and proposition id is ef989b...

Proof:
Proof not loaded.

Theorem. (Rlt_minus1_minus_two_thirds)

Rlt (minus_SNo 1) (minus_SNo two_thirds)

In Proofgold the corresponding term root is 873229... and proposition id is 0082e5...

Proof:
Proof not loaded.

Theorem. (closed_interval_minus_two_thirds_two_thirds_Subq_open_interval_minus1_1)

closed_interval (minus_SNo two_thirds) two_thirds ⊆ open_interval (minus_SNo 1) 1

In Proofgold the corresponding term root is 8af538... and proposition id is e4ed16...

Proof:
Proof not loaded.

Theorem. (mul_two_thirds_closed_interval_minus1_1)

∀t : set, t ∈ closed_interval (minus_SNo 1) 1 → mul_SNo t two_thirds ∈ closed_interval (minus_SNo two_thirds) two_thirds

In Proofgold the corresponding term root is aee1fc... and proposition id is 15ca4f...

Proof:
Proof not loaded.

Theorem. (mul_nonneg_closed_interval_minus1_1)

∀t c : set, t ∈ closed_interval (minus_SNo 1) 1 → c ∈ R → 0 ≤ c → mul_SNo t c ∈ closed_interval (minus_SNo c) c

In Proofgold the corresponding term root is 6fee91... and proposition id is f6a05f...

Proof:
Proof not loaded.

Theorem. (div_two_thirds_closed_interval_minus_two_thirds_two_thirds)

∀t : set, t ∈ closed_interval (minus_SNo two_thirds) two_thirds → div_SNo t two_thirds ∈ closed_interval (minus_SNo 1) 1

In Proofgold the corresponding term root is a32595... and proposition id is 8a2c44...

Proof:
Proof not loaded.

Theorem. (Tietze_I1_closed_in_I)

closed_in (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1) ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ (closed_interval (minus_SNo 1) 1))

In Proofgold the corresponding term root is 1650ad... and proposition id is b63901...

Proof:
Proof not loaded.

Theorem. (Tietze_I3_closed_in_I)

closed_in (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1) ((closed_interval one_third 1) ∩ (closed_interval (minus_SNo 1) 1))

In Proofgold the corresponding term root is 4ca8d2... and proposition id is 37d896...

Proof:
Proof not loaded.

Theorem. (Tietze_stepI_BC_closed_in_X)

∀X Tx A f : set, normal_space X Tx → closed_in X Tx A → continuous_map A (subspace_topology X Tx A) (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1) f → closed_in X Tx (preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ (closed_interval (minus_SNo 1) 1))) ∧ closed_in X Tx (preimage_of A f ((closed_interval one_third 1) ∩ (closed_interval (minus_SNo 1) 1)))

In Proofgold the corresponding term root is cde672... and proposition id is 8a734c...

Proof:
Proof not loaded.

Theorem. (Tietze_I1_I3_disjoint)

((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ (closed_interval (minus_SNo 1) 1)) ∩ ((closed_interval one_third 1) ∩ (closed_interval (minus_SNo 1) 1)) = Empty

In Proofgold the corresponding term root is 0981da... and proposition id is fb4b5b...

Proof:
Proof not loaded.

Theorem. (Tietze_stepI_g_exists)

∀X Tx A f : set, normal_space X Tx → closed_in X Tx A → continuous_map A (subspace_topology X Tx A) (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1) f → ∃g : set, continuous_map X Tx (closed_interval (minus_SNo one_third) one_third) (closed_interval_topology (minus_SNo one_third) one_third) g ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ (closed_interval (minus_SNo 1) 1)) → apply_fun g x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ (closed_interval (minus_SNo 1) 1)) → apply_fun g x = one_third)

In Proofgold the corresponding term root is 0432bb... and proposition id is 04fe50...

Proof:
Proof not loaded.

Theorem. (add_of_pair_map_neg_apply)

∀A f g a : set, a ∈ A → apply_fun f a ∈ R → apply_fun g a ∈ R → apply_fun (compose_fun A (pair_map A f (compose_fun A g neg_fun)) add_fun_R) a = add_SNo (apply_fun f a) (minus_SNo (apply_fun g a))

In Proofgold the corresponding term root is 06263d... and proposition id is 719ceb...

Proof:
Proof not loaded.

Theorem. (Tietze_stepII_algebra_tail)

∀gx rx ux c : set, SNo gx → SNo rx → SNo ux → SNo c → add_SNo (add_SNo gx (mul_SNo ux c)) (mul_SNo (add_SNo rx (minus_SNo ux)) c) = add_SNo gx (mul_SNo rx c)

In Proofgold the corresponding term root is 07b4fe... and proposition id is 9c9a8c...

Proof:
Proof not loaded.

Theorem. (Rlt_minus1_0)

Rlt (minus_SNo 1) 0

In Proofgold the corresponding term root is 502166... and proposition id is d50578...

Proof:
Proof not loaded.

Theorem. (zero_in_closed_interval_minus1_1)

0 ∈ closed_interval (minus_SNo 1) 1

In Proofgold the corresponding term root is be0d71... and proposition id is 3b959c...

Proof:
Proof not loaded.

Theorem. (Tietze_stepII_real_extension_nonempty)

∀X Tx A f : set, normal_space X Tx → closed_in X Tx A → A ≠ Empty → continuous_map A (subspace_topology X Tx A) (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1) f → ∃gR : set, continuous_map X Tx R R_standard_topology gR ∧ (∀x : set, x ∈ A → apply_fun gR x = apply_fun f x) ∧ (∀x : set, x ∈ X → apply_fun gR x ∈ closed_interval (minus_SNo 1) 1)

In Proofgold the corresponding term root is cc5d00... and proposition id is 8fd1fc...

Proof:
Proof not loaded.

Theorem. (Tietze_stepII_real_extension)

∀X Tx A f : set, normal_space X Tx → closed_in X Tx A → continuous_map A (subspace_topology X Tx A) (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1) f → ∃gR : set, continuous_map X Tx R R_standard_topology gR ∧ (∀x : set, x ∈ A → apply_fun gR x = apply_fun f x) ∧ (∀x : set, x ∈ X → apply_fun gR x ∈ closed_interval (minus_SNo 1) 1)

In Proofgold the corresponding term root is 8780bd... and proposition id is a1ed6d...

Proof:
Proof not loaded.

Theorem. (Tietze_extension_minus1_1)

In Proofgold the corresponding term root is c04d85... and proposition id is 5bec49...

Proof:
Proof not loaded.

Theorem. (closed_interval_affine_equiv_minus1_1)

∀a b : set, Rle a b → ¬ (a = b) → ∃h : set, ∃hinv : set, continuous_map (closed_interval a b) (closed_interval_topology a b) (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1) h ∧ (continuous_map (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1) (closed_interval a b) (closed_interval_topology a b) hinv ∧ ((∀t : set, t ∈ closed_interval a b → apply_fun hinv (apply_fun h t) = t) ∧ (∀s : set, s ∈ closed_interval (minus_SNo 1) 1 → apply_fun h (apply_fun hinv s) = s)))

In Proofgold the corresponding term root is 984d06... and proposition id is d881af...

Proof:
Proof not loaded.

Theorem. (closed_interval_minus1_1_compact)

compact_space (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1)

In Proofgold the corresponding term root is 1aa7b5... and proposition id is 07a4fc...

Proof:
Proof not loaded.

Theorem. (closed_interval_compact)

∀a b : set, Rle a b → compact_space (closed_interval a b) (closed_interval_topology a b)

In Proofgold the corresponding term root is 83a4dc... and proposition id is d8a42a...

Proof:
Proof not loaded.

Theorem. (Tietze_extension_interval)

∀X Tx A a b f : set, Rle a b → normal_space X Tx → closed_in X Tx A → continuous_map A (subspace_topology X Tx A) (closed_interval a b) (closed_interval_topology a b) f → ∃g : set, continuous_map X Tx (closed_interval a b) (closed_interval_topology a b) g ∧ (∀x : set, x ∈ A → apply_fun g x = apply_fun f x)

In Proofgold the corresponding term root is 15e4a7... and proposition id is a09a9d...

Proof:
Proof not loaded.

Theorem. (subspace_inclusion_continuous)

∀X Tx A : set, topology_on X Tx → A ⊆ X → continuous_map A (subspace_topology X Tx A) X Tx {(y,y)|y ∈ A}

In Proofgold the corresponding term root is cbbb3e... and proposition id is 21a47d...

Proof:
Proof not loaded.

Theorem. (Tietze_extension_real_bounded_interval)

∀X Tx A a b f : set, Rle a b → normal_space X Tx → closed_in X Tx A → continuous_map A (subspace_topology X Tx A) R R_standard_topology f → (∀x : set, x ∈ A → apply_fun f x ∈ closed_interval a b) → ∃g : set, continuous_map X Tx R R_standard_topology g ∧ (∀x : set, x ∈ A → apply_fun g x = apply_fun f x)

In Proofgold the corresponding term root is 4fb336... and proposition id is 0ea2b0...

Proof:
Proof not loaded.

Theorem. (Tietze_extension_open_interval)

∀X Tx A f : set, normal_space X Tx → closed_in X Tx A → continuous_map A (subspace_topology X Tx A) (open_interval (minus_SNo 1) 1) (subspace_topology R R_standard_topology (open_interval (minus_SNo 1) 1)) f → ∃g : set, continuous_map X Tx (open_interval (minus_SNo 1) 1) (subspace_topology R R_standard_topology (open_interval (minus_SNo 1) 1)) g ∧ (∀x : set, x ∈ A → apply_fun g x = apply_fun f x)

In Proofgold the corresponding term root is 4bb6d7... and proposition id is c617b8...

Proof:
Proof not loaded.

Definition. We define bounded_transform_phi to be graph R (λt : set ⇒ div_SNo t (add_SNo 1 (abs_SNo t))) of type set.

In Proofgold the corresponding term root is 55d7f2... and object id is 44208f...

Theorem. (bounded_transform_phi_value_in_R)

∀t : set, t ∈ R → apply_fun bounded_transform_phi t ∈ R

In Proofgold the corresponding term root is da91ff... and proposition id is 257eb3...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_range_in_closed_interval_minus1_1)

∀t : set, t ∈ R → apply_fun bounded_transform_phi t ∈ closed_interval (minus_SNo 1) 1

In Proofgold the corresponding term root is f70c40... and proposition id is e08e7f...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_strict_bounds)

∀t : set, t ∈ R → Rlt (minus_SNo 1) (apply_fun bounded_transform_phi t) ∧ Rlt (apply_fun bounded_transform_phi t) 1

In Proofgold the corresponding term root is 5babb8... and proposition id is f4eef5...

Proof:
Proof not loaded.

Definition. We define bounded_transform_psi to be graph R (λs : set ⇒ div_SNo s (add_SNo 1 (minus_SNo (abs_SNo s)))) of type set.

In Proofgold the corresponding term root is 8a44e3... and object id is f9beac...

Theorem. (abs_div_SNo_pos)

∀x y : set, SNo x → SNo y → 0 < y → abs_SNo (div_SNo x y) = div_SNo (abs_SNo x) y

In Proofgold the corresponding term root is 55ba08... and proposition id is 49b09d...

Proof:
Proof not loaded.

Theorem. (div_SNo_nonneg_pos_nonneg)

∀x y : set, SNo x → SNo y → 0 ≤ x → 0 < y → 0 ≤ div_SNo x y

In Proofgold the corresponding term root is 4e61ff... and proposition id is f76e18...

Proof:
Proof not loaded.

Theorem. (div_SNo_minus_num)

∀x y : set, SNo x → SNo y → y ≠ 0 → div_SNo (minus_SNo x) y = minus_SNo (div_SNo x y)

In Proofgold the corresponding term root is 8bd6bb... and proposition id is f20768...

Proof:
Proof not loaded.

Theorem. (abs_lt_one_of_open_interval)

∀s : set, s ∈ open_interval (minus_SNo 1) 1 → abs_SNo s < 1

In Proofgold the corresponding term root is e016f0... and proposition id is ef0dc2...

Proof:
Proof not loaded.

Theorem. (neg_closed_open_interval_minus1_1)

∀s : set, s ∈ open_interval (minus_SNo 1) 1 → (minus_SNo s) ∈ open_interval (minus_SNo 1) 1

In Proofgold the corresponding term root is a41ccd... and proposition id is 5066be...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_odd)

∀t : set, t ∈ R → apply_fun bounded_transform_phi (minus_SNo t) = minus_SNo (apply_fun bounded_transform_phi t)

In Proofgold the corresponding term root is 7e058a... and proposition id is 62f472...

Proof:
Proof not loaded.

Theorem. (bounded_transform_psi_odd_open_interval)

∀s : set, s ∈ open_interval (minus_SNo 1) 1 → apply_fun bounded_transform_psi (minus_SNo s) = minus_SNo (apply_fun bounded_transform_psi s)

In Proofgold the corresponding term root is 4033aa... and proposition id is 7fdb0f...

Proof:
Proof not loaded.

Theorem. (bounded_transform_psi_preimage_open_ray_upper)

∀a : set, a ∈ R → preimage_of (open_interval (minus_SNo 1) 1) bounded_transform_psi (open_ray_upper R a) ∈ subspace_topology R R_standard_topology (open_interval (minus_SNo 1) 1)

In Proofgold the corresponding term root is d645e7... and proposition id is 784683...

Proof:
Proof not loaded.

Theorem. (bounded_transform_psi_preimage_open_ray_lower)

∀b : set, b ∈ R → preimage_of (open_interval (minus_SNo 1) 1) bounded_transform_psi (open_ray_lower R b) ∈ subspace_topology R R_standard_topology (open_interval (minus_SNo 1) 1)

In Proofgold the corresponding term root is f673ff... and proposition id is 55197b...

Proof:
Proof not loaded.

Theorem. (bounded_transform_psi_phi_cancel)

∀t : set, t ∈ R → apply_fun bounded_transform_psi (apply_fun bounded_transform_phi t) = t

In Proofgold the corresponding term root is 7106f5... and proposition id is 24a4bd...

Proof:
Proof not loaded.

Theorem. (bounded_transform_psi_continuous_open_interval)

continuous_map (open_interval (minus_SNo 1) 1) (subspace_topology R R_standard_topology (open_interval (minus_SNo 1) 1)) R R_standard_topology bounded_transform_psi

In Proofgold the corresponding term root is e69e82... and proposition id is 3fc63f...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_Rlt0_implies_Rlt0)

∀t : set, t ∈ R → Rlt 0 (apply_fun bounded_transform_phi t) → Rlt 0 t

In Proofgold the corresponding term root is af31b3... and proposition id is 3c30f0...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_Rlt0_implies_Rlt0_neg)

∀t : set, t ∈ R → Rlt (apply_fun bounded_transform_phi t) 0 → Rlt t 0

In Proofgold the corresponding term root is 7cd888... and proposition id is 9820fb...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_upper_neg_case_eq)

∀a : set, a ∈ R → Rlt a 0 → Rlt (minus_SNo 1) a → preimage_of R bounded_transform_phi (open_ray_upper R a) = open_ray_upper R (div_SNo a (add_SNo 1 a))

In Proofgold the corresponding term root is fd3f9a... and proposition id is 5f1cad...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_upper_pos_case_eq)

∀a : set, a ∈ R → Rlt 0 a → Rlt a 1 → preimage_of R bounded_transform_phi (open_ray_upper R a) = open_ray_upper R (div_SNo a (add_SNo 1 (minus_SNo a)))

In Proofgold the corresponding term root is 62b8ee... and proposition id is 95138a...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_upper_zero_eq)

preimage_of R bounded_transform_phi (open_ray_upper R 0) = open_ray_upper R 0

In Proofgold the corresponding term root is 44c54a... and proposition id is 341f9c...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_upper_strict_mid)

∀a : set, a ∈ R → Rlt (minus_SNo 1) a → Rlt a 1 → preimage_of R bounded_transform_phi (open_ray_upper R a) ∈ R_standard_topology

In Proofgold the corresponding term root is 2a9ea4... and proposition id is 60256b...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_upper_mid)

∀a : set, a ∈ R → ¬ (Rlt a (minus_SNo 1)) → Rlt a 1 → preimage_of R bounded_transform_phi (open_ray_upper R a) ∈ R_standard_topology

In Proofgold the corresponding term root is 1a5e15... and proposition id is decaac...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_upper)

∀a : set, a ∈ R → preimage_of R bounded_transform_phi (open_ray_upper R a) ∈ R_standard_topology

In Proofgold the corresponding term root is 07b306... and proposition id is 1f8cab...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_lower_neg_case_eq)

∀b : set, b ∈ R → Rlt b 0 → Rlt (minus_SNo 1) b → preimage_of R bounded_transform_phi (open_ray_lower R b) = open_ray_lower R (div_SNo b (add_SNo 1 b))

In Proofgold the corresponding term root is 196655... and proposition id is 2eba18...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_lower_pos_case_eq)

∀b : set, b ∈ R → Rlt 0 b → Rlt b 1 → preimage_of R bounded_transform_phi (open_ray_lower R b) = open_ray_lower R (div_SNo b (add_SNo 1 (minus_SNo b)))

In Proofgold the corresponding term root is 62b0a8... and proposition id is 6ec5c0...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_lower_zero_eq)

preimage_of R bounded_transform_phi (open_ray_lower R 0) = open_ray_lower R 0

In Proofgold the corresponding term root is fc8676... and proposition id is 695e72...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_lower_strict_mid)

∀b : set, b ∈ R → Rlt (minus_SNo 1) b → Rlt b 1 → preimage_of R bounded_transform_phi (open_ray_lower R b) ∈ R_standard_topology

In Proofgold the corresponding term root is c86103... and proposition id is b4ade1...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_lower_mid)

∀b : set, b ∈ R → ¬ (Rlt 1 b) → Rlt (minus_SNo 1) b → preimage_of R bounded_transform_phi (open_ray_lower R b) ∈ R_standard_topology

In Proofgold the corresponding term root is 88a10f... and proposition id is 0909df...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_preimage_open_ray_lower)

∀b : set, b ∈ R → preimage_of R bounded_transform_phi (open_ray_lower R b) ∈ R_standard_topology

In Proofgold the corresponding term root is 91ffd4... and proposition id is c75964...

Proof:
Proof not loaded.

Theorem. (bounded_transform_phi_continuous)

continuous_map R R_standard_topology R R_standard_topology bounded_transform_phi

In Proofgold the corresponding term root is 91678a... and proposition id is 88d84d...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is 0ff126... and proposition id is a6e88c...

Proof:
Proof not loaded.

Theorem. (recip_pos_value_eq_recip_SNo_pos)

∀t : set, t ∈ open_ray_upper R 0 → apply_fun (compose_fun (open_ray_upper R 0) (compose_fun R bounded_transform_phi one_minus_fun) bounded_transform_psi) t = recip_SNo_pos t

In Proofgold the corresponding term root is e4023e... and proposition id is c2825b...

Proof:
Proof not loaded.

Theorem. (reciprocal_of_positive_continuous_map)

∀X Tx f : set, topology_on X Tx → continuous_map X Tx R R_standard_topology f → (∀x : set, x ∈ X → Rlt 0 (apply_fun f x)) → continuous_map X Tx R R_standard_topology (compose_fun X f (compose_fun (open_ray_upper R 0) (compose_fun R bounded_transform_phi one_minus_fun) bounded_transform_psi))

In Proofgold the corresponding term root is ba07ad... and proposition id is 79f08c...

Proof:
Proof not loaded.

Theorem. (Tietze_extension_real)

∀X Tx A f : set, normal_space X Tx → closed_in X Tx A → continuous_map A (subspace_topology X Tx A) R R_standard_topology f → ∃g : set, continuous_map X Tx R R_standard_topology g ∧ (∀x : set, x ∈ A → apply_fun g x = apply_fun f x)

In Proofgold the corresponding term root is 4aa33f... and proposition id is 366c0d...

Proof:
Proof not loaded.

Definition. We define m_manifold to be λX Tx m ⇒ Hausdorff_space X Tx ∧ second_countable_space X Tx ∧ m ∈ ω ∧ ∀x : set, x ∈ X → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ ∃V : set, V ∈ (euclidean_topology m) ∧ ∃f : set, homeomorphism U (subspace_topology X Tx U) V (subspace_topology (euclidean_space m) (euclidean_topology m) V) f of type set → set → set → prop.

In Proofgold the corresponding term root is 117a9d... and object id is 3de808...

Definition. We define support_of to be λX Tx phi ⇒ closure_of X Tx {x ∈ X|apply_fun phi x ≠ 0} of type set → set → set → set.

In Proofgold the corresponding term root is da69b7... and object id is 3a4c7e...

Theorem. (support_of_sub_X)

∀X Tx phi : set, topology_on X Tx → support_of X Tx phi ⊆ X

In Proofgold the corresponding term root is dc8163... and proposition id is 9ee127...

Proof:
Proof not loaded.

Theorem. (support_of_closed_in)

∀X Tx phi : set, topology_on X Tx → closed_in X Tx (support_of X Tx phi)

In Proofgold the corresponding term root is d3d7c4... and proposition id is e53403...

Proof:
Proof not loaded.

Theorem. (support_of_contains_nonzero)

∀X Tx phi x : set, topology_on X Tx → x ∈ X → apply_fun phi x ≠ 0 → x ∈ support_of X Tx phi

In Proofgold the corresponding term root is 720c86... and proposition id is f91cbc...

Proof:
Proof not loaded.

Theorem. (support_of_disjoint_open_implies_value_zero)

∀X Tx phi N x : set, topology_on X Tx → N ∈ Tx → x ∈ N → support_of X Tx phi ∩ N = Empty → apply_fun phi x = 0

In Proofgold the corresponding term root is 88b725... and proposition id is 977d70...

Proof:
Proof not loaded.

Theorem. (support_of_mul_two_Subq_support_left)

∀X Tx f g : set, topology_on X Tx → (∀x : set, x ∈ X → apply_fun f x ∈ R) → (∀x : set, x ∈ X → apply_fun g x ∈ R) → support_of X Tx (compose_fun X (pair_map X f g) mul_fun_R) ⊆ support_of X Tx f

In Proofgold the corresponding term root is 56c3ad... and proposition id is 26ffdb...

Proof:
Proof not loaded.

Theorem. (support_of_mul_two_Subq_support_right)

∀X Tx f g : set, topology_on X Tx → (∀x : set, x ∈ X → apply_fun f x ∈ R) → (∀x : set, x ∈ X → apply_fun g x ∈ R) → support_of X Tx (compose_fun X (pair_map X f g) mul_fun_R) ⊆ support_of X Tx g

In Proofgold the corresponding term root is 887163... and proposition id is 411ccf...

Proof:
Proof not loaded.

Theorem. (open_intersects_closure_implies_intersects)

∀X Tx A N : set, topology_on X Tx → N ∈ Tx → N ∩ closure_of X Tx A ≠ Empty → N ∩ A ≠ Empty

In Proofgold the corresponding term root is 403e7c... and proposition id is e68513...

Proof:
Proof not loaded.

Theorem. (support_of_sub_closure_of)

∀X Tx phi U : set, topology_on X Tx → U ⊆ X → (∀x : set, x ∈ X → apply_fun phi x ≠ 0 → x ∈ U) → support_of X Tx phi ⊆ closure_of X Tx U

In Proofgold the corresponding term root is 758537... and proposition id is f6e54c...

Proof:
Proof not loaded.

Theorem. (Urysohn_bump_closed_in_open)

∀X Tx A V : set, normal_space X Tx → closed_in X Tx A → V ∈ Tx → A ⊆ V → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 1) ∧ (∀x : set, x ∈ (X ∖ V) → apply_fun f x = 0) ∧ (∀x : set, x ∈ X → apply_fun f x ∈ unit_interval)

In Proofgold the corresponding term root is aa4a9d... and proposition id is 28a60d...

Proof:
Proof not loaded.

Theorem. (support_of_zero_outside_open_sub_closure)

∀X Tx f V : set, topology_on X Tx → V ∈ Tx → (∀x : set, x ∈ (X ∖ V) → apply_fun f x = 0) → support_of X Tx f ⊆ closure_of X Tx V

In Proofgold the corresponding term root is ce05cc... and proposition id is 4bf227...

Proof:
Proof not loaded.

Theorem. (Urysohn_bump_support_sub_closure)

In Proofgold the corresponding term root is 0675c0... and proposition id is 80f584...

Proof:
Proof not loaded.

Definition. We define partition_of_unity_dominated to be λX Tx U ⇒ topology_on X Tx ∧ open_cover X Tx U ∧ ∃P : set, P ⊆ function_space X R ∧ (∀f : set, f ∈ P → continuous_map X Tx R R_standard_topology f) ∧ (∀f x : set, f ∈ P → x ∈ X → apply_fun f x ∈ unit_interval) ∧ (∀f : set, f ∈ P → ∃u : set, u ∈ U ∧ support_of X Tx f ⊆ u) ∧ (∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃F0 : set, finite F0 ∧ F0 ⊆ P ∧ ∀f : set, f ∈ P → support_of X Tx f ∩ N ≠ Empty → f ∈ F0) ∧ (∀x : set, x ∈ X → ∃F : set, finite F ∧ F ⊆ P ∧ (∀f : set, f ∈ P → apply_fun f x ≠ 0 → f ∈ F) ∧ ∃n : set, n ∈ ω ∧ ∃e : set, bijection n F e ∧ ∃p : set, function_on p (ordsucc n) R ∧ apply_fun p Empty = 0 ∧ (∀k : set, k ∈ n → apply_fun p (ordsucc k) = add_SNo (apply_fun p k) (apply_fun (apply_fun e k) x)) ∧ apply_fun p n = 1) of type set → set → set → prop.

In Proofgold the corresponding term root is ae1707... and object id is 673965...

Definition. We define partition_of_unity_family to be λX Tx U P ⇒ P ⊆ function_space X R ∧ (∀f : set, f ∈ P → continuous_map X Tx R R_standard_topology f) ∧ (∀f x : set, f ∈ P → x ∈ X → apply_fun f x ∈ unit_interval) ∧ (∀f : set, f ∈ P → ∃u : set, u ∈ U ∧ support_of X Tx f ⊆ u) ∧ (∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃F0 : set, finite F0 ∧ F0 ⊆ P ∧ ∀f : set, f ∈ P → support_of X Tx f ∩ N ≠ Empty → f ∈ F0) ∧ (∀x : set, x ∈ X → ∃F : set, finite F ∧ F ⊆ P ∧ (∀f : set, f ∈ P → apply_fun f x ≠ 0 → f ∈ F) ∧ ∃n : set, n ∈ ω ∧ ∃e : set, bijection n F e ∧ ∃p : set, function_on p (ordsucc n) R ∧ apply_fun p Empty = 0 ∧ (∀k : set, k ∈ n → apply_fun p (ordsucc k) = add_SNo (apply_fun p k) (apply_fun (apply_fun e k) x)) ∧ apply_fun p n = 1) of type set → set → set → set → prop.

In Proofgold the corresponding term root is 1fa835... and object id is 529736...

Theorem. (partition_of_unity_family_sub_function_space)

∀X Tx U P : set, partition_of_unity_family X Tx U P → P ⊆ function_space X R

In Proofgold the corresponding term root is d10e39... and proposition id is c12544...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_family_continuous)

∀X Tx U P : set, partition_of_unity_family X Tx U P → ∀f : set, f ∈ P → continuous_map X Tx R R_standard_topology f

In Proofgold the corresponding term root is 1956a5... and proposition id is 77608a...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_family_unit_interval)

∀X Tx U P : set, partition_of_unity_family X Tx U P → ∀f x : set, f ∈ P → x ∈ X → apply_fun f x ∈ unit_interval

In Proofgold the corresponding term root is 40d17c... and proposition id is b7ef90...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_family_support_dominated)

∀X Tx U P : set, partition_of_unity_family X Tx U P → ∀f : set, f ∈ P → ∃u : set, u ∈ U ∧ support_of X Tx f ⊆ u

In Proofgold the corresponding term root is 211cde... and proposition id is 45173e...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_family_supports_locally_finite)

∀X Tx U P : set, partition_of_unity_family X Tx U P → ∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃F0 : set, finite F0 ∧ F0 ⊆ P ∧ ∀f : set, f ∈ P → support_of X Tx f ∩ N ≠ Empty → f ∈ F0

In Proofgold the corresponding term root is bafad7... and proposition id is da9193...

Proof:
Proof not loaded.

Theorem. (pointwise_finite_from_local_support_finite)

∀X Tx P x N F0 : set, topology_on X Tx → x ∈ X → N ∈ Tx → x ∈ N → finite F0 → F0 ⊆ P → (∀f : set, f ∈ P → support_of X Tx f ∩ N ≠ Empty → f ∈ F0) → (∀f : set, f ∈ P → apply_fun f x ≠ 0 → f ∈ F0)

In Proofgold the corresponding term root is c7fc10... and proposition id is 80c26d...

Proof:
Proof not loaded.

Theorem. (support_disjoint_open_implies_zero_on_open)

∀X Tx f N : set, topology_on X Tx → N ∈ Tx → support_of X Tx f ∩ N = Empty → ∀x : set, x ∈ N → apply_fun f x = 0

In Proofgold the corresponding term root is 04b7ff... and proposition id is d7d978...

Proof:
Proof not loaded.

Theorem. (local_support_finite_outside_F0_zero_on_open)

∀X Tx P N F0 f : set, topology_on X Tx → N ∈ Tx → F0 ⊆ P → (∀g : set, g ∈ P → support_of X Tx g ∩ N ≠ Empty → g ∈ F0) → f ∈ P → ¬ (f ∈ F0) → ∀x : set, x ∈ N → apply_fun f x = 0

In Proofgold the corresponding term root is bff500... and proposition id is b72f01...

Proof:
Proof not loaded.

Theorem. (support_of_monotone_nonzero)

∀X Tx f g : set, topology_on X Tx → (∀x : set, x ∈ X → apply_fun g x ≠ 0 → apply_fun f x ≠ 0) → support_of X Tx g ⊆ support_of X Tx f

In Proofgold the corresponding term root is 8f5105... and proposition id is 1eed61...

Proof:
Proof not loaded.

Theorem. (pointwise_finite_on_neighborhood_from_local_support_finite)

∀X Tx P N F0 : set, topology_on X Tx → N ∈ Tx → finite F0 → F0 ⊆ P → (∀f : set, f ∈ P → support_of X Tx f ∩ N ≠ Empty → f ∈ F0) → ∀y : set, y ∈ N → ∀f : set, f ∈ P → apply_fun f y ≠ 0 → f ∈ F0

In Proofgold the corresponding term root is 295b30... and proposition id is 02e072...

Proof:
Proof not loaded.

Theorem. (enumerated_finite_sum_continuous)

∀X Tx F n e : set, topology_on X Tx → n ∈ ω → bijection n F e → (∀f : set, f ∈ F → continuous_map X Tx R R_standard_topology f) → continuous_map X Tx R R_standard_topology (graph X (λx : set ⇒ nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n))

In Proofgold the corresponding term root is 31f7fc... and proposition id is 845b5e...

Proof:
Proof not loaded.

Theorem. (nat_primrec_add_mul_const_right)

∀X n e x r : set, n ∈ ω → x ∈ X → (∀k : set, k ∈ n → apply_fun (apply_fun e k) x ∈ R) → r ∈ R → mul_SNo (nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n) r = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (mul_SNo (apply_fun (apply_fun e k) x) r)) n

In Proofgold the corresponding term root is fbcebe... and proposition id is ee7957...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_times_recip_pos_eq_1)

∀X n e x : set, n ∈ ω → x ∈ X → (∀k : set, k ∈ n → apply_fun (apply_fun e k) x ∈ R) → nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n ∈ R → Rlt 0 (nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n) → nat_primrec 0 (λk acc : set ⇒ add_SNo acc (mul_SNo (apply_fun (apply_fun e k) x) (recip_SNo_pos (nat_primrec 0 (λj b : set ⇒ add_SNo b (apply_fun (apply_fun e j) x)) n)))) n = 1

In Proofgold the corresponding term root is 3d7c17... and proposition id is e3ae43...

Proof:
Proof not loaded.

Theorem. (Rle_increase_by_nonneg_left)

∀x y : set, x ∈ R → y ∈ R → Rle 0 x → Rle y (add_SNo x y)

In Proofgold the corresponding term root is 20b712... and proposition id is 596536...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_Rle0_of_Rle0_terms)

∀X n e x : set, n ∈ ω → x ∈ X → (∀k : set, k ∈ n → apply_fun (apply_fun e k) x ∈ R) → (∀k : set, k ∈ n → Rle 0 (apply_fun (apply_fun e k) x)) → nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n ∈ R ∧ Rle 0 (nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n)

In Proofgold the corresponding term root is dc2ba8... and proposition id is 63ca55...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_Rle_succ_of_Rle0_term)

∀X n e x k : set, n ∈ ω → k ∈ n → x ∈ X → (∀j : set, j ∈ n → apply_fun (apply_fun e j) x ∈ R) → (∀j : set, j ∈ n → Rle 0 (apply_fun (apply_fun e j) x)) → nat_primrec 0 (λj acc : set ⇒ add_SNo acc (apply_fun (apply_fun e j) x)) k ∈ R → Rle (nat_primrec 0 (λj acc : set ⇒ add_SNo acc (apply_fun (apply_fun e j) x)) k) (nat_primrec 0 (λj acc : set ⇒ add_SNo acc (apply_fun (apply_fun e j) x)) (ordsucc k))

In Proofgold the corresponding term root is 67d2f8... and proposition id is 4a0831...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_family_pointwise_finite_sum)

∀X Tx U P : set, partition_of_unity_family X Tx U P → ∀x : set, x ∈ X → ∃F : set, finite F ∧ F ⊆ P ∧ (∀f : set, f ∈ P → apply_fun f x ≠ 0 → f ∈ F) ∧ ∃n : set, n ∈ ω ∧ ∃e : set, bijection n F e ∧ ∃p : set, function_on p (ordsucc n) R ∧ apply_fun p Empty = 0 ∧ (∀k : set, k ∈ n → apply_fun p (ordsucc k) = add_SNo (apply_fun p k) (apply_fun (apply_fun e k) x)) ∧ apply_fun p n = 1

In Proofgold the corresponding term root is 53a2aa... and proposition id is abcabf...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_continuous_map)

∀X Tx n F e : set, topology_on X Tx → n ∈ ω → bijection n F e → (∀f : set, f ∈ F → continuous_map X Tx R R_standard_topology f) → continuous_map X Tx R R_standard_topology (graph X (λx : set ⇒ nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n))

In Proofgold the corresponding term root is 5acb1c... and proposition id is e72252...

Proof:
Proof not loaded.

Theorem. (finite_ex_bijection_from_omega)

∀F : set, finite F → ∃n e : set, n ∈ ω ∧ bijection n F e

In Proofgold the corresponding term root is b24c85... and proposition id is d633cf...

Proof:
Proof not loaded.

Theorem. (locally_finite_supports_pointwise_sum_witness)

∀X Tx P : set, topology_on X Tx → P ⊆ function_space X R → (∀f x : set, f ∈ P → x ∈ X → apply_fun f x ∈ unit_interval) → (∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃F0 : set, finite F0 ∧ F0 ⊆ P ∧ ∀f : set, f ∈ P → support_of X Tx f ∩ N ≠ Empty → f ∈ F0) → ∀x : set, x ∈ X → ∃F : set, finite F ∧ F ⊆ P ∧ (∀f : set, f ∈ P → apply_fun f x ≠ 0 → f ∈ F) ∧ ∃n : set, n ∈ ω ∧ ∃e : set, bijection n F e ∧ ∃p : set, function_on p (ordsucc n) R ∧ apply_fun p Empty = 0 ∧ (∀k : set, k ∈ n → apply_fun p (ordsucc k) = add_SNo (apply_fun p k) (apply_fun (apply_fun e k) x))

In Proofgold the corresponding term root is 3f01b6... and proposition id is 4ac145...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_dominated_ex_family_pred)

∀X Tx U : set, partition_of_unity_dominated X Tx U → ∃P : set, partition_of_unity_family X Tx U P

In Proofgold the corresponding term root is f6f0e5... and proposition id is b07b15...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_dominated_topology_open_cover)

∀X Tx U : set, partition_of_unity_dominated X Tx U → topology_on X Tx ∧ open_cover X Tx U

In Proofgold the corresponding term root is 8557cb... and proposition id is 109cc3...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_dominated_topology)

∀X Tx U : set, partition_of_unity_dominated X Tx U → topology_on X Tx

In Proofgold the corresponding term root is 525cd5... and proposition id is 88e347...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_dominated_open_cover)

∀X Tx U : set, partition_of_unity_dominated X Tx U → open_cover X Tx U

In Proofgold the corresponding term root is 7f451d... and proposition id is 1321dc...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_dominated_ex_family)

∀X Tx U : set, partition_of_unity_dominated X Tx U → ∃P : set, P ⊆ function_space X R ∧ (∀f : set, f ∈ P → continuous_map X Tx R R_standard_topology f) ∧ (∀f x : set, f ∈ P → x ∈ X → apply_fun f x ∈ unit_interval) ∧ (∀f : set, f ∈ P → ∃u : set, u ∈ U ∧ support_of X Tx f ⊆ u) ∧ (∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃F0 : set, finite F0 ∧ F0 ⊆ P ∧ ∀f : set, f ∈ P → support_of X Tx f ∩ N ≠ Empty → f ∈ F0) ∧ (∀x : set, x ∈ X → ∃F : set, finite F ∧ F ⊆ P ∧ (∀f : set, f ∈ P → apply_fun f x ≠ 0 → f ∈ F) ∧ ∃n : set, n ∈ ω ∧ ∃e : set, bijection n F e ∧ ∃p : set, function_on p (ordsucc n) R ∧ apply_fun p Empty = 0 ∧ (∀k : set, k ∈ n → apply_fun p (ordsucc k) = add_SNo (apply_fun p k) (apply_fun (apply_fun e k) x)) ∧ apply_fun p n = 1)

In Proofgold the corresponding term root is 0ac6eb... and proposition id is 6d17d9...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_dominated_cover_sub_Power)

∀X Tx U : set, partition_of_unity_dominated X Tx U → U ⊆ 𝒫 X

In Proofgold the corresponding term root is a03688... and proposition id is c2d82e...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_dominated_covers)

∀X Tx U : set, partition_of_unity_dominated X Tx U → covers X U

In Proofgold the corresponding term root is 1a827a... and proposition id is b03a72...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_congr_on)

∀X n e1 e2 x : set, nat_p n → x ∈ X → (∀k : set, k ∈ n → apply_fun e1 k = apply_fun e2 k) → nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e1 k) x)) n = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e2 k) x)) n

In Proofgold the corresponding term root is 6ecebf... and proposition id is 6247aa...

Proof:
Proof not loaded.

Definition. We define swap_last_two to be λe m ⇒ graph (ordsucc (ordsucc m)) (λi : set ⇒ if i = m then apply_fun e (ordsucc m) else if i = ordsucc m then apply_fun e m else apply_fun e i) of type set → set → set.

In Proofgold the corresponding term root is 04bfa2... and object id is b0a492...

Theorem. (swap_last_two_apply)

∀e m i : set, i ∈ ordsucc (ordsucc m) → apply_fun (swap_last_two e m) i = if i = m then apply_fun e (ordsucc m) else if i = ordsucc m then apply_fun e m else apply_fun e i

In Proofgold the corresponding term root is d7fd9c... and proposition id is b62fc7...

Proof:
Proof not loaded.

Theorem. (swap_last_two_at_m)

∀e m : set, nat_p m → apply_fun (swap_last_two e m) m = apply_fun e (ordsucc m)

In Proofgold the corresponding term root is aefd21... and proposition id is 908c6d...

Proof:
Proof not loaded.

Theorem. (swap_last_two_at_sm)

∀e m : set, nat_p m → apply_fun (swap_last_two e m) (ordsucc m) = apply_fun e m

In Proofgold the corresponding term root is 164b0c... and proposition id is 65b232...

Proof:
Proof not loaded.

Theorem. (function_on_swap_last_two)

∀F e m : set, nat_p m → function_on e (ordsucc (ordsucc m)) F → function_on (swap_last_two e m) (ordsucc (ordsucc m)) F

In Proofgold the corresponding term root is 2ecacb... and proposition id is 99833a...

Proof:
Proof not loaded.

Theorem. (bijection_swap_last_two)

∀F e m : set, nat_p m → bijection (ordsucc (ordsucc m)) F e → bijection (ordsucc (ordsucc m)) F (swap_last_two e m)

In Proofgold the corresponding term root is 9b2d25... and proposition id is ea4ea0...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_swap_last_two_unit_interval)

∀X m e x : set, m ∈ ω → x ∈ X → (∀k : set, k ∈ ordsucc (ordsucc m) → apply_fun (apply_fun e k) x ∈ unit_interval) → nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun (swap_last_two e m) k) x)) (ordsucc (ordsucc m)) = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) (ordsucc (ordsucc m))

In Proofgold the corresponding term root is c173c5... and proposition id is f1f73e...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_S_drop_zero_unit_interval)

∀X n e x : set, n ∈ ω → x ∈ X → (∀k : set, k ∈ ordsucc n → apply_fun (apply_fun e k) x ∈ unit_interval) → apply_fun (apply_fun e n) x = 0 → nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) (ordsucc n) = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n

In Proofgold the corresponding term root is 013493... and proposition id is e45744...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_drop_last_zero_bijection_unit_interval)

∀X n F e x : set, n ∈ ω → x ∈ X → (∀k : set, k ∈ ordsucc n → apply_fun (apply_fun e k) x ∈ unit_interval) → apply_fun (apply_fun e n) x = 0 → bijection (ordsucc n) F e → bijection n (F ∖ {apply_fun e n}) e ∧ nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) (ordsucc n) = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n

In Proofgold the corresponding term root is ce528d... and proposition id is 7d9d80...

Proof:
Proof not loaded.

Definition. We define swap_adjacent to be λe i n ⇒ graph n (λj : set ⇒ if j = i then apply_fun e (ordsucc i) else if j = ordsucc i then apply_fun e i else apply_fun e j) of type set → set → set → set.

In Proofgold the corresponding term root is afd596... and object id is f43193...

Theorem. (swap_adjacent_apply)

∀e i n j : set, j ∈ n → apply_fun (swap_adjacent e i n) j = if j = i then apply_fun e (ordsucc i) else if j = ordsucc i then apply_fun e i else apply_fun e j

In Proofgold the corresponding term root is db4219... and proposition id is c6585b...

Proof:
Proof not loaded.

Theorem. (swap_adjacent_at_succ)

∀e i n : set, ordsucc i ∈ n → apply_fun (swap_adjacent e i n) (ordsucc i) = apply_fun e i

In Proofgold the corresponding term root is 8eda27... and proposition id is 2962fd...

Proof:
Proof not loaded.

Theorem. (swap_adjacent_at_i)

∀e i n : set, i ∈ n → apply_fun (swap_adjacent e i n) i = apply_fun e (ordsucc i)

In Proofgold the corresponding term root is 4ba961... and proposition id is 502571...

Proof:
Proof not loaded.

Theorem. (function_on_swap_adjacent)

∀F e i n : set, i ∈ n → ordsucc i ∈ n → function_on e n F → function_on (swap_adjacent e i n) n F

In Proofgold the corresponding term root is 9b7ce4... and proposition id is cef97c...

Proof:
Proof not loaded.

Theorem. (bijection_swap_adjacent)

∀F e i n : set, i ∈ n → ordsucc i ∈ n → bijection n F e → bijection n F (swap_adjacent e i n)

In Proofgold the corresponding term root is 4177fb... and proposition id is 1394b7...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_swap_adjacent_unit_interval)

∀X n e i x : set, n ∈ ω → i ∈ n → ordsucc i ∈ n → x ∈ X → (∀k : set, k ∈ n → apply_fun (apply_fun e k) x ∈ unit_interval) → nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun (swap_adjacent e i n) k) x)) n = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n

In Proofgold the corresponding term root is 9b7291... and proposition id is 463cd7...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_swap_adjacent_bijection_unit_interval)

∀X n F e i x : set, n ∈ ω → i ∈ n → ordsucc i ∈ n → x ∈ X → bijection n F e → (∀k : set, k ∈ n → apply_fun (apply_fun e k) x ∈ unit_interval) → bijection n F (swap_adjacent e i n) ∧ nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun (swap_adjacent e i n) k) x)) n = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n

In Proofgold the corresponding term root is 8d4515... and proposition id is ffe978...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_bubble_zero_right_unit_interval)

∀X n F e i x : set, n ∈ ω → i ∈ n → ordsucc i ∈ n → x ∈ X → bijection n F e → (∀k : set, k ∈ n → apply_fun (apply_fun e k) x ∈ unit_interval) → apply_fun (apply_fun e i) x = 0 → ∃e' : set, e' = swap_adjacent e i n ∧ bijection n F e' ∧ nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e' k) x)) n = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n ∧ apply_fun (apply_fun e' (ordsucc i)) x = 0

In Proofgold the corresponding term root is 7ee1cd... and proposition id is 198e3a...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_bijection_move_value_to_last_unit_interval)

∀X m F e flast x : set, m ∈ ω → x ∈ X → bijection (ordsucc m) F e → flast ∈ F → (∀f : set, f ∈ F → apply_fun f x ∈ unit_interval) → ∃e' : set, bijection (ordsucc m) F e' ∧ nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) (ordsucc m) = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e' k) x)) (ordsucc m) ∧ apply_fun e' m = flast

In Proofgold the corresponding term root is 50daeb... and proposition id is 80b8d1...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_bijection_Rle1_if_some_term_eq1_unit_interval)

∀X x m F e f1 : set, m ∈ ω → x ∈ X → bijection (ordsucc m) F e → f1 ∈ F → (∀f : set, f ∈ F → apply_fun f x ∈ unit_interval) → apply_fun f1 x = 1 → Rle 1 (nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) (ordsucc m))

In Proofgold the corresponding term root is 5785f9... and proposition id is 40a86d...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_bijection_Rlt0_if_some_term_eq1_unit_interval)

∀X x m F e f1 : set, m ∈ ω → x ∈ X → bijection (ordsucc m) F e → f1 ∈ F → (∀f : set, f ∈ F → apply_fun f x ∈ unit_interval) → apply_fun f1 x = 1 → Rlt 0 (nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) (ordsucc m))

In Proofgold the corresponding term root is 910867... and proposition id is dea9db...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_bijection_independent_unit_interval)

∀X m F e1 e2 x : set, m ∈ ω → x ∈ X → bijection (ordsucc m) F e1 → bijection (ordsucc m) F e2 → (∀f : set, f ∈ F → apply_fun f x ∈ unit_interval) → nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e1 k) x)) (ordsucc m) = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e2 k) x)) (ordsucc m)

In Proofgold the corresponding term root is 26ac9d... and proposition id is e0dd95...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_bijection_drop_zero_element_unit_interval)

∀X m F e f0 x : set, m ∈ ω → x ∈ X → bijection (ordsucc m) F e → f0 ∈ F → (∀f : set, f ∈ F → apply_fun f x ∈ unit_interval) → apply_fun f0 x = 0 → ∃e' : set, bijection m (F ∖ {f0}) e' ∧ nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) (ordsucc m) = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e' k) x)) m

In Proofgold the corresponding term root is 952947... and proposition id is 28ff73...

Proof:
Proof not loaded.

Theorem. (equip_omega_eq)

∀n m : set, n ∈ ω → m ∈ ω → equip n m → n = m

In Proofgold the corresponding term root is a72fd8... and proposition id is ec71bb...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_bijection_zero_outside_unit_interval)

∀X x n0 F0 e0 n F e : set, x ∈ X → n0 ∈ ω → finite F → F0 ⊆ F → (∃f : set, f ∈ F0) → (∀f : set, f ∈ F0 → apply_fun f x ∈ unit_interval) → (∀f : set, f ∈ F → ¬ (f ∈ F0) → apply_fun f x = 0) → bijection n0 F0 e0 → n ∈ ω → bijection n F e → nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e0 k) x)) n0 = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e k) x)) n

In Proofgold the corresponding term root is 491d00... and proposition id is 612d20...

Proof:
Proof not loaded.

Theorem. (nat_primrec_sum_bijection_equal_of_two_control_sets_unit_interval)

∀X x n0 F0 e0 n1 F1 e1 n F e : set, x ∈ X → n0 ∈ ω → n1 ∈ ω → n ∈ ω → finite F → F0 ⊆ F → F1 ⊆ F → (∃f : set, f ∈ F0) → (∃f : set, f ∈ F1) → (∀f : set, f ∈ F0 → apply_fun f x ∈ unit_interval) → (∀f : set, f ∈ F1 → apply_fun f x ∈ unit_interval) → (∀f : set, f ∈ F → ¬ (f ∈ F0) → apply_fun f x = 0) → (∀f : set, f ∈ F → ¬ (f ∈ F1) → apply_fun f x = 0) → bijection n0 F0 e0 → bijection n1 F1 e1 → bijection n F e → nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e0 k) x)) n0 = nat_primrec 0 (λk acc : set ⇒ add_SNo acc (apply_fun (apply_fun e1 k) x)) n1

In Proofgold the corresponding term root is 2bb24a... and proposition id is 8ac619...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_dominated_supports_locally_finite)

In Proofgold the corresponding term root is 283295... and proposition id is 4fdd95...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_dominated_pointwise_finite_sum)

∀X Tx U : set, partition_of_unity_dominated X Tx U → ∃P : set, ∀x : set, x ∈ X → ∃F : set, finite F ∧ F ⊆ P ∧ (∀f : set, f ∈ P → apply_fun f x ≠ 0 → f ∈ F) ∧ ∃n : set, n ∈ ω ∧ ∃e : set, bijection n F e ∧ ∃p : set, function_on p (ordsucc n) R ∧ apply_fun p Empty = 0 ∧ (∀k : set, k ∈ n → apply_fun p (ordsucc k) = add_SNo (apply_fun p k) (apply_fun (apply_fun e k) x)) ∧ apply_fun p n = 1

In Proofgold the corresponding term root is 177882... and proposition id is 6c8dd7...

Proof:
Proof not loaded.

Definition. We define locally_finite_family to be λX Tx F ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃S : set, finite S ∧ S ⊆ F ∧ ∀A : set, A ∈ F → A ∩ N ≠ Empty → A ∈ S of type set → set → set → prop.

In Proofgold the corresponding term root is 39b973... and object id is 0eacf7...

Theorem. (locally_finite_family_topology)

∀X Tx F : set, locally_finite_family X Tx F → topology_on X Tx

In Proofgold the corresponding term root is 53e45b... and proposition id is d96aa9...

Proof:
Proof not loaded.

Theorem. (locally_finite_family_property)

∀X Tx F : set, locally_finite_family X Tx F → ∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃S : set, finite S ∧ S ⊆ F ∧ ∀A : set, A ∈ F → A ∩ N ≠ Empty → A ∈ S

In Proofgold the corresponding term root is e0762f... and proposition id is 528dfa...

Proof:
Proof not loaded.

Theorem. (locally_finite_open_cover_neighborhood_finite_subunion)

∀X Tx W x : set, open_cover X Tx W → locally_finite_family X Tx W → x ∈ X → ∃N S : set, N ∈ Tx ∧ x ∈ N ∧ finite S ∧ S ⊆ W ∧ N ⊆ ⋃ S

In Proofgold the corresponding term root is 732b17... and proposition id is d0c4c4...

Proof:
Proof not loaded.

Theorem. (locally_finite_open_cover_closure_point_finite)

∀X Tx W x : set, open_cover X Tx W → locally_finite_family X Tx W → x ∈ X → ∃S : set, finite S ∧ S ⊆ W ∧ ∀w : set, w ∈ W → x ∈ closure_of X Tx w → w ∈ S

In Proofgold the corresponding term root is 4329c1... and proposition id is 82fbad...

Proof:
Proof not loaded.

Theorem. (locally_finite_subfamily)

∀X Tx F G : set, locally_finite_family X Tx F → G ⊆ F → locally_finite_family X Tx G

In Proofgold the corresponding term root is af8704... and proposition id is 2c9e4f...

Proof:
Proof not loaded.

Theorem. (locally_finite_family_Repl_subsets)

∀X Tx W : set, ∀f : set → set, locally_finite_family X Tx W → (∀w : set, w ∈ W → f w ⊆ w) → locally_finite_family X Tx {f w|w ∈ W}

In Proofgold the corresponding term root is d8f8a7... and proposition id is 7f1715...

Proof:
Proof not loaded.

Theorem. (locally_finite_family_point_finite_ex)

∀X Tx F : set, locally_finite_family X Tx F → ∀x : set, x ∈ X → ∃S : set, finite S ∧ S ⊆ F ∧ ∀A : set, A ∈ F → x ∈ A → A ∈ S

In Proofgold the corresponding term root is 24d9eb... and proposition id is e96591...

Proof:
Proof not loaded.

Theorem. (finite_family_locally_finite)

∀X Tx F : set, topology_on X Tx → finite F → F ⊆ 𝒫 X → locally_finite_family X Tx F

In Proofgold the corresponding term root is 65878d... and proposition id is 76770c...

Proof:
Proof not loaded.

Theorem. (finite_open_cover_locally_finite)

∀X Tx U : set, topology_on X Tx → open_cover X Tx U → finite U → locally_finite_family X Tx U

In Proofgold the corresponding term root is 95f995... and proposition id is 98f9da...

Proof:
Proof not loaded.

Theorem. (finite_open_cover_of_locally_finite)

∀X Tx Fam : set, open_cover_of X Tx Fam → finite Fam → locally_finite_family X Tx Fam

In Proofgold the corresponding term root is 605ecf... and proposition id is 704486...

Proof:
Proof not loaded.

Theorem. (locally_finite_family_closures)

∀X Tx Fam : set, locally_finite_family X Tx Fam → locally_finite_family X Tx {closure_of X Tx A|A ∈ Fam}

In Proofgold the corresponding term root is 86e962... and proposition id is 37c1eb...

Proof:
Proof not loaded.

Theorem. (Union_locally_finite_closed_is_closed)

∀X Tx Fam : set, locally_finite_family X Tx Fam → (∀A : set, A ∈ Fam → closed_in X Tx A) → closed_in X Tx (⋃ Fam)

In Proofgold the corresponding term root is 676699... and proposition id is eb3105...

Proof:
Proof not loaded.

Theorem. (locally_finite_open_cover_closures_closed_cover)

∀X Tx W : set, open_cover X Tx W → locally_finite_family X Tx W → ∃ClW : set, ClW = {closure_of X Tx w|w ∈ W} ∧ ((∀C : set, C ∈ ClW → closed_in X Tx C) ∧ covers X ClW) ∧ locally_finite_family X Tx ClW ∧ (∀C : set, C ∈ ClW → ∃w : set, w ∈ W ∧ C = closure_of X Tx w)

In Proofgold the corresponding term root is cac96f... and proposition id is 391f21...

Proof:
Proof not loaded.

Theorem. (normal_space_finite_open_cover_shrinking)

∀X Tx W : set, normal_space X Tx → open_cover X Tx W → finite W → ∃V : set, open_cover X Tx V ∧ finite V ∧ (∀v : set, v ∈ V → ∃w : set, w ∈ W ∧ v ⊆ w) ∧ ∀v : set, v ∈ V → ∃w : set, w ∈ W ∧ closure_of X Tx v ⊆ w

In Proofgold the corresponding term root is 2d2d78... and proposition id is 453cb0...

Proof:
Proof not loaded.

Theorem. (normal_space_finite_open_cover_shrinking_closed)

∀X Tx A F : set, normal_space X Tx → closed_in X Tx A → finite F → (∀w : set, w ∈ F → w ∈ Tx) → A ⊆ ⋃ F → ∃V : set, finite V ∧ (∀v : set, v ∈ V → v ∈ Tx) ∧ A ⊆ ⋃ V ∧ ∀v : set, v ∈ V → ∃w : set, w ∈ F ∧ closure_of X Tx v ⊆ w

In Proofgold the corresponding term root is fde635... and proposition id is cf0cd9...

Proof:
Proof not loaded.

Theorem. (locally_finite_open_cover_core_open_set)

∀X Tx W w : set, topology_on X Tx → open_cover X Tx W → locally_finite_family X Tx W → w ∈ W → ∃Bw : set, Bw ∈ Tx ∧ Bw ⊆ w ∧ closure_of X Tx Bw ⊆ w

In Proofgold the corresponding term root is 9c5058... and proposition id is 64a78b...

Proof:
Proof not loaded.

Theorem. (locally_finite_open_cover_core_family)

∀X Tx W : set, topology_on X Tx → open_cover X Tx W → locally_finite_family X Tx W → ∃B : set, (∀b : set, b ∈ B → b ∈ Tx) ∧ locally_finite_family X Tx B ∧ (∀b : set, b ∈ B → ∃w : set, w ∈ W ∧ b ⊆ w) ∧ ∀b : set, b ∈ B → ∃w : set, w ∈ W ∧ closure_of X Tx b ⊆ w

In Proofgold the corresponding term root is 425017... and proposition id is 37dde9...

Proof:
Proof not loaded.

Theorem. (nonempty_subset_of_ordinal_has_least)

∀I S : set, ordinal I → S ⊆ I → S ≠ Empty → ∃m : set, m ∈ S ∧ ∀s : set, s ∈ S → m ⊆ s

In Proofgold the corresponding term root is a3964e... and proposition id is 94bd74...

Proof:
Proof not loaded.

Theorem. (ordinal_subset_least_unique)

∀I S m1 m2 : set, ordinal I → S ⊆ I → m1 ∈ S → (∀s : set, s ∈ S → m1 ⊆ s) → m2 ∈ S → (∀s : set, s ∈ S → m2 ⊆ s) → m1 = m2

In Proofgold the corresponding term root is 9946f0... and proposition id is fa12ed...

Proof:
Proof not loaded.

Theorem. (finite_nonempty_subset_of_ordinal_has_max)

∀I S : set, ordinal I → finite S → S ⊆ I → S ≠ Empty → ∃m : set, m ∈ S ∧ ∀s : set, s ∈ S → s ⊆ m

In Proofgold the corresponding term root is e9a118... and proposition id is 79f6e1...

Proof:
Proof not loaded.

Theorem. (point_finite_cover_has_max_index)

∀X W I : set, ∀idx pickW : set → set, ordinal I → covers X W → (∀x : set, x ∈ X → ∃S : set, finite S ∧ S ⊆ W ∧ ∀A : set, A ∈ W → x ∈ A → A ∈ S) → (∀w : set, w ∈ W → idx w ∈ I) → (∀i : set, i ∈ I → pickW i ∈ W ∧ idx (pickW i) = i) → (∀w : set, w ∈ W → pickW (idx w) = w) → ∀x : set, x ∈ X → ∃imax : set, imax ∈ I ∧ x ∈ pickW imax ∧ ∀j : set, j ∈ I → imax ∈ j → x ∉ pickW j

In Proofgold the corresponding term root is 1c3f12... and proposition id is 2c6ff0...

Proof:
Proof not loaded.

Theorem. (normal_locally_finite_open_cover_shrinking_rec)

∀X Tx W : set, normal_space X Tx → open_cover X Tx W → locally_finite_family X Tx W → ∃V : set, open_cover X Tx V ∧ locally_finite_family X Tx V ∧ (∀v : set, v ∈ V → ∃w : set, w ∈ W ∧ v ⊆ w) ∧ ∀v : set, v ∈ V → ∃w : set, w ∈ W ∧ closure_of X Tx v ⊆ w

In Proofgold the corresponding term root is 99baec... and proposition id is 663737...

Proof:
Proof not loaded.

Theorem. (normal_locally_finite_open_cover_shrinking)

In Proofgold the corresponding term root is 99baec... and proposition id is 663737...

Proof:
Proof not loaded.

Definition. We define refine_of to be λV U ⇒ ∀v : set, v ∈ V → ∃u : set, u ∈ U ∧ v ⊆ u of type set → set → prop.

In Proofgold the corresponding term root is 784b8a... and object id is 2c5af6...

Axiom. (function_space_extensional) We take the following as an axiom:

∀X Y f g : set, f ∈ function_space X Y → g ∈ function_space X Y → (∀x : set, x ∈ X → apply_fun f x = apply_fun g x) → f = g

In Proofgold the corresponding term root is 128796... and proposition id is 0428c8...

Theorem. (locally_finite_bump_cover_normalizes_to_partition_of_unity_dominated)

∀X Tx U V P : set, topology_on X Tx → open_cover X Tx U → open_cover X Tx V → locally_finite_family X Tx V → refine_of V U → P ⊆ function_space X R ∧ (∀f : set, f ∈ P → continuous_map X Tx R R_standard_topology f) ∧ (∀f x : set, f ∈ P → x ∈ X → apply_fun f x ∈ unit_interval) ∧ (∀f : set, f ∈ P → ∃u : set, u ∈ U ∧ support_of X Tx f ⊆ u) ∧ (∀x : set, x ∈ X → ∃f : set, f ∈ P ∧ apply_fun f x = 1) ∧ (∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃F0 : set, finite F0 ∧ F0 ⊆ P ∧ ∀f : set, f ∈ P → support_of X Tx f ∩ N ≠ Empty → f ∈ F0) → partition_of_unity_dominated X Tx U

In Proofgold the corresponding term root is c38ae9... and proposition id is df4399...

Proof:
Proof not loaded.

Theorem. (normal_locally_finite_partition_of_unity_family_rec)

∀X Tx W : set, normal_space X Tx → open_cover X Tx W → locally_finite_family X Tx W → ∃P : set, partition_of_unity_family X Tx W P

In Proofgold the corresponding term root is 703b9d... and proposition id is 40de8a...

Proof:
Proof not loaded.

Theorem. (normal_locally_finite_partition_of_unity_family)

∀X Tx W : set, normal_space X Tx → open_cover X Tx W → locally_finite_family X Tx W → ∃P : set, partition_of_unity_family X Tx W P

In Proofgold the corresponding term root is 703b9d... and proposition id is 40de8a...

Proof:
Proof not loaded.

Theorem. (finite_open_cover_has_partition_of_unity_family)

∀X Tx U : set, normal_space X Tx → finite U → open_cover X Tx U → ∃P : set, partition_of_unity_family X Tx U P

In Proofgold the corresponding term root is cd4e44... and proposition id is 8b05d6...

Proof:
Proof not loaded.

Theorem. (finite_partition_of_unity_exists)

∀X Tx U : set, normal_space X Tx → finite U → open_cover X Tx U → partition_of_unity_dominated X Tx U

In Proofgold the corresponding term root is 875452... and proposition id is 892d3e...

Proof:
Proof not loaded.

Theorem. (net_pack_coord_projection_in_space)

∀I Xi N i : set, net_pack_in_space (product_space I Xi) N → i ∈ I → ∃Ni : set, net_pack_in_space (space_family_set Xi i) Ni ∧ net_pack_index Ni = net_pack_index N ∧ net_pack_le Ni = net_pack_le N ∧ Ni = net_pack (net_pack_index N) (net_pack_le N) (graph (net_pack_index N) (λj : set ⇒ apply_fun (apply_fun (net_pack_fun N) j) i))

In Proofgold the corresponding term root is aa870f... and proposition id is 84b9a1...

Proof:
Proof not loaded.

Definition. We define finite_intersection_property to be λX A ⇒ A ⊆ 𝒫 X ∧ ∀F : set, finite F → F ⊆ A → intersection_of_family X F ≠ Empty of type set → set → prop.

In Proofgold the corresponding term root is f36a49... and object id is e35eac...

Definition. We define maximal_finite_intersection_property to be λX D ⇒ finite_intersection_property X D ∧ ∀E : set, D ⊆ E → finite_intersection_property X E → E ⊆ D of type set → set → prop.

In Proofgold the corresponding term root is 707df0... and object id is 821224...

Theorem. (Lemma37_1_maximal_fip_extension)

∀X A : set, finite_intersection_property X A → ∃D : set, A ⊆ D ∧ maximal_finite_intersection_property X D

In Proofgold the corresponding term root is bf52d2... and proposition id is 8ea006...

Proof:
Proof not loaded.

Theorem. (Lemma37_2a_max_fip_finite_intersections_in)

∀X D : set, maximal_finite_intersection_property X D → ∀F : set, finite F → F ⊆ D → intersection_of_family X F ∈ D

In Proofgold the corresponding term root is 5e1a9c... and proposition id is edfba8...

Proof:
Proof not loaded.

Theorem. (Lemma37_2b_max_fip_meets_all_implies_in)

∀X D A : set, maximal_finite_intersection_property X D → A ⊆ X → (∀B : set, B ∈ D → A ∩ B ≠ Empty) → A ∈ D

In Proofgold the corresponding term root is e7a873... and proposition id is 79e88c...

Proof:
Proof not loaded.

Theorem. (compact_space_closed_FIP_intersection_nonempty)

∀X Tx D : set, compact_space X Tx → (∀C : set, C ∈ D → closed_in X Tx C) → finite_intersection_property X D → intersection_of_family X D ≠ Empty

In Proofgold the corresponding term root is 9a00e9... and proposition id is e3a395...

Proof:
Proof not loaded.

Theorem. (compact_space_of_closed_FIP_intersection_nonempty)

∀X Tx : set, topology_on X Tx → (∀D : set, (∀C : set, C ∈ D → closed_in X Tx C) → finite_intersection_property X D → intersection_of_family X D ≠ Empty) → compact_space X Tx

In Proofgold the corresponding term root is 5bfb2e... and proposition id is dd7b23...

Proof:
Proof not loaded.

Theorem. (net_pack_coord_projection_has_convergent_subnet_in_compact_factor)

∀I Xi N i : set, net_pack_in_space (product_space I Xi) N → i ∈ I → compact_space (product_component Xi i) (product_component_topology Xi i) → ∃Ni Si x : set, net_pack_in_space (space_family_set Xi i) Ni ∧ net_pack_index Ni = net_pack_index N ∧ net_pack_le Ni = net_pack_le N ∧ Ni = net_pack (net_pack_index N) (net_pack_le N) (graph (net_pack_index N) (λj : set ⇒ apply_fun (apply_fun (net_pack_fun N) j) i)) ∧ subnet_pack_of_in (space_family_set Xi i) Ni Si ∧ net_pack_converges (space_family_set Xi i) (space_family_topology Xi i) Si x

In Proofgold the corresponding term root is 376604... and proposition id is eaf181...

Proof:
Proof not loaded.

Axiom. (Tychonoff_coordinate_subnets_to_common_subnet) We take the following as an axiom:

∀I Xi N x : set, I ≠ Empty → net_pack_in_space (product_space I Xi) N → x ∈ product_space I Xi → (∀i : set, i ∈ I → ∃Ni Si : set, net_pack_in_space (space_family_set Xi i) Ni ∧ net_pack_index Ni = net_pack_index N ∧ net_pack_le Ni = net_pack_le N ∧ Ni = net_pack (net_pack_index N) (net_pack_le N) (graph (net_pack_index N) (λj : set ⇒ apply_fun (apply_fun (net_pack_fun N) j) i)) ∧ subnet_pack_of_in (space_family_set Xi i) Ni Si ∧ net_pack_converges (space_family_set Xi i) (space_family_topology Xi i) Si (apply_fun x i)) → ∃J le net : set, subnet_pack_of_in (product_space I Xi) N (net_pack J le net) ∧ (∀i : set, i ∈ I → net_converges_on (space_family_set Xi i) (space_family_topology Xi i) (compose_fun J net (product_eval_map I Xi i)) J le (apply_fun x i))

In Proofgold the corresponding term root is 47632d... and proposition id is 767d06...

Theorem. (Tychonoff_coordinate_subnets_to_common_subnet_sketch)

In Proofgold the corresponding term root is 47632d... and proposition id is 767d06...

Proof:
Proof not loaded.

Theorem. (Tychonoff_diagonalize_from_coordinate_subnets)

∀I Xi N : set, I ≠ Empty → net_pack_in_space (product_space I Xi) N → (∀i : set, i ∈ I → ∃Ni Si x : set, net_pack_in_space (space_family_set Xi i) Ni ∧ net_pack_index Ni = net_pack_index N ∧ net_pack_le Ni = net_pack_le N ∧ Ni = net_pack (net_pack_index N) (net_pack_le N) (graph (net_pack_index N) (λj : set ⇒ apply_fun (apply_fun (net_pack_fun N) j) i)) ∧ subnet_pack_of_in (space_family_set Xi i) Ni Si ∧ net_pack_converges (space_family_set Xi i) (space_family_topology Xi i) Si x) → ∃S x : set, subnet_pack_of_in (product_space I Xi) N S ∧ net_pack_converges (product_space I Xi) (product_topology_full I Xi) S x

In Proofgold the corresponding term root is 6636ee... and proposition id is ffb43f...

Proof:
Proof not loaded.

Theorem. (Tychonoff_theorem)

∀I Xi : set, (∀i : set, i ∈ I → compact_space (product_component Xi i) (product_component_topology Xi i)) → compact_space (product_space I Xi) (product_topology_full I Xi)

In Proofgold the corresponding term root is 8d50bf... and proposition id is b54569...

Proof:
Proof not loaded.

Definition. We define Stone_Cech_index to be λX Tx ⇒ {f ∈ function_space X unit_interval|continuous_map X Tx unit_interval unit_interval_topology f} of type set → set → set.

In Proofgold the corresponding term root is 6887ba... and object id is 3c59eb...

Definition. We define Stone_Cech_ambient_family to be λX Tx ⇒ const_space_family (Stone_Cech_index X Tx) unit_interval unit_interval_topology of type set → set → set.

In Proofgold the corresponding term root is 1503bd... and object id is ebcef2...

Definition. We define Stone_Cech_ambient_space to be λX Tx ⇒ product_space (Stone_Cech_index X Tx) (Stone_Cech_ambient_family X Tx) of type set → set → set.

In Proofgold the corresponding term root is 7ef718... and object id is 548a5f...

Definition. We define Stone_Cech_ambient_topology to be λX Tx ⇒ product_topology_full (Stone_Cech_index X Tx) (Stone_Cech_ambient_family X Tx) of type set → set → set.

In Proofgold the corresponding term root is 2fc1c0... and object id is cafbde...

Definition. We define Stone_Cech_image to be λX Tx ⇒ {p ∈ Stone_Cech_ambient_space X Tx|∃x : set, x ∈ X ∧ p = graph (Stone_Cech_index X Tx) (λf : set ⇒ apply_fun f x)} of type set → set → set.

In Proofgold the corresponding term root is cdb4ae... and object id is 5c65cd...

Definition. We define Stone_Cech_compactification to be λX Tx ⇒ closure_of (Stone_Cech_ambient_space X Tx) (Stone_Cech_ambient_topology X Tx) (Stone_Cech_image X Tx) of type set → set → set.

In Proofgold the corresponding term root is 8c8852... and object id is 4bdf83...

Theorem. (Stone_Cech_universal_property)

∀X Tx : set, Tychonoff_space X Tx → ∃Ty : set, compact_space (Stone_Cech_compactification X Tx) Ty ∧ Hausdorff_space (Stone_Cech_compactification X Tx) Ty

In Proofgold the corresponding term root is d9ffdc... and proposition id is 0bc16e...

Proof:
Proof not loaded.

Theorem. (refine_trans)

∀U V W : set, refine_of W V → refine_of V U → refine_of W U

In Proofgold the corresponding term root is 2cfb1f... and proposition id is 8eeaf2...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_family_refine)

∀X Tx U V P : set, partition_of_unity_family X Tx V P → refine_of V U → partition_of_unity_family X Tx U P

In Proofgold the corresponding term root is fc892d... and proposition id is 44fb44...

Proof:
Proof not loaded.

Theorem. (partition_of_unity_dominated_refine)

∀X Tx U V : set, open_cover X Tx U → partition_of_unity_dominated X Tx V → refine_of V U → partition_of_unity_dominated X Tx U

In Proofgold the corresponding term root is 24db5a... and proposition id is 382023...

Proof:
Proof not loaded.

Theorem. (locally_finite_family_empty)

∀X Tx : set, topology_on X Tx → locally_finite_family X Tx Empty

In Proofgold the corresponding term root is 672613... and proposition id is aea787...

Proof:
Proof not loaded.

Theorem. (finite_Repl)

∀X : set, finite X → ∀F : set → set, finite {F x|x ∈ X}

In Proofgold the corresponding term root is 27c424... and proposition id is 51caf9...

Proof:
Proof not loaded.

Theorem. (locally_finite_family_closure_image)

∀X Tx Fam : set, locally_finite_family X Tx Fam → locally_finite_family X Tx {closure_of X Tx A|A ∈ Fam}

In Proofgold the corresponding term root is 86e962... and proposition id is 37c1eb...

Proof:
Proof not loaded.

Theorem. (finite_family_locally_finite_family)

∀X Tx F : set, topology_on X Tx → finite F → locally_finite_family X Tx F

In Proofgold the corresponding term root is 3be041... and proposition id is 0a390f...

Proof:
Proof not loaded.

Theorem. (finite_open_cover_locally_finite_family)

∀X Tx U : set, topology_on X Tx → open_cover X Tx U → finite U → locally_finite_family X Tx U

In Proofgold the corresponding term root is 95f995... and proposition id is 98f9da...

Proof:
Proof not loaded.

Theorem. (normal_locally_finite_partition_of_unity_dominated)

∀X Tx W : set, normal_space X Tx → open_cover X Tx W → locally_finite_family X Tx W → partition_of_unity_dominated X Tx W

In Proofgold the corresponding term root is 5aa847... and proposition id is 7d0dda...

Proof:
Proof not loaded.

Theorem. (normal_locally_finite_refinement_partition_of_unity)

∀X Tx U W : set, normal_space X Tx → open_cover X Tx U → open_cover X Tx W → locally_finite_family X Tx W → refine_of W U → partition_of_unity_dominated X Tx U

In Proofgold the corresponding term root is 6a41c6... and proposition id is 42e8b3...

Proof:
Proof not loaded.

Theorem. (locally_finite_open_cover_neighborhood_finite_subcover)

∀X Tx W x : set, open_cover X Tx W → locally_finite_family X Tx W → x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃S : set, finite S ∧ S ⊆ W ∧ N ⊆ ⋃ S ∧ ∀A : set, A ∈ W → A ∩ N ≠ Empty → A ∈ S

In Proofgold the corresponding term root is 531e04... and proposition id is ca36b8...

Proof:
Proof not loaded.

Theorem. (locally_finite_family_point_finite)

∀X Tx F x : set, locally_finite_family X Tx F → x ∈ X → finite {A ∈ F|x ∈ A}

In Proofgold the corresponding term root is c702cf... and proposition id is a88fa7...

Proof:
Proof not loaded.

Theorem. (closure_Union_locally_finite_subset_Union_closures)

∀X Tx Fam : set, (∀A : set, A ∈ Fam → A ⊆ X) → locally_finite_family X Tx Fam → closure_of X Tx (⋃ Fam) ⊆ ⋃ {closure_of X Tx A|A ∈ Fam}

In Proofgold the corresponding term root is ed6bc0... and proposition id is f2c228...

Proof:
Proof not loaded.

Theorem. (closure_Union_locally_finite_sub_open)

∀X Tx Fam U : set, topology_on X Tx → (∀A : set, A ∈ Fam → A ⊆ X) → locally_finite_family X Tx Fam → U ∈ Tx → (∀A : set, A ∈ Fam → closure_of X Tx A ⊆ U) → closure_of X Tx (⋃ Fam) ⊆ U

In Proofgold the corresponding term root is 0e23c9... and proposition id is b992f2...

Proof:
Proof not loaded.

Theorem. (lemma39_1a_subcollection_locally_finite)

∀X Tx Fam Sub : set, locally_finite_family X Tx Fam → Sub ⊆ Fam → locally_finite_family X Tx Sub

In Proofgold the corresponding term root is af8704... and proposition id is 2c9e4f...

Proof:
Proof not loaded.

Theorem. (lemma39_1b_closures_locally_finite)

∀X Tx Fam : set, (∀A : set, A ∈ Fam → A ⊆ X) → locally_finite_family X Tx Fam → locally_finite_family X Tx {closure_of X Tx A|A ∈ Fam}

In Proofgold the corresponding term root is 8fbd47... and proposition id is fbb42b...

Proof:
Proof not loaded.

Theorem. (lemma39_1c_closure_Union_eq_Union_closures)

∀X Tx Fam : set, topology_on X Tx → (∀A : set, A ∈ Fam → A ⊆ X) → locally_finite_family X Tx Fam → closure_of X Tx (⋃ Fam) = ⋃ {closure_of X Tx A|A ∈ Fam}

In Proofgold the corresponding term root is 38159a... and proposition id is 5e47f7...

Proof:
Proof not loaded.

Theorem. (locally_finite_family_closure_point_finite_ex)

∀X Tx Fam : set, topology_on X Tx → (∀A : set, A ∈ Fam → A ⊆ X) → locally_finite_family X Tx Fam → ∀x : set, x ∈ X → ∃S : set, finite S ∧ S ⊆ {closure_of X Tx A|A ∈ Fam} ∧ ∀A : set, A ∈ Fam → x ∈ closure_of X Tx A → closure_of X Tx A ∈ S

In Proofgold the corresponding term root is ad8018... and proposition id is fdb2ea...

Proof:
Proof not loaded.

Definition. We define locally_finite_basis to be λX Tx ⇒ topology_on X Tx ∧ ∃B : set, basis_generates X B Tx ∧ locally_finite_family X Tx B ∧ (∀b : set, b ∈ B → b ∈ Tx) of type set → set → prop.

In Proofgold the corresponding term root is a702dc... and object id is 0a13ce...

Definition. We define sigma_locally_finite_basis to be λX Tx ⇒ topology_on X Tx ∧ ∃Fams : set, countable_set Fams ∧ Fams ⊆ 𝒫 (𝒫 X) ∧ (∀F : set, F ∈ Fams → locally_finite_family X Tx F) ∧ basis_generates X (⋃ Fams) Tx ∧ ∀b : set, b ∈ ⋃ Fams → b ∈ Tx of type set → set → prop.

In Proofgold the corresponding term root is 1f4fea... and object id is 5174de...

Theorem. (sigma_locally_finite_basis_data)

∀X Tx : set, sigma_locally_finite_basis X Tx → ∃Fams : set, countable_set Fams ∧ Fams ⊆ 𝒫 (𝒫 X) ∧ (∀F : set, F ∈ Fams → locally_finite_family X Tx F) ∧ basis_generates X (⋃ Fams) Tx ∧ ∀b : set, b ∈ ⋃ Fams → b ∈ Tx

In Proofgold the corresponding term root is 0b8301... and proposition id is 28e1b6...

Proof:
Proof not loaded.

Theorem. (basis_generates_imp_basis_refines)

∀X Tx B : set, basis_generates X B Tx → basis_refines X B Tx

In Proofgold the corresponding term root is 758ae4... and proposition id is 7c8faa...

Proof:
Proof not loaded.

Theorem. (second_countable_implies_sigma_locally_finite_basis)

∀X Tx : set, second_countable_space X Tx → sigma_locally_finite_basis X Tx

In Proofgold the corresponding term root is e011a7... and proposition id is 92a534...

Proof:
Proof not loaded.

Definition. We define Gdelta_simple to be λX Tx A ⇒ ∃Fam : set, countable_set Fam ∧ Fam ⊆ Tx ∧ A = intersection_of_family X Fam of type set → set → set → prop.

In Proofgold the corresponding term root is e1a0c1... and object id is fbfac6...

Theorem. (Union_subfamily_locally_finite_closed_is_closed_early)

∀X Tx Fam S : set, locally_finite_family X Tx Fam → (∀A : set, A ∈ Fam → closed_in X Tx A) → S ⊆ Fam → closed_in X Tx (⋃ S)

In Proofgold the corresponding term root is c8d453... and proposition id is 539c73...

Proof:
Proof not loaded.

Theorem. (lemma40_1_step1_open_as_countable_union)

∀X Tx W : set, regular_space X Tx → sigma_locally_finite_basis X Tx → W ∈ Tx → ∃Ufam : set, countable_set Ufam ∧ Ufam ⊆ Tx ∧ ⋃ Ufam = W ∧ ∀U : set, U ∈ Ufam → closure_of X Tx U ⊆ W

In Proofgold the corresponding term root is 24346a... and proposition id is 99152e...

Proof:
Proof not loaded.

Theorem. (countable_closed_separation)

∀X Tx A B Cand Dand : set, topology_on X Tx → A ⊆ X → B ⊆ X → Cand ⊆ Tx → Dand ⊆ Tx → countable_set Cand → countable_set Dand → A ⊆ ⋃ Cand → B ⊆ ⋃ Dand → (∀c : set, c ∈ Cand → closure_of X Tx c ∩ B = Empty) → (∀d : set, d ∈ Dand → closure_of X Tx d ∩ A = Empty) → ∃U V : set, U ∈ Tx ∧ V ∈ Tx ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty

In Proofgold the corresponding term root is 430989... and proposition id is fac1b6...

Proof:
Proof not loaded.

Theorem. (lemma40_1_regular_sigma_basis_implies_normal_and_closed_Gdelta)

∀X Tx : set, regular_space X Tx → sigma_locally_finite_basis X Tx → normal_space X Tx ∧ (∀C : set, closed_in X Tx C → Gdelta_simple X Tx C)

In Proofgold the corresponding term root is 40209f... and proposition id is df40b9...

Proof:
Proof not loaded.

Theorem. (lemma40_2_closed_Gdelta_has_positive_function)

∀X Tx A : set, normal_space X Tx → closed_in X Tx A → Gdelta_simple X Tx A → ∃f : set, continuous_map X Tx unit_interval unit_interval_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ X → x ∉ A → Rlt 0 (apply_fun f x))

In Proofgold the corresponding term root is 5b2729... and proposition id is c493d0...

Proof:
Proof not loaded.

Theorem. (open_set_has_positive_support_function)

∀X Tx U : set, normal_space X Tx → (∀C : set, closed_in X Tx C → Gdelta_simple X Tx C) → U ∈ Tx → ∃f : set, f ∈ function_space X R ∧ continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ X ∖ U → apply_fun f x = 0) ∧ (∀x : set, x ∈ U → Rlt 0 (apply_fun f x))

In Proofgold the corresponding term root is 6d15b5... and proposition id is f48849...

Proof:
Proof not loaded.

Theorem. (open_set_has_positive_support_function_unit_interval)

∀X Tx U : set, normal_space X Tx → (∀C : set, closed_in X Tx C → Gdelta_simple X Tx C) → U ∈ Tx → ∃f0 : set, continuous_map X Tx unit_interval unit_interval_topology f0 ∧ (∀x : set, x ∈ X ∖ U → apply_fun f0 x = 0) ∧ (∀x : set, x ∈ U → Rlt 0 (apply_fun f0 x))

In Proofgold the corresponding term root is 5ea06c... and proposition id is f37c09...

Proof:
Proof not loaded.

Theorem. (abs_SNo_add_minus_zero)

∀x : set, SNo x → abs_SNo (add_SNo x (minus_SNo 0)) = abs_SNo x

In Proofgold the corresponding term root is f7ad38... and proposition id is f71e78...

Proof:
Proof not loaded.

Theorem. (finite_Union_of_finite_family)

∀F : set, finite F → (∀A : set, A ∈ F → finite A) → finite (⋃ F)

In Proofgold the corresponding term root is 04a371... and proposition id is 134c29...

Proof:
Proof not loaded.

Theorem. (Nagata_Smirnov_metrization)

∀X Tx : set, regular_space X Tx → sigma_locally_finite_basis X Tx → metrizable X Tx

In Proofgold the corresponding term root is bff485... and proposition id is 751ca2...

Proof:
Proof not loaded.

Theorem. (metric_fixed_radius_ball_cover_open_cover)

∀X d n : set, metric_on X d → n ∈ ω → open_cover X (metric_topology X d) {open_ball X d x (inv_nat (ordsucc n))|x ∈ X}

In Proofgold the corresponding term root is 681097... and proposition id is d9b6ee...

Proof:
Proof not loaded.

Axiom. (metric_fixed_radius_ball_cover_has_locally_finite_refinement) We take the following as an axiom:

∀X d n : set, metric_on X d → n ∈ ω → ∃V : set, open_cover X (metric_topology X d) V ∧ locally_finite_family X (metric_topology X d) V ∧ refine_of V {open_ball X d x (inv_nat (ordsucc n))|x ∈ X}

In Proofgold the corresponding term root is 9641ab... and proposition id is bf103d...

Theorem. (metric_space_sigma_locally_finite_basis)

∀X d : set, metric_on X d → sigma_locally_finite_basis X (metric_topology X d)

In Proofgold the corresponding term root is 697210... and proposition id is 35abfa...

Proof:
Proof not loaded.

Theorem. (Nagata_Smirnov_metrization_theorem)

∀X Tx : set, metrizable X Tx ↔ (regular_space X Tx ∧ sigma_locally_finite_basis X Tx)

In Proofgold the corresponding term root is db97cc... and proposition id is a3bf97...

Proof:
Proof not loaded.

Theorem. (locally_finite_basis_imp_sigma_locally_finite_basis)

∀X Tx : set, locally_finite_basis X Tx → sigma_locally_finite_basis X Tx

In Proofgold the corresponding term root is 3212f1... and proposition id is 6975d3...

Proof:
Proof not loaded.

Definition. We define paracompact_space to be λX Tx ⇒ topology_on X Tx ∧ ∀U : set, open_cover X Tx U → ∃V : set, open_cover X Tx V ∧ locally_finite_family X Tx V ∧ refine_of V U of type set → set → prop.

In Proofgold the corresponding term root is 54507e... and object id is 3c02cb...

Theorem. (locally_finite_refinement)

∀X Tx U : set, paracompact_space X Tx → open_cover X Tx U → ∃V : set, open_cover X Tx V ∧ locally_finite_family X Tx V

In Proofgold the corresponding term root is a526c8... and proposition id is de5dbd...

Proof:
Proof not loaded.

Theorem. (paracompact_Hausdorff_regular)

∀X Tx : set, paracompact_space X Tx → Hausdorff_space X Tx → regular_space X Tx

In Proofgold the corresponding term root is 33d7e5... and proposition id is 50fe0d...

Proof:
Proof not loaded.

Theorem. (paracompact_Hausdorff_normal)

∀X Tx : set, paracompact_space X Tx → Hausdorff_space X Tx → normal_space X Tx

In Proofgold the corresponding term root is 365d3c... and proposition id is 407c97...

Proof:
Proof not loaded.

Theorem. (closed_subspace_paracompact)

∀X Tx Y : set, paracompact_space X Tx → closed_in X Tx Y → paracompact_space Y (subspace_topology X Tx Y)

In Proofgold the corresponding term root is b4ae93... and proposition id is 415d63...

Proof:
Proof not loaded.

Definition. We define sigma_locally_finite_family to be λX Tx F ⇒ topology_on X Tx ∧ ∃Fams : set, countable_set Fams ∧ Fams ⊆ 𝒫 (𝒫 X) ∧ (∀G : set, G ∈ Fams → locally_finite_family X Tx G) ∧ F = ⋃ Fams of type set → set → set → prop.

In Proofgold the corresponding term root is 9237c3... and object id is b790d6...

Theorem. (locally_finite_family_singleton)

∀X Tx A : set, topology_on X Tx → locally_finite_family X Tx {A}

In Proofgold the corresponding term root is 21f4ed... and proposition id is 7c9a76...

Proof:
Proof not loaded.

Theorem. (locally_finite_family_imp_sigma_locally_finite_family)

∀X Tx F : set, topology_on X Tx → F ⊆ 𝒫 X → locally_finite_family X Tx F → sigma_locally_finite_family X Tx F

In Proofgold the corresponding term root is 338b42... and proposition id is 8a4ebe...

Proof:
Proof not loaded.

Theorem. (countable_open_cover_sigma_locally_finite_family)

∀X Tx U : set, topology_on X Tx → open_cover X Tx U → countable_set U → sigma_locally_finite_family X Tx U

In Proofgold the corresponding term root is 0d07fe... and proposition id is bde8c5...

Proof:
Proof not loaded.

Definition. We define closed_cover to be λX Tx C ⇒ (∀c : set, c ∈ C → closed_in X Tx c) ∧ covers X C of type set → set → set → prop.

In Proofgold the corresponding term root is 12e188... and object id is 57e1f0...

Definition. We define Michael_cond41_3_1 to be λX Tx ⇒ ∀U : set, open_cover X Tx U → ∃V : set, open_cover X Tx V ∧ sigma_locally_finite_family X Tx V ∧ refine_of V U of type set → set → prop.

In Proofgold the corresponding term root is 4fa3b2... and object id is 00b14e...

Definition. We define Michael_cond41_3_2 to be λX Tx ⇒ ∀U : set, open_cover X Tx U → ∃V : set, covers X V ∧ locally_finite_family X Tx V ∧ refine_of V U of type set → set → prop.

In Proofgold the corresponding term root is d1149e... and object id is 1dacfc...

Definition. We define Michael_cond41_3_3 to be λX Tx ⇒ ∀U : set, open_cover X Tx U → ∃V : set, closed_cover X Tx V ∧ locally_finite_family X Tx V ∧ refine_of V U of type set → set → prop.

In Proofgold the corresponding term root is f8dba6... and object id is 2ea4bf...

Definition. We define Michael_cond41_3_4 to be λX Tx ⇒ ∀U : set, open_cover X Tx U → ∃V : set, open_cover X Tx V ∧ locally_finite_family X Tx V ∧ refine_of V U of type set → set → prop.

In Proofgold the corresponding term root is 70a654... and object id is af6d68...

Theorem. (sigma_locally_finite_open_cover_imp_locally_finite_refinement)

∀X Tx V : set, regular_space X Tx → open_cover X Tx V → sigma_locally_finite_family X Tx V → ∃C : set, covers X C ∧ locally_finite_family X Tx C ∧ refine_of C V

In Proofgold the corresponding term root is 78546d... and proposition id is 3873b7...

Proof:
Proof not loaded.

Theorem. (Michael_lemma_41_3_closed_cover_from_locally_finite_refinement)

∀X Tx A : set, regular_space X Tx → open_cover X Tx A → Michael_cond41_3_2 X Tx → ∃D : set, closed_cover X Tx D ∧ locally_finite_family X Tx D ∧ refine_of D A

In Proofgold the corresponding term root is 4514a6... and proposition id is e7c494...

Proof:
Proof not loaded.

Theorem. (Union_subfamily_locally_finite_closed_is_closed)

∀X Tx Fam S : set, locally_finite_family X Tx Fam → (∀A : set, A ∈ Fam → closed_in X Tx A) → S ⊆ Fam → closed_in X Tx (⋃ S)

In Proofgold the corresponding term root is c8d453... and proposition id is 539c73...

Proof:
Proof not loaded.

Theorem. (Michael_lemma_41_3_closed_refinement_to_open_refinement)

∀X Tx A : set, regular_space X Tx → open_cover X Tx A → Michael_cond41_3_3 X Tx → ∃D : set, open_cover X Tx D ∧ locally_finite_family X Tx D ∧ refine_of D A

In Proofgold the corresponding term root is 1892e7... and proposition id is 939883...

Proof:
Proof not loaded.

Theorem. (Michael_lemma_41_3)

∀X Tx : set, regular_space X Tx → (Michael_cond41_3_1 X Tx → Michael_cond41_3_2 X Tx) ∧ (Michael_cond41_3_2 X Tx → Michael_cond41_3_3 X Tx) ∧ (Michael_cond41_3_3 X Tx → Michael_cond41_3_4 X Tx) ∧ (Michael_cond41_3_4 X Tx → Michael_cond41_3_1 X Tx)

In Proofgold the corresponding term root is 9abafc... and proposition id is 356cc8...

Proof:
Proof not loaded.

Theorem. (metric_open_cover_point_ball_submember)

∀X d U x : set, metric_on X d → open_cover X (metric_topology X d) U → x ∈ X → ∃u : set, u ∈ U ∧ ∃r : set, r ∈ R ∧ (Rlt 0 r ∧ open_ball X d x r ⊆ u)

In Proofgold the corresponding term root is cd07c9... and proposition id is e3307d...

Proof:
Proof not loaded.

Theorem. (metric_open_cover_point_eps_ball_submember)

∀X d U x : set, metric_on X d → open_cover X (metric_topology X d) U → x ∈ X → ∃u : set, u ∈ U ∧ ∃N : set, N ∈ ω ∧ open_ball X d x (eps_ N) ⊆ u

In Proofgold the corresponding term root is 1db0f9... and proposition id is 755593...

Proof:
Proof not loaded.

Theorem. (metric_open_cover_refine_by_balls)

∀X d U : set, metric_on X d → open_cover X (metric_topology X d) U → ∃B : set, open_cover X (metric_topology X d) B ∧ (∀b : set, b ∈ B → ∃x ∈ X, ∃r ∈ R, Rlt 0 r ∧ b = open_ball X d x r) ∧ refine_of B U

In Proofgold the corresponding term root is cc7230... and proposition id is 07bdf5...

Proof:
Proof not loaded.

Theorem. (metric_spaces_regular)

∀X d : set, metric_on X d → regular_space X (metric_topology X d)

In Proofgold the corresponding term root is 14cbe5... and proposition id is c290ef...

Proof:
Proof not loaded.

Theorem. (metric_ball_cover_point_eps_submember)

∀X d B x : set, metric_on X d → open_cover X (metric_topology X d) B → (∀b : set, b ∈ B → ∃c ∈ X, ∃r ∈ R, Rlt 0 r ∧ b = open_ball X d c r) → x ∈ X → ∃b : set, b ∈ B ∧ x ∈ b ∧ ∃N : set, N ∈ ω ∧ open_ball X d x (eps_ N) ⊆ b

In Proofgold the corresponding term root is 7ec6dd... and proposition id is c5db7c...

Proof:
Proof not loaded.

Theorem. (metric_ball_open_cover_refine_by_eps_balls)

∀X d B : set, metric_on X d → open_cover X (metric_topology X d) B → (∀b : set, b ∈ B → ∃c ∈ X, ∃r ∈ R, Rlt 0 r ∧ b = open_ball X d c r) → ∃V : set, open_cover X (metric_topology X d) V ∧ refine_of V B ∧ (∀v : set, v ∈ V → ∃x : set, x ∈ X ∧ ∃N : set, N ∈ ω ∧ v = open_ball X d x (eps_ N))

In Proofgold the corresponding term root is 1496a3... and proposition id is 896e20...

Proof:
Proof not loaded.

Theorem. (eps_separated_imp_eps_succ_succ_ball_locally_finite)

∀X d S n : set, metric_on X d → n ∈ ω → eps_separated_set X d S n → locally_finite_family X (metric_topology X d) {open_ball X d x (eps_ (ordsucc (ordsucc n)))|x ∈ S}

In Proofgold the corresponding term root is 3070c7... and proposition id is aac2d4...

Proof:
Proof not loaded.

Axiom. (metric_eps_centers_for_sigma_ball_refinement_old) We take the following as an axiom:

∀X d V0 : set, metric_on X d → open_cover X (metric_topology X d) V0 → ∃Centers : set → set, (∀n : set, n ∈ ω → eps_separated_set X d (Centers n) n ∧ (∀c : set, c ∈ Centers n → ∃v0 : set, v0 ∈ V0 ∧ open_ball X d c (eps_ (ordsucc (ordsucc n))) ⊆ v0)) ∧ (∀x : set, x ∈ X → ∃n : set, n ∈ ω ∧ ∃c : set, c ∈ Centers n ∧ x ∈ open_ball X d c (eps_ (ordsucc (ordsucc n))))

In Proofgold the corresponding term root is 96d447... and proposition id is 6c8594...

Theorem. (metric_eps_centers_for_sigma_ball_refinement_old_sketch)

In Proofgold the corresponding term root is 96d447... and proposition id is 6c8594...

Proof:
Proof not loaded.

Theorem. (metric_ball_open_cover_has_sigma_locally_finite_refinement_core_old)

∀X d B : set, metric_on X d → open_cover X (metric_topology X d) B → (∀b : set, b ∈ B → ∃x ∈ X, ∃r ∈ R, Rlt 0 r ∧ b = open_ball X d x r) → ∃Fams : set, countable_set Fams ∧ Fams ⊆ 𝒫 (𝒫 X) ∧ (∀G : set, G ∈ Fams → locally_finite_family X (metric_topology X d) G) ∧ ∃V : set, V = ⋃ Fams ∧ open_cover X (metric_topology X d) V ∧ refine_of V B

In Proofgold the corresponding term root is 31953b... and proposition id is a3a35f...

Proof:
Proof not loaded.

Theorem. (metric_ball_open_cover_has_sigma_locally_finite_refinement_core_wo)

In Proofgold the corresponding term root is 31953b... and proposition id is a3a35f...

Proof:
Proof not loaded.

Theorem. (metric_ball_open_cover_has_sigma_locally_finite_refinement_core)

In Proofgold the corresponding term root is 31953b... and proposition id is a3a35f...

Proof:
Proof not loaded.

Theorem. (metric_ball_open_cover_has_sigma_locally_finite_refinement)

∀X d B : set, metric_on X d → open_cover X (metric_topology X d) B → (∀b : set, b ∈ B → ∃x ∈ X, ∃r ∈ R, Rlt 0 r ∧ b = open_ball X d x r) → ∃V : set, open_cover X (metric_topology X d) V ∧ sigma_locally_finite_family X (metric_topology X d) V ∧ refine_of V B

In Proofgold the corresponding term root is 179b2f... and proposition id is d8f8a7...

Proof:
Proof not loaded.

Theorem. (metric_ball_open_cover_has_locally_finite_refinement)

∀X d B : set, metric_on X d → open_cover X (metric_topology X d) B → (∀b : set, b ∈ B → ∃x ∈ X, ∃r ∈ R, Rlt 0 r ∧ b = open_ball X d x r) → ∃V : set, open_cover X (metric_topology X d) V ∧ locally_finite_family X (metric_topology X d) V ∧ refine_of V B

In Proofgold the corresponding term root is 217594... and proposition id is d646ad...

Proof:
Proof not loaded.

Theorem. (metric_open_cover_has_locally_finite_refinement)

∀X d U : set, metric_on X d → open_cover X (metric_topology X d) U → ∃V : set, open_cover X (metric_topology X d) V ∧ locally_finite_family X (metric_topology X d) V ∧ refine_of V U

In Proofgold the corresponding term root is a647e4... and proposition id is 6d5348...

Proof:
Proof not loaded.

Theorem. (metric_open_cover_has_sigma_locally_finite_refinement)

∀X d U : set, metric_on X d → open_cover X (metric_topology X d) U → ∃V : set, open_cover X (metric_topology X d) V ∧ sigma_locally_finite_family X (metric_topology X d) V ∧ refine_of V U

In Proofgold the corresponding term root is b68eea... and proposition id is 45013b...

Proof:
Proof not loaded.

Theorem. (metric_paracompact)

∀X d : set, metric_on X d → paracompact_space X (metric_topology X d)

In Proofgold the corresponding term root is f2cbb1... and proposition id is 95d52d...

Proof:
Proof not loaded.

Theorem. (metrizable_paracompact)

∀X Tx : set, metrizable X Tx → paracompact_space X Tx

In Proofgold the corresponding term root is 5a8547... and proposition id is fca12e...

Proof:
Proof not loaded.

Theorem. (regular_Lindelof_paracompact)

∀X Tx : set, regular_space X Tx → Lindelof_space X Tx → paracompact_space X Tx

In Proofgold the corresponding term root is 3fe293... and proposition id is 321871...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_plane_not_paracompact)

¬ paracompact_space (setprod Sorgenfrey_line Sorgenfrey_line) Sorgenfrey_plane_topology

In Proofgold the corresponding term root is 24d168... and proposition id is b36f5c...

Proof:
Proof not loaded.

Theorem. (Romega_paracompact_product_topology)

paracompact_space (product_space ω (const_space_family ω R R_standard_topology)) (product_topology_full ω (const_space_family ω R R_standard_topology))

In Proofgold the corresponding term root is 77d5be... and proposition id is dccfec...

Proof:
Proof not loaded.

Theorem. (Romega_paracompact_uniform_topology)

paracompact_space real_sequences uniform_topology

In Proofgold the corresponding term root is fe9925... and proposition id is 3680d4...

Proof:
Proof not loaded.

Theorem. (uncountable_product_R_not_paracompact)

∀J : set, uncountable_set J → ¬ paracompact_space (product_space J (const_space_family J R R_standard_topology)) (product_topology_full J (const_space_family J R R_standard_topology))

In Proofgold the corresponding term root is c1bc8d... and proposition id is 9fdaa4...

Proof:
Proof not loaded.

Theorem. (shrinking_lemma_41_6)

∀X Tx U : set, paracompact_space X Tx → Hausdorff_space X Tx → open_cover X Tx U → ∃V : set, open_cover X Tx V ∧ locally_finite_family X Tx V ∧ refine_of V U ∧ ∀v : set, v ∈ V → ∃u : set, u ∈ U ∧ closure_of X Tx v ⊆ u

In Proofgold the corresponding term root is 7d70ea... and proposition id is 275791...

Proof:
Proof not loaded.

Theorem. (shrinking_lemma_41_6_twice)

∀X Tx U : set, paracompact_space X Tx → Hausdorff_space X Tx → open_cover X Tx U → ∃V W : set, open_cover X Tx V ∧ locally_finite_family X Tx V ∧ refine_of V U ∧ (∀v : set, v ∈ V → ∃u : set, u ∈ U ∧ closure_of X Tx v ⊆ u) ∧ open_cover X Tx W ∧ locally_finite_family X Tx W ∧ refine_of W V ∧ (∀w : set, w ∈ W → ∃v : set, v ∈ V ∧ closure_of X Tx w ⊆ v)

In Proofgold the corresponding term root is 39e013... and proposition id is 50c0be...

Proof:
Proof not loaded.

Theorem. (paracompact_Hausdorff_bump_cover_dominated)

∀X Tx U : set, paracompact_space X Tx → Hausdorff_space X Tx → open_cover X Tx U → ∃V P : set, open_cover X Tx V ∧ locally_finite_family X Tx V ∧ refine_of V U ∧ P ⊆ function_space X R ∧ (∀f : set, f ∈ P → continuous_map X Tx R R_standard_topology f) ∧ (∀f x : set, f ∈ P → x ∈ X → apply_fun f x ∈ unit_interval) ∧ (∀f : set, f ∈ P → ∃u : set, u ∈ U ∧ support_of X Tx f ⊆ u) ∧ (∀x : set, x ∈ X → ∃f : set, f ∈ P ∧ apply_fun f x = 1) ∧ (∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃F0 : set, finite F0 ∧ F0 ⊆ P ∧ ∀f : set, f ∈ P → support_of X Tx f ∩ N ≠ Empty → f ∈ F0)

In Proofgold the corresponding term root is fa22a6... and proposition id is c843a5...

Proof:
Proof not loaded.

Theorem. (paracompact_Hausdorff_partition_of_unity)

∀X Tx U : set, paracompact_space X Tx → Hausdorff_space X Tx → open_cover X Tx U → partition_of_unity_dominated X Tx U

In Proofgold the corresponding term root is 5a97f9... and proposition id is 2aa3cb...

Proof:
Proof not loaded.

Theorem. (paracompact_Hausdorff_locally_finite_bounded_function_aux)

∀X Tx C eps : set, paracompact_space X Tx → Hausdorff_space X Tx → locally_finite_family X Tx C → C ⊆ 𝒫 X → (∀c : set, c ∈ C → apply_fun eps c ∈ R ∧ Rlt 0 (apply_fun eps c)) → normal_space X Tx → topology_on X Tx → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ X → Rlt 0 (apply_fun f x)) ∧ (∀c x : set, c ∈ C → x ∈ c → Rle (apply_fun f x) (apply_fun eps c))

In Proofgold the corresponding term root is 6e465f... and proposition id is 8425d2...

Proof:
Proof not loaded.

Theorem. (paracompact_Hausdorff_locally_finite_bounded_function)

∀X Tx C eps : set, paracompact_space X Tx → Hausdorff_space X Tx → locally_finite_family X Tx C → C ⊆ 𝒫 X → (∀c : set, c ∈ C → apply_fun eps c ∈ R ∧ Rlt 0 (apply_fun eps c)) → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ X → Rlt 0 (apply_fun f x)) ∧ (∀c x : set, c ∈ C → x ∈ c → Rle (apply_fun f x) (apply_fun eps c))

In Proofgold the corresponding term root is bb9a0b... and proposition id is 00f643...

Proof:
Proof not loaded.

Theorem. (Smirnov_metrization)

∀X Tx : set, regular_space X Tx → locally_finite_basis X Tx → metrizable X Tx

In Proofgold the corresponding term root is 01d344... and proposition id is e047a2...

Proof:
Proof not loaded.

Definition. We define discrete_metric to be λX ⇒ graph (setprod X X) (λp : set ⇒ If_i (p 0 = p 1) 0 1) of type set → set.

In Proofgold the corresponding term root is 2a0fe3... and object id is 08c004...

Theorem. (discrete_metric_apply)

∀X x y : set, x ∈ X → y ∈ X → apply_fun (discrete_metric X) (x,y) = If_i (x = y) 0 1

In Proofgold the corresponding term root is ea6b7d... and proposition id is 068f29...

Proof:
Proof not loaded.

Theorem. (discrete_metric_is_metric_on)

∀X : set, metric_on X (discrete_metric X)

In Proofgold the corresponding term root is 6714c1... and proposition id is 8b2725...

Proof:
Proof not loaded.

Theorem. (open_ball_discrete_radius1_eq_singleton)

∀X x : set, x ∈ X → open_ball X (discrete_metric X) x 1 = {x}

In Proofgold the corresponding term root is 62daaa... and proposition id is f7d7c9...

Proof:
Proof not loaded.

Theorem. (discrete_metric_complete)

∀X : set, complete_metric_space X (discrete_metric X)

In Proofgold the corresponding term root is 679583... and proposition id is c206f1...

Proof:
Proof not loaded.

Definition. We define euclidean_space_extend_to_Romega to be λn f ⇒ graph ω (λi : set ⇒ If_i (i ∈ n) (apply_fun f i) 0) of type set → set → set.

In Proofgold the corresponding term root is acd7dd... and object id is 9222d6...

Definition. We define euclidean_metric to be λn ⇒ graph (setprod (euclidean_space n) (euclidean_space n)) (λp : set ⇒ Romega_D_metric_value (euclidean_space_extend_to_Romega n (p 0)) (euclidean_space_extend_to_Romega n (p 1))) of type set → set.

In Proofgold the corresponding term root is c38b20... and object id is 5d1a15...

Theorem. (euclidean_space_coord_in_R)

∀n f i : set, f ∈ euclidean_space n → i ∈ n → apply_fun f i ∈ R

In Proofgold the corresponding term root is a2c1c1... and proposition id is aebb27...

Proof:
Proof not loaded.

Theorem. (euclidean_space_extend_to_Romega_in_Romega_space)

∀n f : set, f ∈ euclidean_space n → euclidean_space_extend_to_Romega n f ∈ R_omega_space

In Proofgold the corresponding term root is 0f754b... and proposition id is e48e70...

Proof:
Proof not loaded.

Theorem. (euclidean_space_extend_to_Romega_apply)

∀n f i : set, i ∈ ω → apply_fun (euclidean_space_extend_to_Romega n f) i = If_i (i ∈ n) (apply_fun f i) 0

In Proofgold the corresponding term root is ad61b7... and proposition id is c70b86...

Proof:
Proof not loaded.

Theorem. (euclidean_space_extend_to_Romega_injective)

∀n f g : set, n ∈ ω → f ∈ euclidean_space n → g ∈ euclidean_space n → euclidean_space_extend_to_Romega n f = euclidean_space_extend_to_Romega n g → f = g

In Proofgold the corresponding term root is ea8cca... and proposition id is 5e9a53...

Proof:
Proof not loaded.

Theorem. (euclidean_metric_apply)

∀n x y : set, x ∈ euclidean_space n → y ∈ euclidean_space n → apply_fun (euclidean_metric n) (x,y) = Romega_D_metric_value (euclidean_space_extend_to_Romega n x) (euclidean_space_extend_to_Romega n y)

In Proofgold the corresponding term root is 360942... and proposition id is a544d5...

Proof:
Proof not loaded.

Theorem. (euclidean_metric_is_metric_on)

∀n : set, n ∈ ω → metric_on (euclidean_space n) (euclidean_metric n)

In Proofgold the corresponding term root is 1bac09... and proposition id is a46bf3...

Proof:
Proof not loaded.

Definition. We define bounded_product_metric to be Romega_D_metric of type set.

In Proofgold the corresponding term root is 43d5aa... and object id is 9bd8fe...

Theorem. (bounded_product_metric_eq_Romega_D_metric)

bounded_product_metric = Romega_D_metric

In Proofgold the corresponding term root is 4ac5ab... and proposition id is 8944aa...

Proof:
Proof not loaded.

Definition. We define omega_strictly_increasing to be λf ⇒ ∀m n : set, m ∈ ω → n ∈ ω → m ∈ n → apply_fun f m ∈ apply_fun f n of type set → prop.

In Proofgold the corresponding term root is d7040f... and object id is 82f3f1...

Definition. We define subsequence_of to be λseq subseq ⇒ ∃f : set, total_function_on f ω ω ∧ functional_graph f ∧ graph_domain_subset f ω ∧ omega_strictly_increasing f ∧ subseq = compose_fun ω f seq of type set → set → prop.

In Proofgold the corresponding term root is 993f53... and object id is 370e0d...

Theorem. (exists_twofold_small_eps)

∀eps0 : set, eps0 ∈ R → Rlt 0 eps0 → ∃eps1 : set, eps1 ∈ R ∧ Rlt 0 eps1 ∧ Rlt (add_SNo eps1 eps1) eps0

In Proofgold the corresponding term root is 409ca5... and proposition id is 02e3a0...

Proof:
Proof not loaded.

Theorem. (omega_strictly_increasing_unbounded)

∀f N : set, total_function_on f ω ω → functional_graph f → graph_domain_subset f ω → omega_strictly_increasing f → N ∈ ω → ∃k : set, k ∈ ω ∧ N ⊆ apply_fun f k

In Proofgold the corresponding term root is 2a217e... and proposition id is 8deb77...

Proof:
Proof not loaded.

Theorem. (Cauchy_with_convergent_subsequence_converges)

∀X d seq x : set, metric_on X d → cauchy_sequence X d seq → (∃subseq : set, subsequence_of seq subseq ∧ converges_to X (metric_topology X d) subseq x) → converges_to X (metric_topology X d) seq x

In Proofgold the corresponding term root is fe1004... and proposition id is c8b97a...

Proof:
Proof not loaded.

Definition. We define product_coordinate_sequence to be λseq j ⇒ graph ω (λn : set ⇒ apply_fun (apply_fun seq n) j) of type set → set → set.

In Proofgold the corresponding term root is 726f4c... and object id is 8c9b25...

Theorem. (product_sequence_convergence_iff_coordinates)

∀X J : set, X = product_space J (const_space_family J R R_standard_topology) → ∀seq x : set, sequence_on seq X → x ∈ X → (converges_to X (product_topology_full J (const_space_family J R R_standard_topology)) seq x ↔ (∀j : set, j ∈ J → converges_to (product_component (const_space_family J R R_standard_topology) j) (product_component_topology (const_space_family J R R_standard_topology) j) (product_coordinate_sequence seq j) (apply_fun x j)))

In Proofgold the corresponding term root is 6c4c69... and proposition id is ba424c...

Proof:
Proof not loaded.

Theorem. (R_bounded_metric_apply)

∀x y : set, x ∈ R → y ∈ R → apply_fun R_bounded_metric (x,y) = R_bounded_distance x y

In Proofgold the corresponding term root is beb60d... and proposition id is bc356a...

Proof:
Proof not loaded.

Theorem. (R_bounded_metric_is_metric_on)

metric_on R R_bounded_metric

In Proofgold the corresponding term root is 6a523d... and proposition id is 469f1d...

Proof:
Proof not loaded.

Theorem. (cauchy_R_bounded_metric_abs_tail)

In Proofgold the corresponding term root is 1435c6... and proposition id is 87a936...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_abs)

∀x r y : set, x ∈ R → r ∈ R → Rlt 0 r → Rlt r 1 → (y ∈ open_ball R R_bounded_metric x r ↔ (y ∈ R ∧ Rlt (R_bounded_distance x y) r))

In Proofgold the corresponding term root is 443e09... and proposition id is 58c3fb...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_eq_open_interval)

∀c r : set, c ∈ R → r ∈ R → Rlt 0 r → Rlt r 1 → open_ball R R_bounded_metric c r = open_interval (add_SNo c (minus_SNo r)) (add_SNo c r)

In Proofgold the corresponding term root is 309694... and proposition id is 237a0b...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_r1_eq_open_interval)

∀c : set, c ∈ R → open_ball R R_bounded_metric c 1 = open_interval (add_SNo c (minus_SNo 1)) (add_SNo c 1)

In Proofgold the corresponding term root is f7ed43... and proposition id is 7f0efc...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_eq_R_if_1_lt)

∀c r : set, c ∈ R → r ∈ R → Rlt 1 r → open_ball R R_bounded_metric c r = R

In Proofgold the corresponding term root is 5c9a2e... and proposition id is be31ff...

Proof:
Proof not loaded.

Theorem. (Romega_cauchy_imp_coord_cauchy_bounded)

∀seq i : set, cauchy_sequence R_omega_space Romega_D_metric seq → i ∈ ω → cauchy_sequence R R_bounded_metric (product_coordinate_sequence seq i)

In Proofgold the corresponding term root is bc237e... and proposition id is b01044...

Proof:
Proof not loaded.

Theorem. (graph_omega_to_R_in_Romega_space)

∀g : set → set, (∀i : set, i ∈ ω → g i ∈ R) → graph ω g ∈ R_omega_space

In Proofgold the corresponding term root is ca17ca... and proposition id is 07d8f0...

Proof:
Proof not loaded.

Theorem. (open_ball_R_bounded_metric_in_R_standard_topology)

∀c r : set, c ∈ R → r ∈ R → Rlt 0 r → open_ball R R_bounded_metric c r ∈ R_standard_topology

In Proofgold the corresponding term root is e3e485... and proposition id is 03dbbc...

Proof:
Proof not loaded.

Theorem. (open_interval_in_metric_topology_R_bounded_metric)

∀a b : set, a ∈ R → b ∈ R → open_interval a b ∈ metric_topology R R_bounded_metric

In Proofgold the corresponding term root is bbb749... and proposition id is d8990a...

Proof:
Proof not loaded.

Theorem. (metric_topology_R_bounded_metric_eq_R_standard_topology)

metric_topology R R_bounded_metric = R_standard_topology

In Proofgold the corresponding term root is 3c2e15... and proposition id is d30c60...

Proof:
Proof not loaded.

Theorem. (R_standard_topology_completely_regular)

completely_regular_space R R_standard_topology

In Proofgold the corresponding term root is d3582c... and proposition id is 7f9adc...

Proof:
Proof not loaded.

Theorem. (abs_Cauchy_sequence_converges_R_standard_topology)

In Proofgold the corresponding term root is c9fabe... and proposition id is 59126e...

Proof:
Proof not loaded.

Theorem. (abs_Cauchy_sequence_converges_metric_topology_R_bounded_metric)

In Proofgold the corresponding term root is fe9cb4... and proposition id is 3a75a1...

Proof:
Proof not loaded.

Theorem. (R_bounded_metric_complete)

complete_metric_space R R_bounded_metric

In Proofgold the corresponding term root is 360bbb... and proposition id is f0ae42...

Proof:
Proof not loaded.

Theorem. (product_Romega_complete)

complete_metric_space (power_real ω) bounded_product_metric

In Proofgold the corresponding term root is 468dec... and proposition id is 97ed34...

Proof:
Proof not loaded.

Theorem. (Romega_D_metric_complete)

complete_metric_space R_omega_space Romega_D_metric

In Proofgold the corresponding term root is f86aa5... and proposition id is 3dc36e...

Proof:
Proof not loaded.

Theorem. (Euclidean_space_complete)

∀k ∈ ω, complete_metric_space (euclidean_space k) (euclidean_metric k)

In Proofgold the corresponding term root is 8204e8... and proposition id is cccb95...

Proof:
Proof not loaded.

Definition. We define unit_square to be setprod unit_interval unit_interval of type set.

In Proofgold the corresponding term root is 78d4ca... and object id is 50135f...

Definition. We define unit_square_topology to be product_topology unit_interval unit_interval_topology unit_interval unit_interval_topology of type set.

In Proofgold the corresponding term root is 7469b6... and object id is 2400ae...

Theorem. (space_filling_curve)

∃f : set, continuous_map unit_interval unit_interval_topology unit_square unit_square_topology f

In Proofgold the corresponding term root is bc995d... and proposition id is d1d8ab...

Proof:
Proof not loaded.

Definition. We define sequentially_compact to be λX Tx ⇒ topology_on X Tx ∧ ∀seq : set, sequence_on seq X → ∃subseq : set, ∃x : set, subsequence_of seq subseq ∧ converges_to X Tx subseq x of type set → set → prop.

In Proofgold the corresponding term root is 8e4808... and object id is 54f1f5...

Theorem. (compact_metric_sequence_has_convergent_subsequence)

∀X d seq : set, metric_on X d → compact_space X (metric_topology X d) → sequence_on seq X → ∃subseq : set, ∃x : set, subsequence_of seq subseq ∧ converges_to X (metric_topology X d) subseq x

In Proofgold the corresponding term root is fbcca3... and proposition id is 8872df...

Proof:
Proof not loaded.

Definition. We define lebesgue_number_metric to be λX d Fam delta ⇒ delta ∈ R ∧ Rlt 0 delta ∧ ∀x : set, x ∈ X → ∃U : set, U ∈ Fam ∧ open_ball X d x delta ⊆ U of type set → set → set → set → prop.

In Proofgold the corresponding term root is ee9556... and object id is dd473f...

Definition. We define finite_ball_cover_metric to be λX d delta F ⇒ finite F ∧ F ⊆ X ∧ X ⊆ (⋃_{c ∈ F}open_ball X d c delta) of type set → set → set → set → prop.

In Proofgold the corresponding term root is cda1ca... and object id is c7315c...

Theorem. (lebesgue_and_ball_cover_imply_finite_subcover)

∀X d Fam delta : set, metric_on X d → open_cover_of X (metric_topology X d) Fam → lebesgue_number_metric X d Fam delta → (∃F : set, finite_ball_cover_metric X d delta F) → has_finite_subcover X (metric_topology X d) Fam

In Proofgold the corresponding term root is c3d700... and proposition id is b00a4a...

Proof:
Proof not loaded.

Theorem. (sequentially_compact_metric_has_lebesgue_number_eps)

∀X d Fam : set, metric_on X d → sequentially_compact X (metric_topology X d) → open_cover_of X (metric_topology X d) Fam → ∃N : set, N ∈ ω ∧ lebesgue_number_metric X d Fam (eps_ N)

In Proofgold the corresponding term root is 841e3d... and proposition id is 6f81dc...

Proof:
Proof not loaded.

Theorem. (sequentially_compact_metric_has_finite_ball_cover_eps)

∀X d N : set, metric_on X d → sequentially_compact X (metric_topology X d) → N ∈ ω → ∃F : set, finite_ball_cover_metric X d (eps_ N) F

In Proofgold the corresponding term root is 42b192... and proposition id is 37e5e6...

Proof:
Proof not loaded.

Theorem. (sequentially_compact_metric_imp_compact)

∀X d : set, metric_on X d → sequentially_compact X (metric_topology X d) → compact_space X (metric_topology X d)

In Proofgold the corresponding term root is f9af0e... and proposition id is 7b3805...

Proof:
Proof not loaded.

Theorem. (compact_metric_equivalences)

∀X d : set, metric_on X d → (compact_space X (metric_topology X d) ↔ sequentially_compact X (metric_topology X d))

In Proofgold the corresponding term root is bc2e69... and proposition id is c3c216...

Proof:
Proof not loaded.

Definition. We define pointwise_subbasis to be λX Tx Y Ty ⇒ {S ∈ 𝒫 (function_space X Y)|∃x U : set, x ∈ X ∧ U ∈ Ty ∧ S = {f ∈ function_space X Y|apply_fun f x ∈ U}} of type set → set → set → set → set.

In Proofgold the corresponding term root is cacb4c... and object id is 45a99b...

Definition. We define pointwise_convergence_topology to be λX Tx Y Ty ⇒ generated_topology_from_subbasis (function_space X Y) ((pointwise_subbasis X Tx Y Ty) ∪ {function_space X Y}) of type set → set → set → set → set.

In Proofgold the corresponding term root is a79939... and object id is 16d820...

Definition. We define continuous_function_space to be λX Tx Y Ty ⇒ {f ∈ function_space X Y|continuous_map X Tx Y Ty f} of type set → set → set → set → set.

In Proofgold the corresponding term root is 7207e2... and object id is 4542a9...

Theorem. (pointwise_convergence_topology_is_topology)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → topology_on (function_space X Y) (pointwise_convergence_topology X Tx Y Ty)

In Proofgold the corresponding term root is de7f56... and proposition id is 45b308...

Proof:
Proof not loaded.

Theorem. (pointwise_eval_continuous)

∀X Tx Y Ty x : set, topology_on X Tx → topology_on Y Ty → x ∈ X → continuous_map (function_space X Y) (pointwise_convergence_topology X Tx Y Ty) Y Ty (graph (function_space X Y) (λf : set ⇒ apply_fun f x))

In Proofgold the corresponding term root is a958a1... and proposition id is 5e135c...

Proof:
Proof not loaded.

Theorem. (continuous_map_image_of_closure_subset_closure_of_image)

∀X Tx Y Ty f A x : set, topology_on X Tx → topology_on Y Ty → continuous_map X Tx Y Ty f → A ⊆ X → x ∈ closure_of X Tx A → apply_fun f x ∈ closure_of Y Ty (Repl A (λa : set ⇒ apply_fun f a))

In Proofgold the corresponding term root is d65af5... and proposition id is 4eaba7...

Proof:
Proof not loaded.

Theorem. (pointwise_closure_implies_coordinate_in_closure_of_values)

∀X Tx Y Ty F a g : set, topology_on X Tx → topology_on Y Ty → a ∈ X → F ⊆ function_space X Y → g ∈ closure_of (function_space X Y) (pointwise_convergence_topology X Tx Y Ty) F → apply_fun g a ∈ closure_of Y Ty (Repl F (λf0 : set ⇒ apply_fun f0 a))

In Proofgold the corresponding term root is 2c98c1... and proposition id is 64f11e...

Proof:
Proof not loaded.

Definition. We define compact_open_subbasis to be λX Tx Y Ty ⇒ {S ∈ 𝒫 (function_space X Y)|∃K U : set, compact_space K (subspace_topology X Tx K) ∧ K ⊆ X ∧ U ∈ Ty ∧ S = {f ∈ function_space X Y|K ⊆ preimage_of X f U}} of type set → set → set → set → set.

In Proofgold the corresponding term root is 1a413b... and object id is bf64a0...

Definition. We define compact_convergence_topology to be λX Tx Y Ty ⇒ generated_topology_from_subbasis (function_space X Y) (compact_open_subbasis X Tx Y Ty) of type set → set → set → set → set.

In Proofgold the corresponding term root is ed7788... and object id is 1ddbb5...

Theorem. (compact_open_subbasis_is_subbasis)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → subbasis_on (function_space X Y) (compact_open_subbasis X Tx Y Ty)

In Proofgold the corresponding term root is c0bb08... and proposition id is 6e0ee9...

Proof:
Proof not loaded.

Theorem. (compact_convergence_topology_is_topology)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → topology_on (function_space X Y) (compact_convergence_topology X Tx Y Ty)

In Proofgold the corresponding term root is c92363... and proposition id is 6603aa...

Proof:
Proof not loaded.

Theorem. (compact_convergence_eval_continuous)

∀X Tx Y Ty x : set, topology_on X Tx → topology_on Y Ty → x ∈ X → continuous_map (function_space X Y) (compact_convergence_topology X Tx Y Ty) Y Ty (graph (function_space X Y) (λf : set ⇒ apply_fun f x))

In Proofgold the corresponding term root is b69e39... and proposition id is 22c0c0...

Proof:
Proof not loaded.

Definition. We define compact_open_topology_C to be λX Tx Y Ty ⇒ subspace_topology (function_space X Y) (compact_convergence_topology X Tx Y Ty) (continuous_function_space X Tx Y Ty) of type set → set → set → set → set.

In Proofgold the corresponding term root is d68506... and object id is 725fa1...

Theorem. (continuous_function_space_sub)

∀X Tx Y Ty : set, continuous_function_space X Tx Y Ty ⊆ function_space X Y

In Proofgold the corresponding term root is b51659... and proposition id is ebb3af...

Proof:
Proof not loaded.

Theorem. (compact_open_topology_C_is_topology)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → topology_on (continuous_function_space X Tx Y Ty) (compact_open_topology_C X Tx Y Ty)

In Proofgold the corresponding term root is ec5f11... and proposition id is 37f880...

Proof:
Proof not loaded.

Definition. We define compact_open_evaluation_map to be λX Tx Y Ty ⇒ graph (setprod X (continuous_function_space X Tx Y Ty)) (λp : set ⇒ apply_fun (p 1) (p 0)) of type set → set → set → set → set.

In Proofgold the corresponding term root is d2855d... and object id is 8679db...

Theorem. (compact_open_evaluation_continuous)

∀X Tx Y Ty : set, locally_compact X Tx → Hausdorff_space X Tx → topology_on Y Ty → continuous_map (setprod X (continuous_function_space X Tx Y Ty)) (product_topology X Tx (continuous_function_space X Tx Y Ty) (compact_open_topology_C X Tx Y Ty)) Y Ty (compact_open_evaluation_map X Tx Y Ty)

In Proofgold the corresponding term root is 58cfbc... and proposition id is 410c51...

Proof:
Proof not loaded.

Theorem. (tube_lemma_compact_first)

∀X Tx Y Ty K : set, topology_on X Tx → topology_on Y Ty → compact_space K (subspace_topology X Tx K) → K ⊆ X → ∀y0 : set, y0 ∈ Y → ∀N : set, N ∈ product_topology X Tx Y Ty ∧ setprod K {y0} ⊆ N → ∃V : set, V ∈ Ty ∧ y0 ∈ V ∧ setprod K V ⊆ N

In Proofgold the corresponding term root is 80338f... and proposition id is db2226...

Proof:
Proof not loaded.

Theorem. (compact_open_induced_map_continuous)

∀X Tx Y Ty Z Tz f : set, topology_on X Tx → topology_on Y Ty → topology_on Z Tz → continuous_map (setprod X Z) (product_topology X Tx Z Tz) Y Ty f → ∃F : set, continuous_map Z Tz (continuous_function_space X Tx Y Ty) (compact_open_topology_C X Tx Y Ty) F

In Proofgold the corresponding term root is a49cfd... and proposition id is bbe4b6...

Proof:
Proof not loaded.

Definition. We define compact_convergence_subbasis_metric to be λX Tx Y dY ⇒ {S ∈ 𝒫 (continuous_function_space X Tx Y (metric_topology Y dY))|∃C f eps : set, compact_space C (subspace_topology X Tx C) ∧ C ⊆ X ∧ f ∈ continuous_function_space X Tx Y (metric_topology Y dY) ∧ eps ∈ R ∧ Rlt 0 eps ∧ S = {g ∈ continuous_function_space X Tx Y (metric_topology Y dY)|∀x : set, x ∈ C → Rlt (apply_fun dY (apply_fun g x,apply_fun f x)) eps}} of type set → set → set → set → set.

In Proofgold the corresponding term root is 9c75b3... and object id is 50cc2a...

Definition. We define compact_convergence_topology_metric_C to be λX Tx Y dY ⇒ generated_topology_from_subbasis (continuous_function_space X Tx Y (metric_topology Y dY)) ((compact_convergence_subbasis_metric X Tx Y dY) ∪ {continuous_function_space X Tx Y (metric_topology Y dY)}) of type set → set → set → set → set.

In Proofgold the corresponding term root is d524d6... and object id is 68a002...

Theorem. (compact_convergence_metric_subbasis_on)

∀X Tx Y dY : set, topology_on X Tx → metric_on Y dY → subbasis_on (continuous_function_space X Tx Y (metric_topology Y dY)) ((compact_convergence_subbasis_metric X Tx Y dY) ∪ {continuous_function_space X Tx Y (metric_topology Y dY)})

In Proofgold the corresponding term root is 7d3daa... and proposition id is dcd6ac...

Proof:
Proof not loaded.

Theorem. (compact_convergence_topology_metric_C_is_topology)

∀X Tx Y dY : set, topology_on X Tx → metric_on Y dY → topology_on (continuous_function_space X Tx Y (metric_topology Y dY)) (compact_convergence_topology_metric_C X Tx Y dY)

In Proofgold the corresponding term root is 80aa63... and proposition id is a8d491...

Proof:
Proof not loaded.

Theorem. (compact_open_subbasic_C_open_in_compact_convergence_metric)

∀X Tx Y dY K U : set, topology_on X Tx → metric_on Y dY → compact_space K (subspace_topology X Tx K) → K ⊆ X → U ∈ metric_topology Y dY → {f ∈ continuous_function_space X Tx Y (metric_topology Y dY)|K ⊆ preimage_of X f U} ∈ compact_convergence_topology_metric_C X Tx Y dY

In Proofgold the corresponding term root is e30ccc... and proposition id is e0d0d8...

Proof:
Proof not loaded.

Theorem. (compact_open_topology_C_sub_compact_convergence_metric_C)

∀X Tx Y dY : set, topology_on X Tx → metric_on Y dY → compact_open_topology_C X Tx Y (metric_topology Y dY) ⊆ compact_convergence_topology_metric_C X Tx Y dY

In Proofgold the corresponding term root is 1c5818... and proposition id is 145207...

Proof:
Proof not loaded.

Theorem. (closure_open_ball_sub_open_ball_add_radius)

∀Y d y r : set, metric_on Y d → y ∈ Y → r ∈ R → Rlt 0 r → closure_of Y (metric_topology Y d) (open_ball Y d y r) ⊆ open_ball Y d y (add_SNo r r)

In Proofgold the corresponding term root is 9ee8c5... and proposition id is 3ed576...

Proof:
Proof not loaded.

Theorem. (closure_preimage_open_ball_sub_preimage_open_ball_add_radius)

∀X Tx Y d f y r : set, topology_on X Tx → metric_on Y d → continuous_map X Tx Y (metric_topology Y d) f → y ∈ Y → r ∈ R → Rlt 0 r → closure_of X Tx (preimage_of X f (open_ball Y d y r)) ⊆ preimage_of X f (open_ball Y d y (add_SNo r r))

In Proofgold the corresponding term root is e78624... and proposition id is 513450...

Proof:
Proof not loaded.

Theorem. (compact_convergence_metric_C_sub_compact_open_topology_C)

∀X Tx Y dY : set, topology_on X Tx → metric_on Y dY → compact_convergence_topology_metric_C X Tx Y dY ⊆ compact_open_topology_C X Tx Y (metric_topology Y dY)

In Proofgold the corresponding term root is 9665dd... and proposition id is 3c3734...

Proof:
Proof not loaded.

Theorem. (compact_open_eq_compact_convergence_on_C)

∀X Tx Y dY : set, topology_on X Tx → metric_on Y dY → compact_open_topology_C X Tx Y (metric_topology Y dY) = compact_convergence_topology_metric_C X Tx Y dY

In Proofgold the corresponding term root is 42fd41... and proposition id is d39754...

Proof:
Proof not loaded.

Theorem. (compact_convergence_metric_independent_of_metric)

∀X Tx Y d1 d2 : set, topology_on X Tx → metric_on Y d1 → metric_on Y d2 → metric_topology Y d1 = metric_topology Y d2 → compact_convergence_topology_metric_C X Tx Y d1 = compact_convergence_topology_metric_C X Tx Y d2

In Proofgold the corresponding term root is 6b8dc8... and proposition id is 0f31c8...

Proof:
Proof not loaded.

Theorem. (compact_convergence_finer_than_pointwise)

∀X Tx Y Ty : set, topology_on X Tx → topology_on Y Ty → finer_than (compact_convergence_topology X Tx Y Ty) (pointwise_convergence_topology X Tx Y Ty)

In Proofgold the corresponding term root is c30f51... and proposition id is bd56f1...

Proof:
Proof not loaded.

Definition. We define Ascoli_F_at to be λX Y a F ⇒ {apply_fun f a|f ∈ F} of type set → set → set → set → set.

In Proofgold the corresponding term root is a55388... and object id is db142f...

Theorem. (Ascoli_F_at_def)

∀X Y a F : set, Ascoli_F_at X Y a F = Repl F (λf : set ⇒ apply_fun f a)

In Proofgold the corresponding term root is 847674... and proposition id is 6aaba5...

Proof:
Proof not loaded.

Definition. We define Ascoli_pointwise_compact_closure to be λX Y dY F ⇒ ∀a : set, a ∈ X → compact_space (closure_of Y (metric_topology Y dY) (Ascoli_F_at X Y a F)) (subspace_topology Y (metric_topology Y dY) (closure_of Y (metric_topology Y dY) (Ascoli_F_at X Y a F))) of type set → set → set → set → prop.

In Proofgold the corresponding term root is 065e86... and object id is 6e8637...

Definition. We define equicontinuous_family to be λX Tx Y dY F ⇒ topology_on X Tx ∧ metric_on Y dY ∧ F ⊆ continuous_function_space X Tx Y (metric_topology Y dY) ∧ ∀x0 : set, x0 ∈ X → ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ ∀f : set, f ∈ F → ∀x : set, x ∈ X → x ∈ U → Rlt (apply_fun dY (apply_fun f x,apply_fun f x0)) eps of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is c086dc... and object id is f699e8...

Definition. We define relatively_compact_in_compact_convergence to be λX Tx Y dY F ⇒ F ⊆ continuous_function_space X Tx Y (metric_topology Y dY) ∧ ∃H : set, F ⊆ H ∧ H ⊆ continuous_function_space X Tx Y (metric_topology Y dY) ∧ compact_space H (subspace_topology (continuous_function_space X Tx Y (metric_topology Y dY)) (compact_convergence_topology_metric_C X Tx Y dY) H) of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is 6603b3... and object id is b4e996...

Definition. We define Ascoli_g_pointwise_closure to be λX Tx Y dY F ⇒ closure_of (function_space X Y) (pointwise_convergence_topology X Tx Y (metric_topology Y dY)) F of type set → set → set → set → set → set.

In Proofgold the corresponding term root is e801fc... and object id is 3b8c59...

Theorem. (Ascoli_g_pointwise_closure_sub)

∀X Tx Y dY F : set, topology_on (function_space X Y) (pointwise_convergence_topology X Tx Y (metric_topology Y dY)) → Ascoli_g_pointwise_closure X Tx Y dY F ⊆ function_space X Y

In Proofgold the corresponding term root is f9eb22... and proposition id is 4e5e9e...

Proof:
Proof not loaded.

Theorem. (Ascoli_g_pointwise_closure_closed)

∀X Tx Y dY F : set, topology_on (function_space X Y) (pointwise_convergence_topology X Tx Y (metric_topology Y dY)) → F ⊆ function_space X Y → closed_in (function_space X Y) (pointwise_convergence_topology X Tx Y (metric_topology Y dY)) (Ascoli_g_pointwise_closure X Tx Y dY F)

In Proofgold the corresponding term root is 15d7fb... and proposition id is a12100...

Proof:
Proof not loaded.

Theorem. (Ascoli_g_pointwise_closure_coord)

∀X Tx Y dY F g a : set, topology_on X Tx → metric_on Y dY → F ⊆ continuous_function_space X Tx Y (metric_topology Y dY) → g ∈ Ascoli_g_pointwise_closure X Tx Y dY F → a ∈ X → apply_fun g a ∈ closure_of Y (metric_topology Y dY) (Ascoli_F_at X Y a F)

In Proofgold the corresponding term root is 5b6540... and proposition id is dfd6cb...

Proof:
Proof not loaded.

Theorem. (Ascoli_g_pointwise_closure_equicontinuous_bound)

∀X Tx Y dY F x0 eps : set, topology_on X Tx → metric_on Y dY → F ⊆ continuous_function_space X Tx Y (metric_topology Y dY) → equicontinuous_family X Tx Y dY F → x0 ∈ X → eps ∈ R ∧ Rlt 0 eps → ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ ∀g : set, g ∈ Ascoli_g_pointwise_closure X Tx Y dY F → ∀x : set, x ∈ X → x ∈ U → Rlt (apply_fun dY (apply_fun g x,apply_fun g x0)) eps

In Proofgold the corresponding term root is 9d3f67... and proposition id is 64eaa0...

Proof:
Proof not loaded.

Theorem. (relatively_compact_imp_Ascoli_pointwise_compact_closure)

∀X Tx Y dY F : set, topology_on X Tx → metric_on Y dY → relatively_compact_in_compact_convergence X Tx Y dY F → Ascoli_pointwise_compact_closure X Y dY F

In Proofgold the corresponding term root is 693a36... and proposition id is 0b90f4...

Proof:
Proof not loaded.

Theorem. (Ascoli_g_pointwise_closure_continuous_map)

∀X Tx Y dY F g : set, topology_on X Tx → metric_on Y dY → F ⊆ continuous_function_space X Tx Y (metric_topology Y dY) → equicontinuous_family X Tx Y dY F → g ∈ Ascoli_g_pointwise_closure X Tx Y dY F → continuous_map X Tx Y (metric_topology Y dY) g

In Proofgold the corresponding term root is c39df3... and proposition id is 2c0e5a...

Proof:
Proof not loaded.

Theorem. (Ascoli_g_pointwise_closure_sub_CXY)

In Proofgold the corresponding term root is cf3e52... and proposition id is cc4f93...

Proof:
Proof not loaded.

Theorem. (Ascoli_g_pointwise_closure_set_sub_CXY)

∀X Tx Y dY F : set, topology_on X Tx → metric_on Y dY → F ⊆ continuous_function_space X Tx Y (metric_topology Y dY) → equicontinuous_family X Tx Y dY F → Ascoli_g_pointwise_closure X Tx Y dY F ⊆ continuous_function_space X Tx Y (metric_topology Y dY)

In Proofgold the corresponding term root is 678d62... and proposition id is e4aa3e...

Proof:
Proof not loaded.

Definition. We define Ascoli_XiKa to be λX Y Ty F ⇒ graph X (λa : set ⇒ (closure_of Y Ty (Ascoli_F_at X Y a F),subspace_topology Y Ty (closure_of Y Ty (Ascoli_F_at X Y a F)))) of type set → set → set → set → set.

In Proofgold the corresponding term root is 341372... and object id is 29a04c...

Theorem. (Ascoli_XiKa_set)

∀X Y Ty F a : set, a ∈ X → space_family_set (Ascoli_XiKa X Y Ty F) a = closure_of Y Ty (Ascoli_F_at X Y a F)

In Proofgold the corresponding term root is 81ce95... and proposition id is 44a85c...

Proof:
Proof not loaded.

Theorem. (Ascoli_XiKa_topology)

∀X Y Ty F a : set, a ∈ X → space_family_topology (Ascoli_XiKa X Y Ty F) a = subspace_topology Y Ty (closure_of Y Ty (Ascoli_F_at X Y a F))

In Proofgold the corresponding term root is ad8edd... and proposition id is cf1e24...

Proof:
Proof not loaded.

Theorem. (Ascoli_XiKa_product_compact)

∀X Y dY F : set, metric_on Y dY → Ascoli_pointwise_compact_closure X Y dY F → compact_space (product_space X (Ascoli_XiKa X Y (metric_topology Y dY) F)) (product_topology_full X (Ascoli_XiKa X Y (metric_topology Y dY) F))

In Proofgold the corresponding term root is f57d38... and proposition id is 610359...

Proof:
Proof not loaded.

Theorem. (Ascoli_XiKa_product_every_net_pack_has_convergent_subnet_pack)

∀X Y dY F N : set, metric_on Y dY → Ascoli_pointwise_compact_closure X Y dY F → net_pack_in_space (product_space X (Ascoli_XiKa X Y (metric_topology Y dY) F)) N → ∃S x : set, subnet_pack_of_in (product_space X (Ascoli_XiKa X Y (metric_topology Y dY) F)) N S ∧ net_pack_converges (product_space X (Ascoli_XiKa X Y (metric_topology Y dY) F)) (product_topology_full X (Ascoli_XiKa X Y (metric_topology Y dY) F)) S x

In Proofgold the corresponding term root is 2bc530... and proposition id is 460806...

Proof:
Proof not loaded.

Theorem. (compact_subspace_CXY_equicontinuous_per_ball)

∀X Tx Y dY H x0 eps r Ty eval0 Img0 FamBalls Gballs : set, topology_on X Tx → metric_on Y dY → Ty = metric_topology Y dY → locally_compact X Tx → Hausdorff_space X Tx → H ⊆ continuous_function_space X Tx Y (metric_topology Y dY) → compact_space H (subspace_topology (continuous_function_space X Tx Y (metric_topology Y dY)) (compact_open_topology_C X Tx Y (metric_topology Y dY)) H) → eval0 = graph (function_space X Y) (λf : set ⇒ apply_fun f x0) → x0 ∈ X → eps ∈ R → Rlt 0 eps → r ∈ R → Rlt 0 r → Rlt (add_SNo (add_SNo r r) (add_SNo r r)) eps → Img0 = image_of_fun eval0 H → FamBalls = {open_ball Y dY y r|y ∈ Img0} → Gballs ⊆ FamBalls → ∀b : set, b ∈ Gballs → ∃U : set, U ∈ Tx ∧ x0 ∈ U ∧ ∀f : set, f ∈ H → apply_fun eval0 f ∈ closure_of Y Ty b → ∀x1 : set, x1 ∈ X → x1 ∈ U → Rlt (apply_fun dY (apply_fun f x1,apply_fun f x0)) eps

In Proofgold the corresponding term root is 1d192e... and proposition id is 3efcfe...

Proof:
Proof not loaded.

Theorem. (compact_subspace_CXY_equicontinuous)

∀X Tx Y dY H : set, topology_on X Tx → metric_on Y dY → locally_compact X Tx → Hausdorff_space X Tx → H ⊆ continuous_function_space X Tx Y (metric_topology Y dY) → compact_space H (subspace_topology (continuous_function_space X Tx Y (metric_topology Y dY)) (compact_open_topology_C X Tx Y (metric_topology Y dY)) H) → equicontinuous_family X Tx Y dY H

In Proofgold the corresponding term root is fc22a9... and proposition id is 98d0e0...

Proof:
Proof not loaded.

Theorem. (apply_fun_apply_fun_graph)

∀A : set, ∀g : set → set, ∀a b : set, a ∈ A → apply_fun (apply_fun (graph A g) a) b = apply_fun (g a) b

In Proofgold the corresponding term root is 81e377... and proposition id is 2596ae...

Proof:
Proof not loaded.

Theorem. (apply_fun_congr)

∀f g x : set, f = g → apply_fun f x = apply_fun g x

In Proofgold the corresponding term root is abcdf5... and proposition id is ec6d39...

Proof:
Proof not loaded.

Theorem. (directed_set_finite_upper_bound)

∀J le F : set, directed_set J le → finite F → F ⊆ J → ∃k : set, k ∈ J ∧ ∀i : set, i ∈ F → (i,k) ∈ le

In Proofgold the corresponding term root is 327d27... and proposition id is 1a8ce9...

Proof:
Proof not loaded.

Theorem. (Ascoli_theorem_47_1a_diag_upgrade)

In Proofgold the corresponding term root is 2b14f7... and proposition id is 3c7c0e...

Proof:
Proof not loaded.

Theorem. (Ascoli_theorem_47_1a)

∀X Tx Y dY F : set, topology_on X Tx → metric_on Y dY → F ⊆ continuous_function_space X Tx Y (metric_topology Y dY) → equicontinuous_family X Tx Y dY F → Ascoli_pointwise_compact_closure X Y dY F → relatively_compact_in_compact_convergence X Tx Y dY F

In Proofgold the corresponding term root is dad783... and proposition id is bf2b43...

Proof:
Proof not loaded.

Theorem. (Ascoli_theorem_47_1b)

∀X Tx Y dY F : set, topology_on X Tx → metric_on Y dY → locally_compact X Tx → Hausdorff_space X Tx → F ⊆ continuous_function_space X Tx Y (metric_topology Y dY) → relatively_compact_in_compact_convergence X Tx Y dY F → equicontinuous_family X Tx Y dY F ∧ Ascoli_pointwise_compact_closure X Y dY F

In Proofgold the corresponding term root is bc111f... and proposition id is acac73...

Proof:
Proof not loaded.

Definition. We define intersection_over_family to be λX Fam ⇒ intersection_of_family X Fam of type set → set → set.

In Proofgold the corresponding term root is 6d7088... and object id is 443aac...

Definition. We define Baire_space to be λX Tx ⇒ topology_on X Tx ∧ ∀U : set, U ⊆ Tx → countable_set U → (∀u : set, u ∈ U → u ∈ Tx ∧ dense_in u X Tx) → dense_in (intersection_over_family X U) X Tx of type set → set → prop.

In Proofgold the corresponding term root is 3d4b75... and object id is b5d45b...

Definition. We define Baire_space_closed to be λX Tx ⇒ topology_on X Tx ∧ ∀Fam : set, countable_set Fam → (∀A : set, A ∈ Fam → closed_in X Tx A ∧ interior_of X Tx A = Empty) → interior_of X Tx (⋃ Fam) = Empty of type set → set → prop.

In Proofgold the corresponding term root is b831de... and object id is d211ce...

Theorem. (union_of_complements_eq_complement_of_intersection_over_family)

∀X U : set, (∀u : set, u ∈ U → u ⊆ X) → ⋃ {X ∖ u|u ∈ U} = X ∖ (intersection_over_family X U)

In Proofgold the corresponding term root is ef4972... and proposition id is 4c79ec...

Proof:
Proof not loaded.

Theorem. (intersection_of_complements_eq_complement_of_Union)

∀X Fam : set, (∀A : set, A ∈ Fam → A ⊆ X) → intersection_over_family X {X ∖ A|A ∈ Fam} = X ∖ ⋃ Fam

In Proofgold the corresponding term root is 535683... and proposition id is 058a7b...

Proof:
Proof not loaded.

Theorem. (Baire_space_closed_imp)

∀X Tx : set, Baire_space_closed X Tx → Baire_space X Tx

In Proofgold the corresponding term root is 98eb1f... and proposition id is dd626b...

Proof:
Proof not loaded.

Theorem. (Baire_space_imp_closed)

∀X Tx : set, Baire_space X Tx → Baire_space_closed X Tx

In Proofgold the corresponding term root is b42199... and proposition id is 91f7fc...

Proof:
Proof not loaded.

Theorem. (Baire_space_closed_iff)

∀X Tx : set, (Baire_space_closed X Tx ↔ Baire_space X Tx)

In Proofgold the corresponding term root is d5ae1a... and proposition id is ec42b2...

Proof:
Proof not loaded.

Theorem. (Baire_space_dense_Gdelta)

∀X Tx : set, Baire_space X Tx → ∀U : set, U ⊆ Tx → countable_set U → (∀u : set, u ∈ U → u ∈ Tx ∧ dense_in u X Tx) → dense_in (intersection_over_family X U) X Tx

In Proofgold the corresponding term root is 1d90ec... and proposition id is 24fee5...

Proof:
Proof not loaded.

Theorem. (not_subset_ex_elem)

∀A B : set, ¬ (B ⊆ A) → ∃x : set, x ∈ B ∧ x ∉ A

In Proofgold the corresponding term root is 30b85f... and proposition id is e0c2d1...

Proof:
Proof not loaded.

Theorem. (subset_of_set_from_closure_sub)

∀X Tx A B : set, topology_on X Tx → A ⊆ X → closure_of X Tx A ⊆ B → A ⊆ B

In Proofgold the corresponding term root is 16f40e... and proposition id is 967204...

Proof:
Proof not loaded.

Theorem. (baire_step_metric_ball)

∀X d A B x : set, metric_on X d → x ∈ X → B ∈ metric_topology X d → x ∈ B → closed_in X (metric_topology X d) A → interior_of X (metric_topology X d) A = Empty → ∃x1 : set, ∃V1 : set, ∃N1 : set, (x1 ∈ B ∧ x1 ∉ A) ∧ (V1 ∈ metric_topology X d ∧ x1 ∈ V1 ∧ closure_of X (metric_topology X d) V1 ⊆ (B ∩ (X ∖ A))) ∧ (N1 ∈ ω ∧ open_ball X d x1 (eps_ N1) ⊆ V1)

In Proofgold the corresponding term root is 3e6680... and proposition id is 77a3ec...

Proof:
Proof not loaded.

Theorem. (baire_step_metric_ball_closure)

∀X d A B x : set, metric_on X d → x ∈ X → B ∈ metric_topology X d → x ∈ B → closed_in X (metric_topology X d) A → interior_of X (metric_topology X d) A = Empty → ∃x1 : set, ∃N1 : set, ∃B1 : set, (x1 ∈ B ∧ x1 ∉ A) ∧ (B1 = open_ball X d x1 (eps_ N1)) ∧ (B1 ∈ metric_topology X d ∧ x1 ∈ B1 ∧ closure_of X (metric_topology X d) B1 ⊆ (B ∩ (X ∖ A)))

In Proofgold the corresponding term root is b2fa1c... and proposition id is 65a9ef...

Proof:
Proof not loaded.

Theorem. (baire_step_metric_ball_bounded_closure)

∀X d A B x a : set, metric_on X d → x ∈ X → B ∈ metric_topology X d → x ∈ B → closed_in X (metric_topology X d) A → interior_of X (metric_topology X d) A = Empty → a ∈ R → Rlt 0 a → ∃x1 : set, ∃r1 : set, ∃B1 : set, (x1 ∈ B ∧ x1 ∉ A) ∧ (r1 ∈ R ∧ Rlt 0 r1 ∧ Rlt r1 a) ∧ (B1 = open_ball X d x1 r1) ∧ (B1 ∈ metric_topology X d ∧ x1 ∈ B1 ∧ closure_of X (metric_topology X d) B1 ⊆ (B ∩ (X ∖ A)))

In Proofgold the corresponding term root is 659a4f... and proposition id is 2a6a47...

Proof:
Proof not loaded.

Theorem. (baire_step_metric_ball_eps_bounded_closure)

∀X d A B x n : set, metric_on X d → x ∈ X → B ∈ metric_topology X d → x ∈ B → closed_in X (metric_topology X d) A → interior_of X (metric_topology X d) A = Empty → n ∈ ω → ∃x1 : set, ∃r1 : set, ∃B1 : set, (x1 ∈ B ∧ x1 ∉ A) ∧ (r1 ∈ R ∧ Rlt 0 r1 ∧ Rlt r1 (eps_ (ordsucc n))) ∧ (B1 = open_ball X d x1 r1) ∧ (B1 ∈ metric_topology X d ∧ x1 ∈ B1 ∧ closure_of X (metric_topology X d) B1 ⊆ (B ∩ (X ∖ A)))

In Proofgold the corresponding term root is 91e5e0... and proposition id is c67394...

Proof:
Proof not loaded.

Theorem. (baire_complete_metric_avoid_closed_seq)

∀X d U0 x0 : set, ∀An : set → set, complete_metric_space X d → (∀n : set, n ∈ ω → closed_in X (metric_topology X d) (An n) ∧ interior_of X (metric_topology X d) (An n) = Empty) → U0 ∈ metric_topology X d → x0 ∈ X → x0 ∈ U0 → ∃x : set, x ∈ U0 ∧ ∀n : set, n ∈ ω → x ∉ (An n)

In Proofgold the corresponding term root is 2c1b71... and proposition id is 5ee9c6...

Proof:
Proof not loaded.

Theorem. (Baire_category_complete_metric_closed_core)

∀X d Fam : set, complete_metric_space X d → countable_set Fam → (∀A : set, A ∈ Fam → closed_in X (metric_topology X d) A ∧ interior_of X (metric_topology X d) A = Empty) → interior_of X (metric_topology X d) (⋃ Fam) = Empty

In Proofgold the corresponding term root is 837199... and proposition id is 77375f...

Proof:
Proof not loaded.

Theorem. (Baire_category_complete_metric_closed)

∀X d : set, complete_metric_space X d → Baire_space_closed X (metric_topology X d)

In Proofgold the corresponding term root is 8892a6... and proposition id is 5d5369...

Proof:
Proof not loaded.

Theorem. (Baire_category_complete_metric)

∀X d : set, complete_metric_space X d → Baire_space X (metric_topology X d)

In Proofgold the corresponding term root is b86bed... and proposition id is 643556...

Proof:
Proof not loaded.

Theorem. (baire_step_regular_open_closure)

∀X Tx A B x : set, regular_space X Tx → x ∈ X → B ∈ Tx → x ∈ B → closed_in X Tx A → interior_of X Tx A = Empty → ∃x1 : set, ∃B1 : set, (x1 ∈ B ∧ x1 ∉ A) ∧ (B1 ∈ Tx ∧ x1 ∈ B1 ∧ closure_of X Tx B1 ⊆ (B ∩ (X ∖ A)))

In Proofgold the corresponding term root is c649f1... and proposition id is 2bada8...

Proof:
Proof not loaded.

Theorem. (accumulation_point_on_in_closure_of_eventually_terms)

∀X Tx net J le A x i0 : set, accumulation_point_of_net_on X Tx net J le x → A ⊆ X → i0 ∈ J → (∀i : set, i ∈ J → (i0,i) ∈ le → apply_fun net i ∈ A) → x ∈ closure_of X Tx A

In Proofgold the corresponding term root is 4022bc... and proposition id is 06e61f...

Proof:
Proof not loaded.

Theorem. (baire_compact_Hausdorff_avoid_closed_seq)

∀X Tx U0 x0 : set, ∀An : set → set, compact_space X Tx → Hausdorff_space X Tx → (∀n : set, n ∈ ω → closed_in X Tx (An n) ∧ interior_of X Tx (An n) = Empty) → U0 ∈ Tx → x0 ∈ X → x0 ∈ U0 → ∃x : set, x ∈ U0 ∧ ∀n : set, n ∈ ω → x ∉ (An n)

In Proofgold the corresponding term root is b4ee4a... and proposition id is a445f2...

Proof:
Proof not loaded.

Theorem. (Baire_category_compact_Hausdorff_closed_core)

∀X Tx Fam : set, compact_space X Tx → Hausdorff_space X Tx → countable_set Fam → (∀A : set, A ∈ Fam → closed_in X Tx A ∧ interior_of X Tx A = Empty) → interior_of X Tx (⋃ Fam) = Empty

In Proofgold the corresponding term root is c800f5... and proposition id is 1d6c13...

Proof:
Proof not loaded.

Theorem. (Baire_category_compact_Hausdorff_closed)

∀X Tx : set, compact_space X Tx → Hausdorff_space X Tx → Baire_space_closed X Tx

In Proofgold the corresponding term root is 389460... and proposition id is 573421...

Proof:
Proof not loaded.

Theorem. (Baire_category_compact_Hausdorff)

∀X Tx : set, compact_space X Tx → Hausdorff_space X Tx → Baire_space X Tx

In Proofgold the corresponding term root is 27a72f... and proposition id is 5d41c2...

Proof:
Proof not loaded.

Theorem. (Baire_category_theorem)

∀X Tx : set, (compact_space X Tx ∧ Hausdorff_space X Tx) ∨ (∃d : set, complete_metric_space X d ∧ Tx = metric_topology X d) → Baire_space X Tx

In Proofgold the corresponding term root is 1aa8b2... and proposition id is 85217d...

Proof:
Proof not loaded.

Theorem. (Baire_open_subspace)

∀X Tx Y : set, Baire_space X Tx → open_in X Tx Y → Baire_space Y (subspace_topology X Tx Y)

In Proofgold the corresponding term root is a1a87d... and proposition id is a11dc6...

Proof:
Proof not loaded.

Definition. We define A_N_eps to be λX Y d fn N eps ⇒ {x ∈ X|∀n : set, n ∈ ω → N ⊆ n → ∀m : set, m ∈ ω → N ⊆ m → Rle (apply_fun d (apply_fun (apply_fun fn n) x,apply_fun (apply_fun fn m) x)) eps} of type set → set → set → set → set → set → set.

In Proofgold the corresponding term root is 00c7fd... and object id is 75e990...

Theorem. (A_N_eps_monotone)

∀X Y d fn N eps1 eps2 : set, eps1 ∈ R → eps2 ∈ R → Rle eps1 eps2 → A_N_eps X Y d fn N eps1 ⊆ A_N_eps X Y d fn N eps2

In Proofgold the corresponding term root is 0f5ae6... and proposition id is fc55cf...

Proof:
Proof not loaded.

Theorem. (metric_distance_above_has_product_ball)

∀Y d a p : set, metric_on_total Y d → a ∈ R → p ∈ preimage_of (setprod Y Y) d (open_ray_upper R a) → ∃r : set, r ∈ R ∧ Rlt 0 r ∧ rectangle_set (open_ball Y d (p 0) r) (open_ball Y d (p 1) r) ⊆ preimage_of (setprod Y Y) d (open_ray_upper R a)

In Proofgold the corresponding term root is 267504... and proposition id is 6cfd82...

Proof:
Proof not loaded.

Theorem. (metric_distance_preimage_open_ray_upper)

∀Y d a : set, metric_on_total Y d → a ∈ R → preimage_of (setprod Y Y) d (open_ray_upper R a) ∈ (product_topology Y (metric_topology Y d) Y (metric_topology Y d))

In Proofgold the corresponding term root is 28fa9f... and proposition id is e84893...

Proof:
Proof not loaded.

Theorem. (metric_distance_below_has_product_ball)

∀Y d b p : set, metric_on_total Y d → b ∈ R → p ∈ preimage_of (setprod Y Y) d (open_ray_lower R b) → ∃r : set, r ∈ R ∧ Rlt 0 r ∧ rectangle_set (open_ball Y d (p 0) r) (open_ball Y d (p 1) r) ⊆ preimage_of (setprod Y Y) d (open_ray_lower R b)

In Proofgold the corresponding term root is c21214... and proposition id is 8c43f1...

Proof:
Proof not loaded.

Theorem. (metric_distance_preimage_open_ray_lower)

∀Y d b : set, metric_on_total Y d → b ∈ R → preimage_of (setprod Y Y) d (open_ray_lower R b) ∈ (product_topology Y (metric_topology Y d) Y (metric_topology Y d))

In Proofgold the corresponding term root is 5cae5b... and proposition id is 5003fe...

Proof:
Proof not loaded.

Theorem. (metric_distance_continuous)

∀Y d : set, metric_on_total Y d → continuous_map (setprod Y Y) (product_topology Y (metric_topology Y d) Y (metric_topology Y d)) R R_standard_topology d

In Proofgold the corresponding term root is 4b7b42... and proposition id is de69c5...

Proof:
Proof not loaded.

Theorem. (A_N_eps_closed_stub)

∀X Tx Y d fn N eps : set, topology_on X Tx → metric_on_total Y d → (∀n : set, n ∈ ω → continuous_map X Tx Y (metric_topology Y d) (apply_fun fn n)) → N ∈ ω → eps ∈ R → closed_in X Tx (A_N_eps X Y d fn N eps)

In Proofgold the corresponding term root is 43a61d... and proposition id is df307c...

Proof:
Proof not loaded.

Definition. We define U_eps to be λX Tx Y d fn eps ⇒ ⋃ {interior_of X Tx (A_N_eps X Y d fn N eps)|N ∈ ω} of type set → set → set → set → set → set → set.

In Proofgold the corresponding term root is 02cd0f... and object id is 816f3e...

Theorem. (U_eps_monotone)

∀X Tx Y d fn eps1 eps2 : set, topology_on X Tx → eps1 ∈ R → eps2 ∈ R → Rle eps1 eps2 → U_eps X Tx Y d fn eps1 ⊆ U_eps X Tx Y d fn eps2

In Proofgold the corresponding term root is f5ac36... and proposition id is 412d48...

Proof:
Proof not loaded.

Theorem. (U_epsE_open_subset_A_N)

∀X Tx Y d fn eps x : set, topology_on X Tx → x ∈ U_eps X Tx Y d fn eps → ∃N : set, N ∈ ω ∧ ∃U : set, U ∈ Tx ∧ x ∈ U ∧ U ⊆ A_N_eps X Y d fn N eps

In Proofgold the corresponding term root is 309593... and proposition id is 2722ce...

Proof:
Proof not loaded.

Theorem. (pointwise_limit_metric_imp_cover_A_N_eps_stub)

∀X Y d fn f eps : set, metric_on_total Y d → (∀n : set, n ∈ ω → function_on (apply_fun fn n) X Y) → function_on f X Y → pointwise_limit_metric X Y d fn f → eps ∈ R → Rlt 0 eps → ∀x : set, x ∈ X → ∃N : set, N ∈ ω ∧ x ∈ A_N_eps X Y d fn N eps

In Proofgold the corresponding term root is a72d51... and proposition id is 8de008...

Proof:
Proof not loaded.

Theorem. (U_eps_open_dense_stub)

∀X Tx Y d fn eps : set, Baire_space X Tx → metric_on_total Y d → (∀n : set, n ∈ ω → continuous_map X Tx Y (metric_topology Y d) (apply_fun fn n)) → (∀N : set, N ∈ ω → closed_in X Tx (A_N_eps X Y d fn N eps)) → (∀x : set, x ∈ X → ∃N : set, N ∈ ω ∧ x ∈ A_N_eps X Y d fn N eps) → eps ∈ R → Rlt 0 eps → open_in X Tx (U_eps X Tx Y d fn eps) ∧ dense_in (U_eps X Tx Y d fn eps) X Tx

In Proofgold the corresponding term root is dd948f... and proposition id is 76eb9b...

Proof:
Proof not loaded.

Theorem. (exists_fourfold_small_eps)

∀eps0 : set, eps0 ∈ R → Rlt 0 eps0 → ∃eps1 : set, eps1 ∈ R ∧ Rlt 0 eps1 ∧ Rlt (add_SNo (add_SNo eps1 eps1) (add_SNo eps1 eps1)) eps0

In Proofgold the corresponding term root is 36665b... and proposition id is fd7e35...

Proof:
Proof not loaded.

Theorem. (pointwise_limit_continuity_points_dense)

∀X Tx Y d fn f : set, metric_on_total Y d → (∀n : set, n ∈ ω → continuous_map X Tx Y (metric_topology Y d) (apply_fun fn n)) → function_on f X Y → pointwise_limit_metric X Y d fn f → Baire_space X Tx → dense_in {x ∈ X|continuous_at_map X Tx Y (metric_topology Y d) f x} X Tx

In Proofgold the corresponding term root is 158870... and proposition id is 7f28da...

Proof:
Proof not loaded.

Definition. We define differentiable_at to be λf x ⇒ function_on f unit_interval R ∧ x ∈ unit_interval ∧ ∃L : set, L ∈ R ∧ ∀eps : set, eps ∈ R → Rlt 0 eps → ∃delta : set, delta ∈ R ∧ Rlt 0 delta ∧ ∀h : set, h ∈ R → ¬ (h = 0) → Rlt 0 (Abs h) → Rlt (Abs h) delta → (add_SNo x h) ∈ unit_interval → Rlt (Abs (add_SNo (div_SNo (add_SNo (apply_fun f (add_SNo x h)) (minus_SNo (apply_fun f x))) h) (minus_SNo L))) eps of type set → set → prop.

In Proofgold the corresponding term root is ba47e8... and object id is 72d84e...

Definition. We define nowhere_differentiable to be λf ⇒ function_on f unit_interval R ∧ ∀x : set, x ∈ unit_interval → ¬ differentiable_at f x of type set → prop.

In Proofgold the corresponding term root is 9924de... and object id is 670b7d...

Definition. We define I_topology to be unit_interval_topology of type set.

In Proofgold the corresponding term root is 59fcc5... and object id is aea1f7...

Theorem. (I_topology_on)

topology_on unit_interval I_topology

In Proofgold the corresponding term root is 14fbd8... and proposition id is a8fa0f...

Proof:
Proof not loaded.

Definition. We define continuous_real_on_I to be λf ⇒ continuous_map unit_interval I_topology R R_standard_topology f of type set → prop.

In Proofgold the corresponding term root is d25dc7... and object id is 821cf9...

Definition. We define C_I_R to be {f ∈ function_space unit_interval R|continuous_real_on_I f} of type set.

In Proofgold the corresponding term root is f4a3f5... and object id is fdcd36...

Axiom. (theorem_49_1_nowhere_differentiable_approx) We take the following as an axiom:

∀h eps : set, continuous_map unit_interval I_topology R R_standard_topology h → eps ∈ R → Rlt 0 eps → ∃g : set, continuous_map unit_interval I_topology R R_standard_topology g ∧ nowhere_differentiable g ∧ ∀x : set, x ∈ unit_interval → Rlt (Abs (add_SNo (apply_fun h x) (minus_SNo (apply_fun g x)))) eps

In Proofgold the corresponding term root is ce9780... and proposition id is 6eeb8f...

Definition. We define diffquot_forward_abs to be λf x h ⇒ Abs (div_SNo (add_SNo (apply_fun f (add_SNo x h)) (minus_SNo (apply_fun f x))) h) of type set → set → set → set.

In Proofgold the corresponding term root is d1efcb... and object id is a20f44...

Definition. We define diffquot_backward_abs to be λf x h ⇒ Abs (div_SNo (add_SNo (apply_fun f (add_SNo x (minus_SNo h))) (minus_SNo (apply_fun f x))) (minus_SNo h)) of type set → set → set → set.

In Proofgold the corresponding term root is 8b3ef7... and object id is affc70...

Definition. We define Delta_gt to be λf x h n ⇒ (add_SNo x h ∈ unit_interval ∧ Rlt n (diffquot_forward_abs f x h)) ∨ (add_SNo x (minus_SNo h) ∈ unit_interval ∧ Rlt n (diffquot_backward_abs f x h)) of type set → set → set → set → prop.

In Proofgold the corresponding term root is 5e8251... and object id is d30b46...

Definition. We define U_n to be λn ⇒ {f ∈ C_I_R|n ∈ ω ∧ 2 ⊆ n ∧ ∃h : set, h ∈ R ∧ Rlt 0 h ∧ Rle h (div_SNo 1 n) ∧ ∀x : set, x ∈ unit_interval → Delta_gt f x h n} of type set → set.

In Proofgold the corresponding term root is 1ee247... and object id is c8438f...

Theorem. (nowhere_differentiable_function_exists)

∃f : set, continuous_map unit_interval I_topology R R_standard_topology f ∧ nowhere_differentiable f

In Proofgold the corresponding term root is 4daa6f... and proposition id is ca477d...

Proof:
Proof not loaded.

Definition. We define cardinality_exact to be λS n ⇒ ordinal n ∧ equip S n of type set → set → prop.

In Proofgold the corresponding term root is b2404a... and object id is 71475a...

Definition. We define cardinality_at_most to be λS n ⇒ ordinal n ∧ ∃k : set, ordinal k ∧ k ⊆ n ∧ equip S k of type set → set → prop.

In Proofgold the corresponding term root is 1a6739... and object id is a55e8a...

Theorem. (finite_ordinal_subset_equip_subordinal)

∀n S : set, nat_p n → S ⊆ n → ∃m : set, nat_p m ∧ (m ⊆ n ∧ equip S m)

In Proofgold the corresponding term root is 096b37... and proposition id is af289c...

Proof:
Proof not loaded.

Theorem. (cardinality_at_most_mono_subset_finite)

∀S1 S2 n : set, n ∈ ω → S1 ⊆ S2 → cardinality_at_most S2 n → cardinality_at_most S1 n

In Proofgold the corresponding term root is 418526... and proposition id is 5f441e...

Proof:
Proof not loaded.

Theorem. (cardinality_at_most_equip_left)

∀S1 S2 n : set, equip S1 S2 → cardinality_at_most S2 n → cardinality_at_most S1 n

In Proofgold the corresponding term root is 1d4f48... and proposition id is 5dc294...

Proof:
Proof not loaded.

Theorem. (cardinality_at_most_image_finite)

∀S n : set, ∀f : set → set, n ∈ ω → cardinality_at_most S (ordsucc n) → cardinality_at_most {f x|x ∈ S} (ordsucc n)

In Proofgold the corresponding term root is ff8248... and proposition id is 436467...

Proof:
Proof not loaded.

Definition. We define collection_has_order_at_m_plus_one to be λX A m ⇒ ordinal m ∧ (∃x : set, x ∈ X ∧ ∃Fam : set, Fam ⊆ A ∧ finite Fam ∧ cardinality_exact Fam (ordsucc m) ∧ ∀U : set, U ∈ Fam → x ∈ U) ∧ ∀x : set, x ∈ X → cardinality_at_most {U ∈ A|x ∈ U} (ordsucc m) of type set → set → set → prop.

In Proofgold the corresponding term root is d8ac77... and object id is 0ba8dc...

Definition. We define collection_has_order_at_most_m_plus_one to be λX A m ⇒ ordinal m ∧ ∀x : set, x ∈ X → cardinality_at_most {U ∈ A|x ∈ U} (ordsucc m) of type set → set → set → prop.

In Proofgold the corresponding term root is d964aa... and object id is 3718f0...

Definition. We define refines_cover to be λB A ⇒ ∀U : set, U ∈ B → ∃V : set, V ∈ A ∧ U ⊆ V of type set → set → prop.

In Proofgold the corresponding term root is 784b8a... and object id is 2c5af6...

Theorem. (refines_cover_ref)

∀A : set, refines_cover A A

In Proofgold the corresponding term root is c1fe61... and proposition id is c17047...

Proof:
Proof not loaded.

Theorem. (refines_cover_tra)

∀A B C : set, refines_cover C B → refines_cover B A → refines_cover C A

In Proofgold the corresponding term root is 2cfb1f... and proposition id is 8eeaf2...

Proof:
Proof not loaded.

Theorem. (eq_subst_mem)

∀x y S : set, x = y → y ∈ S → x ∈ S

In Proofgold the corresponding term root is a86718... and proposition id is 30a5ce...

Proof:
Proof not loaded.

Theorem. (eq_subst_mem_rev)

∀x y S : set, x = y → x ∈ S → y ∈ S

In Proofgold the corresponding term root is 181ae0... and proposition id is f710df...

Proof:
Proof not loaded.

Definition. We define ambient_open_of_subspace_open to be λX Tx Y U ⇒ Eps_i (λV : set ⇒ V ∈ Tx ∧ U = V ∩ Y) of type set → set → set → set → set.

In Proofgold the corresponding term root is a13474... and object id is 5d29fa...

Theorem. (ambient_open_of_subspace_open_spec)

∀X Tx Y U : set, U ∈ subspace_topology X Tx Y → (ambient_open_of_subspace_open X Tx Y U) ∈ Tx ∧ U = (ambient_open_of_subspace_open X Tx Y U) ∩ Y

In Proofgold the corresponding term root is bcd335... and proposition id is cffe5a...

Proof:
Proof not loaded.

Definition. We define covering_dimension to be λX Tx n ⇒ topology_on X Tx ∧ n ∈ ω ∧ ∀A : set, open_cover_of X Tx A → ∃B : set, open_cover_of X Tx B ∧ refines_cover B A ∧ collection_has_order_at_most_m_plus_one X B n of type set → set → set → prop.

In Proofgold the corresponding term root is 46f379... and object id is 8e3294...

Theorem. (covering_dimension_topology_on)

∀X Tx n : set, covering_dimension X Tx n → topology_on X Tx

In Proofgold the corresponding term root is cd3d3d... and proposition id is 928d39...

Proof:
Proof not loaded.

Theorem. (covering_dimension_n_in_omega)

∀X Tx n : set, covering_dimension X Tx n → n ∈ ω

In Proofgold the corresponding term root is 1c9c12... and proposition id is 96275b...

Proof:
Proof not loaded.

Theorem. (covering_dimensionI)

∀X Tx n : set, topology_on X Tx → n ∈ ω → (∀A : set, open_cover_of X Tx A → ∃B : set, open_cover_of X Tx B ∧ refines_cover B A ∧ collection_has_order_at_most_m_plus_one X B n) → covering_dimension X Tx n

In Proofgold the corresponding term root is ac0895... and proposition id is 5ecc9a...

Proof:
Proof not loaded.

Definition. We define finite_dimensional_space to be λX Tx ⇒ topology_on X Tx ∧ ∃m : set, m ∈ ω ∧ ∀A : set, open_cover_of X Tx A → ∃B : set, open_cover_of X Tx B ∧ refines_cover B A ∧ collection_has_order_at_most_m_plus_one X B m of type set → set → prop.

In Proofgold the corresponding term root is 3ca45d... and object id is c15064...

Theorem. (finite_dimensional_spaceI)

∀X Tx : set, topology_on X Tx → (∃m : set, m ∈ ω ∧ ∀A : set, open_cover_of X Tx A → ∃B : set, open_cover_of X Tx B ∧ refines_cover B A ∧ collection_has_order_at_most_m_plus_one X B m) → finite_dimensional_space X Tx

In Proofgold the corresponding term root is ab3728... and proposition id is f5faa4...

Proof:
Proof not loaded.

Theorem. (finite_dimensional_space_topology_on)

∀X Tx : set, finite_dimensional_space X Tx → topology_on X Tx

In Proofgold the corresponding term root is 9649e4... and proposition id is e28737...

Proof:
Proof not loaded.

Theorem. (finite_dimensional_space_exists_bound)

∀X Tx : set, finite_dimensional_space X Tx → ∃m : set, m ∈ ω ∧ ∀A : set, open_cover_of X Tx A → ∃B : set, open_cover_of X Tx B ∧ refines_cover B A ∧ collection_has_order_at_most_m_plus_one X B m

In Proofgold the corresponding term root is 99d7f1... and proposition id is 5a4af6...

Proof:
Proof not loaded.

Theorem. (covering_dimension_implies_finite_dimensional_space)

∀X Tx n : set, covering_dimension X Tx n → finite_dimensional_space X Tx

In Proofgold the corresponding term root is 1da418... and proposition id is 43fa1b...

Proof:
Proof not loaded.

Theorem. (finite_dimensional_space_implies_exists_covering_dimension)

∀X Tx : set, finite_dimensional_space X Tx → ∃m : set, covering_dimension X Tx m

In Proofgold the corresponding term root is 60497d... and proposition id is 8baa8f...

Proof:
Proof not loaded.

Theorem. (euclidean_space_covering_dimension_le)

∀N : set, N ∈ ω → covering_dimension (euclidean_space N) (euclidean_topology N) N

In Proofgold the corresponding term root is e3b2ec... and proposition id is 07f201...

Proof:
Proof not loaded.

Theorem. (compact_manifold_dimension_le)

∀X Tx m : set, m_manifold X Tx m → compact_space X Tx → covering_dimension X Tx m

In Proofgold the corresponding term root is cf510a... and proposition id is 7fac38...

Proof:
Proof not loaded.

Theorem. (Menger_Nobeling_embedding)

∀X Tx m : set, compact_space X Tx → metrizable X Tx → covering_dimension X Tx m → ∃N : set, ∃e : set, embedding_of X Tx (euclidean_space N) (euclidean_topology N) e

In Proofgold the corresponding term root is e06c19... and proposition id is 402ef7...

Proof:
Proof not loaded.

Theorem. (cardinality_at_most_mono_subset)

∀S1 S2 n : set, S1 ⊆ S2 → cardinality_at_most S2 n → cardinality_at_most S1 n

In Proofgold the corresponding term root is d47ad3... and proposition id is 5b84ae...

Proof:
Proof not loaded.

Theorem. (cardinality_at_most_restrict_family_to_subspace_finite)

∀Fam Y n : set, n ∈ ω → cardinality_at_most Fam (ordsucc n) → cardinality_at_most (restrict_family_to_subspace Fam Y) (ordsucc n)

In Proofgold the corresponding term root is 60cb08... and proposition id is 21e00a...

Proof:
Proof not loaded.

Theorem. (collection_has_order_at_most_m_plus_one_restrict_to_subspace)

∀X Y Fam n : set, n ∈ ω → Y ⊆ X → collection_has_order_at_most_m_plus_one X Fam n → collection_has_order_at_most_m_plus_one Y (restrict_family_to_subspace Fam Y) n

In Proofgold the corresponding term root is 7867e1... and proposition id is a11633...

Proof:
Proof not loaded.

Theorem. (collection_has_order_at_most_m_plus_one_mono_family)

∀X Fam1 Fam2 n : set, n ∈ ω → Fam1 ⊆ Fam2 → collection_has_order_at_most_m_plus_one X Fam2 n → collection_has_order_at_most_m_plus_one X Fam1 n

In Proofgold the corresponding term root is c00d42... and proposition id is 939fa7...

Proof:
Proof not loaded.

Theorem. (dimension_closed_subspace_le)

∀X Tx Y n : set, covering_dimension X Tx n → closed_in X Tx Y → covering_dimension Y (subspace_topology X Tx Y) n

In Proofgold the corresponding term root is d756ed... and proposition id is 072031...

Proof:
Proof not loaded.

Theorem. (dimension_union_closed_max)

∀X Tx Y Z n : set, topology_on X Tx → Y ⊆ X → Z ⊆ X → closed_in X Tx Y → closed_in X Tx Z → covering_dimension Y (subspace_topology X Tx Y) n → covering_dimension Z (subspace_topology X Tx Z) n → covering_dimension (Y ∪ Z) (subspace_topology X Tx (Y ∪ Z)) n

In Proofgold the corresponding term root is 6ccdf9... and proposition id is 15ad4c...

Proof:
Proof not loaded.

Theorem. (dimension_finite_union_closed_max)

∀X Tx Fam n : set, topology_on X Tx → n ∈ ω → finite Fam → (∀Y : set, Y ∈ Fam → Y ⊆ X ∧ closed_in X Tx Y ∧ covering_dimension Y (subspace_topology X Tx Y) n) → covering_dimension (⋃ Fam) (subspace_topology X Tx (⋃ Fam)) n

In Proofgold the corresponding term root is 5dd280... and proposition id is 9d5b57...

Proof:
Proof not loaded.

Theorem. (compact_1_manifold_dimension_1)

∀X Tx : set, compact_space X Tx → m_manifold X Tx (Sing Empty) → covering_dimension X Tx (Sing Empty)

In Proofgold the corresponding term root is d1fcb9... and proposition id is b5f87a...

Proof:
Proof not loaded.

Definition. We define two to be 2 of type set.

In Proofgold the corresponding term root is 3bd7f7... and object id is 8c039e...

Theorem. (compact_2_manifold_dimension_le_2)

∀X Tx : set, compact_space X Tx → m_manifold X Tx two → covering_dimension X Tx two

In Proofgold the corresponding term root is 32281e... and proposition id is 8d7b03...

Proof:
Proof not loaded.

Definition. We define arc to be λX Tx ⇒ ∃f : set, homeomorphism unit_interval unit_interval_topology X Tx f of type set → set → prop.

In Proofgold the corresponding term root is 57f7d9... and object id is 995e2e...

Theorem. (arc_is_topological_space)

∀X Tx : set, arc X Tx → topology_on X Tx

In Proofgold the corresponding term root is 704e07... and proposition id is 467ce0...

Proof:
Proof not loaded.

Definition. We define end_points_of_arc to be λX Tx p q ⇒ arc X Tx ∧ p ∈ X ∧ q ∈ X ∧ p ≠ q ∧ connected_space (X ∖ (Sing p)) (subspace_topology X Tx (X ∖ (Sing p))) ∧ connected_space (X ∖ (Sing q)) (subspace_topology X Tx (X ∖ (Sing q))) of type set → set → set → set → prop.

In Proofgold the corresponding term root is 540e96... and object id is 067d5a...

Definition. We define linear_graph to be λG Tg ⇒ Hausdorff_space G Tg ∧ ∃Arcs : set, finite Arcs ∧ (∀A : set, A ∈ Arcs → A ⊆ G ∧ arc A (subspace_topology G Tg A)) ∧ G = ⋃ Arcs ∧ (∀A B : set, A ∈ Arcs → B ∈ Arcs → A ≠ B → ∃p : set, (A ∩ B = Empty ∨ A ∩ B = Sing p)) of type set → set → prop.

In Proofgold the corresponding term root is 19c8c8... and object id is 8cfb5f...

Theorem. (linear_graph_dimension_1)

∀G Tg : set, linear_graph G Tg → covering_dimension G Tg (Sing Empty)

In Proofgold the corresponding term root is 71ca5a... and proposition id is d22dc0...

Proof:
Proof not loaded.

Definition. We define R3_xcoord to be λp ⇒ p 0 of type set → set.

In Proofgold the corresponding term root is f0bb69... and object id is 9211e4...

Definition. We define R3_ycoord to be λp ⇒ p 1 of type set → set.

In Proofgold the corresponding term root is 24bd9a... and object id is 84977b...

Definition. We define R3_zcoord to be λp ⇒ p 2 of type set → set.

In Proofgold the corresponding term root is a18251... and object id is ad7e2f...

Definition. We define R3_dx to be λp q ⇒ add_SNo (R3_xcoord q) (minus_SNo (R3_xcoord p)) of type set → set → set.

In Proofgold the corresponding term root is 782a98... and object id is 8da7b7...

Definition. We define R3_dy to be λp q ⇒ add_SNo (R3_ycoord q) (minus_SNo (R3_ycoord p)) of type set → set → set.

In Proofgold the corresponding term root is 19ebf0... and object id is ad0df3...

Definition. We define R3_dz to be λp q ⇒ add_SNo (R3_zcoord q) (minus_SNo (R3_zcoord p)) of type set → set → set.

In Proofgold the corresponding term root is 09b2cc... and object id is 6443b3...

Definition. We define det2_SNo to be λa b c d ⇒ add_SNo (mul_SNo a d) (minus_SNo (mul_SNo b c)) of type set → set → set → set → set.

In Proofgold the corresponding term root is 0a4633... and object id is a7a32e...

Definition. We define det3_SNo to be λa1 a2 a3 b1 b2 b3 c1 c2 c3 ⇒ add_SNo (mul_SNo a1 (det2_SNo b2 b3 c2 c3)) (add_SNo (minus_SNo (mul_SNo a2 (det2_SNo b1 b3 c1 c3))) (mul_SNo a3 (det2_SNo b1 b2 c1 c2))) of type set → set → set → set → set → set → set → set → set → set.

In Proofgold the corresponding term root is 3fb174... and object id is 0c992a...

Definition. We define collinear_in_R3 to be λp q r ⇒ p ∈ (euclidean_space 3) ∧ q ∈ (euclidean_space 3) ∧ r ∈ (euclidean_space 3) ∧ det2_SNo (R3_dx p q) (R3_dy p q) (R3_dx p r) (R3_dy p r) = 0 ∧ det2_SNo (R3_dx p q) (R3_dz p q) (R3_dx p r) (R3_dz p r) = 0 ∧ det2_SNo (R3_dy p q) (R3_dz p q) (R3_dy p r) (R3_dz p r) = 0 of type set → set → set → prop.

In Proofgold the corresponding term root is 296712... and object id is e4317c...

Definition. We define coplanar_in_R3 to be λp q r s ⇒ p ∈ (euclidean_space 3) ∧ q ∈ (euclidean_space 3) ∧ r ∈ (euclidean_space 3) ∧ s ∈ (euclidean_space 3) ∧ det3_SNo (R3_dx p q) (R3_dy p q) (R3_dz p q) (R3_dx p r) (R3_dy p r) (R3_dz p r) (R3_dx p s) (R3_dy p s) (R3_dz p s) = 0 of type set → set → set → set → prop.

In Proofgold the corresponding term root is 3e1812... and object id is 66cbce...

Definition. We define geometrically_independent to be λS ⇒ ∃N : set, N ∈ ω ∧ S ⊆ euclidean_space N of type set → prop.

In Proofgold the corresponding term root is c8df3d... and object id is 20ba62...

Definition. We define affine_plane to be λS ⇒ Eps_i (λP : set ⇒ ∃N : set, N ∈ ω ∧ S ⊆ euclidean_space N ∧ P ⊆ euclidean_space N) of type set → set.

In Proofgold the corresponding term root is 0470a8... and object id is 33a5a3...

Definition. We define k_plane to be λk P ⇒ k ∈ ω ∧ ∃S : set, geometrically_independent S ∧ finite S ∧ (∃kp1 : set, kp1 = k ∪ (Sing k) ∧ equip S kp1) ∧ P = affine_plane S of type set → set → prop.

In Proofgold the corresponding term root is 27ff47... and object id is 665794...

Definition. We define general_position_RN to be λN A ⇒ N ∈ ω ∧ A ⊆ euclidean_space N ∧ ∀S : set, S ⊆ A → (∀Np1 : set, Np1 = N ∪ (Sing N) → (∃f : set → set, inj S Np1 f) → geometrically_independent S) of type set → set → prop.

In Proofgold the corresponding term root is 675939... and object id is 276f75...

Theorem. (finite_set_approximation_general_position)

∀N : set, ∀pts : set, ∀delta : set, N ∈ ω → finite pts → pts ⊆ euclidean_space N → delta ∈ R → ∃pts' : set, general_position_RN N pts' ∧ finite pts' ∧ equip pts pts'

In Proofgold the corresponding term root is c1ae60... and proposition id is 235ad2...

Proof:
Proof not loaded.

Theorem. (Menger_Nobeling_embedding_full)

∀X Tx m : set, compact_space X Tx → metrizable X Tx → covering_dimension X Tx m → m ∈ ω → ∃N : set, ∃e : set, N = add_nat (mul_nat two m) (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e

In Proofgold the corresponding term root is ea1049... and proposition id is 157acd...

Proof:
Proof not loaded.

Theorem. (compact_subspace_Rn_dimension_le)

∀N X : set, N ∈ ω → X ⊆ (euclidean_space N) → compact_space X (subspace_topology (euclidean_space N) (euclidean_topology N) X) → covering_dimension X (subspace_topology (euclidean_space N) (euclidean_topology N) X) N

In Proofgold the corresponding term root is 085163... and proposition id is 063550...

Proof:
Proof not loaded.

Theorem. (compact_subspace_RN_dimension_le_N)

∀X N : set, N ∈ ω → X ⊆ (euclidean_space N) → compact_space X (subspace_topology (euclidean_space N) (euclidean_topology N) X) → covering_dimension X (subspace_topology (euclidean_space N) (euclidean_topology N) X) N

In Proofgold the corresponding term root is 70ae8f... and proposition id is 9b5d28...

Proof:
Proof not loaded.

Theorem. (compact_m_manifold_dimension_le_m)

∀X Tx m : set, m ∈ ω → compact_space X Tx → m_manifold X Tx m → covering_dimension X Tx m

In Proofgold the corresponding term root is 6c2c76... and proposition id is 7504c6...

Proof:
Proof not loaded.

Theorem. (compact_m_manifold_embeds_R2mp1)

∀X Tx m : set, m ∈ ω → compact_space X Tx → m_manifold X Tx m → ∃N : set, ∃e : set, N = add_nat (mul_nat two m) (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e

In Proofgold the corresponding term root is f18c3f... and proposition id is d640a1...

Proof:
Proof not loaded.

Theorem. (compact_manifold_embeds_in_Euclidean)

∀X Tx : set, ∀m : set, m_manifold X Tx m → compact_space X Tx → ∃N : set, ∃e : set, N ∈ ω ∧ N ≠ Empty ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e

In Proofgold the corresponding term root is 610236... and proposition id is c35b79...

Proof:
Proof not loaded.

Theorem. (compact_metrizable_embeds_iff_finite_dim)

∀X Tx : set, compact_space X Tx → metrizable X Tx → ((∃N : set, ∃e : set, N ∈ ω ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e) ↔ finite_dimensional_space X Tx)

In Proofgold the corresponding term root is bd9934... and proposition id is 0d8416...

Proof:
Proof not loaded.

Definition. We define locally_m_euclidean to be λX Tx m ⇒ m ∈ ω ∧ topology_on X Tx ∧ ∀x : set, x ∈ X → ∃U : set, ∃V : set, ∃f : set, open_in X Tx U ∧ x ∈ U ∧ V ⊆ (euclidean_space m) ∧ open_in (euclidean_space m) (euclidean_topology m) V ∧ homeomorphism U (subspace_topology X Tx U) V (subspace_topology (euclidean_space m) (euclidean_topology m) V) f of type set → set → set → prop.

In Proofgold the corresponding term root is 48a15f... and object id is 3ffa74...

Theorem. (euclidean_space_Hausdorff)

∀m : set, Hausdorff_space (euclidean_space m) (euclidean_topology m)

In Proofgold the corresponding term root is af0d86... and proposition id is 4a57d7...

Proof:
Proof not loaded.

Theorem. (euclidean_space_regular)

∀m : set, regular_space (euclidean_space m) (euclidean_topology m)

In Proofgold the corresponding term root is 002969... and proposition id is 5c1782...

Proof:
Proof not loaded.

Theorem. (euclidean_space_completely_regular)

∀m : set, completely_regular_space (euclidean_space m) (euclidean_topology m)

In Proofgold the corresponding term root is 4f82f3... and proposition id is 5cbca3...

Proof:
Proof not loaded.

Theorem. (euclidean_space_metrizable)

∀n : set, n ∈ ω → metrizable (euclidean_space n) (euclidean_topology n)

In Proofgold the corresponding term root is e63060... and proposition id is 0c1c39...

Proof:
Proof not loaded.

Theorem. (m_manifold_implies_locally_m_euclidean)

∀X Tx m : set, m_manifold X Tx m → locally_m_euclidean X Tx m

In Proofgold the corresponding term root is 262a38... and proposition id is ebc815...

Proof:
Proof not loaded.

Theorem. (euclidean_space_T1)

∀m : set, T1_space (euclidean_space m) (euclidean_topology m)

In Proofgold the corresponding term root is d83ad9... and proposition id is 7c8783...

Proof:
Proof not loaded.

Theorem. (locally_m_euclidean_implies_T1)

∀X Tx m : set, locally_m_euclidean X Tx m → T1_space X Tx

In Proofgold the corresponding term root is 211852... and proposition id is 02e686...

Proof:
Proof not loaded.

Theorem. (ex50_1_discrete_dimension_0)

∀X Tx : set, Tx = discrete_topology X → topology_on X Tx → covering_dimension X Tx Empty

In Proofgold the corresponding term root is f61065... and proposition id is 10d94d...

Proof:
Proof not loaded.

Theorem. (ex50_2_connected_T1_dimension_ge_1)

∀X Tx : set, connected_space X Tx → T1_space X Tx → (∃x y : set, x ∈ X ∧ y ∈ X ∧ x ≠ y) → covering_dimension X Tx Empty → False

In Proofgold the corresponding term root is 4178a4... and proposition id is 7887ea...

Proof:
Proof not loaded.

Definition. We define topologists_sine_curve to be Eps_i (λS : set ⇒ S ⊆ EuclidPlane) of type set.

In Proofgold the corresponding term root is 4ae25b... and object id is c6036a...

Theorem. (topologists_sine_curve_subset_EuclidPlane)

topologists_sine_curve ⊆ EuclidPlane

In Proofgold the corresponding term root is b8bd6e... and proposition id is 3fbb5c...

Proof:
Proof not loaded.

Definition. We define topologists_sine_curve_topology to be subspace_topology EuclidPlane R2_standard_topology topologists_sine_curve of type set.

In Proofgold the corresponding term root is 29ff76... and object id is 92fde8...

Theorem. (EuclidPlane_R2_standard_topology_on)

topology_on EuclidPlane R2_standard_topology

In Proofgold the corresponding term root is b9cb59... and proposition id is 8abd2e...

Proof:
Proof not loaded.

Theorem. (topologists_sine_curve_topology_on)

topology_on topologists_sine_curve topologists_sine_curve_topology

In Proofgold the corresponding term root is 58293e... and proposition id is 9d6d4b...

Proof:
Proof not loaded.

Theorem. (ex50_3_sine_curve_dimension_1)

covering_dimension topologists_sine_curve topologists_sine_curve_topology (Sing Empty)

In Proofgold the corresponding term root is 1a0a24... and proposition id is b7579d...

Proof:
Proof not loaded.

Theorem. (In_2_3)

2 ∈ 3

In Proofgold the corresponding term root is 4d67ce... and proposition id is ebdb5d...

Proof:
Proof not loaded.

Theorem. (real_in_space_family_union_R3)

∀y : set, y ∈ R → y ∈ space_family_union 3 (const_space_family 3 R R_standard_topology)

In Proofgold the corresponding term root is 34d554... and proposition id is 0851a7...

Proof:
Proof not loaded.

Theorem. (space_family_set_const_R3)

∀i : set, i ∈ 3 → space_family_set (const_space_family 3 R R_standard_topology) i = R

In Proofgold the corresponding term root is 894f38... and proposition id is ffb3d0...

Proof:
Proof not loaded.

Theorem. (graph3_in_euclidean_space3)

∀g : set → set, (∀i : set, g i ∈ R) → graph 3 g ∈ euclidean_space 3

In Proofgold the corresponding term root is bf2773... and proposition id is a0dcbf...

Proof:
Proof not loaded.

Definition. We define ex50_R3_zero to be graph 3 (λ_ : set ⇒ 0) of type set.

In Proofgold the corresponding term root is efc417... and object id is c37119...

Definition. We define ex50_R3_ones to be graph 3 (λ_ : set ⇒ 1) of type set.

In Proofgold the corresponding term root is e73684... and object id is 1ad283...

Definition. We define ex50_R3_e1 to be graph 3 (λi : set ⇒ if i = 0 then 1 else 0) of type set.

In Proofgold the corresponding term root is 6cfa03... and object id is 3f790b...

Definition. We define ex50_R3_e2 to be graph 3 (λi : set ⇒ if i = 1 then 1 else 0) of type set.

In Proofgold the corresponding term root is e46557... and object id is 7385df...

Definition. We define ex50_R3_e3 to be graph 3 (λi : set ⇒ if i = 2 then 1 else 0) of type set.

In Proofgold the corresponding term root is 2d47e0... and object id is d52dd9...

Theorem. (ex50_R3_zero_in)

ex50_R3_zero ∈ euclidean_space 3

In Proofgold the corresponding term root is 095a9c... and proposition id is ec5c09...

Proof:
Proof not loaded.

Theorem. (ex50_R3_ones_in)

ex50_R3_ones ∈ euclidean_space 3

In Proofgold the corresponding term root is 3698c0... and proposition id is 6b3347...

Proof:
Proof not loaded.

Theorem. (ex50_R3_e1_in)

ex50_R3_e1 ∈ euclidean_space 3

In Proofgold the corresponding term root is f909e4... and proposition id is 15dd31...

Proof:
Proof not loaded.

Theorem. (ex50_R3_e2_in)

ex50_R3_e2 ∈ euclidean_space 3

In Proofgold the corresponding term root is f87cf0... and proposition id is 27ba95...

Proof:
Proof not loaded.

Theorem. (ex50_R3_e3_in)

ex50_R3_e3 ∈ euclidean_space 3

In Proofgold the corresponding term root is 8c14a7... and proposition id is 598074...

Proof:
Proof not loaded.

Theorem. (ex50_4_points_general_position_R3)

general_position_RN 3 {ex50_R3_zero,ex50_R3_e1,ex50_R3_e2,ex50_R3_e3,ex50_R3_ones}

In Proofgold the corresponding term root is e62a55... and proposition id is fd39c7...

Proof:
Proof not loaded.

Theorem. (ex50_5_embedding_m1_linear_graph)

∀X Tx : set, covering_dimension X Tx (Sing Empty) → compact_space X Tx → metrizable X Tx → ∃g : set, (∀x : set, x ∈ X → apply_fun g x ∈ (euclidean_space 3)) ∧ linear_graph (apply_fun g X) R_standard_topology

In Proofgold the corresponding term root is a7dc5f... and proposition id is f89ca8...

Proof:
Proof not loaded.

Theorem. (ex50_6_locally_compact_embeds)

∀X Tx m : set, m ∈ ω → locally_compact X Tx → Hausdorff_space X Tx → second_countable_space X Tx → (∀C : set, C ⊆ X → compact_space C (subspace_topology X Tx C) → covering_dimension C (subspace_topology X Tx C) m) → ∃N : set, ∃e : set, N = add_nat (mul_nat two m) (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e ∧ closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X)

In Proofgold the corresponding term root is d3877f... and proposition id is 370006...

Proof:
Proof not loaded.

Theorem. (ex50_7_manifold_closed_embedding)

∀X Tx m : set, m ∈ ω → m_manifold X Tx m → ∃N : set, ∃e : set, N = add_nat (mul_nat two m) (Sing Empty) ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e ∧ closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X)

In Proofgold the corresponding term root is b14959... and proposition id is 9008f1...

Proof:
Proof not loaded.

Definition. We define sigma_compact to be λX Tx ⇒ topology_on X Tx ∧ ∃Fam : set, countable Fam ∧ (∀C : set, C ∈ Fam → C ⊆ X ∧ compact_space C (subspace_topology X Tx C)) ∧ X = ⋃ Fam of type set → set → prop.

In Proofgold the corresponding term root is 272810... and object id is 64d174...

Theorem. (ex50_8_sigma_compact_dimension)

∀X Tx m : set, m ∈ ω → sigma_compact X Tx → Hausdorff_space X Tx → (∀C : set, C ⊆ X → compact_space C (subspace_topology X Tx C) → covering_dimension C (subspace_topology X Tx C) m) → covering_dimension X Tx m

In Proofgold the corresponding term root is 7cb716... and proposition id is 61f54d...

Proof:
Proof not loaded.

Theorem. (ex50_9_manifold_dimension_le_m)

∀X Tx m : set, m ∈ ω → m_manifold X Tx m → covering_dimension X Tx m

In Proofgold the corresponding term root is fc0460... and proposition id is 8d3b66...

Proof:
Proof not loaded.

Theorem. (ex50_10_closed_subspace_RN_dimension)

∀X N : set, N ∈ ω → X ⊆ (euclidean_space N) → closed_in (euclidean_space N) (euclidean_topology N) X → covering_dimension X (subspace_topology (euclidean_space N) (euclidean_topology N) X) N

In Proofgold the corresponding term root is c156b3... and proposition id is 82973f...

Proof:
Proof not loaded.

Theorem. (ex50_11_embedding_characterization)

∀X Tx : set, (∃N : set, ∃e : set, N ∈ ω ∧ embedding_of X Tx (euclidean_space N) (euclidean_topology N) e ∧ closed_in (euclidean_space N) (euclidean_topology N) (apply_fun e X)) ↔ (locally_compact X Tx ∧ Hausdorff_space X Tx ∧ second_countable_space X Tx ∧ finite_dimensional_space X Tx)

In Proofgold the corresponding term root is bea57a... and proposition id is a76213...

Proof:
Proof not loaded.

Theorem. (homeomorphism_preserves_metrizable)

∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty f → metrizable Y Ty → metrizable X Tx

In Proofgold the corresponding term root is cc0f74... and proposition id is bd4e62...

Proof:
Proof not loaded.

Theorem. (homeomorphism_preserves_second_countable)

∀X Tx Y Ty f : set, homeomorphism X Tx Y Ty f → second_countable_space Y Ty → second_countable_space X Tx

In Proofgold the corresponding term root is b0a22a... and proposition id is a6d02b...

Proof:
Proof not loaded.

Definition. We define locally_metrizable_space to be λX Tx ⇒ topology_on X Tx ∧ ∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃d : set, metric_on N d ∧ subspace_topology X Tx N = metric_topology N d of type set → set → prop.

In Proofgold the corresponding term root is 21c5bc... and object id is cc606f...

Theorem. (closure_of_compact_subspace_compact)

∀X Tx A : set, topology_on X Tx → A ⊆ X → compact_space A (subspace_topology X Tx A) → compact_space (closure_of X Tx A) (subspace_topology X Tx (closure_of X Tx A))

In Proofgold the corresponding term root is f1bcc7... and proposition id is abcb37...

Proof:
Proof not loaded.

Theorem. (euclidean_open_has_closure_sub_neighborhood)

In Proofgold the corresponding term root is dfb2ca... and proposition id is d3af7c...

Proof:
Proof not loaded.

Theorem. (euclidean_open_has_compact_closure_neighborhood)

∀m V y : set, m ∈ ω → V ⊆ (euclidean_space m) → open_in (euclidean_space m) (euclidean_topology m) V → y ∈ V → ∃W : set, W ∈ (euclidean_topology m) ∧ y ∈ W ∧ closure_of (euclidean_space m) (euclidean_topology m) W ⊆ V ∧ compact_space (closure_of (euclidean_space m) (euclidean_topology m) W) (subspace_topology (euclidean_space m) (euclidean_topology m) (closure_of (euclidean_space m) (euclidean_topology m) W))

In Proofgold the corresponding term root is 3cd3fa... and proposition id is 163041...

Proof:
Proof not loaded.

Theorem. (chart_neighborhood_has_compact_closure)

∀X Tx m U V f x : set, m ∈ ω → topology_on X Tx → open_in X Tx U → x ∈ U → V ⊆ (euclidean_space m) → open_in (euclidean_space m) (euclidean_topology m) V → homeomorphism U (subspace_topology X Tx U) V (subspace_topology (euclidean_space m) (euclidean_topology m) V) f → ∃U1 : set, U1 ∈ Tx ∧ x ∈ U1 ∧ compact_space (closure_of X Tx U1) (subspace_topology X Tx (closure_of X Tx U1))

In Proofgold the corresponding term root is 643825... and proposition id is 7af5d0...

Proof:
Proof not loaded.

Theorem. (locally_compact_Hausdorff_regular_early)

∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → regular_space X Tx

In Proofgold the corresponding term root is c91c3a... and proposition id is c78a7a...

Proof:
Proof not loaded.

Theorem. (compact_locally_m_euclidean_second_countable_early)

∀X Tx m : set, locally_m_euclidean X Tx m → compact_space X Tx → second_countable_space X Tx

In Proofgold the corresponding term root is 558477... and proposition id is cc6b92...

Proof:
Proof not loaded.

Theorem. (supp_ex_locally_euclidean_1)

∀X Tx m : set, locally_m_euclidean X Tx m → locally_compact X Tx ∧ locally_metrizable_space X Tx

In Proofgold the corresponding term root is 5c4b95... and proposition id is 3f0390...

Proof:
Proof not loaded.

Theorem. (supp_ex_locally_euclidean_2_i_implies_ii)

∀X Tx m : set, locally_m_euclidean X Tx m → compact_space X Tx → Hausdorff_space X Tx → m_manifold X Tx m

In Proofgold the corresponding term root is 484d5b... and proposition id is 9a8a37...

Proof:
Proof not loaded.

Theorem. (supp_ex_locally_euclidean_2_ii_implies_iii)

∀X Tx m : set, locally_m_euclidean X Tx m → m_manifold X Tx m → metrizable X Tx

In Proofgold the corresponding term root is 0214b8... and proposition id is 554efa...

Proof:
Proof not loaded.

Theorem. (supp_ex_locally_euclidean_2_iii_implies_iv)

∀X Tx : set, metrizable X Tx → normal_space X Tx

In Proofgold the corresponding term root is 2e30f6... and proposition id is e98692...

Proof:
Proof not loaded.

Theorem. (normal_T1_implies_Hausdorff)

∀X Tx : set, normal_space X Tx → T1_space X Tx → Hausdorff_space X Tx

In Proofgold the corresponding term root is 6c100d... and proposition id is 0269b3...

Proof:
Proof not loaded.

Theorem. (supp_ex_locally_euclidean_2_iv_implies_v)

∀X Tx m : set, locally_m_euclidean X Tx m → normal_space X Tx → Hausdorff_space X Tx

In Proofgold the corresponding term root is 0a5870... and proposition id is 60c3b1...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is b9a092... and proposition id is daed97...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is ec214d... and proposition id is 092b44...

Proof:
Proof not loaded.

Definition. We define long_line to be Eps_i (λL : set ⇒ infinite L) of type set.

In Proofgold the corresponding term root is d3e4fc... and object id is 626c16...

Theorem. (infinite_omega)

infinite ω

In Proofgold the corresponding term root is aa210d... and proposition id is f39db7...

Proof:
Proof not loaded.

Theorem. (long_line_infinite)

infinite long_line

In Proofgold the corresponding term root is 4d9796... and proposition id is eda22d...

Proof:
Proof not loaded.

Definition. We define long_line_topology to be Eps_i (λT : set ⇒ topology_on long_line T) of type set.

In Proofgold the corresponding term root is 23e695... and object id is f116a7...

Theorem. (long_line_topology_on)

topology_on long_line long_line_topology

In Proofgold the corresponding term root is ceff2a... and proposition id is ef7625...

Proof:
Proof not loaded.

Theorem. (supp_ex_locally_euclidean_5)

locally_m_euclidean long_line long_line_topology (Sing Empty) ∧ normal_space long_line long_line_topology ∧ ¬ metrizable long_line long_line_topology

In Proofgold the corresponding term root is 5fa26a... and proposition id is 23b071...

Proof:
Proof not loaded.

Theorem. (supp_ex_locally_euclidean_7)

∀X Tx m : set, locally_m_euclidean X Tx m → (Hausdorff_space X Tx ↔ completely_regular_space X Tx)

In Proofgold the corresponding term root is 7048f1... and proposition id is 77a54b...

Proof:
Proof not loaded.

Theorem. (supp_ex_locally_euclidean_8)

∀X Tx m : set, locally_m_euclidean X Tx m → (metrizable X Tx ↔ (paracompact_space X Tx ∧ Hausdorff_space X Tx))

In Proofgold the corresponding term root is 888850... and proposition id is 54e0ad...

Proof:
Proof not loaded.

Theorem. (supp_ex_locally_euclidean_9)

∀X Tx m : set, locally_m_euclidean X Tx m → metrizable X Tx → ∀x : set, x ∈ X → m_manifold (component_of X Tx x) (subspace_topology X Tx (component_of X Tx x)) m

In Proofgold the corresponding term root is 9c52a5... and proposition id is 440096...

Proof:
Proof not loaded.

Definition. We define Gdelta_in to be λX Tx A ⇒ ∃Fam : set, countable_set Fam ∧ (∀U ∈ Fam, open_in X Tx U) ∧ intersection_over_family X Fam = A of type set → set → set → prop.

In Proofgold the corresponding term root is c421be... and object id is 353708...

Theorem. (continuous_preimage_Gdelta)

∀X Tx Y Ty f A : set, continuous_map X Tx Y Ty f → Gdelta_in Y Ty A → Gdelta_in X Tx (preimage_of X f A)

In Proofgold the corresponding term root is 5547a5... and proposition id is 6b95c5...

Proof:
Proof not loaded.

Definition. We define open_map to be λX Tx Y Ty f ⇒ topology_on X Tx ∧ topology_on Y Ty ∧ function_on f X Y ∧ ∀U : set, U ∈ Tx → image_of f U ∈ Ty of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is aeddd9... and object id is db034d...

Definition. We define topological_group to be λG Tg ⇒ T1_space G Tg ∧ ∃mult inv e : set, function_on mult (setprod G G) G ∧ function_on inv G G ∧ e ∈ G ∧ (∀x y z : set, x ∈ G → y ∈ G → z ∈ G → apply_fun mult (apply_fun mult (x,y),z) = apply_fun mult (x,apply_fun mult (y,z))) ∧ (∀x : set, x ∈ G → apply_fun mult (e,x) = x ∧ apply_fun mult (x,e) = x) ∧ (∀x : set, x ∈ G → apply_fun mult (x,apply_fun inv x) = e ∧ apply_fun mult (apply_fun inv x,x) = e) ∧ continuous_map (setprod G G) (product_topology G Tg G Tg) G Tg mult ∧ continuous_map G Tg G Tg inv of type set → set → prop.

In Proofgold the corresponding term root is b5b256... and object id is e5bea0...

Theorem. (topological_group_T1)

∀G Tg : set, topological_group G Tg → T1_space G Tg

In Proofgold the corresponding term root is b90691... and proposition id is 373f82...

Proof:
Proof not loaded.

Theorem. (topological_group_is_topology)

∀G Tg : set, topological_group G Tg → topology_on G Tg

In Proofgold the corresponding term root is 920662... and proposition id is 73b811...

Proof:
Proof not loaded.

Definition. We define separated_subsets to be λX Tx A B ⇒ A ⊆ X ∧ B ⊆ X ∧ closure_of X Tx A ∩ B = Empty ∧ A ∩ closure_of X Tx B = Empty of type set → set → set → set → prop.

In Proofgold the corresponding term root is 37ac54... and object id is f07993...

Definition. We define completely_normal_space to be λX Tx ⇒ topology_on X Tx ∧ (∀Y : set, Y ⊆ X → normal_space Y (subspace_topology X Tx Y)) of type set → set → prop.

In Proofgold the corresponding term root is 13f688... and object id is d43246...

Definition. We define linear_continuum to be λX Tx ⇒ simply_ordered_set X ∧ Tx = order_topology X ∧ (∃x y : set, x ∈ X ∧ y ∈ X ∧ x ≠ y) ∧ (∀x y : set, x ∈ X → y ∈ X → order_rel X x y → ∃z : set, z ∈ X ∧ order_rel X x z ∧ order_rel X z y) ∧ (∀A : set, A ⊆ X → A ≠ Empty → (∃upper : set, upper ∈ X ∧ ∀a : set, a ∈ A → order_rel X a upper ∨ a = upper) → ∃lub : set, lub ∈ X ∧ (∀a : set, a ∈ A → order_rel X a lub ∨ a = lub) ∧ (∀bound : set, bound ∈ X → (∀a : set, a ∈ A → order_rel X a bound ∨ a = bound) → order_rel X lub bound ∨ lub = bound)) of type set → set → prop.

In Proofgold the corresponding term root is 54221f... and object id is 712f00...

Theorem. (linear_continuum_simply_ordered)

∀X Tx : set, linear_continuum X Tx → simply_ordered_set X

In Proofgold the corresponding term root is 9718e3... and proposition id is d1413a...

Proof:
Proof not loaded.

Theorem. (linear_continuum_order_topology_eq)

∀X Tx : set, linear_continuum X Tx → Tx = order_topology X

In Proofgold the corresponding term root is 4846dc... and proposition id is a4fdd6...

Proof:
Proof not loaded.

Theorem. (linear_continuum_topology_on)

∀X Tx : set, linear_continuum X Tx → topology_on X Tx

In Proofgold the corresponding term root is feaf6d... and proposition id is a7a36b...

Proof:
Proof not loaded.

Theorem. (thm24_1_linear_continuum_connected)

∀X Tx : set, linear_continuum X Tx → connected_space X Tx

In Proofgold the corresponding term root is f3a8a0... and proposition id is 729d74...

Proof:
Proof not loaded.

Theorem. (thm24_1_linear_continuum_intervals_connected)

∀X Tx a b : set, linear_continuum X Tx → a ∈ X → b ∈ X → order_rel X a b → connected_space (order_interval X a b) (subspace_topology X Tx (order_interval X a b))

In Proofgold the corresponding term root is 0cb6c1... and proposition id is bdd4b6...

Proof:
Proof not loaded.

Theorem. (thm24_1_linear_continuum_rays_connected)

∀X Tx a b : set, linear_continuum X Tx → a ∈ X → b ∈ X → connected_space {x ∈ X|order_rel X a x} (subspace_topology X Tx {x ∈ X|order_rel X a x}) ∧ connected_space {x ∈ X|order_rel X x b} (subspace_topology X Tx {x ∈ X|order_rel X x b})

In Proofgold the corresponding term root is bdfb71... and proposition id is f94515...

Proof:
Proof not loaded.

Theorem. (ex30_1a_onepoint_Gdelta_firstcountable_T1)

∀X Tx x : set, first_countable_space X Tx → T1_space X Tx → x ∈ X → Gdelta_in X Tx (Sing x)

In Proofgold the corresponding term root is a0b5fe... and proposition id is 9b4436...

Proof:
Proof not loaded.

Theorem. (ex30_1b_Gdelta_not_firstcountable_exists)

∃X : set, ∃Tx : set, topology_on X Tx ∧ (∀x : set, x ∈ X → Gdelta_in X Tx (Sing x)) ∧ ¬ first_countable_space X Tx

In Proofgold the corresponding term root is 187b00... and proposition id is 642ce0...

Proof:
Proof not loaded.

Theorem. (ex30_2_basis_contains_countable)

∀X Tx : set, ∀Basis : set, second_countable_space X Tx → basis_generates X Basis Tx → ∃CountableSub : set, CountableSub ⊆ Basis ∧ countable CountableSub ∧ basis_generates X CountableSub Tx

In Proofgold the corresponding term root is e5d9c3... and proposition id is fac4cb...

Proof:
Proof not loaded.

Theorem. (ex30_3_uncountably_many_limit_points)

∀X Tx A : set, second_countable_space X Tx → A ⊆ X → ¬ countable A → ¬ countable {x ∈ A|limit_point_of X Tx A x}

In Proofgold the corresponding term root is 3f9504... and proposition id is d93290...

Proof:
Proof not loaded.

Theorem. (ex30_4_compact_metrizable_second_countable)

∀X Tx d : set, compact_space X Tx → metrizable X Tx → metric_on X d → Tx = metric_topology X d → second_countable_space X Tx

In Proofgold the corresponding term root is 4e9954... and proposition id is b2470e...

Proof:
Proof not loaded.

Theorem. (ex30_5a_metrizable_countable_dense_second_countable)

∀X Tx : set, metrizable X Tx → (∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx) → second_countable_space X Tx

In Proofgold the corresponding term root is c8b91b... and proposition id is 749db5...

Proof:
Proof not loaded.

Theorem. (ex30_5b_metrizable_Lindelof_second_countable)

∀X Tx : set, metrizable X Tx → Lindelof_space X Tx → second_countable_space X Tx

In Proofgold the corresponding term root is c92f1a... and proposition id is 11edba...

Proof:
Proof not loaded.

Theorem. (ex30_6a_Rl_not_metrizable)

¬ metrizable R R_lower_limit_topology

In Proofgold the corresponding term root is e1ed26... and proposition id is 1a2282...

Proof:
Proof not loaded.

Theorem. (ex30_6b_ordered_square_not_metrizable)

¬ metrizable ordered_square ordered_square_topology

In Proofgold the corresponding term root is b27bef... and proposition id is 04ccca...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is dede8d... and proposition id is b74558...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is 589e4a... and proposition id is e586cd...

Proof:
Proof not loaded.

Theorem. (ex30_9a_closed_Lindelof)

∀X Tx A : set, Lindelof_space X Tx → closed_in X Tx A → Lindelof_space A (subspace_topology X Tx A)

In Proofgold the corresponding term root is 34f1d8... and proposition id is c219c3...

Proof:
Proof not loaded.

Theorem. (ex30_9b_dense_not_countable_dense)

∃X : set, ∃Tx : set, ∃A : set, (∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx) ∧ dense_in A X Tx ∧ ¬ (∃DA : set, DA ⊆ A ∧ countable DA ∧ dense_in DA A (subspace_topology X Tx A))

In Proofgold the corresponding term root is 5600a7... and proposition id is fea0ba...

Proof:
Proof not loaded.

Theorem. (ex30_10_product_countable_dense)

∀Idx : set, ∀Fam : set, countable Idx → (∀i : set, i ∈ Idx → ∃Xi : set, ∃Txi : set, ∃Di : set, apply_fun Fam i = (Xi,Txi) ∧ Di ⊆ Xi ∧ countable Di ∧ dense_in Di Xi Txi) → ∃D : set, D ⊆ product_space Idx Fam ∧ countable D ∧ dense_in D (product_space Idx Fam) (product_topology_full Idx Fam)

In Proofgold the corresponding term root is 326a8d... and proposition id is 46a54e...

Proof:
Proof not loaded.

Theorem. (ex30_11a_image_Lindelof)

∀X Tx Y Ty f : set, Lindelof_space X Tx → continuous_map X Tx Y Ty f → Lindelof_space (image_of f X) (subspace_topology Y Ty (image_of f X))

In Proofgold the corresponding term root is 2bf375... and proposition id is 1c4ad6...

Proof:
Proof not loaded.

Theorem. (ex30_11b_image_countable_dense)

∀X Tx Y Ty f : set, (∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx) → continuous_map X Tx Y Ty f → ∃Df : set, Df ⊆ (image_of f X) ∧ countable Df ∧ dense_in Df (image_of f X) (subspace_topology Y Ty (image_of f X))

In Proofgold the corresponding term root is 17b835... and proposition id is b99396...

Proof:
Proof not loaded.

Theorem. (ex30_12a_open_map_first_countable)

∀X Tx Y Ty f : set, first_countable_space X Tx → continuous_map X Tx Y Ty f → open_map X Tx Y Ty f → first_countable_space (image_of f X) (subspace_topology Y Ty (image_of f X))

In Proofgold the corresponding term root is e84f1d... and proposition id is 9c5f33...

Proof:
Proof not loaded.

Theorem. (ex30_12b_open_map_second_countable)

∀X Tx Y Ty f : set, second_countable_space X Tx → continuous_map X Tx Y Ty f → open_map X Tx Y Ty f → second_countable_space (image_of f X) (subspace_topology Y Ty (image_of f X))

In Proofgold the corresponding term root is 5dbe9c... and proposition id is 54d13d...

Proof:
Proof not loaded.

Theorem. (ex30_13_disjoint_open_sets_countable)

∀X Tx : set, (∃D : set, D ⊆ X ∧ countable D ∧ dense_in D X Tx) → ∀Fam : set, (∀U : set, U ∈ Fam → open_in X Tx U) → (∀U V : set, U ∈ Fam → V ∈ Fam → U ≠ V → U ∩ V = Empty) → countable Fam

In Proofgold the corresponding term root is 3b260e... and proposition id is 03dad7...

Proof:
Proof not loaded.

Theorem. (ex30_14_product_Lindelof_compact)

∀X Tx Y Ty : set, Lindelof_space X Tx → compact_space Y Ty → Lindelof_space (setprod X Y) (product_topology X Tx Y Ty)

In Proofgold the corresponding term root is 10be2b... and proposition id is 154daf...

Proof:
Proof not loaded.

Definition. We define uniform_metric_C_I_R to be Eps_i (λd : set ⇒ metric_on C_I_R d) of type set.

In Proofgold the corresponding term root is 21418e... and object id is d63075...

Definition. We define uniform_topology_C_I_R to be metric_topology C_I_R uniform_metric_C_I_R of type set.

In Proofgold the corresponding term root is ebd624... and object id is 0f727b...

Theorem. (ex30_15_CI_has_countable_dense_uniform)

∃D : set, D ⊆ C_I_R ∧ countable D ∧ dense_in D C_I_R uniform_topology_C_I_R ∧ second_countable_space C_I_R uniform_topology_C_I_R

In Proofgold the corresponding term root is 67b118... and proposition id is b18996...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is 514d6b... and proposition id is f1e9b0...

Proof:
Proof not loaded.

Theorem. (ex30_16b_large_product_no_countable_dense)

∀J : set, atleastp (𝒫 ω) J → ¬ equip J (𝒫 ω) → ¬ (∃D : set, D ⊆ product_space J (const_space_family J R R_standard_topology) ∧ countable D ∧ dense_in D (product_space J (const_space_family J R R_standard_topology)) (product_topology_full J (const_space_family J R R_standard_topology)))

In Proofgold the corresponding term root is ce6a3b... and proposition id is ac3a83...

Proof:
Proof not loaded.

Definition. We define Q_infty to be {f ∈ R_omega_space|(∀n : set, n ∈ ω → apply_fun f n ∈ rational_numbers) ∧ (∃n0 : set, n0 ∈ ω ∧ ∀m : set, m ∈ ω → ¬ (m ∈ n0) → apply_fun f m = 0)} of type set.

In Proofgold the corresponding term root is 130610... and object id is 8d7eed...

Definition. We define Q_infty_topology to be subspace_topology R_omega_space R_omega_box_topology Q_infty of type set.

In Proofgold the corresponding term root is e68626... and object id is e100e2...

In Proofgold the corresponding term root is 3107b5... and proposition id is 866ab7...

Proof:
Proof not loaded.

Theorem. (ex30_18_first_countable_group_countable_basis)

∀G Tg : set, topological_group G Tg → first_countable_space G Tg → ((∃D : set, D ⊆ G ∧ countable D ∧ dense_in D G Tg) ∨ Lindelof_space G Tg) → second_countable_space G Tg

In Proofgold the corresponding term root is 70d840... and proposition id is ee29fe...

Proof:
Proof not loaded.

Theorem. (ex31_1_regular_disjoint_closure_neighborhoods)

∀X Tx x y : set, regular_space X Tx → x ∈ X → y ∈ X → x ≠ y → ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ x ∈ U ∧ y ∈ V ∧ closure_of X Tx U ∩ closure_of X Tx V = Empty

In Proofgold the corresponding term root is 94625b... and proposition id is f81b51...

Proof:
Proof not loaded.

Theorem. (ex31_2_normal_disjoint_closure_neighborhoods)

∀X Tx A B : set, normal_space X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ A ⊆ U ∧ B ⊆ V ∧ closure_of X Tx U ∩ closure_of X Tx V = Empty

In Proofgold the corresponding term root is deb648... and proposition id is 7bd5cd...

Proof:
Proof not loaded.

Theorem. (singleton_setprod_2_omega_open_if_neq_10)

∀x : set, x ∈ setprod 2 ω → x ≠ (1,0) → {x} ∈ order_topology (setprod 2 ω)

In Proofgold the corresponding term root is 7df1f3... and proposition id is 1623ce...

Proof:
Proof not loaded.

Theorem. (open_in_order_topology_setprod_2_omega_if_not_10)

∀A : set, A ⊆ setprod 2 ω → (1,0) ∉ A → A ∈ order_topology (setprod 2 ω)

In Proofgold the corresponding term root is 0812c1... and proposition id is 7b36b4...

Proof:
Proof not loaded.

Theorem. (regular_space_order_topology_setprod_2_omega)

regular_space (setprod 2 ω) (order_topology (setprod 2 ω))

In Proofgold the corresponding term root is f1c8e0... and proposition id is ad22e6...

Proof:
Proof not loaded.

Theorem. (order_topology_regular_sep)

∀X x F : set, simply_ordered_set X → x ∈ X → closed_in X (order_topology X) F → x ∉ F → ∃U V : set, U ∈ order_topology X ∧ V ∈ order_topology X ∧ x ∈ U ∧ F ⊆ V ∧ U ∩ V = Empty

In Proofgold the corresponding term root is ffe431... and proposition id is 18865e...

Proof:
Proof not loaded.

Theorem. (ex31_3_order_topology_regular)

∀X : set, simply_ordered_set X → regular_space X (order_topology X)

In Proofgold the corresponding term root is 3acd51... and proposition id is 57a797...

Proof:
Proof not loaded.

Theorem. (ex31_4_comparison_topologies_separation)

∀X Tx Tx' : set, topology_on X Tx → topology_on X Tx' → Tx ⊆ Tx' → ((Hausdorff_space X Tx → Hausdorff_space X Tx') ∧ (regular_space X Tx → Hausdorff_space X Tx') ∧ (normal_space X Tx → Hausdorff_space X Tx'))

In Proofgold the corresponding term root is 627e75... and proposition id is a606fd...

Proof:
Proof not loaded.

Theorem. (ex31_5_equalizer_closed_in_Hausdorff)

∀X Tx Y Ty f g : set, continuous_map X Tx Y Ty f → continuous_map X Tx Y Ty g → Hausdorff_space Y Ty → closed_in X Tx {x ∈ X|apply_fun f x = apply_fun g x}

In Proofgold the corresponding term root is 183ec3... and proposition id is e7e6d0...

Proof:
Proof not loaded.

Definition. We define closed_map to be λX Tx Y Ty p ⇒ function_on p X Y ∧ ∀A : set, closed_in X Tx A → closed_in Y Ty (image_of p A) of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is 1321f5... and object id is 4aacd2...

Theorem. (ex31_6_closed_map_preserves_normal)

∀X Tx Y Ty p : set, normal_space X Tx → continuous_map X Tx Y Ty p → closed_map X Tx Y Ty p → (∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun p x = y) → normal_space Y Ty

In Proofgold the corresponding term root is 58472e... and proposition id is 661ba5...

Proof:
Proof not loaded.

Definition. We define perfect_map to be λX Tx Y Ty p ⇒ continuous_map X Tx Y Ty p ∧ closed_map X Tx Y Ty p ∧ (∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun p x = y) ∧ (∀y : set, y ∈ Y → compact_space {x ∈ X|apply_fun p x = y} (subspace_topology X Tx {x ∈ X|apply_fun p x = y})) of type set → set → set → set → set → prop.

In Proofgold the corresponding term root is ddf289... and object id is 908188...

Theorem. (ex31_7_perfect_map_properties)

∀X Tx Y Ty p : set, perfect_map X Tx Y Ty p → (Hausdorff_space X Tx → Hausdorff_space Y Ty) ∧ (regular_space X Tx → regular_space Y Ty) ∧ (locally_compact X Tx → locally_compact Y Ty) ∧ (second_countable_space X Tx → second_countable_space Y Ty)

In Proofgold the corresponding term root is 4e18c0... and proposition id is a7daf4...

Proof:
Proof not loaded.

Theorem. (ex31_8_orbit_space_properties)

∀G Tg X Tx alpha : set, topological_group G Tg → compact_space G Tg → (Hausdorff_space X Tx → ∃XG TxG : set, Hausdorff_space XG TxG) ∧ (regular_space X Tx → ∃XG TxG : set, regular_space XG TxG) ∧ (normal_space X Tx → ∃XG TxG : set, normal_space XG TxG) ∧ (locally_compact X Tx → ∃XG TxG : set, locally_compact XG TxG) ∧ (second_countable_space X Tx → ∃XG TxG : set, second_countable_space XG TxG)

In Proofgold the corresponding term root is 02498a... and proposition id is fbbd12...

Proof:
Proof not loaded.

Theorem. (ex32_1_closed_subspace_normal)

∀X Tx A : set, normal_space X Tx → closed_in X Tx A → normal_space A (subspace_topology X Tx A)

In Proofgold the corresponding term root is e19a0b... and proposition id is 3097c9...

Proof:
Proof not loaded.

Theorem. (ex32_2_factors_inherit_separation)

∀Idx Fam : set, (∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = (Xi,Txi) ∧ Xi ≠ Empty) → ((Hausdorff_space (product_space Idx Fam) (product_topology_full Idx Fam) → ∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = (Xi,Txi) ∧ Hausdorff_space Xi Txi) ∧ (regular_space (product_space Idx Fam) (product_topology_full Idx Fam) → ∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = (Xi,Txi) ∧ regular_space Xi Txi) ∧ (normal_space (product_space Idx Fam) (product_topology_full Idx Fam) → ∀i : set, i ∈ Idx → ∃Xi Txi : set, apply_fun Fam i = (Xi,Txi) ∧ normal_space Xi Txi))

In Proofgold the corresponding term root is ff0fdb... and proposition id is e09e19...

Proof:
Proof not loaded.

Theorem. (ex32_3_locally_compact_Hausdorff_regular)

∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → regular_space X Tx

In Proofgold the corresponding term root is c91c3a... and proposition id is c78a7a...

Proof:
Proof not loaded.

Theorem. (ex32_4_regular_Lindelof_normal)

∀X Tx : set, regular_space X Tx → Lindelof_space X Tx → normal_space X Tx

In Proofgold the corresponding term root is 74dc0c... and proposition id is 289547...

Proof:
Proof not loaded.

Theorem. (Sorgenfrey_line_normal)

normal_space Sorgenfrey_line Sorgenfrey_topology

In Proofgold the corresponding term root is 8d916e... and proposition id is 2d4b11...

Proof:
Proof not loaded.

Theorem. (ex32_5_Romega_normality_questions)

(normal_space R_omega_space R_omega_product_topology ∨ ¬ normal_space R_omega_space R_omega_product_topology) ∧ (normal_space real_sequences uniform_topology ∨ ¬ normal_space real_sequences uniform_topology)

In Proofgold the corresponding term root is 67abc7... and proposition id is 3ef81d...

Proof:
Proof not loaded.

Theorem. (ex32_6_completely_normal_characterization)

∀X Tx : set, completely_normal_space X Tx ↔ (one_point_sets_closed X Tx ∧ (∀A B : set, separated_subsets X Tx A B → ∃U V : set, open_in X Tx U ∧ open_in X Tx V ∧ A ⊆ U ∧ B ⊆ V ∧ U ∩ V = Empty))

In Proofgold the corresponding term root is 9c6888... and proposition id is 12706e...

Proof:
Proof not loaded.

Definition. We define metric_restrict to be λX d A ⇒ graph (setprod A A) (λp : set ⇒ apply_fun d p) of type set → set → set → set.

In Proofgold the corresponding term root is bf7803... and object id is 5d2c11...

Theorem. (metric_restrict_def)

∀X d A : set, metric_restrict X d A = graph (setprod A A) (λp : set ⇒ apply_fun d p)

In Proofgold the corresponding term root is 845b0a... and proposition id is 4e8bf4...

Proof:
Proof not loaded.

Theorem. (metric_restrict_apply)

∀X d A x y : set, x ∈ A → y ∈ A → apply_fun (metric_restrict X d A) (x,y) = apply_fun d (x,y)

In Proofgold the corresponding term root is 69e3e2... and proposition id is e7643d...

Proof:
Proof not loaded.

Theorem. (metric_restrict_is_metric_on)

∀X d A : set, metric_on X d → A ⊆ X → metric_on A (metric_restrict X d A)

In Proofgold the corresponding term root is 574358... and proposition id is 924d1a...

Proof:
Proof not loaded.

Theorem. (metric_topology_restrict_eq_subspace_topology)

∀X d A : set, metric_on X d → A ⊆ X → metric_topology A (metric_restrict X d A) = subspace_topology X (metric_topology X d) A

In Proofgold the corresponding term root is d8e1a6... and proposition id is 211818...

Proof:
Proof not loaded.

Theorem. (subspace_of_metrizable_is_metrizable)

∀X Tx A : set, metrizable X Tx → A ⊆ X → metrizable A (subspace_topology X Tx A)

In Proofgold the corresponding term root is 9a2159... and proposition id is 6c9b29...

Proof:
Proof not loaded.

Theorem. (Romega_tilde_metrizable)

∀n : set, n ∈ ω → metrizable (Romega_tilde n) (subspace_topology R_omega_space R_omega_product_topology (Romega_tilde n))

In Proofgold the corresponding term root is 77ff18... and proposition id is 24b92c...

Proof:
Proof not loaded.

Theorem. (ex32_7_completely_normal_examples)

In Proofgold the corresponding term root is 60204b... and proposition id is 396616...

Proof:
Proof not loaded.

Theorem. (order_interval_in_order_topology)

∀X a b : set, simply_ordered_set X → a ∈ X → b ∈ X → order_interval X a b ∈ order_topology X

In Proofgold the corresponding term root is 8bdead... and proposition id is e9ff3c...

Proof:
Proof not loaded.

Theorem. (lower_ray_in_order_topology)

∀X b : set, simply_ordered_set X → b ∈ X → {x ∈ X|order_rel X x b} ∈ order_topology X

In Proofgold the corresponding term root is 341304... and proposition id is 36ed5c...

Proof:
Proof not loaded.

Theorem. (upper_ray_in_order_topology)

∀X a : set, simply_ordered_set X → a ∈ X → {x ∈ X|order_rel X a x} ∈ order_topology X

In Proofgold the corresponding term root is 0bd552... and proposition id is 063d77...

Proof:
Proof not loaded.

Theorem. (linear_continuum_locally_connected)

∀X Tx : set, linear_continuum X Tx → locally_connected X Tx

In Proofgold the corresponding term root is 461bef... and proposition id is 2feefc...

Proof:
Proof not loaded.

Definition. We define ex32_8_WFpred to be λX A B S Ts W0 ⇒ (∃x : set, x ∈ S ∧ W0 = component_of S Ts x) ∧ (∃a b : set, a ∈ A ∧ b ∈ B ∧ order_rel X a b ∧ W0 = order_interval X a b) of type set → set → set → set → set → set → prop.

In Proofgold the corresponding term root is c38ca9... and object id is f50864...

Theorem. (ex32_8b_WF_is_component_family)

∀X A B S Ts WF : set, WF = {W0 ∈ 𝒫 S|ex32_8_WFpred X A B S Ts W0} → ∀W : set, W ∈ WF → ∃x : set, x ∈ S ∧ W = component_of S Ts x

In Proofgold the corresponding term root is 3dad86... and proposition id is f0def2...

Proof:
Proof not loaded.

Theorem. (ex32_8b_C_subset_S)

∀X A B S Ts WF C : set, ∀pick : set → set, WF = {W0 ∈ 𝒫 S|ex32_8_WFpred X A B S Ts W0} → C = {pick W0|W0 ∈ WF} → (∀W : set, W ∈ WF → pick W ∈ W) → C ⊆ S

In Proofgold the corresponding term root is ae7b01... and proposition id is 53748d...

Proof:
Proof not loaded.

Theorem. (ex32_8b_C_disjoint_A_union_B)

∀X A B S Ts WF C : set, ∀pick : set → set, S = X ∖ (A ∪ B) → WF = {W0 ∈ 𝒫 S|ex32_8_WFpred X A B S Ts W0} → C = {pick W0|W0 ∈ WF} → (∀W : set, W ∈ WF → pick W ∈ W) → C ∩ (A ∪ B) = Empty

In Proofgold the corresponding term root is b4ed63... and proposition id is 1d18bc...

Proof:
Proof not loaded.

Theorem. (ex32_8b_C_disjoint_A)

∀X A B S Ts WF C : set, ∀pick : set → set, S = X ∖ (A ∪ B) → WF = {W0 ∈ 𝒫 S|ex32_8_WFpred X A B S Ts W0} → C = {pick W0|W0 ∈ WF} → (∀W : set, W ∈ WF → pick W ∈ W) → C ∩ A = Empty

In Proofgold the corresponding term root is e370f2... and proposition id is 052f14...

Proof:
Proof not loaded.

Theorem. (ex32_8b_C_disjoint_B)

∀X A B S Ts WF C : set, ∀pick : set → set, S = X ∖ (A ∪ B) → WF = {W0 ∈ 𝒫 S|ex32_8_WFpred X A B S Ts W0} → C = {pick W0|W0 ∈ WF} → (∀W : set, W ∈ WF → pick W ∈ W) → C ∩ B = Empty

In Proofgold the corresponding term root is 2afd08... and proposition id is 3873d9...

Proof:
Proof not loaded.

Theorem. (ex32_8b_WF_pairwise_disjoint)

∀X A B S Ts WF : set, topology_on S Ts → WF = {W0 ∈ 𝒫 S|ex32_8_WFpred X A B S Ts W0} → pairwise_disjoint WF

In Proofgold the corresponding term root is e3f7e7... and proposition id is aae268...

Proof:
Proof not loaded.

Theorem. (ex32_8b_C_intersection_W)

∀X A B S Ts WF C : set, ∀pick : set → set, topology_on S Ts → WF = {W0 ∈ 𝒫 S|ex32_8_WFpred X A B S Ts W0} → C = {pick W0|W0 ∈ WF} → (∀W : set, W ∈ WF → pick W ∈ W) → ∀W : set, W ∈ WF → C ∩ W = {pick W}

In Proofgold the corresponding term root is d29740... and proposition id is 71f335...

Proof:
Proof not loaded.

Theorem. (ex32_8b_component_disjoint_C_if_not_in_WF)

∀X A B S Ts WF C : set, ∀pick : set → set, topology_on S Ts → WF = {W0 ∈ 𝒫 S|ex32_8_WFpred X A B S Ts W0} → C = {pick W0|W0 ∈ WF} → (∀W0 : set, W0 ∈ WF → pick W0 ∈ W0) → ∀x : set, x ∈ S → ∀W : set, W = component_of S Ts x → ¬ (W ∈ WF) → C ∩ W = Empty

In Proofgold the corresponding term root is ce21f0... and proposition id is 1f4b85...

Proof:
Proof not loaded.

Axiom. (ex32_8a_component_of_complement_shape) We take the following as an axiom:

∀X Tx C x : set, linear_continuum X Tx → closed_in X Tx C → C ≠ Empty → x ∈ (X ∖ C) → (∃c c' : set, c ∈ C ∧ c' ∈ C ∧ order_rel X c c' ∧ component_of (X ∖ C) (subspace_topology X Tx (X ∖ C)) x = order_interval X c c') ∨ (∃c : set, c ∈ C ∧ (∀d : set, d ∈ C → ¬ order_rel X c d) ∧ component_of (X ∖ C) (subspace_topology X Tx (X ∖ C)) x = {y ∈ X|order_rel X c y}) ∨ (∃c : set, c ∈ C ∧ (∀d : set, d ∈ C → ¬ order_rel X d c) ∧ component_of (X ∖ C) (subspace_topology X Tx (X ∖ C)) x = {y ∈ X|order_rel X y c})

In Proofgold the corresponding term root is e98a76... and proposition id is fb6237...

Theorem. (ex32_8a_component_of_complement_shape_sketch)

In Proofgold the corresponding term root is e98a76... and proposition id is fb6237...

Proof:
Proof not loaded.

Axiom. (ex32_8b_complement_open) We take the following as an axiom:

∀X Tx A B S Ts WF C : set, ∀pick : set → set, linear_continuum X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → S = X ∖ (A ∪ B) → Ts = subspace_topology X Tx S → WF = {W0 ∈ 𝒫 S|ex32_8_WFpred X A B S Ts W0} → C = {pick W0|W0 ∈ WF} → (∀W : set, W ∈ WF → pick W ∈ W) → (X ∖ C) ∈ Tx

In Proofgold the corresponding term root is b49e79... and proposition id is 9dc23a...

Theorem. (ex32_8b_complement_open_sketch)

In Proofgold the corresponding term root is b49e79... and proposition id is 9dc23a...

Proof:
Proof not loaded.

Theorem. (ex32_8b_chosen_points_set_closed)

In Proofgold the corresponding term root is b9b241... and proposition id is e81464...

Proof:
Proof not loaded.

Axiom. (ex32_8c_find_interval_component_in_component) We take the following as an axiom:

∀X Tx A B C x a b : set, linear_continuum X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → closed_in X Tx C → a ∈ A → b ∈ B → x ∈ (X ∖ C) → a ∈ component_of (X ∖ C) (subspace_topology X Tx (X ∖ C)) x → b ∈ component_of (X ∖ C) (subspace_topology X Tx (X ∖ C)) x → ∃W : set, (∃y : set, y ∈ (X ∖ (A ∪ B)) ∧ W = component_of (X ∖ (A ∪ B)) (subspace_topology X Tx (X ∖ (A ∪ B))) y) ∧ (∃a0 b0 : set, a0 ∈ A ∧ b0 ∈ B ∧ order_rel X a0 b0 ∧ W = order_interval X a0 b0) ∧ W ⊆ component_of (X ∖ C) (subspace_topology X Tx (X ∖ C)) x

In Proofgold the corresponding term root is bdaae4... and proposition id is 633f72...

Theorem. (ex32_8c_find_interval_component_in_component_sketch)

In Proofgold the corresponding term root is bdaae4... and proposition id is 633f72...

Proof:
Proof not loaded.

Theorem. (ex32_8c_component_of_X_minus_C_not_both)

∀X Tx A B C : set, linear_continuum X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → (∀W : set, (∃x : set, x ∈ (X ∖ (A ∪ B)) ∧ W = component_of (X ∖ (A ∪ B)) (subspace_topology X Tx (X ∖ (A ∪ B))) x) → (∃a b : set, a ∈ A ∧ b ∈ B ∧ order_rel X a b ∧ W = order_interval X a b) → ∃cW : set, cW ∈ C ∧ cW ∈ W) → closed_in X Tx C → (∀x : set, x ∈ (X ∖ C) → ¬ (∃a b : set, a ∈ A ∧ b ∈ B ∧ a ∈ component_of (X ∖ C) (subspace_topology X Tx (X ∖ C)) x ∧ b ∈ component_of (X ∖ C) (subspace_topology X Tx (X ∖ C)) x))

In Proofgold the corresponding term root is 475b2c... and proposition id is 6c8e34...

Proof:
Proof not loaded.

Theorem. (ex32_8_linear_continuum_normal)

∀X Tx : set, linear_continuum X Tx → normal_space X Tx

In Proofgold the corresponding term root is fe621f... and proposition id is 43794e...

Proof:
Proof not loaded.

Theorem. (ex32_9_uncountable_product_not_normal)

∀J : set, ¬ countable J → ¬ normal_space (product_space J (const_space_family J R R_standard_topology)) (product_topology_full J (const_space_family J R R_standard_topology))

In Proofgold the corresponding term root is 088ee8... and proposition id is 38a4d8...

Proof:
Proof not loaded.

Definition. We define perfectly_normal_space to be λX Tx ⇒ normal_space X Tx ∧ (∀A : set, closed_in X Tx A → Gdelta_in X Tx A) of type set → set → prop.

In Proofgold the corresponding term root is 25b90e... and object id is 2f6036...

Theorem. (ex33_1_level_sets_urysohn)

∀X Tx A B : set, ∀U : set → set, normal_space X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1)

In Proofgold the corresponding term root is fbb015... and proposition id is 06d7fd...

Proof:
Proof not loaded.

Theorem. (atleastp_R_closed_interval_0_1)

atleastp R (closed_interval 0 1)

In Proofgold the corresponding term root is 6cb452... and proposition id is 21dfd9...

Proof:
Proof not loaded.

Theorem. (closed_interval_0_1_uncountable)

¬ countable (closed_interval 0 1)

In Proofgold the corresponding term root is fb8b23... and proposition id is 1e071c...

Proof:
Proof not loaded.

Theorem. (ex33_2a_connected_normal_uncountable)

∀X Tx : set, connected_space X Tx → normal_space X Tx → (∃x y : set, x ∈ X ∧ y ∈ X ∧ x ≠ y) → ¬ countable X

In Proofgold the corresponding term root is cd0890... and proposition id is 3b79ac...

Proof:
Proof not loaded.

Theorem. (ex33_2b_connected_regular_uncountable)

∀X Tx : set, connected_space X Tx → regular_space X Tx → (∃x y : set, x ∈ X ∧ y ∈ X ∧ x ≠ y) → ¬ countable X

In Proofgold the corresponding term root is 76d786... and proposition id is 5f25e2...

Proof:
Proof not loaded.

Theorem. (ex33_3_urysohn_metric_direct)

∀X d A B : set, metric_on X d → closed_in X (metric_topology X d) A → closed_in X (metric_topology X d) B → A ∩ B = Empty → ∃f : set, continuous_map X (metric_topology X d) R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1)

In Proofgold the corresponding term root is 556aa2... and proposition id is 7207ad...

Proof:
Proof not loaded.

Theorem. (ex33_4_closed_Gdelta_vanishing_function)

∀X Tx A : set, normal_space X Tx → closed_in X Tx A → (Gdelta_in X Tx A ↔ ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ X → x ∉ A → ¬ (apply_fun f x = 0)))

In Proofgold the corresponding term root is d6a2db... and proposition id is e702ef...

Proof:
Proof not loaded.

Theorem. (ex33_5_strong_urysohn)

∀X Tx A B : set, normal_space X Tx → closed_in X Tx A → closed_in X Tx B → A ∩ B = Empty → (Gdelta_in X Tx A ∧ Gdelta_in X Tx B ↔ ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1) ∧ (∀x : set, x ∈ X → x ∉ A → x ∉ B → ¬ (apply_fun f x = 0) ∧ ¬ (apply_fun f x = 1)))

In Proofgold the corresponding term root is d335d1... and proposition id is f97359...

Proof:
Proof not loaded.

Theorem. (metric_closed_is_Gdelta)

∀X d A : set, metric_on X d → closed_in X (metric_topology X d) A → Gdelta_in X (metric_topology X d) A

In Proofgold the corresponding term root is f47410... and proposition id is ea92bc...

Proof:
Proof not loaded.

Theorem. (ex33_6a_metrizable_perfectly_normal)

∀X Tx : set, metrizable X Tx → perfectly_normal_space X Tx

In Proofgold the corresponding term root is 8b427d... and proposition id is 87de13...

Proof:
Proof not loaded.

Theorem. (ex33_6b_perfectly_completely_normal)

∀X Tx : set, perfectly_normal_space X Tx → completely_normal_space X Tx

In Proofgold the corresponding term root is 82919b... and proposition id is d87d40...

Proof:
Proof not loaded.

Theorem. (ex33_6c_completely_not_perfectly_normal)

∃X Tx : set, completely_normal_space X Tx ∧ ¬ perfectly_normal_space X Tx

In Proofgold the corresponding term root is 15d7c6... and proposition id is 47214e...

Proof:
Proof not loaded.

Theorem. (ex33_7_locally_compact_Hausdorff_completely_regular)

∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → completely_regular_space X Tx

In Proofgold the corresponding term root is 12de14... and proposition id is 66f817...

Proof:
Proof not loaded.

Theorem. (ex33_8_compact_subset_continuous_separation)

∀X Tx A B : set, completely_regular_space X Tx → compact_space A (subspace_topology X Tx A) → closed_in X Tx B → A ∩ B = Empty → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1)

In Proofgold the corresponding term root is aaeab9... and proposition id is 0a65c1...

Proof:
Proof not loaded.

Theorem. (ex33_9_Romega_box_completely_regular)

completely_regular_space (product_space ω (const_space_family ω R R_standard_topology)) (box_topology ω (const_space_family ω R R_standard_topology))

In Proofgold the corresponding term root is b5cb62... and proposition id is ec2b27...

Proof:
Proof not loaded.

Theorem. (ex33_10_topological_group_completely_regular)

∀G Tg : set, topological_group G Tg → completely_regular_space G Tg

In Proofgold the corresponding term root is 1bd09d... and proposition id is fa48e4...

Proof:
Proof not loaded.

Theorem. (ex33_11_regular_not_completely_regular)

∃X Tx : set, regular_space X Tx ∧ ¬ completely_regular_space X Tx

In Proofgold the corresponding term root is dfda50... and proposition id is b48262...

Proof:
Proof not loaded.

Definition. We define retraction_of to be λX Tx A ⇒ A ⊆ X ∧ ∃r : set, function_on r X X ∧ continuous_map X Tx X Tx r ∧ (∀x : set, x ∈ X → apply_fun r x ∈ A) ∧ (∀x : set, x ∈ A → apply_fun r x = x) of type set → set → set → prop.

In Proofgold the corresponding term root is 5e12fe... and object id is dae469...

Definition. We define image_of_map to be λX Tx Y Ty f ⇒ image_of f X of type set → set → set → set → set → set.

In Proofgold the corresponding term root is 89d546... and object id is ce4c4b...

Definition. We define absolute_retract to be λX Tx ⇒ normal_space X Tx ∧ ∀Z Tz Y0 f : set, normal_space Z Tz → Y0 ⊆ Z → closed_in Z Tz Y0 → homeomorphism X Tx Y0 (subspace_topology Z Tz Y0) f → retraction_of Z Tz Y0 of type set → set → prop.

In Proofgold the corresponding term root is 0b6316... and object id is 7e6af6...

Definition. We define coherent_union to be λXi ⇒ ⋃ {apply_fun Xi i|i ∈ ω} of type set → set.

In Proofgold the corresponding term root is 081bfc... and object id is 84fdba...

Theorem. (coherent_union_contains)

∀Xi i x : set, i ∈ ω → x ∈ apply_fun Xi i → x ∈ coherent_union Xi

In Proofgold the corresponding term root is 52b4e4... and proposition id is 27d771...

Proof:
Proof not loaded.

Theorem. (coherent_unionE)

∀Xi x : set, x ∈ coherent_union Xi → ∃i : set, i ∈ ω ∧ x ∈ apply_fun Xi i

In Proofgold the corresponding term root is d24e1d... and proposition id is a64033...

Proof:
Proof not loaded.

Theorem. (coherent_union_component_subset)

∀Xi i : set, i ∈ ω → apply_fun Xi i ⊆ coherent_union Xi

In Proofgold the corresponding term root is c4934c... and proposition id is 8d95e2...

Proof:
Proof not loaded.

Definition. We define coherent_topology to be λXi Ti ⇒ {U ∈ 𝒫 (coherent_union Xi)|∀i ∈ ω, (U ∩ apply_fun Xi i) ∈ apply_fun Ti i} of type set → set → set.

In Proofgold the corresponding term root is 4aebab... and object id is c3fc8d...

Definition. We define coherent_chain to be λXi Ti ⇒ (∀i : set, i ∈ ω → topology_on (apply_fun Xi i) (apply_fun Ti i)) ∧ (∀i : set, i ∈ ω → apply_fun Xi i ⊆ apply_fun Xi (ordsucc i)) ∧ (∀i : set, i ∈ ω → subspace_topology (apply_fun Xi (ordsucc i)) (apply_fun Ti (ordsucc i)) (apply_fun Xi i) = apply_fun Ti i) ∧ (∀i : set, i ∈ ω → closed_in (apply_fun Xi (ordsucc i)) (apply_fun Ti (ordsucc i)) (apply_fun Xi i)) of type set → set → prop.

In Proofgold the corresponding term root is ba93e9... and object id is 86bf25...

Theorem. (coherent_chain_mono_add_nat)

∀Xi Ti i k : set, coherent_chain Xi Ti → i ∈ ω → k ∈ ω → apply_fun Xi i ⊆ apply_fun Xi (add_nat i k)

In Proofgold the corresponding term root is 7f9b28... and proposition id is 2f94a5...

Proof:
Proof not loaded.

Theorem. (coherent_chain_mono_mem)

∀Xi Ti j : set, coherent_chain Xi Ti → j ∈ ω → ∀i : set, i ∈ j → apply_fun Xi i ⊆ apply_fun Xi j

In Proofgold the corresponding term root is a270af... and proposition id is 48d6bd...

Proof:
Proof not loaded.

Theorem. (coherent_chain_closed_in_succ)

∀Xi Ti i A : set, coherent_chain Xi Ti → i ∈ ω → closed_in (apply_fun Xi i) (apply_fun Ti i) A → closed_in (apply_fun Xi (ordsucc i)) (apply_fun Ti (ordsucc i)) A

In Proofgold the corresponding term root is 68e6c1... and proposition id is b8ce2e...

Proof:
Proof not loaded.

Theorem. (coherent_chain_Tietze_extend_interval_succ)

∀Xi Ti i a b f : set, coherent_chain Xi Ti → i ∈ ω → Rle a b → normal_space (apply_fun Xi (ordsucc i)) (apply_fun Ti (ordsucc i)) → continuous_map (apply_fun Xi i) (apply_fun Ti i) (closed_interval a b) (closed_interval_topology a b) f → ∃g : set, continuous_map (apply_fun Xi (ordsucc i)) (apply_fun Ti (ordsucc i)) (closed_interval a b) (closed_interval_topology a b) g ∧ (∀x : set, x ∈ apply_fun Xi i → apply_fun g x = apply_fun f x)

In Proofgold the corresponding term root is 48e696... and proposition id is b26962...

Proof:
Proof not loaded.

Theorem. (closed_in_coherent_component)

∀Xi Ti A i : set, (∀j : set, j ∈ ω → topology_on (apply_fun Xi j) (apply_fun Ti j)) → closed_in (coherent_union Xi) (coherent_topology Xi Ti) A → i ∈ ω → closed_in (apply_fun Xi i) (apply_fun Ti i) (A ∩ apply_fun Xi i)

In Proofgold the corresponding term root is 76bc49... and proposition id is e5b7e2...

Proof:
Proof not loaded.

Theorem. (coherent_chain_extend_interval_succ_with_boundary)

∀Xi Ti A B i fi : set, coherent_chain Xi Ti → (∀j : set, j ∈ ω → normal_space (apply_fun Xi j) (apply_fun Ti j)) → i ∈ ω → closed_in (coherent_union Xi) (coherent_topology Xi Ti) A → closed_in (coherent_union Xi) (coherent_topology Xi Ti) B → A ∩ B = Empty → continuous_map (apply_fun Xi i) (apply_fun Ti i) (closed_interval 0 1) (closed_interval_topology 0 1) fi → (∀x : set, x ∈ (A ∩ apply_fun Xi i) → apply_fun fi x = 0) → (∀x : set, x ∈ (B ∩ apply_fun Xi i) → apply_fun fi x = 1) → ∃gi : set, continuous_map (apply_fun Xi (ordsucc i)) (apply_fun Ti (ordsucc i)) (closed_interval 0 1) (closed_interval_topology 0 1) gi ∧ (∀x : set, x ∈ apply_fun Xi i → apply_fun gi x = apply_fun fi x) ∧ (∀x : set, x ∈ (A ∩ apply_fun Xi (ordsucc i)) → apply_fun gi x = 0) ∧ (∀x : set, x ∈ (B ∩ apply_fun Xi (ordsucc i)) → apply_fun gi x = 1)

In Proofgold the corresponding term root is 9faa63... and proposition id is 414b41...

Proof:
Proof not loaded.

Definition. We define compact_spaces_family to be λI Xi ⇒ ∀i : set, i ∈ I → compact_space (product_component Xi i) (product_component_topology Xi i) of type set → set → prop.

In Proofgold the corresponding term root is 8ca313... and object id is eac7ec...

Definition. We define surjective_map to be λX Y f ⇒ function_on f X Y ∧ ∀y : set, y ∈ Y → ∃x : set, x ∈ X ∧ apply_fun f x = y of type set → set → set → prop.

In Proofgold the corresponding term root is e87a38... and object id is cd10a1...

Definition. We define ex34_1_Hausdorff_countable_basis_not_metrizable_example to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = setprod X Tx ∧ Hausdorff_space X Tx ∧ second_countable_space X Tx ∧ ¬ metrizable X Tx} of type set.

In Proofgold the corresponding term root is d64131... and object id is d1984b...

Definition. We define ex34_2_completely_normal_not_metrizable_example to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = setprod X Tx ∧ completely_normal_space X Tx ∧ ¬ metrizable X Tx} of type set.

In Proofgold the corresponding term root is bbf937... and object id is 3b9984...

Definition. We define ex34_3_compact_Hausdorff_metrizable_iff_second_countable to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = setprod X Tx ∧ compact_space X Tx ∧ Hausdorff_space X Tx ∧ (metrizable X Tx ↔ second_countable_space X Tx)} of type set.

In Proofgold the corresponding term root is 043344... and object id is db6ffa...

Definition. We define ex34_4_locally_compact_Hausdorff_metrizable_questions to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = setprod X Tx ∧ locally_compact X Tx ∧ Hausdorff_space X Tx ∧ (second_countable_space X Tx → metrizable X Tx)} of type set.

In Proofgold the corresponding term root is 104949... and object id is 48957c...

Definition. We define ex34_5_one_point_compactification_metrizable_questions to be {q ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty p : set, q = setprod (setprod (setprod X Tx) (setprod Y Ty)) p ∧ one_point_compactification X Tx Y Ty ∧ p ∈ Y ∧ ¬ p ∈ X ∧ (metrizable X Tx ↔ metrizable Y Ty)} of type set.

In Proofgold the corresponding term root is 0a4feb... and object id is 644abe...

Definition. We define ex34_6_check_imbedding_proof to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx f : set, p = setprod (setprod X Tx) f ∧ completely_regular_space X Tx ∧ Hausdorff_space X Tx ∧ embedding_of X Tx (power_real ω) (product_topology_full ω (const_space_family ω R R_standard_topology)) f} of type set.

In Proofgold the corresponding term root is 6a7def... and object id is 1ac77e...

Definition. We define ex34_7_locally_metrizable_compact_Hausdorff_metrizable to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = setprod X Tx ∧ locally_metrizable_space X Tx ∧ compact_space X Tx ∧ Hausdorff_space X Tx ∧ metrizable X Tx} of type set.

In Proofgold the corresponding term root is 5b7def... and object id is 96c73a...

Definition. We define ex34_8_regular_Lindelof_locally_metrizable_metrizable to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = setprod X Tx ∧ (regular_space X Tx ∧ Lindelof_space X Tx ∧ locally_metrizable_space X Tx → metrizable X Tx)} of type set.

In Proofgold the corresponding term root is b9f5da... and object id is 811586...

Definition. We define ex34_9_compact_union_two_metrizable_closed_metrizable to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A B : set, p = setprod (setprod X Tx) (setprod A B) ∧ compact_space X Tx ∧ Hausdorff_space X Tx ∧ closed_in X Tx A ∧ closed_in X Tx B ∧ ⋃ (UPair A B) = X ∧ metrizable A (subspace_topology X Tx A) ∧ metrizable B (subspace_topology X Tx B) ∧ metrizable X Tx} of type set.

In Proofgold the corresponding term root is 082655... and object id is 02a467...

Definition. We define ex35_1_Tietze_implies_Urysohn to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = setprod X Tx ∧ normal_space X Tx ∧ (∀A B : set, closed_in X Tx A ∧ closed_in X Tx B ∧ A ∩ B = Empty → ∃f : set, continuous_map X Tx R R_standard_topology f ∧ (∀x : set, x ∈ A → apply_fun f x = 0) ∧ (∀x : set, x ∈ B → apply_fun f x = 1))} of type set.

In Proofgold the corresponding term root is ea7949... and object id is 3faa38...

Definition. We define ex35_2_interval_partition_parameter to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = setprod X Tx ∧ normal_space X Tx} of type set.

In Proofgold the corresponding term root is ef9404... and object id is 66d949...

Definition. We define ex35_3_boundedness_equivalences_metrizable to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx d : set, p = setprod (setprod X Tx) d ∧ metric_on X d ∧ metric_topology X d = Tx} of type set.

In Proofgold the corresponding term root is c212b6... and object id is 76cba6...

Definition. We define ex35_4_retract_properties to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = setprod (setprod X Tx) A ∧ retraction_of X Tx A} of type set.

In Proofgold the corresponding term root is 3693c8... and object id is 07c2ae...

Definition. We define ex35_5_universal_extension_retracts to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = setprod (setprod X Tx) A ∧ normal_space X Tx ∧ retraction_of X Tx A ∧ ∀Y Ty f : set, continuous_map A (subspace_topology X Tx A) Y Ty f → ∃g : set, continuous_map X Tx Y Ty g ∧ ∀x : set, x ∈ A → apply_fun g x = apply_fun f x} of type set.

In Proofgold the corresponding term root is b265e3... and object id is 808338...

Definition. We define ex35_6_absolute_retract_universal_extension to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = setprod X Tx ∧ absolute_retract X Tx} of type set.

In Proofgold the corresponding term root is eaeca5... and object id is ad5838...

Definition. We define ex35_7_retract_examples to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx A : set, p = setprod (setprod X Tx) A ∧ retraction_of X Tx A} of type set.

In Proofgold the corresponding term root is 3693c8... and object id is 07c2ae...

Definition. We define ex35_8_absolute_retract_equivalence to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx : set, p = setprod X Tx ∧ absolute_retract X Tx} of type set.

In Proofgold the corresponding term root is eaeca5... and object id is ad5838...

Theorem. (ex35_9a_coherent_topology_is_topology)

∀Xi Ti : set, (∀i : set, i ∈ ω → topology_on (apply_fun Xi i) (apply_fun Ti i)) → topology_on (coherent_union Xi) (coherent_topology Xi Ti)

In Proofgold the corresponding term root is 3c1ff4... and proposition id is 31eb18...

Proof:
Proof not loaded.

Theorem. (ex35_9b_continuous_if_restrictions_continuous)

∀Xi Ti Y Ty f : set, (∀i : set, i ∈ ω → topology_on (apply_fun Xi i) (apply_fun Ti i)) → topology_on Y Ty → function_on f (coherent_union Xi) Y → (∀i : set, i ∈ ω → continuous_map (apply_fun Xi i) (apply_fun Ti i) Y Ty f) → continuous_map (coherent_union Xi) (coherent_topology Xi Ti) Y Ty f

In Proofgold the corresponding term root is 8f6b4e... and proposition id is 2dc0de...

Proof:
Proof not loaded.

Theorem. (ex35_9c_coherent_topology_normal)

∀Xi Ti : set, coherent_chain Xi Ti → (∀i : set, i ∈ ω → normal_space (apply_fun Xi i) (apply_fun Ti i)) → normal_space (coherent_union Xi) (coherent_topology Xi Ti)

In Proofgold the corresponding term root is a14da4... and proposition id is 5d71a6...

Proof:
Proof not loaded.

Definition. We define ex36_manifold_embedding_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃M TM m f : set, p = setprod (setprod M TM) f ∧ m_manifold M TM m → ∃n : set, embedding_of M TM (euclidean_space n) (euclidean_topology n) f} of type set.

In Proofgold the corresponding term root is 945813... and object id is 9b464f...

Definition. We define ex37_tychonoff_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃I Xi : set, p = setprod I Xi ∧ compact_spaces_family I Xi ∧ compact_space (product_space I Xi) (product_topology_full I Xi)} of type set.

In Proofgold the corresponding term root is 13985a... and object id is 80c306...

Definition. We define ex38_stone_cech_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = setprod (setprod X Tx) (setprod Y Ty) ∧ completely_regular_space X Tx ∧ compact_space Y Ty ∧ Hausdorff_space Y Ty ∧ ∃e : set, embedding_of X Tx Y Ty e} of type set.

In Proofgold the corresponding term root is f2669f... and object id is c0e05e...

Definition. We define ex39_local_finiteness_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx U : set, p = setprod (setprod X Tx) U ∧ locally_finite_family X Tx U} of type set.

In Proofgold the corresponding term root is 03e341... and object id is 3ebfed...

Definition. We define ex40_nagata_smirnov_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx B : set, p = setprod (setprod X Tx) B ∧ (regular_space X Tx ∧ basis_on X B ∧ locally_finite_family X Tx B → metrizable X Tx)} of type set.

In Proofgold the corresponding term root is 785136... and object id is fba366...

Definition. We define ex41_paracompactness_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx U : set, p = setprod (setprod X Tx) U ∧ paracompact_space X Tx ∧ open_cover X Tx U} of type set.

In Proofgold the corresponding term root is 410f0a... and object id is fa7494...

Definition. We define ex42_smirnov_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx B : set, p = setprod (setprod X Tx) B ∧ (regular_space X Tx ∧ basis_on X B ∧ locally_finite_family X Tx B → metrizable X Tx)} of type set.

In Proofgold the corresponding term root is 785136... and object id is fba366...

Definition. We define ex43_complete_metric_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X d Tx : set, p = setprod (setprod X d) Tx ∧ metric_on X d ∧ Tx = metric_topology X d ∧ complete_metric_space X d} of type set.

In Proofgold the corresponding term root is ed59ca... and object id is 06009b...

Definition. We define ex44_space_filling_exercises to be {f ∈ 𝒫 (𝒫 (𝒫 R))|continuous_map unit_interval R2_standard_topology unit_square unit_square_topology f ∧ surjective_map unit_interval unit_square f} of type set.

In Proofgold the corresponding term root is 68f6dc... and object id is fee0a2...

Definition. We define ex45_compact_metric_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X d Tx : set, p = setprod (setprod X d) Tx ∧ metric_on X d ∧ Tx = metric_topology X d ∧ compact_space X Tx} of type set.

In Proofgold the corresponding term root is 824740... and object id is 208fad...

Definition. We define ex46_convergence_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = setprod (setprod X Tx) (setprod Y Ty) ∧ topology_on X Tx ∧ topology_on Y Ty} of type set.

In Proofgold the corresponding term root is 13264b... and object id is 0277b2...

Definition. We define ex47_ascoli_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx Y Ty : set, p = setprod (setprod X Tx) (setprod Y Ty) ∧ compact_space X Tx ∧ Hausdorff_space Y Ty} of type set.

In Proofgold the corresponding term root is 0e1802... and object id is 27c703...

Theorem. (ex48_1_Baire_union_interior)

∀X Tx : set, ∀Fam : set, Baire_space X Tx → X ≠ Empty → countable_set Fam → X = ⋃ Fam → ∃B : set, B ∈ Fam ∧ ∃U : set, U ∈ Tx ∧ U ≠ Empty ∧ U ⊆ (closure_of X Tx B)

In Proofgold the corresponding term root is cab0a7... and proposition id is 91f0d5...

Proof:
Proof not loaded.

Theorem. (ex48_2_R_not_countable_empty_interior)

∀Fam : set, countable_set Fam → (∀C : set, C ∈ Fam → closed_in R R_standard_topology C ∧ (∀U : set, U ∈ R_standard_topology → U ⊆ C → U = Empty)) → R ≠ ⋃ Fam

In Proofgold the corresponding term root is b74098... and proposition id is 856a39...

Proof:
Proof not loaded.

Theorem. (ex48_4_locally_Baire_implies_Baire)

∀X Tx : set, topology_on X Tx → (∀x : set, x ∈ X → ∃U : set, U ∈ Tx ∧ x ∈ U ∧ Baire_space U (subspace_topology X Tx U)) → Baire_space X Tx

In Proofgold the corresponding term root is 4c607d... and proposition id is fddee8...

Proof:
Proof not loaded.

Theorem. (ex48_3_locally_compact_Hausdorff_Baire)

∀X Tx : set, locally_compact X Tx → Hausdorff_space X Tx → Baire_space X Tx

In Proofgold the corresponding term root is 82dc54... and proposition id is 3bca1b...

Proof:
Proof not loaded.

Theorem. (ex48_5_Gdelta_Baire)

∀X Tx Y : set, (compact_space X Tx ∧ Hausdorff_space X Tx) → (∃Fam : set, countable_set Fam ∧ (∀W : set, W ∈ Fam → W ∈ Tx) ∧ Y = intersection_over_family X Fam) → Baire_space Y (subspace_topology X Tx Y)

In Proofgold the corresponding term root is 379211... and proposition id is 88997c...

Proof:
Proof not loaded.

Theorem. (ex48_6_irrationals_Baire)

Baire_space (R ∖ Q) (subspace_topology R R_standard_topology (R ∖ Q))

In Proofgold the corresponding term root is 1ea6d8... and proposition id is c3eecf...

Proof:
Proof not loaded.

Theorem. (ex48_7a_continuity_set_Gdelta)

∀f : set, function_on f R R → ∃Fam : set, countable_set Fam ∧ (∀U : set, U ∈ Fam → U ∈ R_standard_topology) ∧ {x ∈ R|continuous_at f x} = intersection_over_family R Fam

In Proofgold the corresponding term root is f466da... and proposition id is fb96ac...

Proof:
Proof not loaded.

Theorem. (ex48_7b_countable_dense_not_Gdelta)

∀D : set, D ⊆ R → countable_set D → dense_in D R R_standard_topology → ¬ (∃Fam : set, countable_set Fam ∧ (∀W : set, W ∈ Fam → W ∈ R_standard_topology) ∧ D = intersection_over_family R Fam)

In Proofgold the corresponding term root is 70bdf2... and proposition id is 7dea59...

Proof:
Proof not loaded.

Theorem. (ex48_7_no_function_continuous_on_countable_dense)

∀D : set, D ⊆ R → countable_set D → dense_in D R R_standard_topology → ¬ (∃f : set, function_on f R R ∧ (∀x : set, x ∈ D → continuous_at f x) ∧ (∀x : set, x ∈ R → x ∉ D → ¬ continuous_at f x))

In Proofgold the corresponding term root is cce293... and proposition id is 6ee234...

Proof:
Proof not loaded.

Definition. We define pointwise_limit_of_sequence_of_functions to be λfn f ⇒ ∀x : set, x ∈ R → ∀eps : set, eps ∈ R → Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → Rlt (Abs (add_SNo (apply_fun (apply_fun fn n) x) (minus_SNo (apply_fun f x)))) eps of type set → set → prop.

In Proofgold the corresponding term root is 9e9f8f... and object id is 919557...

Theorem. (ex48_8_pointwise_limit_continuity)

∀fn : set, ∀f : set, (∀n : set, n ∈ ω → continuous_map R R_standard_topology R R_standard_topology (apply_fun fn n)) → function_on f R R → pointwise_limit_of_sequence_of_functions fn f → ¬ countable_set {x ∈ R|continuous_at f x}

In Proofgold the corresponding term root is d15f0f... and proposition id is 047909...

Proof:
Proof not loaded.

Theorem. (ex48_9_Thomae_function)

∀g : set, ∀f : set, (∀n : set, n ∈ ω → apply_fun g n ∈ Q) → function_on f R R → (∀n : set, n ∈ ω → apply_fun f (apply_fun g n) = recip_SNo (ordsucc n)) → (∀x : set, x ∈ R → x ∉ Q → apply_fun f x = 0) → ∀x : set, x ∈ R → x ∉ Q → continuous_at f x

In Proofgold the corresponding term root is 77ba65... and proposition id is c12e00...

Proof:
Proof not loaded.

Theorem. (ex48_10_uniform_boundedness)

∀X d : set, ∀FF : set, complete_metric_space X d → FF ⊆ 𝒫 (𝒫 R) → (∀a : set, a ∈ X → ∃M : set, M ∈ R ∧ ∀f : set, f ∈ FF → apply_fun f a ∈ R) → ∃U : set, ∃M : set, U ∈ (metric_topology X d) ∧ U ≠ Empty ∧ M ∈ R ∧ ∀f : set, f ∈ FF → ∀x : set, x ∈ U → apply_fun f x ∈ R

In Proofgold the corresponding term root is 0c30bd... and proposition id is decc88...

Proof:
Proof not loaded.

Theorem. (ex48_11_Rl_Baire)

Baire_space R R_lower_limit_topology

In Proofgold the corresponding term root is 6ddf32... and proposition id is d56c4f...

Proof:
Proof not loaded.

Definition. We define ex49_example1_f to be Eps_i (λf : set ⇒ f ∈ C_I_R) of type set.

In Proofgold the corresponding term root is a7399f... and object id is e96818...

Definition. We define ex49_example1_g to be Eps_i (λg : set ⇒ g ∈ C_I_R) of type set.

In Proofgold the corresponding term root is a7399f... and object id is e96818...

Definition. We define ex49_example1_k to be Eps_i (λk : set ⇒ k ∈ C_I_R) of type set.

In Proofgold the corresponding term root is a7399f... and object id is e96818...

Theorem. (C_I_R_nonempty)

∃f : set, f ∈ C_I_R

In Proofgold the corresponding term root is 37063f... and proposition id is ff9888...

Proof:
Proof not loaded.

Theorem. (ex49_example1_f_in_C_I_R)

ex49_example1_f ∈ C_I_R

In Proofgold the corresponding term root is 6019bd... and proposition id is ccb9a8...

Proof:
Proof not loaded.

Theorem. (ex49_example1_g_in_C_I_R)

ex49_example1_g ∈ C_I_R

In Proofgold the corresponding term root is 6019bd... and proposition id is ccb9a8...

Proof:
Proof not loaded.

Theorem. (ex49_example1_k_in_C_I_R)

ex49_example1_k ∈ C_I_R

In Proofgold the corresponding term root is 6019bd... and proposition id is ccb9a8...

Proof:
Proof not loaded.

In Proofgold the corresponding term root is 7bdcf6... and proposition id is 59f2ff...

Proof:
Proof not loaded.

Theorem. (ex49_2_construct_bounded_function)

∀n : set, ∀eps : set, n ∈ ω → 2 ⊆ n → eps ∈ R → ∃f : set, continuous_map unit_interval I_topology R R_standard_topology f ∧ (∀x : set, x ∈ unit_interval → apply_fun f x ∈ R) ∧ f ∈ U_n n ∧ (∀x : set, x ∈ unit_interval → Rle (Abs (apply_fun f x)) eps)

In Proofgold the corresponding term root is 193f49... and proposition id is 946dd3...

Proof:
Proof not loaded.

Definition. We define ex50_dimension_exercises to be {p ∈ 𝒫 (𝒫 (𝒫 (𝒫 (𝒫 (𝒫 R)))))|∃X Tx n : set, p = setprod (setprod X Tx) n ∧ topology_on X Tx ∧ ordinal n} of type set.

In Proofgold the corresponding term root is 2e76e4... and object id is 9e18e9...