Primitive. The name Eps_i is a term of type (set → prop) → set.
L2
Axiom. (Eps_i_ax) We take the following as an axiom:
∀P : set → prop, ∀x : set, P x → P (Eps_i P)
In Proofgold the corresponding term root is c3f0de... and proposition id is 756d6e...
L3
Definition. We define True to be ∀p : prop, p → p of type prop.
In Proofgold the corresponding term root is f81b39... and object id is 94e6a7...
L4
Definition. We define False to be ∀p : prop, p of type prop.
In Proofgold the corresponding term root is 1db705... and object id is 266ad5...
L5
Definition. We define not to be λA : prop ⇒ A → False of type prop → prop.
In Proofgold the corresponding term root is f30435... and object id is 0dc3c7...
Notation. We use ¬ as a prefix operator with priority 700 corresponding to applying term not.
L8
Definition. We define and to be λA B : prop ⇒ ∀p : prop, (A → B → p) → p of type prop → prop → prop.
In Proofgold the corresponding term root is 2ba7d0... and object id is 37da83...
Notation. We use ∧ as an infix operator with priority 780 and which associates to the left corresponding to applying term and.
L11
Definition. We define or to be λA B : prop ⇒ ∀p : prop, (A → p) → (B → p) → p of type prop → prop → prop.
In Proofgold the corresponding term root is 957746... and object id is 86ae64...
Notation. We use ∨ as an infix operator with priority 785 and which associates to the left corresponding to applying term or.
L14
Definition. We define iff to be λA B : prop ⇒ and (A → B) (B → A) of type prop → prop → prop.
In Proofgold the corresponding term root is 98aaaf... and object id is 6588e5...
Notation. We use ↔ as an infix operator with priority 805 and no associativity corresponding to applying term iff.
Beginning of Section Eq
L18
Variable A : SType
L19
Definition. We define eq to be λx y : A ⇒ ∀Q : A → A → prop, Q x y → Q y x of type A → A → prop.
L20
Definition. We define neq to be λx y : A ⇒ ¬ eq x y of type A → A → prop.
End of Section Eq
Notation. We use = as an infix operator with priority 502 and no associativity corresponding to applying term eq.
Notation. We use ≠ as an infix operator with priority 502 and no associativity corresponding to applying term neq.
Beginning of Section FE
L26
Variable A B : SType
L27
Axiom. (func_ext) We take the following as an axiom:
∀f g : A → B, (∀x : A, f x = g x) → f = g
End of Section FE
Beginning of Section Ex
L30
Variable A : SType
L31
Definition. We define ex to be λQ : A → prop ⇒ ∀P : prop, (∀x : A, Q x → P) → P of type (A → prop) → prop.
End of Section Ex
Notation. We use ∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex.
L35
Axiom. (prop_ext) We take the following as an axiom:
∀p q : prop, iff p q → p = q
In Proofgold the corresponding term root is 00ac81... and proposition id is 1d1e56...
Primitive. The name In is a term of type set → set → prop.
Notation. We use ∈ as an infix operator with priority 500 and no associativity corresponding to applying term In. Furthermore, we may write ∀ x ∈ A, B to mean ∀ x : set, x ∈ A → B.
L37
Definition. We define Subq to be λA B ⇒ ∀x ∈ A, x ∈ B of type set → set → prop.
In Proofgold the corresponding term root is 81c0ef... and object id is 317007...
Notation. We use ⊆ as an infix operator with priority 500 and no associativity corresponding to applying term Subq. Furthermore, we may write ∀ x ⊆ A, B to mean ∀ x : set, x ⊆ A → B.
L38
Axiom. (set_ext) We take the following as an axiom:
∀X Y : set, X ⊆ Y → Y ⊆ X → X = Y
In Proofgold the corresponding term root is b3b698... and proposition id is 21238a...
L39
Axiom. (In_ind) We take the following as an axiom:
∀P : set → prop, (∀X : set, (∀x ∈ X, P x) → P X) → ∀X : set, P X
In Proofgold the corresponding term root is 045211... and proposition id is 20faa5...
Notation. We use ∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using ex and handling ∈ or ⊆ ascriptions using and.
Primitive. The name Empty is a term of type set.
L42
Axiom. (EmptyAx) We take the following as an axiom:
¬ ∃x : set, x ∈ Empty
In Proofgold the corresponding term root is 4da931... and proposition id is c7b0d7...
Primitive. The name ⋃ is a term of type set → set.
L45
Axiom. (UnionEq) We take the following as an axiom:
∀X x, x ∈ ⋃ X ↔ ∃Y, x ∈ Y ∧ Y ∈ X
In Proofgold the corresponding term root is bf4c26... and proposition id is 40660f...
Primitive. The name 𝒫 is a term of type set → set.
L48
Axiom. (PowerEq) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X ↔ Y ⊆ X
In Proofgold the corresponding term root is 746e5a... and proposition id is d67d71...
Primitive. The name Repl is a term of type set → (set → set) → set.
Notation. {B| x ∈ A} is notation for Repl A (λ x . B).
L51
Axiom. (ReplEq) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} ↔ ∃x ∈ A, y = F x
In Proofgold the corresponding term root is 4f2357... and proposition id is 471122...
L52
Definition. We define TransSet to be λU : set ⇒ ∀x ∈ U, x ⊆ U of type set → prop.
In Proofgold the corresponding term root is 538bb7... and object id is e69f92...
L53
Definition. We define Union_closed to be λU : set ⇒ ∀X : set, X ∈ U → ⋃ X ∈ U of type set → prop.
In Proofgold the corresponding term root is 57561d... and object id is e45ed9...
L54
Definition. We define Power_closed to be λU : set ⇒ ∀X : set, X ∈ U → 𝒫 X ∈ U of type set → prop.
In Proofgold the corresponding term root is 8b9cc0... and object id is d2954e...
L55
Definition. We define Repl_closed to be λU : set ⇒ ∀X : set, X ∈ U → ∀F : set → set, (∀x : set, x ∈ X → F x ∈ U) → {F x|x ∈ X} ∈ U of type set → prop.
In Proofgold the corresponding term root is 5574bb... and object id is 513534...
L57
Definition. We define ZF_closed to be λU : set ⇒ Union_closed U ∧ Power_closed U ∧ Repl_closed U of type set → prop.
In Proofgold the corresponding term root is 1bd4aa... and object id is e1eccf...
Primitive. The name UnivOf is a term of type set → set.
L62
Axiom. (UnivOf_In) We take the following as an axiom:
∀N : set, N ∈ UnivOf N
In Proofgold the corresponding term root is efcc16... and proposition id is e805d4...
L63
Axiom. (UnivOf_TransSet) We take the following as an axiom:
∀N : set, TransSet (UnivOf N)
In Proofgold the corresponding term root is 053575... and proposition id is cf4f5c...
L64
Axiom. (UnivOf_ZF_closed) We take the following as an axiom:
∀N : set, ZF_closed (UnivOf N)
In Proofgold the corresponding term root is 770e35... and proposition id is 0e7363...
L65
Axiom. (UnivOf_Min) We take the following as an axiom:
∀N U : set, N ∈ U → TransSet U → ZF_closed U → UnivOf N ⊆ U
In Proofgold the corresponding term root is 042dda... and proposition id is 1f7933...
L69
Theorem. (andI)
∀A B : prop, A → B → A ∧ B
In Proofgold the corresponding term root is 3c4f59... and proposition id is 3bf198...
Proof:
Proof not loaded.
L73
Theorem. (orIL)
∀A B : prop, A → A ∨ B
In Proofgold the corresponding term root is 55be8c... and proposition id is 382673...
Proof:
Proof not loaded.
L77
Theorem. (orIR)
∀A B : prop, B → A ∨ B
In Proofgold the corresponding term root is 3c63fc... and proposition id is 8d8e40...
Proof:
Proof not loaded.
L81
Theorem. (iffI)
∀A B : prop, (A → B) → (B → A) → (A ↔ B)
In Proofgold the corresponding term root is 7516b9... and proposition id is 4d966a...
Proof:
Proof not loaded.
L85
Theorem. (pred_ext)
∀P Q : set → prop, (∀x, P x ↔ Q x) → P = Q
In Proofgold the corresponding term root is e9e64b... and proposition id is d4b2d3...
Proof:
Proof not loaded.
L91
Definition. We define nIn to be λx X ⇒ ¬ In x X of type set → set → prop.
In Proofgold the corresponding term root is 36808c... and object id is 253451...
Notation. We use ∉ as an infix operator with priority 502 and no associativity corresponding to applying term nIn.
L96
Theorem. (EmptyE)
∀x : set, x ∉ Empty
In Proofgold the corresponding term root is 93a77a... and proposition id is 8349ab...
Proof:
Proof not loaded.
L102
Theorem. (PowerI)
∀X Y : set, Y ⊆ X → Y ∈ 𝒫 X
In Proofgold the corresponding term root is 7143d7... and proposition id is 3c48d3...
Proof:
Proof not loaded.
L106
Theorem. (Subq_Empty)
∀X : set, Empty ⊆ X
In Proofgold the corresponding term root is a517cd... and proposition id is 2faec3...
Proof:
Proof not loaded.
L110
Theorem. (Empty_In_Power)
∀X : set, Empty ∈ 𝒫 X
In Proofgold the corresponding term root is d2656b... and proposition id is 487855...
Proof:
Proof not loaded.
L114
Theorem. (xm)
∀P : prop, P ∨ ¬ P
In Proofgold the corresponding term root is 62e4c3... and proposition id is 7b6737...
Proof:
Proof not loaded.
L165
Theorem. (FalseE)
False → ∀p : prop, p
In Proofgold the corresponding term root is 0dcfe3... and proposition id is f9a230...
Proof:
Proof not loaded.
L169
Theorem. (andEL)
∀A B : prop, A ∧ B → A
In Proofgold the corresponding term root is c2f006... and proposition id is 718be4...
Proof:
Proof not loaded.
L173
Theorem. (andER)
∀A B : prop, A ∧ B → B
In Proofgold the corresponding term root is f66b82... and proposition id is 5caea9...
Proof:
Proof not loaded.
Beginning of Section PropN
L179
Variable P1 P2 P3 : prop
L180
Theorem. (and3I)
P1 → P2 → P3 → P1 ∧ P2 ∧ P3
In Proofgold the corresponding term root is 23ac9c... and proposition id is cb999b...
Proof:
Proof not loaded.
L184
Theorem. (and3E)
P1 ∧ P2 ∧ P3 → (∀p : prop, (P1 → P2 → P3 → p) → p)
In Proofgold the corresponding term root is 445f76... and proposition id is 50fabe...
Proof:
Proof not loaded.
L188
Theorem. (or3I1)
P1 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is e44f34... and proposition id is ef9071...
Proof:
Proof not loaded.
L192
Theorem. (or3I2)
P2 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is 8df6e1... and proposition id is 827a91...
Proof:
Proof not loaded.
L196
Theorem. (or3I3)
P3 → P1 ∨ P2 ∨ P3
In Proofgold the corresponding term root is 017a00... and proposition id is 3b7f5a...
Proof:
Proof not loaded.
L200
Theorem. (or3E)
P1 ∨ P2 ∨ P3 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → p)
In Proofgold the corresponding term root is 5d069d... and proposition id is 2120c9...
Proof:
Proof not loaded.
L204
Variable P4 : prop
L206
Theorem. (and4I)
P1 → P2 → P3 → P4 → P1 ∧ P2 ∧ P3 ∧ P4
In Proofgold the corresponding term root is ed32cb... and proposition id is 906aaf...
Proof:
Proof not loaded.
L210
Variable P5 : prop
L212
Theorem. (and5I)
P1 → P2 → P3 → P4 → P5 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5
In Proofgold the corresponding term root is 041ced... and proposition id is 11fc47...
Proof:
Proof not loaded.
L216
Variable P6 : prop
L218
Theorem. (and6I)
P1 → P2 → P3 → P4 → P5 → P6 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6
In Proofgold the corresponding term root is 9817c7... and proposition id is a5dc29...
Proof:
Proof not loaded.
L222
Variable P7 : prop
L224
Theorem. (and7I)
P1 → P2 → P3 → P4 → P5 → P6 → P7 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5 ∧ P6 ∧ P7
In Proofgold the corresponding term root is 6a4d8c... and proposition id is c89388...
Proof:
Proof not loaded.
End of Section PropN
L230
Theorem. (not_or_and_demorgan)
∀A B : prop, ¬ (A ∨ B) → ¬ A ∧ ¬ B
In Proofgold the corresponding term root is 8733b0... and proposition id is 27db91...
Proof:
Proof not loaded.
L238
Theorem. (not_ex_all_demorgan_i)
∀P : set → prop, (¬ ∃x, P x) → ∀x, ¬ P x
In Proofgold the corresponding term root is 73523e... and proposition id is 075ddf...
Proof:
Proof not loaded.
L244
Theorem. (iffEL)
∀A B : prop, (A ↔ B) → A → B
In Proofgold the corresponding term root is 0e94c6... and proposition id is 9a07f6...
Proof:
Proof not loaded.
L248
Theorem. (iffER)
∀A B : prop, (A ↔ B) → B → A
In Proofgold the corresponding term root is 54702a... and proposition id is f5dfc0...
Proof:
Proof not loaded.
L252
Theorem. (iff_refl)
∀A : prop, A ↔ A
In Proofgold the corresponding term root is 01338d... and proposition id is 7b71b5...
Proof:
Proof not loaded.
L256
Theorem. (iff_sym)
∀A B : prop, (A ↔ B) → (B ↔ A)
In Proofgold the corresponding term root is 00bac1... and proposition id is 11f51b...
Proof:
Proof not loaded.
L265
Theorem. (iff_trans)
∀A B C : prop, (A ↔ B) → (B ↔ C) → (A ↔ C)
In Proofgold the corresponding term root is 363285... and proposition id is c3b8f1...
Proof:
Proof not loaded.
L278
Theorem. (eq_i_tra)
∀x y z, x = y → y = z → x = z
In Proofgold the corresponding term root is 193bbf... and proposition id is 7ecbc4...
Proof:
Proof not loaded.
L282
Theorem. (neq_i_sym)
∀x y, x ≠ y → y ≠ x
In Proofgold the corresponding term root is 5c0ec9... and proposition id is 1a9898...
Proof:
Proof not loaded.
L286
Theorem. (Eps_i_ex)
∀P : set → prop, (∃x, P x) → P (Eps_i P)
In Proofgold the corresponding term root is be36f0... and proposition id is 6a2832...
Proof:
Proof not loaded.
L292
Theorem. (prop_ext_2)
∀p q : prop, (p → q) → (q → p) → p = q
In Proofgold the corresponding term root is a9ede2... and proposition id is a56fc7...
Proof:
Proof not loaded.
L298
Theorem. (Subq_ref)
∀X : set, X ⊆ X
In Proofgold the corresponding term root is 4c5395... and proposition id is 2efbac...
Proof:
Proof not loaded.
L302
Theorem. (Subq_tra)
∀X Y Z : set, X ⊆ Y → Y ⊆ Z → X ⊆ Z
In Proofgold the corresponding term root is 79accb... and proposition id is fed662...
Proof:
Proof not loaded.
L306
Theorem. (Empty_Subq_eq)
∀X : set, X ⊆ Empty → X = Empty
In Proofgold the corresponding term root is 2dd959... and proposition id is 988691...
Proof:
Proof not loaded.
L314
Theorem. (Empty_eq)
∀X : set, (∀x, x ∉ X) → X = Empty
In Proofgold the corresponding term root is 3de0fd... and proposition id is 8b915b...
Proof:
Proof not loaded.
L324
Theorem. (UnionI)
∀X x Y : set, x ∈ Y → Y ∈ X → x ∈ ⋃ X
In Proofgold the corresponding term root is 51b549... and proposition id is bade05...
Proof:
Proof not loaded.
L337
Theorem. (UnionE)
∀X x : set, x ∈ ⋃ X → ∃Y : set, x ∈ Y ∧ Y ∈ X
In Proofgold the corresponding term root is db4bc1... and proposition id is ab781b...
Proof:
Proof not loaded.
L341
Theorem. (UnionE_impred)
∀X x : set, x ∈ ⋃ X → ∀p : prop, (∀Y : set, x ∈ Y → Y ∈ X → p) → p
In Proofgold the corresponding term root is 8efd57... and proposition id is d621b2...
Proof:
Proof not loaded.
L349
Theorem. (PowerE)
∀X Y : set, Y ∈ 𝒫 X → Y ⊆ X
In Proofgold the corresponding term root is 149030... and proposition id is 647070...
Proof:
Proof not loaded.
L353
Theorem. (Self_In_Power)
∀X : set, X ∈ 𝒫 X
In Proofgold the corresponding term root is f81ef2... and proposition id is 8336ff...
Proof:
Proof not loaded.
L357
Theorem. (dneg)
∀P : prop, ¬ ¬ P → P
In Proofgold the corresponding term root is 852f34... and proposition id is b7cc63...
Proof:
Proof not loaded.
L366
Theorem. (not_all_ex_demorgan_i)
∀P : set → prop, ¬ (∀x, P x) → ∃x, ¬ P x
In Proofgold the corresponding term root is 9bfa06... and proposition id is 4b1474...
Proof:
Proof not loaded.
L376
Theorem. (eq_or_nand)
or = (λx y : prop ⇒ ¬ (¬ x ∧ ¬ y))
In Proofgold the corresponding term root is 24ac29... and proposition id is d212ef...
Proof:
Proof not loaded.
L393
Definition. We define exactly1of2 to be λA B : prop ⇒ A ∧ ¬ B ∨ ¬ A ∧ B of type prop → prop → prop.
In Proofgold the corresponding term root is 163602... and object id is a0b944...
L395
Theorem. (exactly1of2_I1)
∀A B : prop, A → ¬ B → exactly1of2 A B
In Proofgold the corresponding term root is 8e1e45... and proposition id is 4f94fa...
Proof:
Proof not loaded.
L405
Theorem. (exactly1of2_I2)
∀A B : prop, ¬ A → B → exactly1of2 A B
In Proofgold the corresponding term root is c9fd90... and proposition id is 6c1ad9...
Proof:
Proof not loaded.
L415
Theorem. (exactly1of2_E)
∀A B : prop, exactly1of2 A B → ∀p : prop, (A → ¬ B → p) → (¬ A → B → p) → p
In Proofgold the corresponding term root is 29b901... and proposition id is c67434...
Proof:
Proof not loaded.
L430
Theorem. (exactly1of2_or)
∀A B : prop, exactly1of2 A B → A ∨ B
In Proofgold the corresponding term root is 18af73... and proposition id is e092f6...
Proof:
Proof not loaded.
L438
Theorem. (ReplI)
∀A : set, ∀F : set → set, ∀x : set, x ∈ A → F x ∈ {F x|x ∈ A}
In Proofgold the corresponding term root is b146c5... and proposition id is f84982...
Proof:
Proof not loaded.
L448
Theorem. (ReplE)
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∃x ∈ A, y = F x
In Proofgold the corresponding term root is 311cdf... and proposition id is d07e78...
Proof:
Proof not loaded.
L452
Theorem. (ReplE_impred)
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∀p : prop, (∀x : set, x ∈ A → y = F x → p) → p
In Proofgold the corresponding term root is 31e90a... and proposition id is 34fefd...
Proof:
Proof not loaded.
L461
Theorem. (ReplE')
∀X, ∀f : set → set, ∀p : set → prop, (∀x ∈ X, p (f x)) → ∀y ∈ {f x|x ∈ X}, p y
In Proofgold the corresponding term root is e374fe... and proposition id is d4b7b0...
Proof:
Proof not loaded.
L468
Theorem. (Repl_Empty)
∀F : set → set, {F x|x ∈ Empty} = Empty
In Proofgold the corresponding term root is 2af2ac... and proposition id is 0835bc...
Proof:
Proof not loaded.
L479
Theorem. (ReplEq_ext_sub)
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} ⊆ {G x|x ∈ X}
In Proofgold the corresponding term root is eb4c1d... and proposition id is 56aad0...
Proof:
Proof not loaded.
L494
Theorem. (ReplEq_ext)
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} = {G x|x ∈ X}
In Proofgold the corresponding term root is 81eb90... and proposition id is 781846...
Proof:
Proof not loaded.
L503
Theorem. (Repl_inv_eq)
∀P : set → prop, ∀f g : set → set, (∀x, P x → g (f x) = x) → ∀X, (∀x ∈ X, P x) → {g y|y ∈ {f x|x ∈ X}} = X
In Proofgold the corresponding term root is 968c7a... and proposition id is 9ce73d...
Proof:
Proof not loaded.
L527
Theorem. (Repl_invol_eq)
∀P : set → prop, ∀f : set → set, (∀x, P x → f (f x) = x) → ∀X, (∀x ∈ X, P x) → {f y|y ∈ {f x|x ∈ X}} = X
In Proofgold the corresponding term root is e15a6b... and proposition id is 5faff0...
Proof:
Proof not loaded.
L536
Definition. We define If_i to be (λp x y ⇒ Eps_i (λz : set ⇒ p ∧ z = x ∨ ¬ p ∧ z = y)) of type prop → set → set → set.
In Proofgold the corresponding term root is b8ff52... and object id is 76c4c0...
Notation. if cond then T else E is notation corresponding to If_i type cond T E where type is the inferred type of T.
L538
Theorem. (If_i_correct)
∀p : prop, ∀x y : set, p ∧ (if p then x else y) = x ∨ ¬ p ∧ (if p then x else y) = y
In Proofgold the corresponding term root is 2a17ae... and proposition id is b5c26b...
Proof:
Proof not loaded.
L560
Theorem. (If_i_0)
∀p : prop, ∀x y : set, ¬ p → (if p then x else y) = y
In Proofgold the corresponding term root is fde1a8... and proposition id is a9abd3...
Proof:
Proof not loaded.
L571
Theorem. (If_i_1)
∀p : prop, ∀x y : set, p → (if p then x else y) = x
In Proofgold the corresponding term root is b1aa3f... and proposition id is a93c2c...
Proof:
Proof not loaded.
L582
Theorem. (If_i_or)
∀p : prop, ∀x y : set, (if p then x else y) = x ∨ (if p then x else y) = y
In Proofgold the corresponding term root is 3e1439... and proposition id is 4de742...
Proof:
Proof not loaded.
L591
Definition. We define UPair to be λy z ⇒ {if Empty ∈ X then y else z|X ∈ 𝒫 (𝒫 Empty)} of type set → set → set.
In Proofgold the corresponding term root is 742438... and object id is dc3134...
Notation. {x,y} is notation for UPair x y.
L594
Theorem. (UPairE)
∀x y z : set, x ∈ {y,z} → x = y ∨ x = z
In Proofgold the corresponding term root is 5009cc... and proposition id is e67fc5...
Proof:
Proof not loaded.
L614
Theorem. (UPairI1)
∀y z : set, y ∈ {y,z}
In Proofgold the corresponding term root is b1eea2... and proposition id is bf3649...
Proof:
Proof not loaded.
L625
Theorem. (UPairI2)
∀y z : set, z ∈ {y,z}
In Proofgold the corresponding term root is 27455b... and proposition id is a0a5b8...
Proof:
Proof not loaded.
L638
Definition. We define Sing to be λx ⇒ {x,x} of type set → set.
In Proofgold the corresponding term root is bd01a8... and object id is b7dfc1...
Notation. {x} is notation for Sing x.
L640
Theorem. (SingI)
∀x : set, x ∈ {x}
In Proofgold the corresponding term root is 6591bd... and proposition id is 2f93be...
Proof:
Proof not loaded.
L644
Theorem. (SingE)
∀x y : set, y ∈ {x} → y = x
In Proofgold the corresponding term root is 9341fe... and proposition id is 3b0dd8...
Proof:
Proof not loaded.
L650
Definition. We define binunion to be λX Y ⇒ ⋃ {X,Y} of type set → set → set.
In Proofgold the corresponding term root is 5e1ac4... and object id is e14989...
Notation. We use ∪ as an infix operator with priority 345 and which associates to the left corresponding to applying term binunion.
L653
Theorem. (binunionI1)
∀X Y z : set, z ∈ X → z ∈ X ∪ Y
In Proofgold the corresponding term root is 753b6e... and proposition id is e43c8d...
Proof:
Proof not loaded.
L663
Theorem. (binunionI2)
∀X Y z : set, z ∈ Y → z ∈ X ∪ Y
In Proofgold the corresponding term root is b8821c... and proposition id is 959535...
Proof:
Proof not loaded.
L673
Theorem. (binunionE)
∀X Y z : set, z ∈ X ∪ Y → z ∈ X ∨ z ∈ Y
In Proofgold the corresponding term root is 5a3195... and proposition id is 1ceeb0...
Proof:
Proof not loaded.
L696
Theorem. (binunionE')
∀X Y z, ∀p : prop, (z ∈ X → p) → (z ∈ Y → p) → (z ∈ X ∪ Y → p)
In Proofgold the corresponding term root is e5b951... and proposition id is 570dd1...
Proof:
Proof not loaded.
L703
Theorem. (binunion_asso)
∀X Y Z : set, X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z
In Proofgold the corresponding term root is e0f701... and proposition id is 61dfba...
Proof:
Proof not loaded.
L729
Theorem. (binunion_com_Subq)
∀X Y : set, X ∪ Y ⊆ Y ∪ X
In Proofgold the corresponding term root is ee8f5a... and proposition id is d84076...
Proof:
Proof not loaded.
L737
Theorem. (binunion_com)
∀X Y : set, X ∪ Y = Y ∪ X
In Proofgold the corresponding term root is e20f4f... and proposition id is bed33e...
Proof:
Proof not loaded.
L743
Theorem. (binunion_idl)
∀X : set, Empty ∪ X = X
In Proofgold the corresponding term root is 8dc616... and proposition id is 52cd24...
Proof:
Proof not loaded.
L752
Theorem. (binunion_idr)
∀X : set, X ∪ Empty = X
In Proofgold the corresponding term root is 67e82f... and proposition id is d5a005...
Proof:
Proof not loaded.
L758
Theorem. (binunion_Subq_1)
∀X Y : set, X ⊆ X ∪ Y
In Proofgold the corresponding term root is b2c658... and proposition id is a9b3a8...
Proof:
Proof not loaded.
L762
Theorem. (binunion_Subq_2)
∀X Y : set, Y ⊆ X ∪ Y
In Proofgold the corresponding term root is 3cf229... and proposition id is 1507ce...
Proof:
Proof not loaded.
L766
Theorem. (binunion_Subq_min)
∀X Y Z : set, X ⊆ Z → Y ⊆ Z → X ∪ Y ⊆ Z
In Proofgold the corresponding term root is cc32a5... and proposition id is d64da4...
Proof:
Proof not loaded.
L777
Theorem. (Subq_binunion_eq)
∀X Y, (X ⊆ Y) = (X ∪ Y = Y)
In Proofgold the corresponding term root is f8ee53... and proposition id is c2d393...
Proof:
Proof not loaded.
L793
Definition. We define SetAdjoin to be λX y ⇒ X ∪ {y} of type set → set → set.
In Proofgold the corresponding term root is 1f3a09... and object id is 5bdda8...
Notation. We now use the set enumeration notation {...,...,...} in general. If 0 elements are given, then Empty is used to form the corresponding term. If 1 element is given, then Sing is used to form the corresponding term. If 2 elements are given, then UPair is used to form the corresponding term. If more than elements are given, then SetAdjoin is used to reduce to the case with one fewer elements.
L797
Definition. We define famunion to be λX F ⇒ ⋃ {F x|x ∈ X} of type set → (set → set) → set.
In Proofgold the corresponding term root is b3e3bf... and object id is 241349...
Notation. We use ⋃ x [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using famunion.
L802
Theorem. (famunionI)
∀X : set, ∀F : (set → set), ∀x y : set, x ∈ X → y ∈ F x → y ∈ ⋃x ∈ XF x
In Proofgold the corresponding term root is 0c23d0... and proposition id is e4f490...
Proof:
Proof not loaded.
L806
Theorem. (famunionE)
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∃x ∈ X, y ∈ F x
In Proofgold the corresponding term root is 48683a... and proposition id is ede6e5...
Proof:
Proof not loaded.
L827
Theorem. (famunionE_impred)
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∀p : prop, (∀x, x ∈ X → y ∈ F x → p) → p
In Proofgold the corresponding term root is ab2266... and proposition id is 562023...
Proof:
Proof not loaded.
L835
Theorem. (famunion_Empty)
∀F : set → set, (⋃x ∈ EmptyF x) = Empty
In Proofgold the corresponding term root is fc1bdd... and proposition id is 22bd10...
Proof:
Proof not loaded.
L842
Theorem. (famunion_Subq)
∀X, ∀f g : set → set, (∀x ∈ X, f x ⊆ g x) → famunion X f ⊆ famunion X g
In Proofgold the corresponding term root is 5f0b79... and proposition id is b1b158...
Proof:
Proof not loaded.
L852
Theorem. (famunion_ext)
∀X, ∀f g : set → set, (∀x ∈ X, f x = g x) → famunion X f = famunion X g
In Proofgold the corresponding term root is 8dc4c3... and proposition id is 5bc238...
Proof:
Proof not loaded.
Beginning of Section SepSec
L863
Variable X : set
L864
Variable P : set → prop
L865
Let z : set ≝ Eps_i (λz ⇒ z ∈ X ∧ P z)
L866
Let F : set → set ≝ λx ⇒ if P x then x else z
L868
Definition. We define Sep to be if (∃z ∈ X, P z) then {F x|x ∈ X} else Empty of type set.
In Proofgold the corresponding term root is f336a4... and object id is 7008ec...
End of Section SepSec
Notation. {x ∈ A | B} is notation for Sep A (λ x . B).
L872
Theorem. (SepI)
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ X → P x → x ∈ {x ∈ X|P x}
In Proofgold the corresponding term root is ea56bb... and proposition id is 147a6a...
Proof:
Proof not loaded.
L910
Theorem. (SepE)
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X ∧ P x
In Proofgold the corresponding term root is 102830... and proposition id is 00899a...
Proof:
Proof not loaded.
L961
Theorem. (SepE1)
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X
In Proofgold the corresponding term root is d83445... and proposition id is 6399b0...
Proof:
Proof not loaded.
L965
Theorem. (SepE2)
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → P x
In Proofgold the corresponding term root is aa61ca... and proposition id is 104227...
Proof:
Proof not loaded.
L969
Theorem. (Sep_Empty)
∀P : set → prop, {x ∈ Empty|P x} = Empty
In Proofgold the corresponding term root is f36949... and proposition id is e3473a...
Proof:
Proof not loaded.
L975
Theorem. (Sep_Subq)
∀X : set, ∀P : set → prop, {x ∈ X|P x} ⊆ X
In Proofgold the corresponding term root is 97d080... and proposition id is c17d45...
Proof:
Proof not loaded.
L979
Theorem. (Sep_In_Power)
∀X : set, ∀P : set → prop, {x ∈ X|P x} ∈ 𝒫 X
In Proofgold the corresponding term root is 612c94... and proposition id is fba53e...
Proof:
Proof not loaded.
L985
Definition. We define ReplSep to be λX P F ⇒ {F x|x ∈ {z ∈ X|P z}} of type set → (set → prop) → (set → set) → set.
In Proofgold the corresponding term root is ec807b... and object id is 9bdb56...
Notation. {B| x ∈ A, C} is notation for ReplSep A (λ x . C) (λ x . B).
L987
Theorem. (ReplSepI)
∀X : set, ∀P : set → prop, ∀F : set → set, ∀x : set, x ∈ X → P x → F x ∈ {F x|x ∈ X, P x}
In Proofgold the corresponding term root is 35b8ec... and proposition id is aa7b2b...
Proof:
Proof not loaded.
L991
Theorem. (ReplSepE)
∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∃x : set, x ∈ X ∧ P x ∧ y = F x
In Proofgold the corresponding term root is 415ff1... and proposition id is b1bc76...
Proof:
Proof not loaded.
L1010
Theorem. (ReplSepE_impred)
∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∀p : prop, (∀x ∈ X, P x → y = F x → p) → p
In Proofgold the corresponding term root is 5d26e1... and proposition id is 063cea...
Proof:
Proof not loaded.
L1023
Definition. We define binintersect to be λX Y ⇒ {x ∈ X|x ∈ Y} of type set → set → set.
In Proofgold the corresponding term root is b2abd2... and object id is 2c68d8...
Notation. We use ∩ as an infix operator with priority 340 and which associates to the left corresponding to applying term binintersect.
L1027
Theorem. (binintersectI)
∀X Y z, z ∈ X → z ∈ Y → z ∈ X ∩ Y
In Proofgold the corresponding term root is fd656d... and proposition id is 9081a0...
Proof:
Proof not loaded.
L1031
Theorem. (binintersectE)
∀X Y z, z ∈ X ∩ Y → z ∈ X ∧ z ∈ Y
In Proofgold the corresponding term root is 463b5a... and proposition id is ca0636...
Proof:
Proof not loaded.
L1035
Theorem. (binintersectE1)
∀X Y z, z ∈ X ∩ Y → z ∈ X
In Proofgold the corresponding term root is b12123... and proposition id is 51005e...
Proof:
Proof not loaded.
L1039
Theorem. (binintersectE2)
∀X Y z, z ∈ X ∩ Y → z ∈ Y
In Proofgold the corresponding term root is 39656f... and proposition id is 599de4...
Proof:
Proof not loaded.
L1043
Theorem. (binintersect_Subq_1)
∀X Y : set, X ∩ Y ⊆ X
In Proofgold the corresponding term root is f296ad... and proposition id is 303833...
Proof:
Proof not loaded.
L1047
Theorem. (binintersect_Subq_2)
∀X Y : set, X ∩ Y ⊆ Y
In Proofgold the corresponding term root is fcd5f5... and proposition id is 8b9fd2...
Proof:
Proof not loaded.
L1051
Theorem. (binintersect_Subq_eq_1)
∀X Y, X ⊆ Y → X ∩ Y = X
In Proofgold the corresponding term root is 277eec... and proposition id is ec1cc3...
Proof:
Proof not loaded.
L1062
Theorem. (binintersect_Subq_max)
∀X Y Z : set, Z ⊆ X → Z ⊆ Y → Z ⊆ X ∩ Y
In Proofgold the corresponding term root is 95624d... and proposition id is 64636b...
Proof:
Proof not loaded.
L1073
Theorem. (binintersect_com_Subq)
∀X Y : set, X ∩ Y ⊆ Y ∩ X
In Proofgold the corresponding term root is 410255... and proposition id is 5977b8...
Proof:
Proof not loaded.
L1079
Theorem. (binintersect_com)
∀X Y : set, X ∩ Y = Y ∩ X
In Proofgold the corresponding term root is 1a2c6f... and proposition id is ea8a1c...
Proof:
Proof not loaded.
L1087
Definition. We define setminus to be λX Y ⇒ Sep X (λx ⇒ x ∉ Y) of type set → set → set.
In Proofgold the corresponding term root is c68e5a... and object id is 913da1...
Notation. We use ∖ as an infix operator with priority 350 and no associativity corresponding to applying term setminus.
L1091
Theorem. (setminusI)
∀X Y z, (z ∈ X) → (z ∉ Y) → z ∈ X ∖ Y
In Proofgold the corresponding term root is 0feeb8... and proposition id is 80db6a...
Proof:
Proof not loaded.
L1095
Theorem. (setminusE)
∀X Y z, (z ∈ X ∖ Y) → z ∈ X ∧ z ∉ Y
In Proofgold the corresponding term root is 6774f4... and proposition id is 71763a...
Proof:
Proof not loaded.
L1099
Theorem. (setminusE1)
∀X Y z, (z ∈ X ∖ Y) → z ∈ X
In Proofgold the corresponding term root is 064bd5... and proposition id is d74baf...
Proof:
Proof not loaded.
L1103
Theorem. (setminusE2)
∀X Y z, (z ∈ X ∖ Y) → z ∉ Y
In Proofgold the corresponding term root is 3e6969... and proposition id is 36b273...
Proof:
Proof not loaded.
L1107
Theorem. (setminus_Subq)
∀X Y : set, X ∖ Y ⊆ X
In Proofgold the corresponding term root is 5a497a... and proposition id is 66f982...
Proof:
Proof not loaded.
L1111
Theorem. (setminus_In_Power)
∀A U, A ∖ U ∈ 𝒫 A
In Proofgold the corresponding term root is 062275... and proposition id is f503e1...
Proof:
Proof not loaded.
L1115
Theorem. (binunion_remove1_eq)
∀X, ∀x ∈ X, X = (X ∖ {x}) ∪ {x}
In Proofgold the corresponding term root is 20fce6... and proposition id is eb0c40...
Proof:
Proof not loaded.
L1140
Theorem. (In_irref)
∀x, x ∉ x
In Proofgold the corresponding term root is 12629a... and proposition id is 7bfd46...
Proof:
Proof not loaded.
L1149
Theorem. (In_no2cycle)
∀x y, x ∈ y → y ∈ x → False
In Proofgold the corresponding term root is 6d0cb3... and proposition id is 344f30...
Proof:
Proof not loaded.
L1161
Definition. We define ordsucc to be λx : set ⇒ x ∪ {x} of type set → set.
In Proofgold the corresponding term root is 65d883... and object id is 1452f7...
L1162
Theorem. (ordsuccI1)
∀x : set, x ⊆ ordsucc x
In Proofgold the corresponding term root is 5a2f15... and proposition id is 367d80...
Proof:
Proof not loaded.
L1167
Theorem. (ordsuccI2)
∀x : set, x ∈ ordsucc x
In Proofgold the corresponding term root is 2f6efd... and proposition id is 2be6be...
Proof:
Proof not loaded.
L1171
Theorem. (ordsuccE)
∀x y : set, y ∈ ordsucc x → y ∈ x ∨ y = x
In Proofgold the corresponding term root is 01c4ee... and proposition id is 3849f5...
Proof:
Proof not loaded.
Notation. Natural numbers 0,1,2,... are notation for the terms formed using Empty as 0 and forming successors with ordsucc.
L1181
Theorem. (neq_0_ordsucc)
∀a : set, 0 ≠ ordsucc a
In Proofgold the corresponding term root is 1b20cb... and proposition id is 04c410...
Proof:
Proof not loaded.
L1189
Theorem. (neq_ordsucc_0)
∀a : set, ordsucc a ≠ 0
In Proofgold the corresponding term root is 9b391e... and proposition id is 346ee0...
Proof:
Proof not loaded.
L1193
Theorem. (ordsucc_inj)
∀a b : set, ordsucc a = ordsucc b → a = b
In Proofgold the corresponding term root is 40cad6... and proposition id is 4869a1...
Proof:
Proof not loaded.
L1214
Theorem. (In_0_1)
0 ∈ 1
In Proofgold the corresponding term root is e984b0... and proposition id is 3be80a...
Proof:
Proof not loaded.
L1218
Theorem. (In_0_2)
0 ∈ 2
In Proofgold the corresponding term root is 56dc2c... and proposition id is 5a202e...
Proof:
Proof not loaded.
L1222
Theorem. (In_1_2)
1 ∈ 2
In Proofgold the corresponding term root is 8704b3... and proposition id is 431e9e...
Proof:
Proof not loaded.
L1226
Definition. We define nat_p to be λn : set ⇒ ∀p : set → prop, p 0 → (∀x : set, p x → p (ordsucc x)) → p n of type set → prop.
In Proofgold the corresponding term root is 458be3... and object id is cfd8eb...
L1228
Theorem. (nat_0)
nat_p 0
In Proofgold the corresponding term root is 1617bc... and proposition id is 457a72...
Proof:
Proof not loaded.
L1232
Theorem. (nat_ordsucc)
∀n : set, nat_p n → nat_p (ordsucc n)
In Proofgold the corresponding term root is 87f001... and proposition id is 57e940...
Proof:
Proof not loaded.
L1236
Theorem. (nat_1)
nat_p 1
In Proofgold the corresponding term root is 2f83fa... and proposition id is 672d0d...
Proof:
Proof not loaded.
L1240
Theorem. (nat_2)
nat_p 2
In Proofgold the corresponding term root is a8be15... and proposition id is 9112e9...
Proof:
Proof not loaded.
L1244
Theorem. (nat_0_in_ordsucc)
∀n, nat_p n → 0 ∈ ordsucc n
In Proofgold the corresponding term root is 9b50b9... and proposition id is 251f3d...
Proof:
Proof not loaded.
L1256
Theorem. (nat_ordsucc_in_ordsucc)
∀n, nat_p n → ∀m ∈ n, ordsucc m ∈ ordsucc n
In Proofgold the corresponding term root is 819399... and proposition id is 4c2c95...
Proof:
Proof not loaded.
L1282
Theorem. (nat_ind)
∀p : set → prop, p 0 → (∀n, nat_p n → p n → p (ordsucc n)) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is 5d06f5... and proposition id is ecdce8...
Proof:
Proof not loaded.
L1307
Theorem. (nat_complete_ind)
∀p : set → prop, (∀n, nat_p n → (∀m ∈ n, p m) → p n) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is d32e5b... and proposition id is 1e1608...
Proof:
Proof not loaded.
L1337
Theorem. (nat_inv_impred)
∀p : set → prop, p 0 → (∀n, nat_p n → p (ordsucc n)) → ∀n, nat_p n → p n
In Proofgold the corresponding term root is eddc5f... and proposition id is 175e22...
Proof:
Proof not loaded.
L1341
Theorem. (nat_inv)
∀n, nat_p n → n = 0 ∨ ∃x, nat_p x ∧ n = ordsucc x
In Proofgold the corresponding term root is 5d4296... and proposition id is 4f4167...
Proof:
Proof not loaded.
L1349
Theorem. (nat_p_trans)
∀n, nat_p n → ∀m ∈ n, nat_p m
In Proofgold the corresponding term root is e88449... and proposition id is 03b874...
Proof:
Proof not loaded.
L1370
Theorem. (nat_trans)
∀n, nat_p n → ∀m ∈ n, m ⊆ n
In Proofgold the corresponding term root is 6bc6e1... and proposition id is cad6fe...
Proof:
Proof not loaded.
L1396
Theorem. (nat_ordsucc_trans)
∀n, nat_p n → ∀m ∈ ordsucc n, m ⊆ n
In Proofgold the corresponding term root is e42e58... and proposition id is bc868d...
Proof:
Proof not loaded.
L1412
Definition. We define surj to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type set → set → (set → set) → prop.
In Proofgold the corresponding term root is 4b68e4... and object id is f022f9...
L1418
Theorem. (form100_63_surjCantor)
∀A : set, ∀f : set → set, ¬ surj A (𝒫 A) f
In Proofgold the corresponding term root is eb4dfa... and proposition id is feecd4...
Proof:
Proof not loaded.
L1444
Definition. We define inj to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) of type set → set → (set → set) → prop.
In Proofgold the corresponding term root is 264b04... and object id is 442958...
L1450
Theorem. (form100_63_injCantor)
∀A : set, ∀f : set → set, ¬ inj (𝒫 A) A f
In Proofgold the corresponding term root is 4f3dc8... and proposition id is 2f2854...
Proof:
Proof not loaded.
L1481
Theorem. (injI)
∀X Y, ∀f : set → set, (∀x ∈ X, f x ∈ Y) → (∀x z ∈ X, f x = f z → x = z) → inj X Y f
In Proofgold the corresponding term root is 8739d9... and proposition id is 71c936...
Proof:
Proof not loaded.
L1489
Theorem. (inj_comp)
∀X Y Z : set, ∀f g : set → set, inj X Y f → inj Y Z g → inj X Z (λx ⇒ g (f x))
In Proofgold the corresponding term root is 649bd9... and proposition id is 05a66c...
Proof:
Proof not loaded.
L1509
Definition. We define bij to be λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type set → set → (set → set) → prop.
In Proofgold the corresponding term root is 76bef6... and object id is f7d93b...
L1517
Theorem. (bijI)
∀X Y, ∀f : set → set, (∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → bij X Y f
In Proofgold the corresponding term root is a9064f... and proposition id is 3229f9...
Proof:
Proof not loaded.
L1532
Theorem. (bijE)
∀X Y, ∀f : set → set, bij X Y f → ∀p : prop, ((∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → p) → p
In Proofgold the corresponding term root is 2cf278... and proposition id is cf0759...
Proof:
Proof not loaded.
L1546
Theorem. (bij_inj)
∀X Y, ∀f : set → set, bij X Y f → inj X Y f
In Proofgold the corresponding term root is e3635e... and proposition id is 00097c...
Proof:
Proof not loaded.
L1550
Theorem. (bij_id)
∀X, bij X X (λx ⇒ x)
In Proofgold the corresponding term root is a8cefb... and proposition id is 707f37...
Proof:
Proof not loaded.
L1561
Theorem. (bij_comp)
∀X Y Z : set, ∀f g : set → set, bij X Y f → bij Y Z g → bij X Z (λx ⇒ g (f x))
In Proofgold the corresponding term root is abf851... and proposition id is 4c9045...
Proof:
Proof not loaded.
L1593
Theorem. (bij_surj)
∀X Y, ∀f : set → set, bij X Y f → surj X Y f
In Proofgold the corresponding term root is e7a049... and proposition id is 7fa9a5...
Proof:
Proof not loaded.
L1603
Definition. We define inv to be λX f ⇒ λy : set ⇒ Eps_i (λx ⇒ x ∈ X ∧ f x = y) of type set → (set → set) → set → set.
In Proofgold the corresponding term root is 896fa9... and object id is f3a015...
L1605
Theorem. (surj_rinv)
∀X Y, ∀f : set → set, (∀w ∈ Y, ∃u ∈ X, f u = w) → ∀y ∈ Y, inv X f y ∈ X ∧ f (inv X f y) = y
In Proofgold the corresponding term root is ebf371... and proposition id is 6489d2...
Proof:
Proof not loaded.
L1614
Theorem. (inj_linv)
∀X, ∀f : set → set, (∀u v ∈ X, f u = f v → u = v) → ∀x ∈ X, inv X f (f x) = x
In Proofgold the corresponding term root is 18ecce... and proposition id is d16557...
Proof:
Proof not loaded.
L1629
Theorem. (bij_inv)
∀X Y, ∀f : set → set, bij X Y f → bij Y X (inv X f)
In Proofgold the corresponding term root is ff1332... and proposition id is 69c537...
Proof:
Proof not loaded.
L1670
Definition. We define atleastp to be λX Y : set ⇒ ∃f : set → set, inj X Y f of type set → set → prop.
In Proofgold the corresponding term root is 9bb9c2... and object id is 13545c...
L1673
Theorem. (atleastp_tra)
∀X Y Z, atleastp X Y → atleastp Y Z → atleastp X Z
In Proofgold the corresponding term root is 0e83bf... and proposition id is c412bf...
Proof:
Proof not loaded.
L1687
Theorem. (Subq_atleastp)
∀X Y, X ⊆ Y → atleastp X Y
In Proofgold the corresponding term root is 507ac6... and proposition id is 3a427f...
Proof:
Proof not loaded.
L1700
Definition. We define equip to be λX Y : set ⇒ ∃f : set → set, bij X Y f of type set → set → prop.
In Proofgold the corresponding term root is eb4419... and object id is 7f2bf6...
L1703
Theorem. (equip_atleastp)
∀X Y, equip X Y → atleastp X Y
In Proofgold the corresponding term root is c9d4bd... and proposition id is 9a412a...
Proof:
Proof not loaded.
L1713
Theorem. (equip_ref)
∀X, equip X X
In Proofgold the corresponding term root is cc7e80... and proposition id is c41fb1...
Proof:
Proof not loaded.
L1720
Theorem. (equip_sym)
∀X Y, equip X Y → equip Y X
In Proofgold the corresponding term root is f28954... and proposition id is 272f04...
Proof:
Proof not loaded.
L1729
Theorem. (equip_tra)
∀X Y Z, equip X Y → equip Y Z → equip X Z
In Proofgold the corresponding term root is 322bfb... and proposition id is d616c1...
Proof:
Proof not loaded.
L1740
Theorem. (equip_0_Empty)
∀X, equip X 0 → X = 0
In Proofgold the corresponding term root is 83ab2f... and proposition id is 4c101c...
Proof:
Proof not loaded.
L1753
Theorem. (equip_adjoin_ordsucc)
∀N X y, y ∉ X → equip N X → equip (ordsucc N) (X ∪ {y})
In Proofgold the corresponding term root is eab190... and proposition id is e8bc03...
Proof:
Proof not loaded.
L1843
Theorem. (equip_ordsucc_remove1)
∀X N, ∀x ∈ X, equip X (ordsucc N) → equip (X ∖ {x}) N
In Proofgold the corresponding term root is 997b32... and proposition id is 9c2236...
Proof:
Proof not loaded.
Beginning of Section SchroederBernstein
L2039
Theorem. (KnasterTarski_set)
∀A, ∀F : set → set, (∀U ∈ 𝒫 A, F U ∈ 𝒫 A) → (∀U V ∈ 𝒫 A, U ⊆ V → F U ⊆ F V) → ∃Y ∈ 𝒫 A, F Y = Y
In Proofgold the corresponding term root is 2ffb9f... and proposition id is 31ef9d...
Proof:
Proof not loaded.
L2080
Theorem. (image_In_Power)
∀A B, ∀f : set → set, (∀x ∈ A, f x ∈ B) → ∀U ∈ 𝒫 A, {f x|x ∈ U} ∈ 𝒫 B
In Proofgold the corresponding term root is 6799e8... and proposition id is 43ed06...
Proof:
Proof not loaded.
L2093
Theorem. (image_monotone)
∀f : set → set, ∀U V, U ⊆ V → {f x|x ∈ U} ⊆ {f x|x ∈ V}
In Proofgold the corresponding term root is ac6fe8... and proposition id is f923c6...
Proof:
Proof not loaded.
L2104
Theorem. (setminus_antimonotone)
∀A U V, U ⊆ V → A ∖ V ⊆ A ∖ U
In Proofgold the corresponding term root is 67bd0e... and proposition id is f9b735...
Proof:
Proof not loaded.
L2112
Theorem. (SchroederBernstein)
∀A B, ∀f g : set → set, inj A B f → inj B A g → equip A B
In Proofgold the corresponding term root is e247a8... and proposition id is 03a2cd...
Proof:
Proof not loaded.
L2266
Theorem. (atleastp_antisym_equip)
∀A B, atleastp A B → atleastp B A → equip A B
In Proofgold the corresponding term root is e218ed... and proposition id is 2c48a8...
Proof:
Proof not loaded.
End of Section SchroederBernstein
Beginning of Section PigeonHole
L2281
Theorem. (PigeonHole_nat)
∀n, nat_p n → ∀f : set → set, (∀i ∈ ordsucc n, f i ∈ n) → ¬ (∀i j ∈ ordsucc n, f i = f j → i = j)
In Proofgold the corresponding term root is d756b7... and proposition id is 604a84...
Proof:
Proof not loaded.
L2428
Theorem. (Pigeonhole_not_atleastp_ordsucc)
∀n, nat_p n → ¬ atleastp (ordsucc n) n
In Proofgold the corresponding term root is 8a6bdc... and proposition id is 4fb58b...
Proof:
Proof not loaded.
End of Section PigeonHole
L2440
Theorem. (Union_ordsucc_eq)
∀n, nat_p n → ⋃ (ordsucc n) = n
In Proofgold the corresponding term root is 87308c... and proposition id is e8cdeb...
Proof:
Proof not loaded.
L2463
Theorem. (cases_1)
∀i ∈ 1, ∀p : set → prop, p 0 → p i
In Proofgold the corresponding term root is 706605... and proposition id is c6f0c0...
Proof:
Proof not loaded.
L2472
Theorem. (cases_2)
∀i ∈ 2, ∀p : set → prop, p 0 → p 1 → p i
In Proofgold the corresponding term root is 4693b2... and proposition id is 74563a...
Proof:
Proof not loaded.
L2481
Theorem. (neq_0_1)
0 ≠ 1
In Proofgold the corresponding term root is 9630fd... and proposition id is 74867a...
Proof:
Proof not loaded.
L2485
Theorem. (neq_1_0)
1 ≠ 0
In Proofgold the corresponding term root is 3a0478... and proposition id is 952fed...
Proof:
Proof not loaded.
L2489
Theorem. (neq_0_2)
0 ≠ 2
In Proofgold the corresponding term root is 055580... and proposition id is 9c9a3f...
Proof:
Proof not loaded.
L2493
Theorem. (neq_2_0)
2 ≠ 0
In Proofgold the corresponding term root is 4d54d8... and proposition id is 00b357...
Proof:
Proof not loaded.
L2497
Definition. We define ordinal to be λalpha : set ⇒ TransSet alpha ∧ ∀beta ∈ alpha, TransSet beta of type set → prop.
In Proofgold the corresponding term root is ee28d9... and object id is 53ee8f...
L2499
Theorem. (ordinal_TransSet)
∀alpha : set, ordinal alpha → TransSet alpha
In Proofgold the corresponding term root is 0dbaa3... and proposition id is c9e708...
Proof:
Proof not loaded.
L2503
Theorem. (ordinal_Empty)
ordinal Empty
In Proofgold the corresponding term root is ce6bea... and proposition id is 7d95d1...
Proof:
Proof not loaded.
L2516
Theorem. (ordinal_Hered)
∀alpha : set, ordinal alpha → ∀beta ∈ alpha, ordinal beta
In Proofgold the corresponding term root is a7e78b... and proposition id is 43b0f3...
Proof:
Proof not loaded.
L2536
Theorem. (TransSet_ordsucc)
∀X : set, TransSet X → TransSet (ordsucc X)
In Proofgold the corresponding term root is fc2157... and proposition id is ebf9e3...
Proof:
Proof not loaded.
L2557
Theorem. (ordinal_ordsucc)
∀alpha : set, ordinal alpha → ordinal (ordsucc alpha)
In Proofgold the corresponding term root is 0ba97f... and proposition id is ab0f7f...
Proof:
Proof not loaded.
L2577
Theorem. (nat_p_ordinal)
∀n : set, nat_p n → ordinal n
In Proofgold the corresponding term root is 274fc4... and proposition id is 47cd49...
Proof:
Proof not loaded.
L2588
Theorem. (ordinal_1)
ordinal 1
In Proofgold the corresponding term root is c94d41... and proposition id is e325a5...
Proof:
Proof not loaded.
L2592
Theorem. (ordinal_2)
ordinal 2
In Proofgold the corresponding term root is a57075... and proposition id is e64a0d...
Proof:
Proof not loaded.
L2596
Theorem. (TransSet_ordsucc_In_Subq)
∀X : set, TransSet X → ∀x ∈ X, ordsucc x ⊆ X
In Proofgold the corresponding term root is d5d882... and proposition id is 3c2e45...
Proof:
Proof not loaded.
L2613
Theorem. (ordinal_ordsucc_In_Subq)
∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ⊆ alpha
In Proofgold the corresponding term root is 57345e... and proposition id is b00304...
Proof:
Proof not loaded.
L2619
Theorem. (ordinal_trichotomy_or)
∀alpha beta : set, ordinal alpha → ordinal beta → alpha ∈ beta ∨ alpha = beta ∨ beta ∈ alpha
In Proofgold the corresponding term root is 078a34... and proposition id is 21f2d6...
Proof:
Proof not loaded.
L2687
Theorem. (ordinal_trichotomy_or_impred)
∀alpha beta : set, ordinal alpha → ordinal beta → ∀p : prop, (alpha ∈ beta → p) → (alpha = beta → p) → (beta ∈ alpha → p) → p
In Proofgold the corresponding term root is 7c0f0b... and proposition id is bb6d2d...
Proof:
Proof not loaded.
L2692
Theorem. (ordinal_In_Or_Subq)
∀alpha beta, ordinal alpha → ordinal beta → alpha ∈ beta ∨ beta ⊆ alpha
In Proofgold the corresponding term root is 70fe0d... and proposition id is 7b99b0...
Proof:
Proof not loaded.
L2709
Theorem. (ordinal_linear)
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta ∨ beta ⊆ alpha
In Proofgold the corresponding term root is 3f30a5... and proposition id is 3b1fce...
Proof:
Proof not loaded.
L2722
Theorem. (ordinal_ordsucc_In_eq)
∀alpha beta, ordinal alpha → beta ∈ alpha → ordsucc beta ∈ alpha ∨ alpha = ordsucc beta
In Proofgold the corresponding term root is 500378... and proposition id is 0c19e7...
Proof:
Proof not loaded.
L2739
Theorem. (ordinal_lim_or_succ)
∀alpha, ordinal alpha → (∀beta ∈ alpha, ordsucc beta ∈ alpha) ∨ (∃beta ∈ alpha, alpha = ordsucc beta)
In Proofgold the corresponding term root is 4450c0... and proposition id is a12697...
Proof:
Proof not loaded.
L2754
Theorem. (ordinal_ordsucc_In)
∀alpha, ordinal alpha → ∀beta ∈ alpha, ordsucc beta ∈ ordsucc alpha
In Proofgold the corresponding term root is f4b1ed... and proposition id is 8ab1dc...
Proof:
Proof not loaded.
L2769
Theorem. (ordinal_famunion)
∀X, ∀F : set → set, (∀x ∈ X, ordinal (F x)) → ordinal (⋃x ∈ XF x)
In Proofgold the corresponding term root is fc0359... and proposition id is b3f1e1...
Proof:
Proof not loaded.
L2800
Theorem. (ordinal_binintersect)
∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∩ beta)
In Proofgold the corresponding term root is 43d507... and proposition id is 025328...
Proof:
Proof not loaded.
L2814
Theorem. (ordinal_binunion)
∀alpha beta, ordinal alpha → ordinal beta → ordinal (alpha ∪ beta)
In Proofgold the corresponding term root is 861ae3... and proposition id is b24989...
Proof:
Proof not loaded.
L2829
Theorem. (ordinal_ind)
∀p : set → prop, (∀alpha, ordinal alpha → (∀beta ∈ alpha, p beta) → p alpha) → ∀alpha, ordinal alpha → p alpha
In Proofgold the corresponding term root is bb5575... and proposition id is 973cd8...
Proof:
Proof not loaded.
L2849
Theorem. (least_ordinal_ex)
∀p : set → prop, (∃alpha, ordinal alpha ∧ p alpha) → ∃alpha, ordinal alpha ∧ p alpha ∧ ∀beta ∈ alpha, ¬ p beta
In Proofgold the corresponding term root is df6c77... and proposition id is ab6fe1...
Proof:
Proof not loaded.
L2871
Theorem. (equip_Sing_1)
∀x, equip {x} 1
In Proofgold the corresponding term root is be1ab2... and proposition id is 5169f4...
Proof:
Proof not loaded.
L2890
Theorem. (TransSet_In_ordsucc_Subq)
∀x y, TransSet y → x ∈ ordsucc y → x ⊆ y
In Proofgold the corresponding term root is f80c6e... and proposition id is 341993...
Proof:
Proof not loaded.
L2896
Theorem. (exandE_i)
∀P Q : set → prop, (∃x, P x ∧ Q x) → ∀r : prop, (∀x, P x → Q x → r) → r
In Proofgold the corresponding term root is 083f82... and proposition id is b1b26e...
Proof:
Proof not loaded.
L2901
Theorem. (exandE_ii)
∀P Q : (set → set) → prop, (∃x : set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set, P x → Q x → p) → p
In Proofgold the corresponding term root is c3e995... and proposition id is ed9d65...
Proof:
Proof not loaded.
L2908
Theorem. (exandE_iii)
∀P Q : (set → set → set) → prop, (∃x : set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set, P x → Q x → p) → p
In Proofgold the corresponding term root is 028a8f... and proposition id is 80b3b0...
Proof:
Proof not loaded.
L2915
Theorem. (exandE_iiii)
∀P Q : (set → set → set → set) → prop, (∃x : set → set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set → set, P x → Q x → p) → p
In Proofgold the corresponding term root is 7807d4... and proposition id is d1536a...
Proof:
Proof not loaded.
Beginning of Section Descr_ii
L2924
Variable P : (set → set) → prop
L2926
Definition. We define Descr_ii to be λx : set ⇒ Eps_i (λy ⇒ ∀h : set → set, P h → h x = y) of type set → set.
In Proofgold the corresponding term root is 3bae39... and object id is 3a4ad2...
L2927
Hypothesis Pex : ∃f : set → set, P f
L2928
Hypothesis Puniq : ∀f g : set → set, P f → P g → f = g
L2929
Theorem. (Descr_ii_prop)
P Descr_ii
In Proofgold the corresponding term root is 45bfc0... and proposition id is e705d1...
Proof:
Proof not loaded.
End of Section Descr_ii
Beginning of Section Descr_iii
L2949
Variable P : (set → set → set) → prop
L2951
Definition. We define Descr_iii to be λx y : set ⇒ Eps_i (λz ⇒ ∀h : set → set → set, P h → h x y = z) of type set → set → set.
In Proofgold the corresponding term root is ca5fc1... and object id is 877e40...
L2952
Hypothesis Pex : ∃f : set → set → set, P f
L2953
Hypothesis Puniq : ∀f g : set → set → set, P f → P g → f = g
L2954
Theorem. (Descr_iii_prop)
P Descr_iii
In Proofgold the corresponding term root is 261c3d... and proposition id is 160421...
Proof:
Proof not loaded.
End of Section Descr_iii
Beginning of Section Descr_Vo1
L2976
Variable P : Vo 1 → prop
L2978
Definition. We define Descr_Vo1 to be λx : set ⇒ ∀h : set → prop, P h → h x of type Vo 1.
In Proofgold the corresponding term root is 615c0a... and object id is a4ecdd...
L2979
Hypothesis Pex : ∃f : Vo 1, P f
L2980
Hypothesis Puniq : ∀f g : Vo 1, P f → P g → f = g
L2981
Theorem. (Descr_Vo1_prop)
P Descr_Vo1
In Proofgold the corresponding term root is f550ef... and proposition id is 2213e2...
Proof:
Proof not loaded.
End of Section Descr_Vo1
Beginning of Section If_ii
L3000
Variable p : prop
L3001
Variable f g : set → set
L3003
Definition. We define If_ii to be λx ⇒ if p then f x else g x of type set → set.
In Proofgold the corresponding term root is 17057f... and object id is e29015...
L3004
Theorem. (If_ii_1)
p → If_ii = f
In Proofgold the corresponding term root is 31521e... and proposition id is 342a1f...
Proof:
Proof not loaded.
L3011
Theorem. (If_ii_0)
¬ p → If_ii = g
In Proofgold the corresponding term root is 6acab0... and proposition id is 36418b...
Proof:
Proof not loaded.
End of Section If_ii
Beginning of Section If_iii
L3021
Variable p : prop
L3022
Variable f g : set → set → set
L3024
Definition. We define If_iii to be λx y ⇒ if p then f x y else g x y of type set → set → set.
In Proofgold the corresponding term root is 331476... and object id is dd7e2b...
L3025
Theorem. (If_iii_1)
p → If_iii = f
In Proofgold the corresponding term root is 2d62ee... and proposition id is 48bc10...
Proof:
Proof not loaded.
L3035
Theorem. (If_iii_0)
¬ p → If_iii = g
In Proofgold the corresponding term root is 9b77d9... and proposition id is d11201...
Proof:
Proof not loaded.
End of Section If_iii
Beginning of Section EpsilonRec_i
L3048
Variable F : set → (set → set) → set
L3049
Definition. We define In_rec_i_G to be λX Y ⇒ ∀R : set → set → prop, (∀X : set, ∀f : set → set, (∀x ∈ X, R x (f x)) → R X (F X f)) → R X Y of type set → set → prop.
In Proofgold the corresponding term root is fde509... and object id is a81e18...
L3055
Definition. We define In_rec_i to be λX ⇒ Eps_i (In_rec_i_G X) of type set → set.
In Proofgold the corresponding term root is fac413... and object id is 93aac4...
L3056
Theorem. (In_rec_i_G_c)
∀X : set, ∀f : set → set, (∀x ∈ X, In_rec_i_G x (f x)) → In_rec_i_G X (F X f)
In Proofgold the corresponding term root is 4da597... and proposition id is bc452e...
Proof:
Proof not loaded.
L3072
Theorem. (In_rec_i_G_inv)
∀X : set, ∀Y : set, In_rec_i_G X Y → ∃f : set → set, (∀x ∈ X, In_rec_i_G x (f x)) ∧ Y = F X f
In Proofgold the corresponding term root is ba1774... and proposition id is 5cfac2...
Proof:
Proof not loaded.
L3095
Hypothesis Fr : ∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
L3097
Theorem. (In_rec_i_G_f)
∀X : set, ∀Y Z : set, In_rec_i_G X Y → In_rec_i_G X Z → Y = Z
In Proofgold the corresponding term root is 25f7ff... and proposition id is 02b4e4...
Proof:
Proof not loaded.
L3128
Theorem. (In_rec_i_G_In_rec_i)
∀X : set, In_rec_i_G X (In_rec_i X)
In Proofgold the corresponding term root is 095407... and proposition id is dc2359...
Proof:
Proof not loaded.
L3139
Theorem. (In_rec_i_G_In_rec_i_d)
∀X : set, In_rec_i_G X (F X In_rec_i)
In Proofgold the corresponding term root is b480fe... and proposition id is 5b92ff...
Proof:
Proof not loaded.
L3147
Theorem. (In_rec_i_eq)
∀X : set, In_rec_i X = F X In_rec_i
In Proofgold the corresponding term root is 110653... and proposition id is e8653b...
Proof:
Proof not loaded.
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
L3154
Variable F : set → (set → (set → set)) → (set → set)
L3155
Definition. We define In_rec_G_ii to be λX Y ⇒ ∀R : set → (set → set) → prop, (∀X : set, ∀f : set → (set → set), (∀x ∈ X, R x (f x)) → R X (F X f)) → R X Y of type set → (set → set) → prop.
In Proofgold the corresponding term root is 6edba9... and object id is b31dbf...
L3161
Definition. We define In_rec_ii to be λX ⇒ Descr_ii (In_rec_G_ii X) of type set → (set → set).
In Proofgold the corresponding term root is f3c9ab... and object id is 3841bd...
L3162
Theorem. (In_rec_G_ii_c)
∀X : set, ∀f : set → (set → set), (∀x ∈ X, In_rec_G_ii x (f x)) → In_rec_G_ii X (F X f)
In Proofgold the corresponding term root is 381613... and proposition id is 49fbeb...
Proof:
Proof not loaded.
L3178
Theorem. (In_rec_G_ii_inv)
∀X : set, ∀Y : (set → set), In_rec_G_ii X Y → ∃f : set → (set → set), (∀x ∈ X, In_rec_G_ii x (f x)) ∧ Y = F X f
In Proofgold the corresponding term root is 336cfe... and proposition id is b4a180...
Proof:
Proof not loaded.
L3201
Hypothesis Fr : ∀X : set, ∀g h : set → (set → set), (∀x ∈ X, g x = h x) → F X g = F X h
L3203
Theorem. (In_rec_G_ii_f)
∀X : set, ∀Y Z : (set → set), In_rec_G_ii X Y → In_rec_G_ii X Z → Y = Z
In Proofgold the corresponding term root is 64b6e8... and proposition id is a3826b...
Proof:
Proof not loaded.
L3234
Theorem. (In_rec_G_ii_In_rec_ii)
∀X : set, In_rec_G_ii X (In_rec_ii X)
In Proofgold the corresponding term root is b2f81f... and proposition id is 2893f4...
Proof:
Proof not loaded.
L3247
Theorem. (In_rec_G_ii_In_rec_ii_d)
∀X : set, In_rec_G_ii X (F X In_rec_ii)
In Proofgold the corresponding term root is 2ff1ca... and proposition id is f12072...
Proof:
Proof not loaded.
L3255
Theorem. (In_rec_ii_eq)
∀X : set, In_rec_ii X = F X In_rec_ii
In Proofgold the corresponding term root is f7aff6... and proposition id is 7bb1e9...
Proof:
Proof not loaded.
End of Section EpsilonRec_ii
Beginning of Section EpsilonRec_iii
L3262
Variable F : set → (set → (set → set → set)) → (set → set → set)
L3263
Definition. We define In_rec_G_iii to be λX Y ⇒ ∀R : set → (set → set → set) → prop, (∀X : set, ∀f : set → (set → set → set), (∀x ∈ X, R x (f x)) → R X (F X f)) → R X Y of type set → (set → set → set) → prop.
In Proofgold the corresponding term root is 533cc7... and object id is 7a2f6b...
L3269
Definition. We define In_rec_iii to be λX ⇒ Descr_iii (In_rec_G_iii X) of type set → (set → set → set).
In Proofgold the corresponding term root is 9b3a85... and object id is 651077...
L3270
Theorem. (In_rec_G_iii_c)
∀X : set, ∀f : set → (set → set → set), (∀x ∈ X, In_rec_G_iii x (f x)) → In_rec_G_iii X (F X f)
In Proofgold the corresponding term root is 4ea572... and proposition id is a7d90f...
Proof:
Proof not loaded.
L3286
Theorem. (In_rec_G_iii_inv)
∀X : set, ∀Y : (set → set → set), In_rec_G_iii X Y → ∃f : set → (set → set → set), (∀x ∈ X, In_rec_G_iii x (f x)) ∧ Y = F X f
In Proofgold the corresponding term root is 5f6e0c... and proposition id is f47dae...
Proof:
Proof not loaded.
L3309
Hypothesis Fr : ∀X : set, ∀g h : set → (set → set → set), (∀x ∈ X, g x = h x) → F X g = F X h
L3311
Theorem. (In_rec_G_iii_f)
∀X : set, ∀Y Z : (set → set → set), In_rec_G_iii X Y → In_rec_G_iii X Z → Y = Z
In Proofgold the corresponding term root is 574d71... and proposition id is 40978a...
Proof:
Proof not loaded.
L3342
Theorem. (In_rec_G_iii_In_rec_iii)
∀X : set, In_rec_G_iii X (In_rec_iii X)
In Proofgold the corresponding term root is d8c3ba... and proposition id is 37f9a7...
Proof:
Proof not loaded.
L3355
Theorem. (In_rec_G_iii_In_rec_iii_d)
∀X : set, In_rec_G_iii X (F X In_rec_iii)
In Proofgold the corresponding term root is 60c6d7... and proposition id is c05f77...
Proof:
Proof not loaded.
L3363
Theorem. (In_rec_iii_eq)
∀X : set, In_rec_iii X = F X In_rec_iii
In Proofgold the corresponding term root is 5f0574... and proposition id is a0709d...
Proof:
Proof not loaded.
End of Section EpsilonRec_iii
Beginning of Section NatRec
L3370
Variable z : set
L3371
Variable f : set → set → set
L3372
Let F : set → (set → set) → set ≝ λn g ⇒ if ⋃ n ∈ n then f (⋃ n) (g (⋃ n)) else z
L3373
Definition. We define nat_primrec to be In_rec_i F of type set → set.
In Proofgold the corresponding term root is 3be1c5... and object id is 1a64d5...
L3374
Theorem. (nat_primrec_r)
∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
In Proofgold the corresponding term root is c955cd... and proposition id is 9ba887...
Proof:
Proof not loaded.
L3394
Theorem. (nat_primrec_0)
nat_primrec 0 = z
In Proofgold the corresponding term root is 416eaf... and proposition id is be11a2...
Proof:
Proof not loaded.
L3402
Theorem. (nat_primrec_S)
∀n : set, nat_p n → nat_primrec (ordsucc n) = f n (nat_primrec n)
In Proofgold the corresponding term root is 797185... and proposition id is fe4c88...
Proof:
Proof not loaded.
End of Section NatRec
Beginning of Section NatAdd
L3418
Definition. We define add_nat to be λn m : set ⇒ nat_primrec n (λ_ r ⇒ ordsucc r) m of type set → set → set.
In Proofgold the corresponding term root is afa8ae... and object id is 29768b...
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
L3421
Theorem. (add_nat_0R)
∀n : set, n + 0 = n
In Proofgold the corresponding term root is 402bcb... and proposition id is 23277f...
Proof:
Proof not loaded.
L3426
Theorem. (add_nat_SR)
∀n m : set, nat_p m → n + ordsucc m = ordsucc (n + m)
In Proofgold the corresponding term root is a53523... and proposition id is e61e18...
Proof:
Proof not loaded.
L3431
Theorem. (add_nat_p)
∀n : set, nat_p n → ∀m : set, nat_p m → nat_p (n + m)
In Proofgold the corresponding term root is a68eaa... and proposition id is 4c1d5d...
Proof:
Proof not loaded.
L3450
Theorem. (add_nat_1_1_2)
1 + 1 = 2
In Proofgold the corresponding term root is 174b87... and proposition id is 77f23f...
Proof:
Proof not loaded.
L3459
Theorem. (add_nat_asso)
∀n : set, nat_p n → ∀m : set, nat_p m → ∀k : set, nat_p k → (n + m) + k = n + (m + k)
In Proofgold the corresponding term root is c81c09... and proposition id is eec294...
Proof:
Proof not loaded.
L3481
Theorem. (add_nat_0L)
∀m : set, nat_p m → 0 + m = m
In Proofgold the corresponding term root is 8ceeb9... and proposition id is 84fbe4...
Proof:
Proof not loaded.
L3495
Theorem. (add_nat_SL)
∀n : set, nat_p n → ∀m : set, nat_p m → ordsucc n + m = ordsucc (n + m)
In Proofgold the corresponding term root is 9c442a... and proposition id is e6efb7...
Proof:
Proof not loaded.
L3514
Theorem. (add_nat_com)
∀n : set, nat_p n → ∀m : set, nat_p m → n + m = m + n
In Proofgold the corresponding term root is 40e952... and proposition id is 4ab179...
Proof:
Proof not loaded.
L3532
Theorem. (add_nat_In_R)
∀m, nat_p m → ∀k ∈ m, ∀n, nat_p n → k + n ∈ m + n
In Proofgold the corresponding term root is eeaa55... and proposition id is 85b3ac...
Proof:
Proof not loaded.
L3550
Theorem. (add_nat_In_L)
∀n, nat_p n → ∀m, nat_p m → ∀k ∈ m, n + k ∈ n + m
In Proofgold the corresponding term root is 0572fd... and proposition id is eaa26c...
Proof:
Proof not loaded.
L3558
Theorem. (add_nat_Subq_R)
∀k, nat_p k → ∀m, nat_p m → k ⊆ m → ∀n, nat_p n → k + n ⊆ m + n
In Proofgold the corresponding term root is 0d6a07... and proposition id is d59a02...
Proof:
Proof not loaded.
L3580
Theorem. (add_nat_Subq_L)
∀n, nat_p n → ∀k, nat_p k → ∀m, nat_p m → k ⊆ m → n + k ⊆ n + m
In Proofgold the corresponding term root is b7d52c... and proposition id is 586cfc...
Proof:
Proof not loaded.
L3590
Theorem. (add_nat_Subq_R')
∀m, nat_p m → ∀n, nat_p n → m ⊆ m + n
In Proofgold the corresponding term root is ed3e7e... and proposition id is cff680...
Proof:
Proof not loaded.
L3604
Theorem. (nat_Subq_add_ex)
∀n, nat_p n → ∀m, nat_p m → n ⊆ m → ∃k, nat_p k ∧ m = k + n
In Proofgold the corresponding term root is a1e406... and proposition id is 2c7fa4...
Proof:
Proof not loaded.
L3642
Theorem. (add_nat_cancel_R)
∀k, nat_p k → ∀m, nat_p m → ∀n, nat_p n → k + n = m + n → k = m
In Proofgold the corresponding term root is acfe90... and proposition id is 1c3ee5...
Proof:
Proof not loaded.
End of Section NatAdd
Beginning of Section NatMul
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
L3669
Definition. We define mul_nat to be λn m : set ⇒ nat_primrec 0 (λ_ r ⇒ n + r) m of type set → set → set.
In Proofgold the corresponding term root is 35a1ef... and object id is 80ccc6...
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
L3672
Theorem. (mul_nat_0R)
∀n : set, n * 0 = 0
In Proofgold the corresponding term root is b2f56c... and proposition id is 1e5b2c...
Proof:
Proof not loaded.
L3677
Theorem. (mul_nat_SR)
∀n m, nat_p m → n * ordsucc m = n + n * m
In Proofgold the corresponding term root is 37df54... and proposition id is 6dd6fc...
Proof:
Proof not loaded.
L3682
Theorem. (mul_nat_1R)
∀n, n * 1 = n
In Proofgold the corresponding term root is cab8eb... and proposition id is a3051d...
Proof:
Proof not loaded.
L3691
Theorem. (mul_nat_p)
∀n : set, nat_p n → ∀m : set, nat_p m → nat_p (n * m)
In Proofgold the corresponding term root is d63438... and proposition id is e910cb...
Proof:
Proof not loaded.
L3708
Theorem. (mul_nat_0L)
∀m : set, nat_p m → 0 * m = 0
In Proofgold the corresponding term root is db8a35... and proposition id is 6a2dd6...
Proof:
Proof not loaded.
L3723
Theorem. (mul_nat_SL)
∀n : set, nat_p n → ∀m : set, nat_p m → ordsucc n * m = n * m + m
In Proofgold the corresponding term root is 5cf7ec... and proposition id is 24e4eb...
Proof:
Proof not loaded.
L3755
Theorem. (mul_nat_com)
∀n : set, nat_p n → ∀m : set, nat_p m → n * m = m * n
In Proofgold the corresponding term root is 4525ef... and proposition id is 5d6ce4...
Proof:
Proof not loaded.
L3775
Theorem. (mul_add_nat_distrL)
∀n : set, nat_p n → ∀m : set, nat_p m → ∀k : set, nat_p k → n * (m + k) = n * m + n * k
In Proofgold the corresponding term root is 7758f1... and proposition id is d57b02...
Proof:
Proof not loaded.
L3806
Theorem. (mul_nat_asso)
∀n : set, nat_p n → ∀m : set, nat_p m → ∀k : set, nat_p k → (n * m) * k = n * (m * k)
In Proofgold the corresponding term root is 564886... and proposition id is 229d2c...
Proof:
Proof not loaded.
L3835
Theorem. (mul_nat_Subq_R)
∀m n, nat_p m → nat_p n → m ⊆ n → ∀k, nat_p k → m * k ⊆ n * k
In Proofgold the corresponding term root is d99b10... and proposition id is 4324d1...
Proof:
Proof not loaded.
L3857
Theorem. (mul_nat_Subq_L)
∀m n, nat_p m → nat_p n → m ⊆ n → ∀k, nat_p k → k * m ⊆ k * n
In Proofgold the corresponding term root is 929864... and proposition id is df9ccb...
Proof:
Proof not loaded.
L3865
Theorem. (mul_nat_0_or_Subq)
∀m, nat_p m → ∀n, nat_p n → n = 0 ∨ m ⊆ m * n
In Proofgold the corresponding term root is bdf0eb... and proposition id is 2bbf0e...
Proof:
Proof not loaded.
L3880
Theorem. (mul_nat_0_inv)
∀m, nat_p m → ∀n, nat_p n → m * n = 0 → m = 0 ∨ n = 0
In Proofgold the corresponding term root is 2da221... and proposition id is f3fbb4...
Proof:
Proof not loaded.
L3901
Theorem. (mul_nat_0m_1n_In)
∀m, nat_p m → ∀n, nat_p n → 0 ∈ m → 1 ∈ n → m ∈ m * n
In Proofgold the corresponding term root is 611f85... and proposition id is 5d96a9...
Proof:
Proof not loaded.
L3937
Theorem. (nat_le1_cases)
∀m, nat_p m → m ⊆ 1 → m = 0 ∨ m = 1
In Proofgold the corresponding term root is 591dbf... and proposition id is 5d5690...
Proof:
Proof not loaded.
L3954
Definition. We define Pi_nat to be λf n ⇒ nat_primrec 1 (λi r ⇒ r * f i) n of type (set → set) → set → set.
In Proofgold the corresponding term root is a0b865... and object id is 70d0e8...
L3957
Theorem. (Pi_nat_0)
∀f : set → set, Pi_nat f 0 = 1
In Proofgold the corresponding term root is 6af04c... and proposition id is b7ebea...
Proof:
Proof not loaded.
L3962
Theorem. (Pi_nat_S)
∀f : set → set, ∀n, nat_p n → Pi_nat f (ordsucc n) = Pi_nat f n * f n
In Proofgold the corresponding term root is 15f21a... and proposition id is b5fbbf...
Proof:
Proof not loaded.
L3968
Theorem. (Pi_nat_p)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, nat_p (f i)) → nat_p (Pi_nat f n)
In Proofgold the corresponding term root is 1e8072... and proposition id is 58005c...
Proof:
Proof not loaded.
L3993
Theorem. (Pi_nat_0_inv)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, nat_p (f i)) → Pi_nat f n = 0 → (∃i ∈ n, f i = 0)
In Proofgold the corresponding term root is 688c96... and proposition id is 98d7e0...
Proof:
Proof not loaded.
L4036
Definition. We define exp_nat to be λn m : set ⇒ nat_primrec 1 (λ_ r ⇒ n * r) m of type set → set → set.
In Proofgold the corresponding term root is 37c531... and object id is 8e35a1...
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_nat.
L4040
Theorem. (exp_nat_0)
∀n, n ^ 0 = 1
In Proofgold the corresponding term root is e32008... and proposition id is 856b46...
Proof:
Proof not loaded.
L4046
Theorem. (exp_nat_S)
∀n m, nat_p m → n ^ (ordsucc m) = n * n ^ m
In Proofgold the corresponding term root is e9c4ce... and proposition id is caaf40...
Proof:
Proof not loaded.
L4052
Theorem. (exp_nat_p)
∀n, nat_p n → ∀m, nat_p m → nat_p (n ^ m)
In Proofgold the corresponding term root is 1b73e0... and proposition id is 4f4027...
Proof:
Proof not loaded.
L4063
Theorem. (exp_nat_1)
∀n, n ^ 1 = n
In Proofgold the corresponding term root is 5858d8... and proposition id is 3e9f77...
Proof:
Proof not loaded.
End of Section NatMul
Beginning of Section form100_52
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_nat.
L4080
Theorem. (Subq_Sing0_1)
{0} ⊆ 1
In Proofgold the corresponding term root is cd7864... and proposition id is 1806bf...
Proof:
Proof not loaded.
L4084
Theorem. (Subq_1_Sing0)
1 ⊆ {0}
In Proofgold the corresponding term root is aeae9a... and proposition id is abab2d...
Proof:
Proof not loaded.
L4098
Theorem. (eq_1_Sing0)
1 = {0}
In Proofgold the corresponding term root is 5e5996... and proposition id is 264f38...
Proof:
Proof not loaded.
L4102
Theorem. (Power_0_Sing_0)
𝒫 0 = {0}
In Proofgold the corresponding term root is 6526a1... and proposition id is a39fd0...
Proof:
Proof not loaded.
L4123
Theorem. (equip_finite_Power)
∀n, nat_p n → ∀X, equip X n → equip (𝒫 X) (2 ^ n)
In Proofgold the corresponding term root is 0058ef... and proposition id is b4d9c8...
Proof:
Proof not loaded.
End of Section form100_52
L4496
Theorem. (ZF_closed_E)
∀U, ZF_closed U → ∀p : prop, (Union_closed U → Power_closed U → Repl_closed U → p) → p
In Proofgold the corresponding term root is d795d4... and proposition id is c78c79...
Proof:
Proof not loaded.
L4505
Theorem. (ZF_Union_closed)
∀U, ZF_closed U → ∀X ∈ U, ⋃ X ∈ U
In Proofgold the corresponding term root is a7cdf1... and proposition id is 250496...
Proof:
Proof not loaded.
L4510
Theorem. (ZF_Power_closed)
∀U, ZF_closed U → ∀X ∈ U, 𝒫 X ∈ U
In Proofgold the corresponding term root is 5d781f... and proposition id is 7411dd...
Proof:
Proof not loaded.
L4515
Theorem. (ZF_Repl_closed)
∀U, ZF_closed U → ∀X ∈ U, ∀F : set → set, (∀x ∈ X, F x ∈ U) → {F x|x ∈ X} ∈ U
In Proofgold the corresponding term root is 312268... and proposition id is 18f8ac...
Proof:
Proof not loaded.
L4520
Theorem. (ZF_UPair_closed)
∀U, ZF_closed U → ∀x y ∈ U, {x,y} ∈ U
In Proofgold the corresponding term root is 54256a... and proposition id is ef42d3...
Proof:
Proof not loaded.
L4604
Theorem. (ZF_Sing_closed)
∀U, ZF_closed U → ∀x ∈ U, {x} ∈ U
In Proofgold the corresponding term root is 5d8a83... and proposition id is d02d66...
Proof:
Proof not loaded.
L4609
Theorem. (ZF_binunion_closed)
∀U, ZF_closed U → ∀X Y ∈ U, (X ∪ Y) ∈ U
In Proofgold the corresponding term root is 8a6240... and proposition id is 1bacfd...
Proof:
Proof not loaded.
L4614
Theorem. (ZF_ordsucc_closed)
∀U, ZF_closed U → ∀x ∈ U, ordsucc x ∈ U
In Proofgold the corresponding term root is 68a6ba... and proposition id is 946e61...
Proof:
Proof not loaded.
L4619
Theorem. (nat_p_UnivOf_Empty)
∀n : set, nat_p n → n ∈ UnivOf Empty
In Proofgold the corresponding term root is c885b4... and proposition id is 645aee...
Proof:
Proof not loaded.
L4633
Definition. We define ω to be {n ∈ UnivOf Empty|nat_p n} of type set.
In Proofgold the corresponding term root is 6fc30a... and object id is ad9ad8...
L4634
Theorem. (omega_nat_p)
∀n ∈ ω, nat_p n
In Proofgold the corresponding term root is f44a30... and proposition id is c2c61b...
Proof:
Proof not loaded.
L4638
Theorem. (nat_p_omega)
∀n : set, nat_p n → n ∈ ω
In Proofgold the corresponding term root is 6a1b44... and proposition id is d5080f...
Proof:
Proof not loaded.
L4647
Theorem. (omega_ordsucc)
∀n ∈ ω, ordsucc n ∈ ω
In Proofgold the corresponding term root is 246d63... and proposition id is 17fb64...
Proof:
Proof not loaded.
L4655
Theorem. (form100_22_v2)
∀f : set → set, ¬ inj (𝒫 ω) ω f
In Proofgold the corresponding term root is eabc14... and proposition id is 5eb9bd...
Proof:
Proof not loaded.
L4659
Theorem. (form100_22_v3)
∀g : set → set, ¬ surj ω (𝒫 ω) g
In Proofgold the corresponding term root is 4de6f8... and proposition id is b20e6a...
Proof:
Proof not loaded.
L4684
Theorem. (form100_22_v1)
¬ equip ω (𝒫 ω)
In Proofgold the corresponding term root is 7ab652... and proposition id is a76888...
Proof:
Proof not loaded.
L4693
Theorem. (omega_TransSet)
TransSet ω
In Proofgold the corresponding term root is 8e89e6... and proposition id is c79e75...
Proof:
Proof not loaded.
L4706
Theorem. (omega_ordinal)
ordinal ω
In Proofgold the corresponding term root is 3d621a... and proposition id is 177484...
Proof:
Proof not loaded.
L4720
Theorem. (ordsucc_omega_ordinal)
ordinal (ordsucc ω)
In Proofgold the corresponding term root is 66659c... and proposition id is ecfad7...
Proof:
Proof not loaded.
L4724
Definition. We define finite to be λX ⇒ ∃n ∈ ω, equip X n of type set → prop.
In Proofgold the corresponding term root is 04632d... and object id is 4c54a5...
L4726
Theorem. (nat_finite)
∀n, nat_p n → finite n
In Proofgold the corresponding term root is a27dbe... and proposition id is 9b83bf...
Proof:
Proof not loaded.
L4734
Theorem. (finite_ind)
∀p : set → prop, p Empty → (∀X y, finite X → y ∉ X → p X → p (X ∪ {y})) → ∀X, finite X → p X
In Proofgold the corresponding term root is c53cb1... and proposition id is 19baca...
Proof:
Proof not loaded.
L4826
Theorem. (finite_Empty)
finite 0
In Proofgold the corresponding term root is c073c6... and proposition id is ec2c27...
Proof:
Proof not loaded.
L4833
Theorem. (Sing_finite)
∀x, finite {x}
In Proofgold the corresponding term root is f7016a... and proposition id is 281486...
Proof:
Proof not loaded.
L4841
Theorem. (adjoin_finite)
∀X y, finite X → finite (X ∪ {y})
In Proofgold the corresponding term root is def317... and proposition id is e6aa6d...
Proof:
Proof not loaded.
L4924
Theorem. (binunion_finite)
∀X, finite X → ∀Y, finite Y → finite (X ∪ Y)
In Proofgold the corresponding term root is fc26bb... and proposition id is 3365a6...
Proof:
Proof not loaded.
L4939
Theorem. (famunion_nat_finite)
∀X : set → set, ∀n, nat_p n → (∀i ∈ n, finite (X i)) → finite (⋃i ∈ nX i)
In Proofgold the corresponding term root is 7f709a... and proposition id is f84ed0...
Proof:
Proof not loaded.
L4978
Theorem. (Subq_finite)
∀X, finite X → ∀Y, Y ⊆ X → finite Y
In Proofgold the corresponding term root is 6b062c... and proposition id is e4e3ee...
Proof:
Proof not loaded.
L5031
Definition. We define infinite to be λA ⇒ ¬ finite A of type set → prop.
In Proofgold the corresponding term root is 313e7b... and object id is fad8ea...
Beginning of Section InfinitePrimes
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
L5038
Definition. We define divides_nat to be λm n ⇒ m ∈ ω ∧ n ∈ ω ∧ ∃k ∈ ω, m * k = n of type set → set → prop.
In Proofgold the corresponding term root is 1b17d5... and object id is 355928...
L5041
Theorem. (divides_nat_ref)
∀n, nat_p n → divides_nat n n
In Proofgold the corresponding term root is 98e754... and proposition id is 913818...
Proof:
Proof not loaded.
L5055
Theorem. (divides_nat_tra)
∀k m n, divides_nat k m → divides_nat m n → divides_nat k n
In Proofgold the corresponding term root is f799b9... and proposition id is e4e38e...
Proof:
Proof not loaded.
L5093
Definition. We define prime_nat to be λn ⇒ n ∈ ω ∧ 1 ∈ n ∧ ∀k ∈ ω, divides_nat k n → k = 1 ∨ k = n of type set → prop.
In Proofgold the corresponding term root is 729ee8... and object id is a54ca4...
L5096
Theorem. (divides_nat_mulR)
∀m n ∈ ω, divides_nat m (m * n)
In Proofgold the corresponding term root is eec687... and proposition id is ceb239...
Proof:
Proof not loaded.
L5109
Theorem. (divides_nat_mulL)
∀m n ∈ ω, divides_nat n (m * n)
In Proofgold the corresponding term root is cc5e38... and proposition id is d5d991...
Proof:
Proof not loaded.
L5116
Theorem. (Pi_nat_divides)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, nat_p (f i)) → (∀i ∈ n, divides_nat (f i) (Pi_nat f n))
In Proofgold the corresponding term root is 5e46a8... and proposition id is 2325b8...
Proof:
Proof not loaded.
L5150
Definition. We define composite_nat to be λn ⇒ n ∈ ω ∧ ∃k m ∈ ω, 1 ∈ k ∧ 1 ∈ m ∧ k * m = n of type set → prop.
In Proofgold the corresponding term root is 38d880... and object id is 3dafd8...
L5153
Theorem. (prime_nat_or_composite_nat)
∀n ∈ ω, 1 ∈ n → prime_nat n ∨ composite_nat n
In Proofgold the corresponding term root is a8970f... and proposition id is a1c7fc...
Proof:
Proof not loaded.
L5235
Theorem. (prime_nat_divisor_ex)
∀n, nat_p n → 1 ∈ n → ∃p, prime_nat p ∧ divides_nat p n
In Proofgold the corresponding term root is 2a24d5... and proposition id is 979fd7...
Proof:
Proof not loaded.
L5277
Theorem. (nat_1In_not_divides_ordsucc)
∀m n, 1 ∈ m → divides_nat m n → ¬ divides_nat m (ordsucc n)
In Proofgold the corresponding term root is 96fffb... and proposition id is 8d635d...
Proof:
Proof not loaded.
L5344
Definition. We define primes to be {n ∈ ω|prime_nat n} of type set.
In Proofgold the corresponding term root is f3caf8... and object id is 98dde1...
L5346
Theorem. (form100_11_infinite_primes)
infinite primes
In Proofgold the corresponding term root is 34eb3c... and proposition id is 13f23d...
Proof:
Proof not loaded.
End of Section InfinitePrimes
Beginning of Section InfiniteRamsey
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
L5477
Theorem. (atleastp_omega_infinite)
∀X, atleastp ω X → infinite X
In Proofgold the corresponding term root is 0b0e2a... and proposition id is c1147e...
Proof:
Proof not loaded.
L5503
Theorem. (infinite_remove1)
∀X, infinite X → ∀y, infinite (X ∖ {y})
In Proofgold the corresponding term root is f236a9... and proposition id is 7a03dc...
Proof:
Proof not loaded.
L5536
Theorem. (infinite_Finite_Subq_ex)
∀X, infinite X → ∀n, nat_p n → ∃Y ⊆ X, equip Y n
In Proofgold the corresponding term root is d43352... and proposition id is da00fc...
Proof:
Proof not loaded.
L5675
Theorem. (infiniteRamsey_lem)
∀X, ∀f g f' : set → set, infinite X → (∀Z ⊆ X, infinite Z → f Z ⊆ Z ∧ infinite (f Z)) → (∀Z ⊆ X, infinite Z → g Z ∈ Z ∧ g Z ∉ f Z) → f' 0 = f X → (∀m, nat_p m → f' (ordsucc m) = f (f' m)) → (∀m, nat_p m → f' m ⊆ X ∧ infinite (f' m)) ∧ (∀m m' ∈ ω, m ⊆ m' → f' m' ⊆ f' m) ∧ (∀m m' ∈ ω, g (f' m) = g (f' m') → m = m')
In Proofgold the corresponding term root is c8e826... and proposition id is c1cc5f...
Proof:
Proof not loaded.
L5801
Theorem. (infiniteRamsey)
∀c, nat_p c → ∀n, nat_p n → ∀X, infinite X → ∀C : set → set, (∀Y ⊆ X, equip Y n → C Y ∈ c) → ∃H ⊆ X, ∃i ∈ c, infinite H ∧ ∀Y ⊆ H, equip Y n → C Y = i
In Proofgold the corresponding term root is f92f8f... and proposition id is 865fc1...
Proof:
Proof not loaded.
End of Section InfiniteRamsey
L6649
Definition. We define Inj1 to be In_rec_i (λX f ⇒ {0} ∪ {f x|x ∈ X}) of type set → set.
In Proofgold the corresponding term root is 8f0026... and object id is 7736ca...
L6652
Theorem. (Inj1_eq)
∀X : set, Inj1 X = {0} ∪ {Inj1 x|x ∈ X}
In Proofgold the corresponding term root is 73d189... and proposition id is b80457...
Proof:
Proof not loaded.
(*** Injection of set into itself that will correspond to x |-> (1,x) after pairing is defined. ***)
L6669
Theorem. (Inj1I1)
∀X : set, 0 ∈ Inj1 X
In Proofgold the corresponding term root is 3c2cc4... and proposition id is 4ccac3...
Proof:
Proof not loaded.
L6677
Theorem. (Inj1I2)
∀X x : set, x ∈ X → Inj1 x ∈ Inj1 X
In Proofgold the corresponding term root is b6f23b... and proposition id is 086475...
Proof:
Proof not loaded.
L6686
Theorem. (Inj1E)
∀X y : set, y ∈ Inj1 X → y = 0 ∨ ∃x ∈ X, y = Inj1 x
In Proofgold the corresponding term root is d4177b... and proposition id is 259217...
Proof:
Proof not loaded.
L6701
Theorem. (Inj1NE1)
∀x : set, Inj1 x ≠ 0
In Proofgold the corresponding term root is 5e3197... and proposition id is 09bd10...
Proof:
Proof not loaded.
L6711
Theorem. (Inj1NE2)
∀x : set, Inj1 x ∉ {0}
In Proofgold the corresponding term root is 9e94b1... and proposition id is 3c4dd0...
Proof:
Proof not loaded.
L6717
Definition. We define Inj0 to be λX ⇒ {Inj1 x|x ∈ X} of type set → set.
In Proofgold the corresponding term root is 814321... and object id is 7d54be...
L6720
Theorem. (Inj0I)
∀X x : set, x ∈ X → Inj1 x ∈ Inj0 X
In Proofgold the corresponding term root is 7fc2e9... and proposition id is 565378...
Proof:
Proof not loaded.
(*** Injection of set into itself that will correspond to x |-> (0,x) after pairing is defined. ***)
L6724
Theorem. (Inj0E)
∀X y : set, y ∈ Inj0 X → ∃x : set, x ∈ X ∧ y = Inj1 x
In Proofgold the corresponding term root is 416684... and proposition id is 855213...
Proof:
Proof not loaded.
L6728
Definition. We define Unj to be In_rec_i (λX f ⇒ {f x|x ∈ X ∖ {0}}) of type set → set.
In Proofgold the corresponding term root is f202c1... and object id is 2e1675...
L6731
Theorem. (Unj_eq)
∀X : set, Unj X = {Unj x|x ∈ X ∖ {0}}
In Proofgold the corresponding term root is 3b1aa6... and proposition id is b4c852...
Proof:
Proof not loaded.
(*** Unj : Left inverse of Inj1 and Inj0 ***)
L6748
Theorem. (Unj_Inj1_eq)
∀X : set, Unj (Inj1 X) = X
In Proofgold the corresponding term root is e2cdfc... and proposition id is ae00e1...
Proof:
Proof not loaded.
L6800
Theorem. (Inj1_inj)
∀X Y : set, Inj1 X = Inj1 Y → X = Y
In Proofgold the corresponding term root is ce8b66... and proposition id is 3e918a...
Proof:
Proof not loaded.
L6810
Theorem. (Unj_Inj0_eq)
∀X : set, Unj (Inj0 X) = X
In Proofgold the corresponding term root is f3e39c... and proposition id is 3808ad...
Proof:
Proof not loaded.
L6858
Theorem. (Inj0_inj)
∀X Y : set, Inj0 X = Inj0 Y → X = Y
In Proofgold the corresponding term root is 662f53... and proposition id is 29f945...
Proof:
Proof not loaded.
L6868
Theorem. (Inj0_0)
Inj0 0 = 0
In Proofgold the corresponding term root is 171f47... and proposition id is 57f5af...
Proof:
Proof not loaded.
L6872
Theorem. (Inj0_Inj1_neq)
∀X Y : set, Inj0 X ≠ Inj1 Y
In Proofgold the corresponding term root is 21bb03... and proposition id is 2b14d8...
Proof:
Proof not loaded.
L6887
Definition. We define setsum to be λX Y ⇒ {Inj0 x|x ∈ X} ∪ {Inj1 y|y ∈ Y} of type set → set → set.
In Proofgold the corresponding term root is afe37e... and object id is 5b7e09...
Notation. We use + as an infix operator with priority 450 and which associates to the left corresponding to applying term setsum.
(*** setsum ***)
L6892
Theorem. (Inj0_setsum)
∀X Y x : set, x ∈ X → Inj0 x ∈ X + Y
In Proofgold the corresponding term root is 1c6408... and proposition id is 246ac5...
Proof:
Proof not loaded.
L6902
Theorem. (Inj1_setsum)
∀X Y y : set, y ∈ Y → Inj1 y ∈ X + Y
In Proofgold the corresponding term root is d47a0f... and proposition id is 2b4f94...
Proof:
Proof not loaded.
L6912
Theorem. (setsum_Inj_inv)
∀X Y z : set, z ∈ X + Y → (∃x ∈ X, z = Inj0 x) ∨ (∃y ∈ Y, z = Inj1 y)
In Proofgold the corresponding term root is 645a6c... and proposition id is a36400...
Proof:
Proof not loaded.
L6924
Theorem. (Inj0_setsum_0L)
∀X : set, 0 + X = Inj0 X
In Proofgold the corresponding term root is 0be1e7... and proposition id is 36fabd...
Proof:
Proof not loaded.
L6957
Theorem. (Inj1_setsum_1L)
∀X : set, 1 + X = Inj1 X
In Proofgold the corresponding term root is f8d721... and proposition id is 45b1c8...
Proof:
Proof not loaded.
Beginning of Section pair_setsum
L7010
Let pair ≝ setsum
L7011
Definition. We define proj0 to be λZ ⇒ {Unj z|z ∈ Z, ∃x : set, Inj0 x = z} of type set → set.
In Proofgold the corresponding term root is 21a488... and object id is 9c5e4e...
L7012
Definition. We define proj1 to be λZ ⇒ {Unj z|z ∈ Z, ∃y : set, Inj1 y = z} of type set → set.
In Proofgold the corresponding term root is 7aba21... and object id is 7fd81e...
L7013
Theorem. (Inj0_pair_0_eq)
Inj0 = pair 0
In Proofgold the corresponding term root is 0fe601... and proposition id is 490c2a...
Proof:
Proof not loaded.
L7021
Theorem. (Inj1_pair_1_eq)
Inj1 = pair 1
In Proofgold the corresponding term root is 0bbde8... and proposition id is 8272d7...
Proof:
Proof not loaded.
L7029
Theorem. (pairI0)
∀X Y x, x ∈ X → pair 0 x ∈ pair X Y
In Proofgold the corresponding term root is 21a414... and proposition id is 34813b...
Proof:
Proof not loaded.
L7034
Theorem. (pairI1)
∀X Y y, y ∈ Y → pair 1 y ∈ pair X Y
In Proofgold the corresponding term root is bdaab5... and proposition id is b5248b...
Proof:
Proof not loaded.
L7039
Theorem. (pairE)
∀X Y z, z ∈ pair X Y → (∃x ∈ X, z = pair 0 x) ∨ (∃y ∈ Y, z = pair 1 y)
In Proofgold the corresponding term root is aac610... and proposition id is 0afdeb...
Proof:
Proof not loaded.
L7045
Theorem. (pairE0)
∀X Y x, pair 0 x ∈ pair X Y → x ∈ X
In Proofgold the corresponding term root is 042f30... and proposition id is f50b25...
Proof:
Proof not loaded.
L7071
Theorem. (pairE1)
∀X Y y, pair 1 y ∈ pair X Y → y ∈ Y
In Proofgold the corresponding term root is 505cfc... and proposition id is 30a181...
Proof:
Proof not loaded.
L7099
Theorem. (proj0I)
∀w u : set, pair 0 u ∈ w → u ∈ proj0 w
In Proofgold the corresponding term root is be55b9... and proposition id is 840c8d...
Proof:
Proof not loaded.
L7114
Theorem. (proj0E)
∀w u : set, u ∈ proj0 w → pair 0 u ∈ w
In Proofgold the corresponding term root is b9afa9... and proposition id is 5b7f58...
Proof:
Proof not loaded.
L7138
Theorem. (proj1I)
∀w u : set, pair 1 u ∈ w → u ∈ proj1 w
In Proofgold the corresponding term root is fbddb7... and proposition id is 595b3c...
Proof:
Proof not loaded.
L7153
Theorem. (proj1E)
∀w u : set, u ∈ proj1 w → pair 1 u ∈ w
In Proofgold the corresponding term root is 06e32e... and proposition id is c3c3e6...
Proof:
Proof not loaded.
L7177
Theorem. (proj0_pair_eq)
∀X Y : set, proj0 (pair X Y) = X
In Proofgold the corresponding term root is d546f3... and proposition id is 0c12fb...
Proof:
Proof not loaded.
L7197
Theorem. (proj1_pair_eq)
∀X Y : set, proj1 (pair X Y) = Y
In Proofgold the corresponding term root is 9f98a8... and proposition id is e0cb0f...
Proof:
Proof not loaded.
L7219
Definition. We define Sigma to be λX Y ⇒ ⋃x ∈ X{pair x y|y ∈ Y x} of type set → (set → set) → set.
In Proofgold the corresponding term root is 8acbb2... and object id is d64bcb...
Notation. We use ∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
(*** Sigma X Y = {(x,y) | x in X, y in Y x} ***)
L7225
Theorem. (Sigma_eta_proj0_proj1)
∀X : set, ∀Y : set → set, ∀z ∈ (∑x ∈ X, Y x), pair (proj0 z) (proj1 z) = z ∧ proj0 z ∈ X ∧ proj1 z ∈ Y (proj0 z)
In Proofgold the corresponding term root is afe962... and proposition id is 77ba8e...
Proof:
Proof not loaded.
L7253
Theorem. (proj0_Sigma)
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj0 z ∈ X
In Proofgold the corresponding term root is ae12a7... and proposition id is 506869...
Proof:
Proof not loaded.
L7261
Theorem. (proj1_Sigma)
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj1 z ∈ Y (proj0 z)
In Proofgold the corresponding term root is 553db6... and proposition id is d79917...
Proof:
Proof not loaded.
L7269
Theorem. (pair_Sigma)
∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, pair x y ∈ ∑x ∈ X, Y x
In Proofgold the corresponding term root is 0d8645... and proposition id is 1456cb...
Proof:
Proof not loaded.
L7284
Theorem. (pair_Sigma_E1)
∀X : set, ∀Y : set → set, ∀x y : set, pair x y ∈ (∑x ∈ X, Y x) → y ∈ Y x
In Proofgold the corresponding term root is c294a7... and proposition id is 81d8b0...
Proof:
Proof not loaded.
L7295
Theorem. (Sigma_E)
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → ∃x ∈ X, ∃y ∈ Y x, z = pair x y
In Proofgold the corresponding term root is 5b1bd2... and proposition id is 8ecfab...
Proof:
Proof not loaded.
L7313
Definition. We define setprod to be λX Y : set ⇒ ∑x ∈ X, Y of type set → set → set.
In Proofgold the corresponding term root is fc0b60... and object id is 107b6b...
Notation. We use ⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
L7317
Let lam : set → (set → set) → set ≝ Sigma
L7319
Definition. We define ap to be λf x ⇒ {proj1 z|z ∈ f, ∃y : set, z = pair x y} of type set → set → set.
In Proofgold the corresponding term root is c7aaa6... and object id is a4d82c...
(*** lam X F = {(x,y) | x :e X, y :e F x} = \/_{x :e X} {(x,y) | y :e (F x)} = Sigma X F ***)
Notation. When x is a set, a term x y is notation for ap x y.
Notation. λ x ∈ A ⇒ B is notation for the set Sigma A (λ x : set ⇒ B).
Notation. We now use n-tuple notation (a0,...,an-1) for n ≥ 2 for λ i ∈ n . if i = 0 then a0 else ... if i = n-2 then an-2 else an-1.
(*** ap f x = {proj1 z | z :e f, exists y, z = pair x y}} ***)
L7323
Theorem. (lamI)
∀X : set, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, pair x y ∈ λx ∈ X ⇒ F x
In Proofgold the corresponding term root is 0d8645... and proposition id is 1456cb...
Proof:
Proof not loaded.
L7327
Theorem. (lamE)
∀X : set, ∀F : set → set, ∀z : set, z ∈ (λx ∈ X ⇒ F x) → ∃x ∈ X, ∃y ∈ F x, z = pair x y
In Proofgold the corresponding term root is 5b1bd2... and proposition id is 8ecfab...
Proof:
Proof not loaded.
L7331
Theorem. (apI)
∀f x y, pair x y ∈ f → y ∈ f x
In Proofgold the corresponding term root is 64667c... and proposition id is 770816...
Proof:
Proof not loaded.
L7343
Theorem. (apE)
∀f x y, y ∈ f x → pair x y ∈ f
In Proofgold the corresponding term root is 85693a... and proposition id is c08565...
Proof:
Proof not loaded.
L7371
Theorem. (beta)
∀X : set, ∀F : set → set, ∀x : set, x ∈ X → (λx ∈ X ⇒ F x) x = F x
In Proofgold the corresponding term root is 95991d... and proposition id is b5182d...
Proof:
Proof not loaded.
L7391
Theorem. (proj0_ap_0)
∀u, proj0 u = u 0
In Proofgold the corresponding term root is 588f37... and proposition id is d4f351...
Proof:
Proof not loaded.
L7411
Theorem. (proj1_ap_1)
∀u, proj1 u = u 1
In Proofgold the corresponding term root is d116b3... and proposition id is 2328b0...
Proof:
Proof not loaded.
L7431
Theorem. (pair_ap_0)
∀x y : set, (pair x y) 0 = x
In Proofgold the corresponding term root is ac34b9... and proposition id is bb0477...
Proof:
Proof not loaded.
L7438
Theorem. (pair_ap_1)
∀x y : set, (pair x y) 1 = y
In Proofgold the corresponding term root is 2274fc... and proposition id is cce3d2...
Proof:
Proof not loaded.
L7445
Theorem. (ap0_Sigma)
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 0) ∈ X
In Proofgold the corresponding term root is 861a79... and proposition id is 1affcc...
Proof:
Proof not loaded.
L7451
Theorem. (ap1_Sigma)
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 1) ∈ (Y (z 0))
In Proofgold the corresponding term root is ad3bfe... and proposition id is 520af5...
Proof:
Proof not loaded.
L7458
Definition. We define pair_p to be λu : set ⇒ pair (u 0) (u 1) = u of type set → prop.
In Proofgold the corresponding term root is ade2bb... and object id is 5c8a35...
L7461
Theorem. (pair_p_I)
∀x y, pair_p (pair x y)
In Proofgold the corresponding term root is 06d9a2... and proposition id is 8f2fde...
Proof:
Proof not loaded.
L7469
Theorem. (Subq_2_UPair01)
2 ⊆ {0,1}
In Proofgold the corresponding term root is d90884... and proposition id is 94203e...
Proof:
Proof not loaded.
L7487
Theorem. (tuple_pair)
∀x y : set, pair x y = (x,y)
In Proofgold the corresponding term root is c122f6... and proposition id is 18198e...
Proof:
Proof not loaded.
L7562
Definition. We define Pi to be λX Y ⇒ {f ∈ 𝒫 (∑x ∈ X, ⋃ (Y x))|∀x ∈ X, f x ∈ Y x} of type set → (set → set) → set.
In Proofgold the corresponding term root is 40f1c3... and object id is a7b1b0...
Notation. We use ∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
L7566
Theorem. (PiI)
∀X : set, ∀Y : set → set, ∀f : set, (∀u ∈ f, pair_p u ∧ u 0 ∈ X) → (∀x ∈ X, f x ∈ Y x) → f ∈ ∏x ∈ X, Y x
In Proofgold the corresponding term root is 2b7ef4... and proposition id is 1f8f6f...
Proof:
Proof not loaded.
L7600
Theorem. (lam_Pi)
∀X : set, ∀Y : set → set, ∀F : set → set, (∀x ∈ X, F x ∈ Y x) → (λx ∈ X ⇒ F x) ∈ (∏x ∈ X, Y x)
In Proofgold the corresponding term root is c0b88f... and proposition id is 3733f8...
Proof:
Proof not loaded.
L7639
Theorem. (ap_Pi)
∀X : set, ∀Y : set → set, ∀f : set, ∀x : set, f ∈ (∏x ∈ X, Y x) → x ∈ X → f x ∈ Y x
In Proofgold the corresponding term root is 7c3617... and proposition id is cb962e...
Proof:
Proof not loaded.
L7645
Definition. We define setexp to be λX Y : set ⇒ ∏y ∈ Y, X of type set → set → set.
In Proofgold the corresponding term root is 1de7fc... and object id is 021ae0...
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
L7649
Theorem. (pair_tuple_fun)
pair = (λx y ⇒ (x,y))
In Proofgold the corresponding term root is b02dfa... and proposition id is ed37d0...
Proof:
Proof not loaded.
Beginning of Section Tuples
L7658
Variable x0 x1 : set
L7659
Theorem. (tuple_2_0_eq)
(x0,x1) 0 = x0
In Proofgold the corresponding term root is 42d452... and proposition id is ad9037...
Proof:
Proof not loaded.
L7664
Theorem. (tuple_2_1_eq)
(x0,x1) 1 = x1
In Proofgold the corresponding term root is 596569... and proposition id is 2c0ad1...
Proof:
Proof not loaded.
End of Section Tuples
L7671
Theorem. (ReplEq_setprod_ext)
∀X Y, ∀F G : set → set → set, (∀x ∈ X, ∀y ∈ Y, F x y = G x y) → {F (w 0) (w 1)|w ∈ X ⨯ Y} = {G (w 0) (w 1)|w ∈ X ⨯ Y}
In Proofgold the corresponding term root is c486ab... and proposition id is f689a0...
Proof:
Proof not loaded.
L7682
Theorem. (lamI2)
∀X, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, (x,y) ∈ λx ∈ X ⇒ F x
In Proofgold the corresponding term root is d4a9d9... and proposition id is de381f...
Proof:
Proof not loaded.
L7688
Theorem. (tuple_2_Sigma)
∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, (x,y) ∈ ∑x ∈ X, Y x
In Proofgold the corresponding term root is d4a9d9... and proposition id is de381f...
Proof:
Proof not loaded.
L7692
Theorem. (tuple_2_setprod)
∀X : set, ∀Y : set, ∀x ∈ X, ∀y ∈ Y, (x,y) ∈ X ⨯ Y
In Proofgold the corresponding term root is e907b8... and proposition id is 1f9d7f...
Proof:
Proof not loaded.
End of Section pair_setsum
Notation. We use ∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Sigma.
Notation. We use ⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term setprod.
Notation. We use ∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using Pi.
Notation. We use :^: as an infix operator with priority 430 and which associates to the left corresponding to applying term setexp.
L7707
Definition. We define DescrR_i_io_1 to be λR ⇒ Eps_i (λx ⇒ (∃y : set → prop, R x y) ∧ (∀y z : set → prop, R x y → R x z → y = z)) of type (set → (set → prop) → prop) → set.
In Proofgold the corresponding term root is 1d3fd4... and object id is 82aecc...
L7709
Definition. We define DescrR_i_io_2 to be λR ⇒ Descr_Vo1 (λy ⇒ R (DescrR_i_io_1 R) y) of type (set → (set → prop) → prop) → set → prop.
In Proofgold the corresponding term root is 768eb2... and object id is 3f82bc...
L7710
Theorem. (DescrR_i_io_12)
∀R : set → (set → prop) → prop, (∃x, (∃y : set → prop, R x y) ∧ (∀y z : set → prop, R x y → R x z → y = z)) → R (DescrR_i_io_1 R) (DescrR_i_io_2 R)
In Proofgold the corresponding term root is a4fae2... and proposition id is da4f02...
Proof:
Proof not loaded.
L7720
Definition. We define PNoEq_ to be λalpha p q ⇒ ∀beta ∈ alpha, p beta ↔ q beta of type set → (set → prop) → (set → prop) → prop.
In Proofgold the corresponding term root is d7d959... and object id is 144b65...
(*** Conway describes this way of formalizing in ZF in an appendix to Part Zero of his book,
but rejects formalization in favor of "Mathematician's Liberation." ***)
L7726
Theorem. (PNoEq_ref_)
∀alpha, ∀p : set → prop, PNoEq_ alpha p p
In Proofgold the corresponding term root is c27ebb... and proposition id is df4cf6...
Proof:
Proof not loaded.
L7732
Theorem. (PNoEq_sym_)
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q p
In Proofgold the corresponding term root is b813f3... and proposition id is bfda0b...
Proof:
Proof not loaded.
L7740
Theorem. (PNoEq_tra_)
∀alpha, ∀p q r : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q r → PNoEq_ alpha p r
In Proofgold the corresponding term root is dded4c... and proposition id is fc92f3...
Proof:
Proof not loaded.
L7749
Theorem. (PNoEq_antimon_)
∀p q : set → prop, ∀alpha, ordinal alpha → ∀beta ∈ alpha, PNoEq_ alpha p q → PNoEq_ beta p q
In Proofgold the corresponding term root is 8868e6... and proposition id is 094ae2...
Proof:
Proof not loaded.
L7759
Definition. We define PNoLt_ to be λalpha p q ⇒ ∃beta ∈ alpha, PNoEq_ beta p q ∧ ¬ p beta ∧ q beta of type set → (set → prop) → (set → prop) → prop.
In Proofgold the corresponding term root is 34de68... and object id is c95964...
L7762
Theorem. (PNoLt_E_)
∀alpha, ∀p q : set → prop, PNoLt_ alpha p q → ∀R : prop, (∀beta, beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → q beta → R) → R
In Proofgold the corresponding term root is 9ba543... and proposition id is 45aec8...
Proof:
Proof not loaded.
L7773
Theorem. (PNoLt_irref_)
∀alpha, ∀p : set → prop, ¬ PNoLt_ alpha p p
In Proofgold the corresponding term root is fcb502... and proposition id is 987dff...
Proof:
Proof not loaded.
L7780
Theorem. (PNoLt_mon_)
∀p q : set → prop, ∀alpha, ordinal alpha → ∀beta ∈ alpha, PNoLt_ beta p q → PNoLt_ alpha p q
In Proofgold the corresponding term root is b1c3cf... and proposition id is d8d0f0...
Proof:
Proof not loaded.
L7791
Theorem. (PNoLt_trichotomy_or_)
∀p q : set → prop, ∀alpha, ordinal alpha → PNoLt_ alpha p q ∨ PNoEq_ alpha p q ∨ PNoLt_ alpha q p
In Proofgold the corresponding term root is 664408... and proposition id is 8d3824...
Proof:
Proof not loaded.
L7866
Definition. We define PNoLt to be λalpha p beta q ⇒ PNoLt_ (alpha ∩ beta) p q ∨ alpha ∈ beta ∧ PNoEq_ alpha p q ∧ q alpha ∨ beta ∈ alpha ∧ PNoEq_ beta p q ∧ ¬ p beta of type set → (set → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 8f57a0... and object id is f2094c...
L7871
Theorem. (PNoLtI1)
∀alpha beta, ∀p q : set → prop, PNoLt_ (alpha ∩ beta) p q → PNoLt alpha p beta q
In Proofgold the corresponding term root is c5735c... and proposition id is ecbc71...
Proof:
Proof not loaded.
L7880
Theorem. (PNoLtI2)
∀alpha beta, ∀p q : set → prop, alpha ∈ beta → PNoEq_ alpha p q → q alpha → PNoLt alpha p beta q
In Proofgold the corresponding term root is 419b53... and proposition id is 030093...
Proof:
Proof not loaded.
L7892
Theorem. (PNoLtI3)
∀alpha beta, ∀p q : set → prop, beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → PNoLt alpha p beta q
In Proofgold the corresponding term root is c392cb... and proposition id is 410442...
Proof:
Proof not loaded.
L7904
Theorem. (PNoLtE)
∀alpha beta, ∀p q : set → prop, PNoLt alpha p beta q → ∀R : prop, (PNoLt_ (alpha ∩ beta) p q → R) → (alpha ∈ beta → PNoEq_ alpha p q → q alpha → R) → (beta ∈ alpha → PNoEq_ beta p q → ¬ p beta → R) → R
In Proofgold the corresponding term root is f7e6e3... and proposition id is af36ce...
Proof:
Proof not loaded.
L7919
Theorem. (PNoLt_irref)
∀alpha, ∀p : set → prop, ¬ PNoLt alpha p alpha p
In Proofgold the corresponding term root is ff1a80... and proposition id is d4c379...
Proof:
Proof not loaded.
L7930
Theorem. (PNoLt_trichotomy_or)
∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q ∨ PNoLt beta q alpha p
In Proofgold the corresponding term root is 4f39e4... and proposition id is 90b387...
Proof:
Proof not loaded.
L7997
Theorem. (PNoLtEq_tra)
∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoLt alpha p beta q → PNoEq_ beta q r → PNoLt alpha p beta r
In Proofgold the corresponding term root is 2e082a... and proposition id is 2f2a82...
Proof:
Proof not loaded.
L8051
Theorem. (PNoEqLt_tra)
∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoEq_ alpha p q → PNoLt alpha q beta r → PNoLt alpha p beta r
In Proofgold the corresponding term root is fe144b... and proposition id is f07654...
Proof:
Proof not loaded.
L8109
Theorem. (PNoLt_tra)
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLt alpha p beta q → PNoLt beta q gamma r → PNoLt alpha p gamma r
In Proofgold the corresponding term root is b72127... and proposition id is 8eff85...
Proof:
Proof not loaded.
L8424
Definition. We define PNoLe to be λalpha p beta q ⇒ PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q of type set → (set → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is ac6311... and object id is 770cf7...
L8427
Theorem. (PNoLeI1)
∀alpha beta, ∀p q : set → prop, PNoLt alpha p beta q → PNoLe alpha p beta q
In Proofgold the corresponding term root is 1f0528... and proposition id is 674a2c...
Proof:
Proof not loaded.
L8435
Theorem. (PNoLeI2)
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoLe alpha p alpha q
In Proofgold the corresponding term root is b45332... and proposition id is 0b2157...
Proof:
Proof not loaded.
L8445
Theorem. (PNoLe_ref)
∀alpha, ∀p : set → prop, PNoLe alpha p alpha p
In Proofgold the corresponding term root is 1b91bd... and proposition id is 613386...
Proof:
Proof not loaded.
L8451
Theorem. (PNoLe_antisym)
∀alpha beta, ordinal alpha → ordinal beta → ∀p q : set → prop, PNoLe alpha p beta q → PNoLe beta q alpha p → alpha = beta ∧ PNoEq_ alpha p q
In Proofgold the corresponding term root is 934d7a... and proposition id is 4e3594...
Proof:
Proof not loaded.
L8497
Theorem. (PNoLtLe_tra)
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLt alpha p beta q → PNoLe beta q gamma r → PNoLt alpha p gamma r
In Proofgold the corresponding term root is 156586... and proposition id is 09246b...
Proof:
Proof not loaded.
L8514
Theorem. (PNoLeLt_tra)
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLe alpha p beta q → PNoLt beta q gamma r → PNoLt alpha p gamma r
In Proofgold the corresponding term root is a260ca... and proposition id is ad6971...
Proof:
Proof not loaded.
L8533
Theorem. (PNoEqLe_tra)
∀alpha beta, ordinal alpha → ordinal beta → ∀p q r : set → prop, PNoEq_ alpha p q → PNoLe alpha q beta r → PNoLe alpha p beta r
In Proofgold the corresponding term root is 7d3ff8... and proposition id is 002ebf...
Proof:
Proof not loaded.
L8552
Theorem. (PNoLe_tra)
∀alpha beta gamma, ordinal alpha → ordinal beta → ordinal gamma → ∀p q r : set → prop, PNoLe alpha p beta q → PNoLe beta q gamma r → PNoLe alpha p gamma r
In Proofgold the corresponding term root is 70b92b... and proposition id is 4b557d...
Proof:
Proof not loaded.
L8573
Definition. We define PNo_downc to be λL alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, L beta q ∧ PNoLe alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 93dab1... and object id is f6c962...
L8576
Definition. We define PNo_upc to be λR alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, R beta q ∧ PNoLe beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 543712... and object id is b97935...
L8578
Theorem. (PNoLe_downc)
∀L : set → (set → prop) → prop, ∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNo_downc L alpha p → PNoLe beta q alpha p → PNo_downc L beta q
In Proofgold the corresponding term root is 6b1f0d... and proposition id is cd130e...
Proof:
Proof not loaded.
L8598
Theorem. (PNo_downc_ref)
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, L alpha p → PNo_downc L alpha p
In Proofgold the corresponding term root is 347aa9... and proposition id is 69dfcb...
Proof:
Proof not loaded.
L8609
Theorem. (PNo_upc_ref)
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, R alpha p → PNo_upc R alpha p
In Proofgold the corresponding term root is 403942... and proposition id is c4bf71...
Proof:
Proof not loaded.
L8620
Theorem. (PNoLe_upc)
∀R : set → (set → prop) → prop, ∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNo_upc R alpha p → PNoLe alpha p beta q → PNo_upc R beta q
In Proofgold the corresponding term root is 111b74... and proposition id is eb9ed5...
Proof:
Proof not loaded.
L8640
Definition. We define PNoLt_pwise to be λL R ⇒ ∀gamma, ordinal gamma → ∀p : set → prop, L gamma p → ∀delta, ordinal delta → ∀q : set → prop, R delta q → PNoLt gamma p delta q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → prop.
In Proofgold the corresponding term root is ae448b... and object id is 123808...
L8643
Theorem. (PNoLt_pwise_downc_upc)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → PNoLt_pwise (PNo_downc L) (PNo_upc R)
In Proofgold the corresponding term root is 2b8e3b... and proposition id is 2f244c...
Proof:
Proof not loaded.
L8684
Definition. We define PNo_rel_strict_upperbd to be λL alpha p ⇒ ∀beta ∈ alpha, ∀q : set → prop, PNo_downc L beta q → PNoLt beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is dc1665... and object id is 3e1663...
L8688
Definition. We define PNo_rel_strict_lowerbd to be λR alpha p ⇒ ∀beta ∈ alpha, ∀q : set → prop, PNo_upc R beta q → PNoLt alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is e5a4b6... and object id is ff3150...
L8691
Definition. We define PNo_rel_strict_imv to be λL R alpha p ⇒ PNo_rel_strict_upperbd L alpha p ∧ PNo_rel_strict_lowerbd R alpha p of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 64ce99... and object id is 5b35fb...
L8693
Theorem. (PNoEq_rel_strict_upperbd)
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_upperbd L alpha p → PNo_rel_strict_upperbd L alpha q
In Proofgold the corresponding term root is bb65cf... and proposition id is db4b82...
Proof:
Proof not loaded.
L8710
Theorem. (PNo_rel_strict_upperbd_antimon)
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_upperbd L alpha p → PNo_rel_strict_upperbd L beta p
In Proofgold the corresponding term root is 9986af... and proposition id is ed53da...
Proof:
Proof not loaded.
L8751
Theorem. (PNoEq_rel_strict_lowerbd)
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R alpha q
In Proofgold the corresponding term root is 76080e... and proposition id is 614e58...
Proof:
Proof not loaded.
L8768
Theorem. (PNo_rel_strict_lowerbd_antimon)
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R beta p
In Proofgold the corresponding term root is 27f253... and proposition id is 344b63...
Proof:
Proof not loaded.
L8809
Theorem. (PNoEq_rel_strict_imv)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R alpha q
In Proofgold the corresponding term root is e3f251... and proposition id is 7ae13d...
Proof:
Proof not loaded.
L8819
Theorem. (PNo_rel_strict_imv_antimon)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, ∀beta ∈ alpha, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R beta p
In Proofgold the corresponding term root is 2434dc... and proposition id is 87e0ce...
Proof:
Proof not loaded.
L8829
Definition. We define PNo_rel_strict_uniq_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R alpha p ∧ ∀q : set → prop, PNo_rel_strict_imv L R alpha q → PNoEq_ alpha p q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 23c9d9... and object id is 56ed1b...
L8832
Definition. We define PNo_rel_strict_split_imv to be λL R alpha p ⇒ PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∧ delta ≠ alpha) ∧ PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∨ delta = alpha) of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 217d17... and object id is dd9cee...
L8836
Theorem. (PNo_extend0_eq)
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∧ delta ≠ alpha)
In Proofgold the corresponding term root is 2d441e... and proposition id is 5ed08e...
Proof:
Proof not loaded.
L8853
Theorem. (PNo_extend1_eq)
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∨ delta = alpha)
In Proofgold the corresponding term root is 8eb99c... and proposition id is 4cac1a...
Proof:
Proof not loaded.
L8870
Theorem. (PNo_rel_imv_ex)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → (∃p : set → prop, PNo_rel_strict_uniq_imv L R alpha p) ∨ (∃tau ∈ alpha, ∃p : set → prop, PNo_rel_strict_split_imv L R tau p)
In Proofgold the corresponding term root is 3e3224... and proposition id is 749b63...
Proof:
Proof not loaded.
L9947
Definition. We define PNo_lenbdd to be λalpha L ⇒ ∀beta, ∀p : set → prop, L beta p → beta ∈ alpha of type set → (set → (set → prop) → prop) → prop.
In Proofgold the corresponding term root is 009fa7... and object id is a452a6...
L9950
Theorem. (PNo_lenbdd_strict_imv_extend0)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∧ delta ≠ alpha)
In Proofgold the corresponding term root is 17a114... and proposition id is 69327e...
Proof:
Proof not loaded.
L10087
Theorem. (PNo_lenbdd_strict_imv_extend1)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_imv L R (ordsucc alpha) (λdelta ⇒ p delta ∨ delta = alpha)
In Proofgold the corresponding term root is ad69ec... and proposition id is 5ff993...
Proof:
Proof not loaded.
L10229
Theorem. (PNo_lenbdd_strict_imv_split)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∀p : set → prop, PNo_rel_strict_imv L R alpha p → PNo_rel_strict_split_imv L R alpha p
In Proofgold the corresponding term root is 76ae4f... and proposition id is f98998...
Proof:
Proof not loaded.
L10245
Theorem. (PNo_rel_imv_bdd_ex)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta ∈ ordsucc alpha, ∃p : set → prop, PNo_rel_strict_split_imv L R beta p
In Proofgold the corresponding term root is 9a98e4... and proposition id is 25bcd2...
Proof:
Proof not loaded.
L10279
Definition. We define PNo_strict_upperbd to be λL alpha p ⇒ ∀beta, ordinal beta → ∀q : set → prop, L beta q → PNoLt beta q alpha p of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 6913bd... and object id is 3723ab...
L10283
Definition. We define PNo_strict_lowerbd to be λR alpha p ⇒ ∀beta, ordinal beta → ∀q : set → prop, R beta q → PNoLt alpha p beta q of type (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 87fac4... and object id is 3606d4...
L10286
Definition. We define PNo_strict_imv to be λL R alpha p ⇒ PNo_strict_upperbd L alpha p ∧ PNo_strict_lowerbd R alpha p of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is ee97f6... and object id is bc6c55...
L10288
Theorem. (PNoEq_strict_upperbd)
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_upperbd L alpha p → PNo_strict_upperbd L alpha q
In Proofgold the corresponding term root is 9b7b8e... and proposition id is f28b04...
Proof:
Proof not loaded.
L10303
Theorem. (PNoEq_strict_lowerbd)
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_lowerbd R alpha p → PNo_strict_lowerbd R alpha q
In Proofgold the corresponding term root is 4db3f9... and proposition id is 67fa73...
Proof:
Proof not loaded.
L10318
Theorem. (PNoEq_strict_imv)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p q : set → prop, PNoEq_ alpha p q → PNo_strict_imv L R alpha p → PNo_strict_imv L R alpha q
In Proofgold the corresponding term root is 1ec352... and proposition id is 5f582f...
Proof:
Proof not loaded.
L10328
Theorem. (PNo_strict_upperbd_imp_rel_strict_upperbd)
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_upperbd L alpha p → PNo_rel_strict_upperbd L beta p
In Proofgold the corresponding term root is 2898f0... and proposition id is 3cdd3b...
Proof:
Proof not loaded.
L10409
Theorem. (PNo_strict_lowerbd_imp_rel_strict_lowerbd)
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_lowerbd R alpha p → PNo_rel_strict_lowerbd R beta p
In Proofgold the corresponding term root is 442af5... and proposition id is 04acc2...
Proof:
Proof not loaded.
L10488
Theorem. (PNo_strict_imv_imp_rel_strict_imv)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀beta ∈ ordsucc alpha, ∀p : set → prop, PNo_strict_imv L R alpha p → PNo_rel_strict_imv L R beta p
In Proofgold the corresponding term root is f4fc71... and proposition id is 46a6aa...
Proof:
Proof not loaded.
L10507
Theorem. (PNo_rel_split_imv_imp_strict_imv)
∀L R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, PNo_rel_strict_split_imv L R alpha p → PNo_strict_imv L R alpha p
In Proofgold the corresponding term root is 72be36... and proposition id is adfdb0...
Proof:
Proof not loaded.
L10753
Theorem. (PNo_lenbdd_strict_imv_ex)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta ∈ ordsucc alpha, ∃p : set → prop, PNo_strict_imv L R beta p
In Proofgold the corresponding term root is 9485d6... and proposition id is a864a5...
Proof:
Proof not loaded.
L10779
Definition. We define PNo_least_rep to be λL R beta p ⇒ ordinal beta ∧ PNo_strict_imv L R beta p ∧ ∀gamma ∈ beta, ∀q : set → prop, ¬ PNo_strict_imv L R gamma q of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 0711e2... and object id is e4c707...
L10785
Definition. We define PNo_least_rep2 to be λL R beta p ⇒ PNo_least_rep L R beta p ∧ ∀x, x ∉ beta → ¬ p x of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
In Proofgold the corresponding term root is 1f6dc5... and object id is 09d320...
L10787
Theorem. (PNo_strict_imv_pred_eq)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → ∀p q : set → prop, PNo_least_rep L R alpha p → PNo_strict_imv L R alpha q → ∀beta ∈ alpha, p beta ↔ q beta
In Proofgold the corresponding term root is 5e7822... and proposition id is 72e14d...
Proof:
Proof not loaded.
L10883
Theorem. (PNo_lenbdd_least_rep2_exuniq2)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → ∃beta, (∃p : set → prop, PNo_least_rep2 L R beta p) ∧ (∀p q : set → prop, PNo_least_rep2 L R beta p → PNo_least_rep2 L R beta q → p = q)
In Proofgold the corresponding term root is 5060a9... and proposition id is 8b62c5...
Proof:
Proof not loaded.
L10971
Definition. We define PNo_bd to be λL R ⇒ DescrR_i_io_1 (PNo_least_rep2 L R) of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set.
In Proofgold the corresponding term root is ed76e7... and object id is aa75ef...
L10974
Definition. We define PNo_pred to be λL R ⇒ DescrR_i_io_2 (PNo_least_rep2 L R) of type (set → (set → prop) → prop) → (set → (set → prop) → prop) → set → prop.
In Proofgold the corresponding term root is b2d51d... and object id is 2efd18...
L10976
Theorem. (PNo_bd_pred_lem)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_least_rep2 L R (PNo_bd L R) (PNo_pred L R)
In Proofgold the corresponding term root is 749b8f... and proposition id is 5de4a3...
Proof:
Proof not loaded.
L10988
Theorem. (PNo_bd_pred)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_least_rep L R (PNo_bd L R) (PNo_pred L R)
In Proofgold the corresponding term root is 42224b... and proposition id is 3309f6...
Proof:
Proof not loaded.
L10999
Theorem. (PNo_bd_In)
∀L R : set → (set → prop) → prop, PNoLt_pwise L R → ∀alpha, ordinal alpha → PNo_lenbdd alpha L → PNo_lenbdd alpha R → PNo_bd L R ∈ ordsucc alpha
In Proofgold the corresponding term root is ad857b... and proposition id is 4719e1...
Proof:
Proof not loaded.
Beginning of Section TaggedSets
L11033
Let tag : set → set ≝ λalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
L11035
Theorem. (not_TransSet_Sing1)
¬ TransSet {1}
In Proofgold the corresponding term root is 8f2d67... and proposition id is be9a70...
Proof:
Proof not loaded.
L11043
Theorem. (not_ordinal_Sing1)
¬ ordinal {1}
In Proofgold the corresponding term root is 99a3f6... and proposition id is db7977...
Proof:
Proof not loaded.
L11047
Theorem. (tagged_not_ordinal)
∀y, ¬ ordinal (y ')
In Proofgold the corresponding term root is f9488c... and proposition id is aaf6e4...
Proof:
Proof not loaded.
L11059
Theorem. (tagged_notin_ordinal)
∀alpha y, ordinal alpha → (y ') ∉ alpha
In Proofgold the corresponding term root is b603ba... and proposition id is c0e2c3...
Proof:
Proof not loaded.
L11066
Theorem. (tagged_eqE_Subq)
∀alpha beta, ordinal alpha → alpha ' = beta ' → alpha ⊆ beta
In Proofgold the corresponding term root is cf032f... and proposition id is ef37f5...
Proof:
Proof not loaded.
L11087
Theorem. (tagged_eqE_eq)
∀alpha beta, ordinal alpha → ordinal beta → alpha ' = beta ' → alpha = beta
In Proofgold the corresponding term root is 1f7dc9... and proposition id is ec1608...
Proof:
Proof not loaded.
L11095
Theorem. (tagged_ReplE)
∀alpha beta, ordinal alpha → ordinal beta → beta ' ∈ {gamma '|gamma ∈ alpha} → beta ∈ alpha
In Proofgold the corresponding term root is a1d43c... and proposition id is 70c250...
Proof:
Proof not loaded.
L11108
Theorem. (ordinal_notin_tagged_Repl)
∀alpha Y, ordinal alpha → alpha ∉ {y '|y ∈ Y}
In Proofgold the corresponding term root is 7f35b5... and proposition id is e7ff3b...
Proof:
Proof not loaded.
L11122
Definition. We define SNoElts_ to be λalpha ⇒ alpha ∪ {beta '|beta ∈ alpha} of type set → set.
In Proofgold the corresponding term root is c0ec73... and object id is 6f0c36...
L11124
Theorem. (SNoElts_mon)
∀alpha beta, alpha ⊆ beta → SNoElts_ alpha ⊆ SNoElts_ beta
In Proofgold the corresponding term root is 49a08e... and proposition id is 859611...
Proof:
Proof not loaded.
L11148
Definition. We define SNo_ to be λalpha x ⇒ x ⊆ SNoElts_ alpha ∧ ∀beta ∈ alpha, exactly1of2 (beta ' ∈ x) (beta ∈ x) of type set → set → prop.
In Proofgold the corresponding term root is 4ab7e4... and object id is c16a73...
L11152
Definition. We define PSNo to be λalpha p ⇒ {beta ∈ alpha|p beta} ∪ {beta '|beta ∈ alpha, ¬ p beta} of type set → (set → prop) → set.
In Proofgold the corresponding term root is 3657ae... and object id is 0a8351...
L11154
Theorem. (PNoEq_PSNo)
∀alpha, ordinal alpha → ∀p : set → prop, PNoEq_ alpha (λbeta ⇒ beta ∈ PSNo alpha p) p
In Proofgold the corresponding term root is c12e02... and proposition id is 688518...
Proof:
Proof not loaded.
L11181
Theorem. (SNo_PSNo)
∀alpha, ordinal alpha → ∀p : set → prop, SNo_ alpha (PSNo alpha p)
In Proofgold the corresponding term root is 96e4e2... and proposition id is f03db0...
Proof:
Proof not loaded.
L11261
Theorem. (SNo_PSNo_eta_)
∀alpha x, ordinal alpha → SNo_ alpha x → x = PSNo alpha (λbeta ⇒ beta ∈ x)
In Proofgold the corresponding term root is 1642af... and proposition id is 630119...
Proof:
Proof not loaded.
L11328
Definition. We define SNo to be λx ⇒ ∃alpha, ordinal alpha ∧ SNo_ alpha x of type set → prop.
In Proofgold the corresponding term root is 11faa7... and object id is 116b76...
L11329
Theorem. (SNo_SNo)
∀alpha, ordinal alpha → ∀z, SNo_ alpha z → SNo z
In Proofgold the corresponding term root is 929b7a... and proposition id is a59341...
Proof:
Proof not loaded.
L11340
Definition. We define SNoLev to be λx ⇒ Eps_i (λalpha ⇒ ordinal alpha ∧ SNo_ alpha x) of type set → set.
In Proofgold the corresponding term root is 293b77... and object id is 83dce3...
L11341
Theorem. (SNoLev_uniq_Subq)
∀x alpha beta, ordinal alpha → ordinal beta → SNo_ alpha x → SNo_ beta x → alpha ⊆ beta
In Proofgold the corresponding term root is 2ca114... and proposition id is 65843d...
Proof:
Proof not loaded.
L11374
Theorem. (SNoLev_uniq)
∀x alpha beta, ordinal alpha → ordinal beta → SNo_ alpha x → SNo_ beta x → alpha = beta
In Proofgold the corresponding term root is 68f71d... and proposition id is 5d96d4...
Proof:
Proof not loaded.
L11381
Theorem. (SNoLev_prop)
∀x, SNo x → ordinal (SNoLev x) ∧ SNo_ (SNoLev x) x
In Proofgold the corresponding term root is a8afcc... and proposition id is 21761c...
Proof:
Proof not loaded.
L11387
Theorem. (SNoLev_ordinal)
∀x, SNo x → ordinal (SNoLev x)
In Proofgold the corresponding term root is 7cacc5... and proposition id is f0e9f7...
Proof:
Proof not loaded.
L11391
Theorem. (SNoLev_)
∀x, SNo x → SNo_ (SNoLev x) x
In Proofgold the corresponding term root is 4a1033... and proposition id is 49cd5a...
Proof:
Proof not loaded.
L11395
Theorem. (SNo_PSNo_eta)
∀x, SNo x → x = PSNo (SNoLev x) (λbeta ⇒ beta ∈ x)
In Proofgold the corresponding term root is 6ffefe... and proposition id is 8c7870...
Proof:
Proof not loaded.
L11405
Theorem. (SNoLev_PSNo)
∀alpha, ordinal alpha → ∀p : set → prop, SNoLev (PSNo alpha p) = alpha
In Proofgold the corresponding term root is 07f1eb... and proposition id is 9c4890...
Proof:
Proof not loaded.
L11422
Theorem. (SNo_Subq)
∀x y, SNo x → SNo y → SNoLev x ⊆ SNoLev y → (∀alpha ∈ SNoLev x, alpha ∈ x ↔ alpha ∈ y) → x ⊆ y
In Proofgold the corresponding term root is a244d7... and proposition id is a428af...
Proof:
Proof not loaded.
L11467
Definition. We define SNoEq_ to be λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) of type set → set → set → prop.
In Proofgold the corresponding term root is 5f11e2... and object id is 12b5f6...
L11470
Theorem. (SNoEq_I)
∀alpha x y, (∀beta ∈ alpha, beta ∈ x ↔ beta ∈ y) → SNoEq_ alpha x y
In Proofgold the corresponding term root is a718b1... and proposition id is fdcb78...
Proof:
Proof not loaded.
L11474
Theorem. (SNo_eq)
∀x y, SNo x → SNo y → SNoLev x = SNoLev y → SNoEq_ (SNoLev x) x y → x = y
In Proofgold the corresponding term root is b08bdc... and proposition id is bf4ac3...
Proof:
Proof not loaded.
End of Section TaggedSets
L11494
Definition. We define SNoLt to be λx y ⇒ PNoLt (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y) of type set → set → prop.
In Proofgold the corresponding term root is 46e7ed... and object id is 8c8aa9...
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
L11497
Definition. We define SNoLe to be λx y ⇒ PNoLe (SNoLev x) (λbeta ⇒ beta ∈ x) (SNoLev y) (λbeta ⇒ beta ∈ y) of type set → set → prop.
In Proofgold the corresponding term root is ddf7d3... and object id is 54f405...
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
L11501
Theorem. (SNoLtLe)
∀x y, x < y → x ≤ y
In Proofgold the corresponding term root is c0fef1... and proposition id is f09833...
Proof:
Proof not loaded.
L11508
Theorem. (SNoLeE)
∀x y, SNo x → SNo y → x ≤ y → x < y ∨ x = y
In Proofgold the corresponding term root is 1c9c62... and proposition id is 23cc47...
Proof:
Proof not loaded.
L11522
Theorem. (SNoEq_sym_)
∀alpha x y, SNoEq_ alpha x y → SNoEq_ alpha y x
In Proofgold the corresponding term root is f8c61d... and proposition id is f756d2...
Proof:
Proof not loaded.
L11527
Theorem. (SNoEq_tra_)
∀alpha x y z, SNoEq_ alpha x y → SNoEq_ alpha y z → SNoEq_ alpha x z
In Proofgold the corresponding term root is 0fff96... and proposition id is 246d04...
Proof:
Proof not loaded.
L11532
Theorem. (SNoLtE)
∀x y, SNo x → SNo y → x < y → ∀p : prop, (∀z, SNo z → SNoLev z ∈ SNoLev x ∩ SNoLev y → SNoEq_ (SNoLev z) z x → SNoEq_ (SNoLev z) z y → x < z → z < y → SNoLev z ∉ x → SNoLev z ∈ y → p) → (SNoLev x ∈ SNoLev y → SNoEq_ (SNoLev x) x y → SNoLev x ∈ y → p) → (SNoLev y ∈ SNoLev x → SNoEq_ (SNoLev y) x y → SNoLev y ∉ x → p) → p
In Proofgold the corresponding term root is 958711... and proposition id is d12fc6...
Proof:
Proof not loaded.
L11635
Theorem. (SNoLtI2)
∀x y, SNoLev x ∈ SNoLev y → SNoEq_ (SNoLev x) x y → SNoLev x ∈ y → x < y
In Proofgold the corresponding term root is 282b01... and proposition id is 82b82c...
Proof:
Proof not loaded.
L11650
Theorem. (SNoLtI3)
∀x y, SNoLev y ∈ SNoLev x → SNoEq_ (SNoLev y) x y → SNoLev y ∉ x → x < y
In Proofgold the corresponding term root is 006afc... and proposition id is 938f67...
Proof:
Proof not loaded.
L11664
Theorem. (SNoLt_irref)
∀x, ¬ SNoLt x x
In Proofgold the corresponding term root is d5ecf9... and proposition id is 06fac5...
Proof:
Proof not loaded.
L11669
Theorem. (SNoLt_trichotomy_or)
∀x y, SNo x → SNo y → x < y ∨ x = y ∨ y < x
In Proofgold the corresponding term root is 35a4c8... and proposition id is 8ad056...
Proof:
Proof not loaded.
L11680
Theorem. (SNoLt_trichotomy_or_impred)
∀x y, SNo x → SNo y → ∀p : prop, (x < y → p) → (x = y → p) → (y < x → p) → p
In Proofgold the corresponding term root is ba04ef... and proposition id is 7588e3...
Proof:
Proof not loaded.
L11693
Theorem. (SNoLt_tra)
∀x y z, SNo x → SNo y → SNo z → x < y → y < z → x < z
In Proofgold the corresponding term root is ac0e24... and proposition id is ccfdf7...
Proof:
Proof not loaded.
L11699
Theorem. (SNoLe_ref)
∀x, SNoLe x x
In Proofgold the corresponding term root is c940c8... and proposition id is 83bd2d...
Proof:
Proof not loaded.
L11703
Theorem. (SNoLe_antisym)
∀x y, SNo x → SNo y → x ≤ y → y ≤ x → x = y
In Proofgold the corresponding term root is f9f757... and proposition id is 3abee4...
Proof:
Proof not loaded.
L11711
Theorem. (SNoLtLe_tra)
∀x y z, SNo x → SNo y → SNo z → x < y → y ≤ z → x < z
In Proofgold the corresponding term root is 3bc661... and proposition id is ba8545...
Proof:
Proof not loaded.
L11716
Theorem. (SNoLeLt_tra)
∀x y z, SNo x → SNo y → SNo z → x ≤ y → y < z → x < z
In Proofgold the corresponding term root is b2178f... and proposition id is 4a77a2...
Proof:
Proof not loaded.
L11721
Theorem. (SNoLe_tra)
∀x y z, SNo x → SNo y → SNo z → x ≤ y → y ≤ z → x ≤ z
In Proofgold the corresponding term root is a0c5c6... and proposition id is 031989...
Proof:
Proof not loaded.
L11726
Theorem. (SNoLtLe_or)
∀x y, SNo x → SNo y → x < y ∨ y ≤ x
In Proofgold the corresponding term root is f595ac... and proposition id is 8965d2...
Proof:
Proof not loaded.
L11735
Theorem. (SNoLt_PSNo_PNoLt)
∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PSNo alpha p < PSNo beta q → PNoLt alpha p beta q
In Proofgold the corresponding term root is 6e122d... and proposition id is 535161...
Proof:
Proof not loaded.
L11758
Theorem. (PNoLt_SNoLt_PSNo)
∀alpha beta, ∀p q : set → prop, ordinal alpha → ordinal beta → PNoLt alpha p beta q → PSNo alpha p < PSNo beta q
In Proofgold the corresponding term root is 11b6ed... and proposition id is 6196d9...
Proof:
Proof not loaded.
L11781
Definition. We define SNoCut to be λL R ⇒ PSNo (PNo_bd (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ L) (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ R)) (PNo_pred (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ L) (λalpha p ⇒ ordinal alpha ∧ PSNo alpha p ∈ R)) of type set → set → set.
In Proofgold the corresponding term root is ec849e... and object id is 295bc4...
L11784
Definition. We define SNoCutP to be λL R ⇒ (∀x ∈ L, SNo x) ∧ (∀y ∈ R, SNo y) ∧ (∀x ∈ L, ∀y ∈ R, x < y) of type set → set → prop.
In Proofgold the corresponding term root is c083d8... and object id is 775127...
L11789
Theorem. (SNoCutP_SNoCut)
∀L R, SNoCutP L R → SNo (SNoCut L R) ∧ SNoLev (SNoCut L R) ∈ ordsucc ((⋃x ∈ Lordsucc (SNoLev x)) ∪ (⋃y ∈ Rordsucc (SNoLev y))) ∧ (∀x ∈ L, x < SNoCut L R) ∧ (∀y ∈ R, SNoCut L R < y) ∧ (∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z)
In Proofgold the corresponding term root is 9398b7... and proposition id is 66456c...
Proof:
Proof not loaded.
L12178
Theorem. (SNoCutP_SNoCut_impred)
∀L R, SNoCutP L R → ∀p : prop, (SNo (SNoCut L R) → SNoLev (SNoCut L R) ∈ ordsucc ((⋃x ∈ Lordsucc (SNoLev x)) ∪ (⋃y ∈ Rordsucc (SNoLev y))) → (∀x ∈ L, x < SNoCut L R) → (∀y ∈ R, SNoCut L R < y) → (∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z) → p) → p
In Proofgold the corresponding term root is 02c4ae... and proposition id is 5bcc8f...
Proof:
Proof not loaded.
L12193
Theorem. (SNoCutP_L_0)
∀L, (∀x ∈ L, SNo x) → SNoCutP L 0
In Proofgold the corresponding term root is 47329c... and proposition id is ba61e1...
Proof:
Proof not loaded.
L12204
Theorem. (SNoCutP_0_0)
SNoCutP 0 0
In Proofgold the corresponding term root is 3b144e... and proposition id is a5144d...
Proof:
Proof not loaded.
L12208
Definition. We define SNoS_ to be λalpha ⇒ {x ∈ 𝒫 (SNoElts_ alpha)|∃beta ∈ alpha, SNo_ beta x} of type set → set.
In Proofgold the corresponding term root is d5069a... and object id is 4680ce...
L12210
Theorem. (SNoS_E)
∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, ∃beta ∈ alpha, SNo_ beta x
In Proofgold the corresponding term root is 970ce2... and proposition id is bb731c...
Proof:
Proof not loaded.
Beginning of Section TaggedSets2
L12220
Let tag : set → set ≝ λalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
L12222
Theorem. (SNoS_I)
∀alpha, ordinal alpha → ∀x, ∀beta ∈ alpha, SNo_ beta x → x ∈ SNoS_ alpha
In Proofgold the corresponding term root is f5227d... and proposition id is 04f24b...
Proof:
Proof not loaded.
L12247
Theorem. (SNoS_I2)
∀x y, SNo x → SNo y → SNoLev x ∈ SNoLev y → x ∈ SNoS_ (SNoLev y)
In Proofgold the corresponding term root is 7a14c7... and proposition id is b4042d...
Proof:
Proof not loaded.
L12253
Theorem. (SNoS_Subq)
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta → SNoS_ alpha ⊆ SNoS_ beta
In Proofgold the corresponding term root is 6828e7... and proposition id is d928b8...
Proof:
Proof not loaded.
L12263
Theorem. (SNoLev_uniq2)
∀alpha, ordinal alpha → ∀x, SNo_ alpha x → SNoLev x = alpha
In Proofgold the corresponding term root is fbb461... and proposition id is f48d47...
Proof:
Proof not loaded.
L12276
Theorem. (SNoS_E2)
∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, ∀p : prop, (SNoLev x ∈ alpha → ordinal (SNoLev x) → SNo x → SNo_ (SNoLev x) x → p) → p
In Proofgold the corresponding term root is deb031... and proposition id is 62b490...
Proof:
Proof not loaded.
L12300
Theorem. (SNoS_In_neq)
∀w, SNo w → ∀x ∈ SNoS_ (SNoLev w), x ≠ w
In Proofgold the corresponding term root is ab85fd... and proposition id is d5dbd7...
Proof:
Proof not loaded.
L12316
Theorem. (SNoS_SNoLev)
∀z, SNo z → z ∈ SNoS_ (ordsucc (SNoLev z))
In Proofgold the corresponding term root is a9ba1f... and proposition id is 8efb88...
Proof:
Proof not loaded.
L12324
Definition. We define SNoL to be λz ⇒ {x ∈ SNoS_ (SNoLev z)|x < z} of type set → set.
In Proofgold the corresponding term root is 8cd812... and object id is 48bcf8...
L12326
Definition. We define SNoR to be λz ⇒ {y ∈ SNoS_ (SNoLev z)|z < y} of type set → set.
In Proofgold the corresponding term root is 73b2b9... and object id is 0cfe87...
L12327
Theorem. (SNoCutP_SNoL_SNoR)
∀z, SNo z → SNoCutP (SNoL z) (SNoR z)
In Proofgold the corresponding term root is 524866... and proposition id is 7aba57...
Proof:
Proof not loaded.
L12375
Theorem. (SNoL_E)
∀x, SNo x → ∀w ∈ SNoL x, ∀p : prop, (SNo w → SNoLev w ∈ SNoLev x → w < x → p) → p
In Proofgold the corresponding term root is cead4f... and proposition id is 7ccd8b...
Proof:
Proof not loaded.
L12393
Theorem. (SNoR_E)
∀x, SNo x → ∀z ∈ SNoR x, ∀p : prop, (SNo z → SNoLev z ∈ SNoLev x → x < z → p) → p
In Proofgold the corresponding term root is efca05... and proposition id is 7dca04...
Proof:
Proof not loaded.
L12411
Theorem. (SNoL_SNoS_)
∀z, SNoL z ⊆ SNoS_ (SNoLev z)
In Proofgold the corresponding term root is 73f5b2... and proposition id is 409713...
Proof:
Proof not loaded.
L12415
Theorem. (SNoR_SNoS_)
∀z, SNoR z ⊆ SNoS_ (SNoLev z)
In Proofgold the corresponding term root is 16bdfc... and proposition id is 2fe1c0...
Proof:
Proof not loaded.
L12419
Theorem. (SNoL_SNoS)
∀x, SNo x → ∀w ∈ SNoL x, w ∈ SNoS_ (SNoLev x)
In Proofgold the corresponding term root is 6e3a22... and proposition id is ded78a...
Proof:
Proof not loaded.
L12429
Theorem. (SNoR_SNoS)
∀x, SNo x → ∀z ∈ SNoR x, z ∈ SNoS_ (SNoLev x)
In Proofgold the corresponding term root is 98051c... and proposition id is e13f49...
Proof:
Proof not loaded.
L12439
Theorem. (SNoL_I)
∀z, SNo z → ∀x, SNo x → SNoLev x ∈ SNoLev z → x < z → x ∈ SNoL z
In Proofgold the corresponding term root is ddd91b... and proposition id is 4f3d04...
Proof:
Proof not loaded.
L12449
Theorem. (SNoR_I)
∀z, SNo z → ∀y, SNo y → SNoLev y ∈ SNoLev z → z < y → y ∈ SNoR z
In Proofgold the corresponding term root is 97693c... and proposition id is 383e88...
Proof:
Proof not loaded.
L12459
Theorem. (SNo_eta)
∀z, SNo z → z = SNoCut (SNoL z) (SNoR z)
In Proofgold the corresponding term root is 3cc369... and proposition id is dc9273...
Proof:
Proof not loaded.
L12533
Theorem. (SNoCutP_SNo_SNoCut)
∀L R, SNoCutP L R → SNo (SNoCut L R)
In Proofgold the corresponding term root is 9621aa... and proposition id is ae46e8...
Proof:
Proof not loaded.
L12542
Theorem. (SNoCutP_SNoCut_L)
∀L R, SNoCutP L R → ∀x ∈ L, x < SNoCut L R
In Proofgold the corresponding term root is 984a30... and proposition id is 4bca14...
Proof:
Proof not loaded.
L12550
Theorem. (SNoCutP_SNoCut_R)
∀L R, SNoCutP L R → ∀y ∈ R, SNoCut L R < y
In Proofgold the corresponding term root is 9161db... and proposition id is 7c3d12...
Proof:
Proof not loaded.
L12557
Theorem. (SNoCutP_SNoCut_fst)
∀L R, SNoCutP L R → ∀z, SNo z → (∀x ∈ L, x < z) → (∀y ∈ R, z < y) → SNoLev (SNoCut L R) ⊆ SNoLev z ∧ SNoEq_ (SNoLev (SNoCut L R)) (SNoCut L R) z
In Proofgold the corresponding term root is b52eb7... and proposition id is 8362f8...
Proof:
Proof not loaded.
L12569
Theorem. (SNoCut_Le)
∀L1 R1 L2 R2, SNoCutP L1 R1 → SNoCutP L2 R2 → (∀w ∈ L1, w < SNoCut L2 R2) → (∀z ∈ R2, SNoCut L1 R1 < z) → SNoCut L1 R1 ≤ SNoCut L2 R2
In Proofgold the corresponding term root is bb8cdc... and proposition id is 423731...
Proof:
Proof not loaded.
L12672
Theorem. (SNoCut_ext)
∀L1 R1 L2 R2, SNoCutP L1 R1 → SNoCutP L2 R2 → (∀w ∈ L1, w < SNoCut L2 R2) → (∀z ∈ R1, SNoCut L2 R2 < z) → (∀w ∈ L2, w < SNoCut L1 R1) → (∀z ∈ R2, SNoCut L1 R1 < z) → SNoCut L1 R1 = SNoCut L2 R2
In Proofgold the corresponding term root is c1078e... and proposition id is f66618...
Proof:
Proof not loaded.
L12695
Theorem. (SNoLt_SNoL_or_SNoR_impred)
∀x y, SNo x → SNo y → x < y → ∀p : prop, (∀z ∈ SNoL y, z ∈ SNoR x → p) → (x ∈ SNoL y → p) → (y ∈ SNoR x → p) → p
In Proofgold the corresponding term root is f3b15a... and proposition id is ffc0bf...
Proof:
Proof not loaded.
L12717
Theorem. (SNoL_or_SNoR_impred)
∀x y, SNo x → SNo y → ∀p : prop, (x = y → p) → (∀z ∈ SNoL y, z ∈ SNoR x → p) → (x ∈ SNoL y → p) → (y ∈ SNoR x → p) → (∀z ∈ SNoR y, z ∈ SNoL x → p) → (x ∈ SNoR y → p) → (y ∈ SNoL x → p) → p
In Proofgold the corresponding term root is f8a5e2... and proposition id is 68ba98...
Proof:
Proof not loaded.
L12743
Theorem. (SNoL_SNoCutP_ex)
∀L R, SNoCutP L R → ∀w ∈ SNoL (SNoCut L R), ∃w' ∈ L, w ≤ w'
In Proofgold the corresponding term root is 3b4695... and proposition id is 6d8af6...
Proof:
Proof not loaded.
L12783
Theorem. (SNoR_SNoCutP_ex)
∀L R, SNoCutP L R → ∀z ∈ SNoR (SNoCut L R), ∃z' ∈ R, z' ≤ z
In Proofgold the corresponding term root is ca7be2... and proposition id is bb0b53...
Proof:
Proof not loaded.
L12823
Theorem. (ordinal_SNo_)
∀alpha, ordinal alpha → SNo_ alpha alpha
In Proofgold the corresponding term root is 4731fd... and proposition id is 7754e5...
Proof:
Proof not loaded.
L12846
Theorem. (ordinal_SNo)
∀alpha, ordinal alpha → SNo alpha
In Proofgold the corresponding term root is ac39bc... and proposition id is 1eea33...
Proof:
Proof not loaded.
L12855
Theorem. (ordinal_SNoLev)
∀alpha, ordinal alpha → SNoLev alpha = alpha
In Proofgold the corresponding term root is 56f8b0... and proposition id is fea5fc...
Proof:
Proof not loaded.
L12864
Theorem. (ordinal_SNoLev_max)
∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ alpha → z < alpha
In Proofgold the corresponding term root is 945794... and proposition id is 51ee47...
Proof:
Proof not loaded.
L12911
Theorem. (ordinal_SNoL)
∀alpha, ordinal alpha → SNoL alpha = SNoS_ alpha
In Proofgold the corresponding term root is fef608... and proposition id is 95823e...
Proof:
Proof not loaded.
L12943
Theorem. (ordinal_SNoR)
∀alpha, ordinal alpha → SNoR alpha = Empty
In Proofgold the corresponding term root is 07c69a... and proposition id is f853d9...
Proof:
Proof not loaded.
L12965
Theorem. (nat_p_SNo)
∀n, nat_p n → SNo n
In Proofgold the corresponding term root is 0644f3... and proposition id is 5fd930...
Proof:
Proof not loaded.
L12973
Theorem. (omega_SNo)
∀n ∈ ω, SNo n
In Proofgold the corresponding term root is d32269... and proposition id is d0f9c6...
Proof:
Proof not loaded.
L12980
Theorem. (omega_SNoS_omega)
ω ⊆ SNoS_ ω
In Proofgold the corresponding term root is 4ba265... and proposition id is 2ccf7a...
Proof:
Proof not loaded.
L12988
Theorem. (ordinal_In_SNoLt)
∀alpha, ordinal alpha → ∀beta ∈ alpha, beta < alpha
In Proofgold the corresponding term root is e37d00... and proposition id is ced78e...
Proof:
Proof not loaded.
L13004
Theorem. (ordinal_SNoLev_max_2)
∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ ordsucc alpha → z ≤ alpha
In Proofgold the corresponding term root is 72c5eb... and proposition id is c9a731...
Proof:
Proof not loaded.
L13083
Theorem. (ordinal_Subq_SNoLe)
∀alpha beta, ordinal alpha → ordinal beta → alpha ⊆ beta → alpha ≤ beta
In Proofgold the corresponding term root is 7558ba... and proposition id is 15c58a...
Proof:
Proof not loaded.
L13105
Theorem. (ordinal_SNoLt_In)
∀alpha beta, ordinal alpha → ordinal beta → alpha < beta → alpha ∈ beta
In Proofgold the corresponding term root is d6f066... and proposition id is de775b...
Proof:
Proof not loaded.
L13118
Theorem. (omega_nonneg)
∀m ∈ ω, 0 ≤ m
In Proofgold the corresponding term root is ee67c9... and proposition id is eaf06b...
Proof:
Proof not loaded.
L13124
Theorem. (SNo_0)
SNo 0
In Proofgold the corresponding term root is c8204b... and proposition id is 94a310...
Proof:
Proof not loaded.
L13128
Theorem. (SNo_1)
SNo 1
In Proofgold the corresponding term root is ac50d7... and proposition id is 9a733a...
Proof:
Proof not loaded.
L13132
Theorem. (SNo_2)
SNo 2
In Proofgold the corresponding term root is 487591... and proposition id is c524a4...
Proof:
Proof not loaded.
L13136
Theorem. (SNoLev_0)
SNoLev 0 = 0
In Proofgold the corresponding term root is 45a0f7... and proposition id is 3527a4...
Proof:
Proof not loaded.
L13140
Theorem. (SNoCut_0_0)
SNoCut 0 0 = 0
In Proofgold the corresponding term root is 6a1f81... and proposition id is abbffb...
Proof:
Proof not loaded.
L13165
Theorem. (SNoL_0)
SNoL 0 = 0
In Proofgold the corresponding term root is c01c2b... and proposition id is fdaa69...
Proof:
Proof not loaded.
L13180
Theorem. (SNoR_0)
SNoR 0 = 0
In Proofgold the corresponding term root is 7329c7... and proposition id is 510751...
Proof:
Proof not loaded.
L13195
Theorem. (SNoL_1)
SNoL 1 = 1
In Proofgold the corresponding term root is 84d688... and proposition id is 15aa79...
Proof:
Proof not loaded.
L13231
Theorem. (SNoR_1)
SNoR 1 = 0
In Proofgold the corresponding term root is e6a83a... and proposition id is 8da427...
Proof:
Proof not loaded.
L13235
Theorem. (SNo_max_SNoLev)
∀x, SNo x → (∀y ∈ SNoS_ (SNoLev x), y < x) → SNoLev x = x
In Proofgold the corresponding term root is ff019e... and proposition id is 3be246...
Proof:
Proof not loaded.
L13282
Theorem. (SNo_max_ordinal)
∀x, SNo x → (∀y ∈ SNoS_ (SNoLev x), y < x) → ordinal x
In Proofgold the corresponding term root is 0b230c... and proposition id is f28b9a...
Proof:
Proof not loaded.
L13291
Theorem. (pos_low_eq_one)
∀x, SNo x → 0 < x → SNoLev x ⊆ 1 → x = 1
In Proofgold the corresponding term root is 7c84ef... and proposition id is db40ad...
Proof:
Proof not loaded.
L13338
Definition. We define SNo_extend0 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta ∈ x ∧ delta ≠ SNoLev x) of type set → set.
In Proofgold the corresponding term root is 997d91... and object id is ff4161...
L13340
Definition. We define SNo_extend1 to be λx ⇒ PSNo (ordsucc (SNoLev x)) (λdelta ⇒ delta ∈ x ∨ delta = SNoLev x) of type set → set.
In Proofgold the corresponding term root is 464e47... and object id is 8c551b...
L13341
Theorem. (SNo_extend0_SNo_)
∀x, SNo x → SNo_ (ordsucc (SNoLev x)) (SNo_extend0 x)
In Proofgold the corresponding term root is d41d3e... and proposition id is c8f8ea...
Proof:
Proof not loaded.
L13349
Theorem. (SNo_extend1_SNo_)
∀x, SNo x → SNo_ (ordsucc (SNoLev x)) (SNo_extend1 x)
In Proofgold the corresponding term root is b7b866... and proposition id is 73bbf2...
Proof:
Proof not loaded.
L13357
Theorem. (SNo_extend0_SNo)
∀x, SNo x → SNo (SNo_extend0 x)
In Proofgold the corresponding term root is b353a2... and proposition id is 76562c...
Proof:
Proof not loaded.
L13364
Theorem. (SNo_extend1_SNo)
∀x, SNo x → SNo (SNo_extend1 x)
In Proofgold the corresponding term root is fa323a... and proposition id is 5345a7...
Proof:
Proof not loaded.
L13371
Theorem. (SNo_extend0_SNoLev)
∀x, SNo x → SNoLev (SNo_extend0 x) = ordsucc (SNoLev x)
In Proofgold the corresponding term root is 26efcd... and proposition id is 5aa0af...
Proof:
Proof not loaded.
L13378
Theorem. (SNo_extend1_SNoLev)
∀x, SNo x → SNoLev (SNo_extend1 x) = ordsucc (SNoLev x)
In Proofgold the corresponding term root is 33cf60... and proposition id is 529347...
Proof:
Proof not loaded.
L13385
Theorem. (SNo_extend0_nIn)
∀x, SNo x → SNoLev x ∉ SNo_extend0 x
In Proofgold the corresponding term root is 217cff... and proposition id is 0dfd17...
Proof:
Proof not loaded.
L13408
Theorem. (SNo_extend1_In)
∀x, SNo x → SNoLev x ∈ SNo_extend1 x
In Proofgold the corresponding term root is ac9570... and proposition id is 8ca372...
Proof:
Proof not loaded.
L13420
Theorem. (SNo_extend0_SNoEq)
∀x, SNo x → SNoEq_ (SNoLev x) (SNo_extend0 x) x
In Proofgold the corresponding term root is bd4a27... and proposition id is 129986...
Proof:
Proof not loaded.
L13444
Theorem. (SNo_extend1_SNoEq)
∀x, SNo x → SNoEq_ (SNoLev x) (SNo_extend1 x) x
In Proofgold the corresponding term root is ba974f... and proposition id is d6f8bb...
Proof:
Proof not loaded.
L13468
Theorem. (SNoLev_0_eq_0)
∀x, SNo x → SNoLev x = 0 → x = 0
In Proofgold the corresponding term root is fe0bc4... and proposition id is dfcf39...
Proof:
Proof not loaded.
L13478
Definition. We define eps_ to be λn ⇒ {0} ∪ {(ordsucc m) '|m ∈ n} of type set → set.
In Proofgold the corresponding term root is 5e992a... and object id is ffee2b...
L13481
Theorem. (eps_ordinal_In_eq_0)
∀n alpha, ordinal alpha → alpha ∈ eps_ n → alpha = 0
In Proofgold the corresponding term root is 3046fd... and proposition id is 1190f3...
Proof:
Proof not loaded.
(*** eps_ n is the Surreal Number 1/2^n, without needing to define division or exponents first ***)
L13496
Theorem. (eps_0_1)
eps_ 0 = 1
In Proofgold the corresponding term root is 5bbce1... and proposition id is d073f0...
Proof:
Proof not loaded.
L13511
Theorem. (SNo__eps_)
∀n ∈ ω, SNo_ (ordsucc n) (eps_ n)
In Proofgold the corresponding term root is e9c4d9... and proposition id is 1c1428...
Proof:
Proof not loaded.
L13609
Theorem. (SNo_eps_)
∀n ∈ ω, SNo (eps_ n)
In Proofgold the corresponding term root is 9fb700... and proposition id is 006b0f...
Proof:
Proof not loaded.
L13616
Theorem. (SNo_eps_1)
SNo (eps_ 1)
In Proofgold the corresponding term root is 17c276... and proposition id is 8b78d7...
Proof:
Proof not loaded.
L13620
Theorem. (SNoLev_eps_)
∀n ∈ ω, SNoLev (eps_ n) = ordsucc n
In Proofgold the corresponding term root is 6891e3... and proposition id is 75efd1...
Proof:
Proof not loaded.
L13627
Theorem. (SNo_eps_SNoS_omega)
∀n ∈ ω, eps_ n ∈ SNoS_ ω
In Proofgold the corresponding term root is 736d2a... and proposition id is 52da18...
Proof:
Proof not loaded.
L13634
Theorem. (SNo_eps_decr)
∀n ∈ ω, ∀m ∈ n, eps_ n < eps_ m
In Proofgold the corresponding term root is d6a46a... and proposition id is 67c76a...
Proof:
Proof not loaded.
L13672
Theorem. (SNo_eps_pos)
∀n ∈ ω, 0 < eps_ n
In Proofgold the corresponding term root is 8b75de... and proposition id is 760b4a...
Proof:
Proof not loaded.
L13689
Theorem. (SNo_pos_eps_Lt)
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc n), 0 < x → eps_ n < x
In Proofgold the corresponding term root is 72974f... and proposition id is 771b0e...
Proof:
Proof not loaded.
L13736
Theorem. (SNo_pos_eps_Le)
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc (ordsucc n)), 0 < x → eps_ n ≤ x
In Proofgold the corresponding term root is f1e297... and proposition id is 6658de...
Proof:
Proof not loaded.
L13782
Theorem. (eps_SNo_eq)
∀n, nat_p n → ∀x ∈ SNoS_ (ordsucc n), 0 < x → SNoEq_ (SNoLev x) (eps_ n) x → ∃m ∈ n, x = eps_ m
In Proofgold the corresponding term root is 8c2587... and proposition id is 8515ac...
Proof:
Proof not loaded.
L13840
Theorem. (eps_SNoCutP)
∀n ∈ ω, SNoCutP {0} {eps_ m|m ∈ n}
In Proofgold the corresponding term root is 545873... and proposition id is 30edf9...
Proof:
Proof not loaded.
L13867
Theorem. (eps_SNoCut)
∀n ∈ ω, eps_ n = SNoCut {0} {eps_ m|m ∈ n}
In Proofgold the corresponding term root is 06d6c5... and proposition id is eba0c6...
Proof:
Proof not loaded.
End of Section TaggedSets2
L14047
Theorem. (SNo_etaE)
∀z, SNo z → ∀p : prop, (∀L R, SNoCutP L R → (∀x ∈ L, SNoLev x ∈ SNoLev z) → (∀y ∈ R, SNoLev y ∈ SNoLev z) → z = SNoCut L R → p) → p
In Proofgold the corresponding term root is c24473... and proposition id is 109f1d...
Proof:
Proof not loaded.
L14158
Theorem. (SNo_ind)
∀P : set → prop, (∀L R, SNoCutP L R → (∀x ∈ L, P x) → (∀y ∈ R, P y) → P (SNoCut L R)) → ∀z, SNo z → P z
In Proofgold the corresponding term root is 5b4c39... and proposition id is 94cbe6...
Proof:
Proof not loaded.
Beginning of Section SurrealRecI
L14211
Variable F : set → (set → set) → set
(*** surreal recursion ***)
L14212
Let default : set ≝ Eps_i (λ_ ⇒ True)
L14213
Let G : set → (set → set → set) → set → set ≝ λalpha g ⇒ If_ii (ordinal alpha) (λz : set ⇒ if z ∈ SNoS_ (ordsucc alpha) then F z (λw ⇒ g (SNoLev w) w) else default) (λz : set ⇒ default)
L14223
Definition. We define SNo_rec_i to be λz ⇒ In_rec_ii G (SNoLev z) z of type set → set.
In Proofgold the corresponding term root is be45df... and object id is 230ba8...
L14225
Hypothesis Fr : ∀z, SNo z → ∀g h : set → set, (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
L14228
Theorem. (SNo_rec_i_eq)
∀z, SNo z → SNo_rec_i z = F z SNo_rec_i
In Proofgold the corresponding term root is d5ffa8... and proposition id is f547c8...
Proof:
Proof not loaded.
End of Section SurrealRecI
Beginning of Section SurrealRecII
L14305
Variable F : set → (set → (set → set)) → (set → set)
L14306
Let default : (set → set) ≝ Descr_ii (λ_ ⇒ True)
L14307
Let G : set → (set → set → (set → set)) → set → (set → set) ≝ λalpha g ⇒ If_iii (ordinal alpha) (λz : set ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) (λz : set ⇒ default)
L14316
Definition. We define SNo_rec_ii to be λz ⇒ In_rec_iii G (SNoLev z) z of type set → (set → set).
In Proofgold the corresponding term root is e148e6... and object id is c3f218...
L14318
Hypothesis Fr : ∀z, SNo z → ∀g h : set → (set → set), (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
L14321
Theorem. (SNo_rec_ii_eq)
∀z, SNo z → SNo_rec_ii z = F z SNo_rec_ii
In Proofgold the corresponding term root is 39e866... and proposition id is 1c5373...
Proof:
Proof not loaded.
End of Section SurrealRecII
Beginning of Section SurrealRec2
L14398
Variable F : set → set → (set → set → set) → set
L14399
Let G : set → (set → set → set) → set → (set → set) → set ≝ λw f z g ⇒ F w z (λx y ⇒ if x = w then g y else f x y)
L14401
Let H : set → (set → set → set) → set → set ≝ λw f z ⇒ if SNo z then SNo_rec_i (G w f) z else Empty
L14404
Definition. We define SNo_rec2 to be SNo_rec_ii H of type set → set → set.
In Proofgold the corresponding term root is 7d10ab... and object id is 951110...
L14406
Hypothesis Fr : ∀w, SNo w → ∀z, SNo z → ∀g h : set → set → set, (∀x ∈ SNoS_ (SNoLev w), ∀y, SNo y → g x y = h x y) → (∀y ∈ SNoS_ (SNoLev z), g w y = h w y) → F w z g = F w z h
L14411
Theorem. (SNo_rec2_G_prop)
∀w, SNo w → ∀f k : set → set → set, (∀x ∈ SNoS_ (SNoLev w), f x = k x) → ∀z, SNo z → ∀g h : set → set, (∀u ∈ SNoS_ (SNoLev z), g u = h u) → G w f z g = G w k z h
In Proofgold the corresponding term root is 0f875d... and proposition id is de8737...
Proof:
Proof not loaded.
L14448
Theorem. (SNo_rec2_eq_1)
∀w, SNo w → ∀f : set → set → set, ∀z, SNo z → SNo_rec_i (G w f) z = G w f z (SNo_rec_i (G w f))
In Proofgold the corresponding term root is 12a51a... and proposition id is 0dc50e...
Proof:
Proof not loaded.
L14462
Theorem. (SNo_rec2_eq)
∀w, SNo w → ∀z, SNo z → SNo_rec2 w z = F w z SNo_rec2
In Proofgold the corresponding term root is e9ea41... and proposition id is ac9f2b...
Proof:
Proof not loaded.
End of Section SurrealRec2
L14559
Theorem. (SNo_ordinal_ind)
∀P : set → prop, (∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, P x) → (∀x, SNo x → P x)
In Proofgold the corresponding term root is 5b75fb... and proposition id is dc2c6d...
Proof:
Proof not loaded.
L14575
Theorem. (SNo_ordinal_ind2)
∀P : set → set → prop, (∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, P x y) → (∀x y, SNo x → SNo y → P x y)
In Proofgold the corresponding term root is ba3315... and proposition id is 4cd5ac...
Proof:
Proof not loaded.
L14603
Theorem. (SNo_ordinal_ind3)
∀P : set → set → set → prop, (∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma, ordinal gamma → ∀x ∈ SNoS_ alpha, ∀y ∈ SNoS_ beta, ∀z ∈ SNoS_ gamma, P x y z) → (∀x y z, SNo x → SNo y → SNo z → P x y z)
In Proofgold the corresponding term root is fccdb6... and proposition id is dece8c...
Proof:
Proof not loaded.
L14640
Theorem. (SNoLev_ind)
∀P : set → prop, (∀x, SNo x → (∀w ∈ SNoS_ (SNoLev x), P w) → P x) → (∀x, SNo x → P x)
In Proofgold the corresponding term root is 8b2b42... and proposition id is 851a88...
Proof:
Proof not loaded.
L14663
Theorem. (SNoLev_ind2)
∀P : set → set → prop, (∀x y, SNo x → SNo y → (∀w ∈ SNoS_ (SNoLev x), P w y) → (∀z ∈ SNoS_ (SNoLev y), P x z) → (∀w ∈ SNoS_ (SNoLev x), ∀z ∈ SNoS_ (SNoLev y), P w z) → P x y) → ∀x y, SNo x → SNo y → P x y
In Proofgold the corresponding term root is 87eb9a... and proposition id is 800d4c...
Proof:
Proof not loaded.
L14709
Theorem. (SNoLev_ind3)
∀P : set → set → set → prop, (∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), P u y z) → (∀v ∈ SNoS_ (SNoLev y), P x v z) → (∀w ∈ SNoS_ (SNoLev z), P x y w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), P u v z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), P u y w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P x v w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P u v w) → P x y z) → ∀x y z, SNo x → SNo y → SNo z → P x y z
In Proofgold the corresponding term root is f1d779... and proposition id is ee16b9...
Proof:
Proof not loaded.
L14788
Theorem. (SNo_omega)
SNo ω
In Proofgold the corresponding term root is f65848... and proposition id is db7b5f...
Proof:
Proof not loaded.
L14792
Theorem. (SNoLt_0_1)
0 < 1
In Proofgold the corresponding term root is dd6047... and proposition id is cb6392...
Proof:
Proof not loaded.
L14798
Theorem. (SNoLt_0_2)
0 < 2
In Proofgold the corresponding term root is 7be36f... and proposition id is f46934...
Proof:
Proof not loaded.
L14804
Theorem. (SNoLt_1_2)
1 < 2
In Proofgold the corresponding term root is cc44e8... and proposition id is 19d5e1...
Proof:
Proof not loaded.
L14810
Theorem. (restr_SNo_)
∀x, SNo x → ∀alpha ∈ SNoLev x, SNo_ alpha (x ∩ SNoElts_ alpha)
In Proofgold the corresponding term root is d78444... and proposition id is e0eca8...
Proof:
Proof not loaded.
L14850
Theorem. (restr_SNo)
∀x, SNo x → ∀alpha ∈ SNoLev x, SNo (x ∩ SNoElts_ alpha)
In Proofgold the corresponding term root is 930b56... and proposition id is d141ed...
Proof:
Proof not loaded.
L14859
Theorem. (restr_SNoLev)
∀x, SNo x → ∀alpha ∈ SNoLev x, SNoLev (x ∩ SNoElts_ alpha) = alpha
In Proofgold the corresponding term root is cae8b5... and proposition id is 8b1853...
Proof:
Proof not loaded.
L14868
Theorem. (restr_SNoEq)
∀x, SNo x → ∀alpha ∈ SNoLev x, SNoEq_ alpha (x ∩ SNoElts_ alpha) x
In Proofgold the corresponding term root is 829f8b... and proposition id is 704f5a...
Proof:
Proof not loaded.
L14884
Theorem. (SNo_extend0_restr_eq)
∀x, SNo x → x = SNo_extend0 x ∩ SNoElts_ (SNoLev x)
In Proofgold the corresponding term root is 2c202e... and proposition id is 9827a9...
Proof:
Proof not loaded.
L14909
Theorem. (SNo_extend1_restr_eq)
∀x, SNo x → x = SNo_extend1 x ∩ SNoElts_ (SNoLev x)
In Proofgold the corresponding term root is 75bbe8... and proposition id is 275196...
Proof:
Proof not loaded.
Beginning of Section SurrealMinus
L14939
Definition. We define minus_SNo to be SNo_rec_i (λx m ⇒ SNoCut {m z|z ∈ SNoR x} {m w|w ∈ SNoL x}) of type set → set.
In Proofgold the corresponding term root is 268a6c... and object id is 8a6d85...
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
L14944
Theorem. (minus_SNo_eq)
∀x, SNo x → - x = SNoCut {- z|z ∈ SNoR x} {- w|w ∈ SNoL x}
In Proofgold the corresponding term root is c9793e... and proposition id is 570cba...
Proof:
Proof not loaded.
L14975
Theorem. (minus_SNo_prop1)
∀x, SNo x → SNo (- x) ∧ (∀u ∈ SNoL x, - x < - u) ∧ (∀u ∈ SNoR x, - u < - x) ∧ SNoCutP {- z|z ∈ SNoR x} {- w|w ∈ SNoL x}
In Proofgold the corresponding term root is 7c0e82... and proposition id is f0b75f...
Proof:
Proof not loaded.
L15166
Theorem. (SNo_minus_SNo)
∀x, SNo x → SNo (- x)
In Proofgold the corresponding term root is 5cf7d9... and proposition id is 744cc2...
Proof:
Proof not loaded.
L15173
Theorem. (minus_SNo_Lt_contra)
∀x y, SNo x → SNo y → x < y → - y < - x
In Proofgold the corresponding term root is 5ce52d... and proposition id is 3ebf6e...
Proof:
Proof not loaded.
L15242
Theorem. (minus_SNo_Le_contra)
∀x y, SNo x → SNo y → x ≤ y → - y ≤ - x
In Proofgold the corresponding term root is c1b225... and proposition id is f8c896...
Proof:
Proof not loaded.
L15260
Theorem. (minus_SNo_SNoCutP)
∀x, SNo x → SNoCutP {- z|z ∈ SNoR x} {- w|w ∈ SNoL x}
In Proofgold the corresponding term root is 99c6fc... and proposition id is 5ad5e4...
Proof:
Proof not loaded.
L15268
Theorem. (minus_SNo_SNoCutP_gen)
∀L R, SNoCutP L R → SNoCutP {- z|z ∈ R} {- w|w ∈ L}
In Proofgold the corresponding term root is e7650d... and proposition id is 687fea...
Proof:
Proof not loaded.
L15319
Theorem. (minus_SNo_Lev_lem1)
∀alpha, ordinal alpha → ∀x ∈ SNoS_ alpha, SNoLev (- x) ⊆ SNoLev x
In Proofgold the corresponding term root is 14986a... and proposition id is afc115...
Proof:
Proof not loaded.
L15490
Theorem. (minus_SNo_Lev_lem2)
∀x, SNo x → SNoLev (- x) ⊆ SNoLev x
In Proofgold the corresponding term root is 879020... and proposition id is 19cf80...
Proof:
Proof not loaded.
L15501
Theorem. (minus_SNo_invol)
∀x, SNo x → - - x = x
In Proofgold the corresponding term root is 44bfe8... and proposition id is cc64ee...
Proof:
Proof not loaded.
L15569
Theorem. (minus_SNo_Lev)
∀x, SNo x → SNoLev (- x) = SNoLev x
In Proofgold the corresponding term root is f65499... and proposition id is 476ae5...
Proof:
Proof not loaded.
L15580
Theorem. (minus_SNo_SNo_)
∀alpha, ordinal alpha → ∀x, SNo_ alpha x → SNo_ alpha (- x)
In Proofgold the corresponding term root is 84605f... and proposition id is 14bd6d...
Proof:
Proof not loaded.
L15597
Theorem. (minus_SNo_SNoS_)
∀alpha, ordinal alpha → ∀x, x ∈ SNoS_ alpha → - x ∈ SNoS_ alpha
In Proofgold the corresponding term root is 12f1a5... and proposition id is e10c92...
Proof:
Proof not loaded.
L15610
Theorem. (minus_SNoCut_eq_lem)
∀v, SNo v → ∀L R, SNoCutP L R → v = SNoCut L R → - v = SNoCut {- z|z ∈ R} {- w|w ∈ L}
In Proofgold the corresponding term root is 29069a... and proposition id is 11eb03...
Proof:
Proof not loaded.
L15720
Theorem. (minus_SNoCut_eq)
∀L R, SNoCutP L R → - SNoCut L R = SNoCut {- z|z ∈ R} {- w|w ∈ L}
In Proofgold the corresponding term root is 083ffe... and proposition id is 54736f...
Proof:
Proof not loaded.
L15725
Theorem. (minus_SNo_Lt_contra1)
∀x y, SNo x → SNo y → - x < y → - y < x
In Proofgold the corresponding term root is 226757... and proposition id is 298729...
Proof:
Proof not loaded.
L15738
Theorem. (minus_SNo_Lt_contra2)
∀x y, SNo x → SNo y → x < - y → y < - x
In Proofgold the corresponding term root is c6f2da... and proposition id is f41968...
Proof:
Proof not loaded.
L15751
Theorem. (mordinal_SNoLev_min_2)
∀alpha, ordinal alpha → ∀z, SNo z → SNoLev z ∈ ordsucc alpha → - alpha ≤ z
In Proofgold the corresponding term root is 2766c7... and proposition id is a96d68...
Proof:
Proof not loaded.
L15767
Theorem. (minus_SNo_SNoS_omega)
∀x ∈ SNoS_ ω, - x ∈ SNoS_ ω
In Proofgold the corresponding term root is daeba5... and proposition id is 0c19f5...
Proof:
Proof not loaded.
L15783
Theorem. (SNoL_minus_SNoR)
∀x, SNo x → SNoL (- x) = {- w|w ∈ SNoR x}
In Proofgold the corresponding term root is 084b2d... and proposition id is 8b2b92...
Proof:
Proof not loaded.
End of Section SurrealMinus
Beginning of Section SurrealAdd
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
L15831
Definition. We define add_SNo to be SNo_rec2 (λx y a ⇒ SNoCut ({a w y|w ∈ SNoL x} ∪ {a x w|w ∈ SNoL y}) ({a z y|z ∈ SNoR x} ∪ {a x z|z ∈ SNoR y})) of type set → set → set.
In Proofgold the corresponding term root is 127d04... and object id is f62f9a...
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
L15834
Theorem. (add_SNo_eq)
∀x, SNo x → ∀y, SNo y → x + y = SNoCut ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y}) ({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y})
In Proofgold the corresponding term root is 0c1128... and proposition id is 437a9c...
Proof:
Proof not loaded.
L15890
Theorem. (add_SNo_prop1)
∀x y, SNo x → SNo y → SNo (x + y) ∧ (∀u ∈ SNoL x, u + y < x + y) ∧ (∀u ∈ SNoR x, x + y < u + y) ∧ (∀u ∈ SNoL y, x + u < x + y) ∧ (∀u ∈ SNoR y, x + y < x + u) ∧ SNoCutP ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y}) ({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y})
In Proofgold the corresponding term root is f3ece7... and proposition id is 5a879f...
Proof:
Proof not loaded.
L16351
Theorem. (SNo_add_SNo)
∀x y, SNo x → SNo y → SNo (x + y)
In Proofgold the corresponding term root is c4a9a6... and proposition id is a57095...
Proof:
Proof not loaded.
L16361
Theorem. (SNo_add_SNo_3)
∀x y z, SNo x → SNo y → SNo z → SNo (x + y + z)
In Proofgold the corresponding term root is f6ef07... and proposition id is f2d78c...
Proof:
Proof not loaded.
L16370
Theorem. (SNo_add_SNo_3c)
∀x y z, SNo x → SNo y → SNo z → SNo (x + y + - z)
In Proofgold the corresponding term root is 104eb9... and proposition id is 833be8...
Proof:
Proof not loaded.
L16378
Theorem. (SNo_add_SNo_4)
∀x y z w, SNo x → SNo y → SNo z → SNo w → SNo (x + y + z + w)
In Proofgold the corresponding term root is d71f0e... and proposition id is 42037e...
Proof:
Proof not loaded.
L16383
Theorem. (add_SNo_Lt1)
∀x y z, SNo x → SNo y → SNo z → x < z → x + y < z + y
In Proofgold the corresponding term root is 92232f... and proposition id is cb0738...
Proof:
Proof not loaded.
L16448
Theorem. (add_SNo_Le1)
∀x y z, SNo x → SNo y → SNo z → x ≤ z → x + y ≤ z + y
In Proofgold the corresponding term root is d8f37e... and proposition id is abcf72...
Proof:
Proof not loaded.
L16460
Theorem. (add_SNo_Lt2)
∀x y z, SNo x → SNo y → SNo z → y < z → x + y < x + z
In Proofgold the corresponding term root is 6ad464... and proposition id is 6f0941...
Proof:
Proof not loaded.
L16526
Theorem. (add_SNo_Le2)
∀x y z, SNo x → SNo y → SNo z → y ≤ z → x + y ≤ x + z
In Proofgold the corresponding term root is 14975e... and proposition id is 3f52ba...
Proof:
Proof not loaded.
L16538
Theorem. (add_SNo_Lt3a)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x < z → y ≤ w → x + y < z + w
In Proofgold the corresponding term root is b2215f... and proposition id is 18ff91...
Proof:
Proof not loaded.
L16551
Theorem. (add_SNo_Lt3b)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x ≤ z → y < w → x + y < z + w
In Proofgold the corresponding term root is 5b4f92... and proposition id is 8fe0d9...
Proof:
Proof not loaded.
L16564
Theorem. (add_SNo_Lt3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x < z → y < w → x + y < z + w
In Proofgold the corresponding term root is 7141a8... and proposition id is 55bdab...
Proof:
Proof not loaded.
L16579
Theorem. (add_SNo_Le3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x ≤ z → y ≤ w → x + y ≤ z + w
In Proofgold the corresponding term root is 0be637... and proposition id is be66ca...
Proof:
Proof not loaded.
L16592
Theorem. (add_SNo_SNoCutP)
∀x y, SNo x → SNo y → SNoCutP ({w + y|w ∈ SNoL x} ∪ {x + w|w ∈ SNoL y}) ({z + y|z ∈ SNoR x} ∪ {x + z|z ∈ SNoR y})
In Proofgold the corresponding term root is e2b7d7... and proposition id is f86b1a...
Proof:
Proof not loaded.
L16600
Theorem. (add_SNo_com)
∀x y, SNo x → SNo y → x + y = y + x
In Proofgold the corresponding term root is 548814... and proposition id is f017e5...
Proof:
Proof not loaded.
L16701
Theorem. (add_SNo_0L)
∀x, SNo x → 0 + x = x
In Proofgold the corresponding term root is 107598... and proposition id is 446bad...
Proof:
Proof not loaded.
L16764
Theorem. (add_SNo_0R)
∀x, SNo x → x + 0 = x
In Proofgold the corresponding term root is d8a186... and proposition id is 1a7fa1...
Proof:
Proof not loaded.
L16770
Theorem. (add_SNo_minus_SNo_linv)
∀x, SNo x → - x + x = 0
In Proofgold the corresponding term root is 424719... and proposition id is 44e251...
Proof:
Proof not loaded.
L16945
Theorem. (add_SNo_minus_SNo_rinv)
∀x, SNo x → x + - x = 0
In Proofgold the corresponding term root is 9e8c44... and proposition id is b2f9cf...
Proof:
Proof not loaded.
L16955
Theorem. (add_SNo_ordinal_SNoCutP)
∀alpha, ordinal alpha → ∀beta, ordinal beta → SNoCutP ({x + beta|x ∈ SNoS_ alpha} ∪ {alpha + x|x ∈ SNoS_ beta}) Empty
In Proofgold the corresponding term root is 2823e4... and proposition id is 507eab...
Proof:
Proof not loaded.
L16995
Theorem. (add_SNo_ordinal_eq)
∀alpha, ordinal alpha → ∀beta, ordinal beta → alpha + beta = SNoCut ({x + beta|x ∈ SNoS_ alpha} ∪ {alpha + x|x ∈ SNoS_ beta}) Empty
In Proofgold the corresponding term root is cfe1f5... and proposition id is ee8e7c...
Proof:
Proof not loaded.
L17027
Theorem. (add_SNo_ordinal_ordinal)
∀alpha, ordinal alpha → ∀beta, ordinal beta → ordinal (alpha + beta)
In Proofgold the corresponding term root is e9309a... and proposition id is ee9c23...
Proof:
Proof not loaded.
L17092
Theorem. (add_SNo_ordinal_SL)
∀alpha, ordinal alpha → ∀beta, ordinal beta → ordsucc alpha + beta = ordsucc (alpha + beta)
In Proofgold the corresponding term root is d55bd8... and proposition id is 61ae57...
Proof:
Proof not loaded.
L17244
Theorem. (add_SNo_ordinal_SR)
∀alpha, ordinal alpha → ∀beta, ordinal beta → alpha + ordsucc beta = ordsucc (alpha + beta)
In Proofgold the corresponding term root is e7378a... and proposition id is 0c1a6d...
Proof:
Proof not loaded.
L17265
Theorem. (add_SNo_ordinal_InL)
∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma ∈ alpha, gamma + beta ∈ alpha + beta
In Proofgold the corresponding term root is 09a829... and proposition id is 5acda8...
Proof:
Proof not loaded.
L17289
Theorem. (add_SNo_ordinal_InR)
∀alpha, ordinal alpha → ∀beta, ordinal beta → ∀gamma ∈ beta, alpha + gamma ∈ alpha + beta
In Proofgold the corresponding term root is d291f4... and proposition id is 0f42b5...
Proof:
Proof not loaded.
L17306
Theorem. (add_nat_add_SNo)
∀n m ∈ ω, add_nat n m = n + m
In Proofgold the corresponding term root is 1701fc... and proposition id is 9df659...
Proof:
Proof not loaded.
L17335
Theorem. (add_SNo_In_omega)
∀n m ∈ ω, n + m ∈ ω
In Proofgold the corresponding term root is f45cbb... and proposition id is eb6375...
Proof:
Proof not loaded.
L17344
Theorem. (add_SNo_1_1_2)
1 + 1 = 2
In Proofgold the corresponding term root is bbc448... and proposition id is 2b8298...
Proof:
Proof not loaded.
L17349
Theorem. (add_SNo_SNoL_interpolate)
∀x y, SNo x → SNo y → ∀u ∈ SNoL (x + y), (∃v ∈ SNoL x, u ≤ v + y) ∨ (∃v ∈ SNoL y, u ≤ x + v)
In Proofgold the corresponding term root is 971363... and proposition id is 3cafc1...
Proof:
Proof not loaded.
L17506
Theorem. (add_SNo_SNoR_interpolate)
∀x y, SNo x → SNo y → ∀u ∈ SNoR (x + y), (∃v ∈ SNoR x, v + y ≤ u) ∨ (∃v ∈ SNoR y, x + v ≤ u)
In Proofgold the corresponding term root is 928f71... and proposition id is 31fa94...
Proof:
Proof not loaded.
L17663
Theorem. (add_SNo_assoc)
∀x y z, SNo x → SNo y → SNo z → x + (y + z) = (x + y) + z
In Proofgold the corresponding term root is 39b331... and proposition id is bc15ad...
Proof:
Proof not loaded.
L18079
Theorem. (add_SNo_minus_R2)
∀x y, SNo x → SNo y → (x + y) + - y = x
In Proofgold the corresponding term root is 3e7b33... and proposition id is 91f4ef...
Proof:
Proof not loaded.
L18088
Theorem. (add_SNo_minus_R2')
∀x y, SNo x → SNo y → (x + - y) + y = x
In Proofgold the corresponding term root is 4f9468... and proposition id is 43b4d1...
Proof:
Proof not loaded.
L18094
Theorem. (add_SNo_minus_L2)
∀x y, SNo x → SNo y → - x + (x + y) = y
In Proofgold the corresponding term root is 2cd65e... and proposition id is 598ee6...
Proof:
Proof not loaded.
L18103
Theorem. (add_SNo_minus_L2')
∀x y, SNo x → SNo y → x + (- x + y) = y
In Proofgold the corresponding term root is 632a4d... and proposition id is a8450d...
Proof:
Proof not loaded.
L18109
Theorem. (add_SNo_cancel_L)
∀x y z, SNo x → SNo y → SNo z → x + y = x + z → y = z
In Proofgold the corresponding term root is 014096... and proposition id is 781256...
Proof:
Proof not loaded.
L18122
Theorem. (add_SNo_cancel_R)
∀x y z, SNo x → SNo y → SNo z → x + y = z + y → x = z
In Proofgold the corresponding term root is 3de43c... and proposition id is e62801...
Proof:
Proof not loaded.
L18136
Theorem. (minus_SNo_0)
- 0 = 0
In Proofgold the corresponding term root is 77bc67... and proposition id is 3fcc57...
Proof:
Proof not loaded.
L18144
Theorem. (minus_add_SNo_distr)
∀x y, SNo x → SNo y → - (x + y) = (- x) + (- y)
In Proofgold the corresponding term root is 57ec9d... and proposition id is da8d5a...
Proof:
Proof not loaded.
L18174
Theorem. (minus_add_SNo_distr_3)
∀x y z, SNo x → SNo y → SNo z → - (x + y + z) = - x + - y + - z
In Proofgold the corresponding term root is b074ac... and proposition id is 94dbf3...
Proof:
Proof not loaded.
L18182
Theorem. (add_SNo_Lev_bd)
∀x y, SNo x → SNo y → SNoLev (x + y) ⊆ SNoLev x + SNoLev y
In Proofgold the corresponding term root is 0f257c... and proposition id is 0680ef...
Proof:
Proof not loaded.
L18546
Theorem. (add_SNo_SNoS_omega)
∀x y ∈ SNoS_ ω, x + y ∈ SNoS_ ω
In Proofgold the corresponding term root is 931103... and proposition id is 6d16aa...
Proof:
Proof not loaded.
L18576
Theorem. (add_SNo_Lt1_cancel)
∀x y z, SNo x → SNo y → SNo z → x + y < z + y → x < z
In Proofgold the corresponding term root is e5c11e... and proposition id is 2ec8ae...
Proof:
Proof not loaded.
L18592
Theorem. (add_SNo_Lt2_cancel)
∀x y z, SNo x → SNo y → SNo z → x + y < x + z → y < z
In Proofgold the corresponding term root is 9ecfc8... and proposition id is d4afbd...
Proof:
Proof not loaded.
L18599
Theorem. (add_SNo_Le1_cancel)
∀x y z, SNo x → SNo y → SNo z → x + y ≤ z + y → x ≤ z
In Proofgold the corresponding term root is 01a9b8... and proposition id is 8bfac1...
Proof:
Proof not loaded.
L18615
Theorem. (add_SNo_assoc_4)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y + z + w = (x + y + z) + w
In Proofgold the corresponding term root is b9919a... and proposition id is f2ec8c...
Proof:
Proof not loaded.
L18625
Theorem. (add_SNo_com_3_0_1)
∀x y z, SNo x → SNo y → SNo z → x + y + z = y + x + z
In Proofgold the corresponding term root is 53adb5... and proposition id is 3c9cc8...
Proof:
Proof not loaded.
L18635
Theorem. (add_SNo_com_3b_1_2)
∀x y z, SNo x → SNo y → SNo z → (x + y) + z = (x + z) + y
In Proofgold the corresponding term root is 2ccf2b... and proposition id is f2dd80...
Proof:
Proof not loaded.
L18645
Theorem. (add_SNo_com_4_inner_mid)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y) + (z + w) = (x + z) + (y + w)
In Proofgold the corresponding term root is 635cb7... and proposition id is e25ee6...
Proof:
Proof not loaded.
L18657
Theorem. (add_SNo_rotate_3_1)
∀x y z, SNo x → SNo y → SNo z → x + y + z = z + x + y
In Proofgold the corresponding term root is 51654e... and proposition id is 340640...
Proof:
Proof not loaded.
L18671
Theorem. (add_SNo_rotate_4_1)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y + z + w = w + x + y + z
In Proofgold the corresponding term root is 4bc7aa... and proposition id is ce0a6b...
Proof:
Proof not loaded.
L18679
Theorem. (add_SNo_rotate_5_1)
∀x y z w v, SNo x → SNo y → SNo z → SNo w → SNo v → x + y + z + w + v = v + x + y + z + w
In Proofgold the corresponding term root is 4b4b64... and proposition id is 0714aa...
Proof:
Proof not loaded.
L18687
Theorem. (add_SNo_rotate_5_2)
∀x y z w v, SNo x → SNo y → SNo z → SNo w → SNo v → x + y + z + w + v = w + v + x + y + z
In Proofgold the corresponding term root is 33b5cc... and proposition id is fec9d0...
Proof:
Proof not loaded.
L18695
Theorem. (add_SNo_minus_SNo_prop2)
∀x y, SNo x → SNo y → x + - x + y = y
In Proofgold the corresponding term root is 632a4d... and proposition id is a8450d...
Proof:
Proof not loaded.
L18704
Theorem. (add_SNo_minus_SNo_prop3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y + z) + (- z + w) = x + y + w
In Proofgold the corresponding term root is 038fdd... and proposition id is 0d4f24...
Proof:
Proof not loaded.
L18715
Theorem. (add_SNo_minus_SNo_prop5)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y + - z) + (z + w) = x + y + w
In Proofgold the corresponding term root is bdfd3f... and proposition id is eeb587...
Proof:
Proof not loaded.
L18723
Theorem. (add_SNo_minus_Lt1)
∀x y z, SNo x → SNo y → SNo z → x + - y < z → x < z + y
In Proofgold the corresponding term root is 643876... and proposition id is 4033dc...
Proof:
Proof not loaded.
L18733
Theorem. (add_SNo_minus_Lt2)
∀x y z, SNo x → SNo y → SNo z → z < x + - y → z + y < x
In Proofgold the corresponding term root is 76c543... and proposition id is 362145...
Proof:
Proof not loaded.
L18744
Theorem. (add_SNo_minus_Lt1b)
∀x y z, SNo x → SNo y → SNo z → x < z + y → x + - y < z
In Proofgold the corresponding term root is 5012b1... and proposition id is 0b5cca...
Proof:
Proof not loaded.
L18754
Theorem. (add_SNo_minus_Lt2b)
∀x y z, SNo x → SNo y → SNo z → z + y < x → z < x + - y
In Proofgold the corresponding term root is 24d1a6... and proposition id is 684192...
Proof:
Proof not loaded.
L18764
Theorem. (add_SNo_minus_Lt1b3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y < w + z → x + y + - z < w
In Proofgold the corresponding term root is 218e4f... and proposition id is 8ea4a7...
Proof:
Proof not loaded.
L18773
Theorem. (add_SNo_minus_Lt2b3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → w + z < x + y → w < x + y + - z
In Proofgold the corresponding term root is d9462b... and proposition id is 4bf575...
Proof:
Proof not loaded.
L18782
Theorem. (add_SNo_minus_Lt_lem)
∀x y z u v w, SNo x → SNo y → SNo z → SNo u → SNo v → SNo w → x + y + w < u + v + z → x + y + - z < u + v + - w
In Proofgold the corresponding term root is da9f29... and proposition id is d29d2a...
Proof:
Proof not loaded.
L18812
Theorem. (add_SNo_minus_Le2)
∀x y z, SNo x → SNo y → SNo z → z ≤ x + - y → z + y ≤ x
In Proofgold the corresponding term root is 85af84... and proposition id is 86b40b...
Proof:
Proof not loaded.
L18825
Theorem. (add_SNo_minus_Le2b)
∀x y z, SNo x → SNo y → SNo z → z + y ≤ x → z ≤ x + - y
In Proofgold the corresponding term root is 688ef9... and proposition id is d815dd...
Proof:
Proof not loaded.
L18838
Theorem. (add_SNo_Lt_subprop2)
∀x y z w u v, SNo x → SNo y → SNo z → SNo w → SNo u → SNo v → x + u < z + v → y + v < w + u → x + y < z + w
In Proofgold the corresponding term root is 848909... and proposition id is 7f1b50...
Proof:
Proof not loaded.
L18869
Theorem. (add_SNo_Lt_subprop3a)
∀x y z w u a, SNo x → SNo y → SNo z → SNo w → SNo u → SNo a → x + z < w + a → y + a < u → x + y + z < w + u
In Proofgold the corresponding term root is 43322d... and proposition id is cde0c9...
Proof:
Proof not loaded.
L18892
Theorem. (add_SNo_Lt_subprop3b)
∀x y w u v a, SNo x → SNo y → SNo w → SNo u → SNo v → SNo a → x + a < w + v → y < a + u → x + y < w + u + v
In Proofgold the corresponding term root is 9d5c37... and proposition id is b197b1...
Proof:
Proof not loaded.
L18909
Theorem. (add_SNo_Lt_subprop3c)
∀x y z w u a b c, SNo x → SNo y → SNo z → SNo w → SNo u → SNo a → SNo b → SNo c → x + a < b + c → y + c < u → b + z < w + a → x + y + z < w + u
In Proofgold the corresponding term root is 184614... and proposition id is f8cf2b...
Proof:
Proof not loaded.
L18943
Theorem. (add_SNo_Lt_subprop3d)
∀x y w u v a b c, SNo x → SNo y → SNo w → SNo u → SNo v → SNo a → SNo b → SNo c → x + a < b + v → y < c + u → b + c < w + a → x + y < w + u + v
In Proofgold the corresponding term root is a447c6... and proposition id is a1520a...
Proof:
Proof not loaded.
L19003
Theorem. (ordinal_ordsucc_SNo_eq)
∀alpha, ordinal alpha → ordsucc alpha = 1 + alpha
In Proofgold the corresponding term root is f0596d... and proposition id is 36bf6d...
Proof:
Proof not loaded.
L19011
Theorem. (add_SNo_3a_2b)
∀x y z w u, SNo x → SNo y → SNo z → SNo w → SNo u → (x + y + z) + (w + u) = (u + y + z) + (w + x)
In Proofgold the corresponding term root is d5c6f9... and proposition id is f24a27...
Proof:
Proof not loaded.
L19024
Theorem. (add_SNo_1_ordsucc)
∀n ∈ ω, n + 1 = ordsucc n
In Proofgold the corresponding term root is ec6cad... and proposition id is 85f670...
Proof:
Proof not loaded.
L19034
Theorem. (add_SNo_eps_Lt)
∀x, SNo x → ∀n ∈ ω, x < x + eps_ n
In Proofgold the corresponding term root is dfe40b... and proposition id is d492a4...
Proof:
Proof not loaded.
L19042
Theorem. (add_SNo_eps_Lt')
∀x y, SNo x → SNo y → ∀n ∈ ω, x < y → x < y + eps_ n
In Proofgold the corresponding term root is 281ca0... and proposition id is 943874...
Proof:
Proof not loaded.
L19050
Theorem. (SNoLt_minus_pos)
∀x y, SNo x → SNo y → x < y → 0 < y + - x
In Proofgold the corresponding term root is fff649... and proposition id is 2c799f...
Proof:
Proof not loaded.
L19057
Theorem. (add_SNo_omega_In_cases)
∀m, ∀n ∈ ω, ∀k, nat_p k → m ∈ n + k → m ∈ n ∨ m + - n ∈ k
In Proofgold the corresponding term root is 3eee93... and proposition id is 584cf6...
Proof:
Proof not loaded.
L19083
Theorem. (add_SNo_Lt4)
∀x y z w u v, SNo x → SNo y → SNo z → SNo w → SNo u → SNo v → x < w → y < u → z < v → x + y + z < w + u + v
In Proofgold the corresponding term root is 7e45aa... and proposition id is c85ce6...
Proof:
Proof not loaded.
L19092
Theorem. (add_SNo_3_3_3_Lt1)
∀x y z w u, SNo x → SNo y → SNo z → SNo w → SNo u → x + y < z + w → x + y + u < z + w + u
In Proofgold the corresponding term root is 5889f8... and proposition id is fa11ed...
Proof:
Proof not loaded.
L19101
Theorem. (add_SNo_3_2_3_Lt1)
∀x y z w u, SNo x → SNo y → SNo z → SNo w → SNo u → y + x < z + w → x + u + y < z + w + u
In Proofgold the corresponding term root is c05b73... and proposition id is e4013b...
Proof:
Proof not loaded.
L19109
Theorem. (add_SNo_minus_Lt12b3)
∀x y z w u v, SNo x → SNo y → SNo z → SNo w → SNo u → SNo v → x + y + v < w + u + z → x + y + - z < w + u + - v
In Proofgold the corresponding term root is da9f29... and proposition id is d29d2a...
Proof:
Proof not loaded.
L19132
Theorem. (add_SNo_minus_Le1b)
∀x y z, SNo x → SNo y → SNo z → x ≤ z + y → x + - y ≤ z
In Proofgold the corresponding term root is c8c038... and proposition id is 6a5e5e...
Proof:
Proof not loaded.
L19142
Theorem. (add_SNo_minus_Le1b3)
∀x y z w, SNo x → SNo y → SNo z → SNo w → x + y ≤ w + z → x + y + - z ≤ w
In Proofgold the corresponding term root is 11c341... and proposition id is 6a25dd...
Proof:
Proof not loaded.
L19151
Theorem. (add_SNo_minus_Le12b3)
∀x y z w u v, SNo x → SNo y → SNo z → SNo w → SNo u → SNo v → x + y + v ≤ w + u + z → x + y + - z ≤ w + u + - v
In Proofgold the corresponding term root is 055445... and proposition id is d88d85...
Proof:
Proof not loaded.
End of Section SurrealAdd
Beginning of Section SurrealAbs
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
L19184
Definition. We define abs_SNo to be λx ⇒ if 0 ≤ x then x else - x of type set → set.
In Proofgold the corresponding term root is 34f6dc... and object id is 2990e5...
L19185
Theorem. (nonneg_abs_SNo)
∀x, 0 ≤ x → abs_SNo x = x
In Proofgold the corresponding term root is 330ed1... and proposition id is 1ed323...
Proof:
Proof not loaded.
L19190
Theorem. (not_nonneg_abs_SNo)
∀x, ¬ (0 ≤ x) → abs_SNo x = - x
In Proofgold the corresponding term root is bc755d... and proposition id is b29f70...
Proof:
Proof not loaded.
L19195
Theorem. (pos_abs_SNo)
∀x, 0 < x → abs_SNo x = x
In Proofgold the corresponding term root is 32e626... and proposition id is 88568f...
Proof:
Proof not loaded.
L19201
Theorem. (neg_abs_SNo)
∀x, SNo x → x < 0 → abs_SNo x = - x
In Proofgold the corresponding term root is d1245a... and proposition id is f858cb...
Proof:
Proof not loaded.
L19211
Theorem. (SNo_abs_SNo)
∀x, SNo x → SNo (abs_SNo x)
In Proofgold the corresponding term root is 7266a7... and proposition id is 1c9c46...
Proof:
Proof not loaded.
L19219
Theorem. (abs_SNo_minus)
∀x, SNo x → abs_SNo (- x) = abs_SNo x
In Proofgold the corresponding term root is 977106... and proposition id is 2c0d4c...
Proof:
Proof not loaded.
L19251
Theorem. (abs_SNo_dist_swap)
∀x y, SNo x → SNo y → abs_SNo (x + - y) = abs_SNo (y + - x)
In Proofgold the corresponding term root is 73d709... and proposition id is fd708c...
Proof:
Proof not loaded.
End of Section SurrealAbs
Beginning of Section SurrealMul
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
L19278
Definition. We define mul_SNo to be SNo_rec2 (λx y m ⇒ SNoCut ({m (w 0) y + m x (w 1) + - m (w 0) (w 1)|w ∈ SNoL x ⨯ SNoL y} ∪ {m (z 0) y + m x (z 1) + - m (z 0) (z 1)|z ∈ SNoR x ⨯ SNoR y}) ({m (w 0) y + m x (w 1) + - m (w 0) (w 1)|w ∈ SNoL x ⨯ SNoR y} ∪ {m (z 0) y + m x (z 1) + - m (z 0) (z 1)|z ∈ SNoR x ⨯ SNoL y})) of type set → set → set.
In Proofgold the corresponding term root is 48d054... and object id is b14730...
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
L19289
Theorem. (mul_SNo_eq)
∀x, SNo x → ∀y, SNo y → x * y = SNoCut ({(w 0) * y + x * (w 1) + - (w 0) * (w 1)|w ∈ SNoL x ⨯ SNoL y} ∪ {(z 0) * y + x * (z 1) + - (z 0) * (z 1)|z ∈ SNoR x ⨯ SNoR y}) ({(w 0) * y + x * (w 1) + - (w 0) * (w 1)|w ∈ SNoL x ⨯ SNoR y} ∪ {(z 0) * y + x * (z 1) + - (z 0) * (z 1)|z ∈ SNoR x ⨯ SNoL y})
In Proofgold the corresponding term root is 726bd9... and proposition id is 3525c5...
Proof:
Proof not loaded.
L19449
Theorem. (mul_SNo_eq_2)
∀x y, SNo x → SNo y → ∀p : prop, (∀L R, (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p
In Proofgold the corresponding term root is 97f72a... and proposition id is d69cc6...
Proof:
Proof not loaded.
L19565
Theorem. (mul_SNo_prop_1)
∀x, SNo x → ∀y, SNo y → ∀p : prop, (SNo (x * y) → (∀u ∈ SNoL x, ∀v ∈ SNoL y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoR x, ∀v ∈ SNoR y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoL x, ∀v ∈ SNoR y, x * y + u * v < u * y + x * v) → (∀u ∈ SNoR x, ∀v ∈ SNoL y, x * y + u * v < u * y + x * v) → p) → p
In Proofgold the corresponding term root is 3018e4... and proposition id is 5e4d16...
Proof:
Proof not loaded.
L20606
Theorem. (SNo_mul_SNo)
∀x y, SNo x → SNo y → SNo (x * y)
In Proofgold the corresponding term root is 7fa2e2... and proposition id is 9fd082...
Proof:
Proof not loaded.
L20611
Theorem. (SNo_mul_SNo_lem)
∀x y u v, SNo x → SNo y → SNo u → SNo v → SNo (u * y + x * v + - (u * v))
In Proofgold the corresponding term root is 632e95... and proposition id is 3a9abb...
Proof:
Proof not loaded.
L20625
Theorem. (SNo_mul_SNo_3)
∀x y z, SNo x → SNo y → SNo z → SNo (x * y * z)
In Proofgold the corresponding term root is 4fc87d... and proposition id is 8f5450...
Proof:
Proof not loaded.
L20634
Theorem. (mul_SNo_eq_3)
∀x y, SNo x → SNo y → ∀p : prop, (∀L R, SNoCutP L R → (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p
In Proofgold the corresponding term root is 517939... and proposition id is 720ba0...
Proof:
Proof not loaded.
L20875
Theorem. (mul_SNo_Lt)
∀x y u v, SNo x → SNo y → SNo u → SNo v → u < x → v < y → u * y + x * v < x * y + u * v
In Proofgold the corresponding term root is 428b35... and proposition id is c23201...
Proof:
Proof not loaded.
L21231
Theorem. (mul_SNo_Le)
∀x y u v, SNo x → SNo y → SNo u → SNo v → u ≤ x → v ≤ y → u * y + x * v ≤ x * y + u * v
In Proofgold the corresponding term root is d9a12c... and proposition id is acb620...
Proof:
Proof not loaded.
L21250
Theorem. (mul_SNo_SNoL_interpolate)
∀x y, SNo x → SNo y → ∀u ∈ SNoL (x * y), (∃v ∈ SNoL x, ∃w ∈ SNoL y, u + v * w ≤ v * y + x * w) ∨ (∃v ∈ SNoR x, ∃w ∈ SNoR y, u + v * w ≤ v * y + x * w)
In Proofgold the corresponding term root is 7d1b52... and proposition id is f460fc...
Proof:
Proof not loaded.
L21432
Theorem. (mul_SNo_SNoL_interpolate_impred)
∀x y, SNo x → SNo y → ∀u ∈ SNoL (x * y), ∀p : prop, (∀v ∈ SNoL x, ∀w ∈ SNoL y, u + v * w ≤ v * y + x * w → p) → (∀v ∈ SNoR x, ∀w ∈ SNoR y, u + v * w ≤ v * y + x * w → p) → p
In Proofgold the corresponding term root is 395b74... and proposition id is ad6f96...
Proof:
Proof not loaded.
L21454
Theorem. (mul_SNo_SNoR_interpolate)
∀x y, SNo x → SNo y → ∀u ∈ SNoR (x * y), (∃v ∈ SNoL x, ∃w ∈ SNoR y, v * y + x * w ≤ u + v * w) ∨ (∃v ∈ SNoR x, ∃w ∈ SNoL y, v * y + x * w ≤ u + v * w)
In Proofgold the corresponding term root is e15949... and proposition id is b67a35...
Proof:
Proof not loaded.
L21636
Theorem. (mul_SNo_SNoR_interpolate_impred)
∀x y, SNo x → SNo y → ∀u ∈ SNoR (x * y), ∀p : prop, (∀v ∈ SNoL x, ∀w ∈ SNoR y, v * y + x * w ≤ u + v * w → p) → (∀v ∈ SNoR x, ∀w ∈ SNoL y, v * y + x * w ≤ u + v * w → p) → p
In Proofgold the corresponding term root is e5cae5... and proposition id is d1fff1...
Proof:
Proof not loaded.
L21658
Theorem. (mul_SNo_Subq_lem)
∀x y X Y Z W, ∀U U', (∀u, u ∈ U → (∀q : prop, (∀w0 ∈ X, ∀w1 ∈ Y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ Z, ∀z1 ∈ W, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ X, ∀w1 ∈ Y, w0 * y + x * w1 + - w0 * w1 ∈ U') → (∀w0 ∈ Z, ∀w1 ∈ W, w0 * y + x * w1 + - w0 * w1 ∈ U') → U ⊆ U'
In Proofgold the corresponding term root is cc8343... and proposition id is e8bfa2...
Proof:
Proof not loaded.
L21681
Theorem. (mul_SNo_zeroR)
∀x, SNo x → x * 0 = 0
In Proofgold the corresponding term root is d279f4... and proposition id is a4f4a6...
Proof:
Proof not loaded.
L21723
Theorem. (mul_SNo_oneR)
∀x, SNo x → x * 1 = x
In Proofgold the corresponding term root is 1d2f1d... and proposition id is 5477a4...
Proof:
Proof not loaded.
L21851
Theorem. (mul_SNo_com)
∀x y, SNo x → SNo y → x * y = y * x
In Proofgold the corresponding term root is 21f7d8... and proposition id is 83d5fa...
Proof:
Proof not loaded.
L22010
Theorem. (mul_SNo_minus_distrL)
∀x y, SNo x → SNo y → (- x) * y = - x * y
In Proofgold the corresponding term root is 9beaa0... and proposition id is 3d0cd1...
Proof:
Proof not loaded.
L22402
Theorem. (mul_SNo_minus_distrR)
∀x y, SNo x → SNo y → x * (- y) = - (x * y)
In Proofgold the corresponding term root is ab775a... and proposition id is 2e7475...
Proof:
Proof not loaded.
L22409
Theorem. (mul_SNo_distrR)
∀x y z, SNo x → SNo y → SNo z → (x + y) * z = x * z + y * z
In Proofgold the corresponding term root is 6a23e5... and proposition id is 5f0382...
Proof:
Proof not loaded.
L23722
Theorem. (mul_SNo_distrL)
∀x y z, SNo x → SNo y → SNo z → x * (y + z) = x * y + x * z
In Proofgold the corresponding term root is 4e7eba... and proposition id is 2a4834...
Proof:
Proof not loaded.
Beginning of Section mul_SNo_assoc_lems
L23737
Variable M : set → set → set
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term M.
L23739
Hypothesis SNo_M : ∀x y, SNo x → SNo y → SNo (x * y)
L23740
Hypothesis DL : ∀x y z, SNo x → SNo y → SNo z → x * (y + z) = x * y + x * z
L23741
Hypothesis DR : ∀x y z, SNo x → SNo y → SNo z → (x + y) * z = x * z + y * z
L23742
Hypothesis IL : ∀x y, SNo x → SNo y → ∀u ∈ SNoL (x * y), ∀p : prop, (∀v ∈ SNoL x, ∀w ∈ SNoL y, u + v * w ≤ v * y + x * w → p) → (∀v ∈ SNoR x, ∀w ∈ SNoR y, u + v * w ≤ v * y + x * w → p) → p
L23747
Hypothesis IR : ∀x y, SNo x → SNo y → ∀u ∈ SNoR (x * y), ∀p : prop, (∀v ∈ SNoL x, ∀w ∈ SNoR y, v * y + x * w ≤ u + v * w → p) → (∀v ∈ SNoR x, ∀w ∈ SNoL y, v * y + x * w ≤ u + v * w → p) → p
L23752
Hypothesis M_Lt : ∀x y u v, SNo x → SNo y → SNo u → SNo v → u < x → v < y → u * y + x * v < x * y + u * v
L23754
Hypothesis M_Le : ∀x y u v, SNo x → SNo y → SNo u → SNo v → u ≤ x → v ≤ y → u * y + x * v ≤ x * y + u * v
L23756
Theorem. (mul_SNo_assoc_lem1)
∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), u * (y * z) = (u * y) * z) → (∀v ∈ SNoS_ (SNoLev y), x * (v * z) = (x * v) * z) → (∀w ∈ SNoS_ (SNoLev z), x * (y * w) = (x * y) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), u * (v * z) = (u * v) * z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), u * (y * w) = (u * y) * w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), x * (v * w) = (x * v) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), u * (v * w) = (u * v) * w) → ∀L, (∀u ∈ L, ∀q : prop, (∀v ∈ SNoL x, ∀w ∈ SNoL (y * z), u = v * (y * z) + x * w + - v * w → q) → (∀v ∈ SNoR x, ∀w ∈ SNoR (y * z), u = v * (y * z) + x * w + - v * w → q) → q) → ∀u ∈ L, u < (x * y) * z
In Proofgold the corresponding term root is fca036... and proposition id is b70da8...
Proof:
Proof not loaded.
L24216
Theorem. (mul_SNo_assoc_lem2)
∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), u * (y * z) = (u * y) * z) → (∀v ∈ SNoS_ (SNoLev y), x * (v * z) = (x * v) * z) → (∀w ∈ SNoS_ (SNoLev z), x * (y * w) = (x * y) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), u * (v * z) = (u * v) * z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), u * (y * w) = (u * y) * w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), x * (v * w) = (x * v) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), u * (v * w) = (u * v) * w) → ∀R, (∀u ∈ R, ∀q : prop, (∀v ∈ SNoL x, ∀w ∈ SNoR (y * z), u = v * (y * z) + x * w + - v * w → q) → (∀v ∈ SNoR x, ∀w ∈ SNoL (y * z), u = v * (y * z) + x * w + - v * w → q) → q) → ∀u ∈ R, (x * y) * z < u
In Proofgold the corresponding term root is 749b30... and proposition id is f249d3...
Proof:
Proof not loaded.
End of Section mul_SNo_assoc_lems
L24701
Theorem. (mul_SNo_assoc)
∀x y z, SNo x → SNo y → SNo z → x * (y * z) = (x * y) * z
In Proofgold the corresponding term root is 71a504... and proposition id is d322c1...
Proof:
Proof not loaded.
L24916
Theorem. (mul_nat_mul_SNo)
∀n m ∈ ω, mul_nat n m = n * m
In Proofgold the corresponding term root is da39d3... and proposition id is eb6335...
Proof:
Proof not loaded.
L24958
Theorem. (mul_SNo_In_omega)
∀n m ∈ ω, n * m ∈ ω
In Proofgold the corresponding term root is 7b6023... and proposition id is 8435a4...
Proof:
Proof not loaded.
L24964
Theorem. (mul_SNo_zeroL)
∀x, SNo x → 0 * x = 0
In Proofgold the corresponding term root is ddc2f5... and proposition id is aaf0da...
Proof:
Proof not loaded.
L24970
Theorem. (mul_SNo_oneL)
∀x, SNo x → 1 * x = x
In Proofgold the corresponding term root is 6f8b10... and proposition id is 1b25c9...
Proof:
Proof not loaded.
L24976
Theorem. (mul_SNo_rotate_3_1)
∀x y z, SNo x → SNo y → SNo z → x * y * z = z * x * y
In Proofgold the corresponding term root is 920309... and proposition id is 15ee6c...
Proof:
Proof not loaded.
L24990
Theorem. (pos_mul_SNo_Lt)
∀x y z, SNo x → 0 < x → SNo y → SNo z → y < z → x * y < x * z
In Proofgold the corresponding term root is 674213... and proposition id is 0e949e...
Proof:
Proof not loaded.
L25008
Theorem. (nonneg_mul_SNo_Le)
∀x y z, SNo x → 0 ≤ x → SNo y → SNo z → y ≤ z → x * y ≤ x * z
In Proofgold the corresponding term root is b8880b... and proposition id is ae94b7...
Proof:
Proof not loaded.
L25026
Theorem. (neg_mul_SNo_Lt)
∀x y z, SNo x → x < 0 → SNo y → SNo z → z < y → x * y < x * z
In Proofgold the corresponding term root is 713502... and proposition id is 63326a...
Proof:
Proof not loaded.
L25044
Theorem. (pos_mul_SNo_Lt')
∀x y z, SNo x → SNo y → SNo z → 0 < z → x < y → x * z < y * z
In Proofgold the corresponding term root is a08fe5... and proposition id is 436be0...
Proof:
Proof not loaded.
L25051
Theorem. (mul_SNo_Lt1_pos_Lt)
∀x y, SNo x → SNo y → x < 1 → 0 < y → x * y < y
In Proofgold the corresponding term root is c5e733... and proposition id is 9c395c...
Proof:
Proof not loaded.
L25058
Theorem. (nonneg_mul_SNo_Le')
∀x y z, SNo x → SNo y → SNo z → 0 ≤ z → x ≤ y → x * z ≤ y * z
In Proofgold the corresponding term root is 1577b1... and proposition id is e85848...
Proof:
Proof not loaded.
L25065
Theorem. (mul_SNo_Le1_nonneg_Le)
∀x y, SNo x → SNo y → x ≤ 1 → 0 ≤ y → x * y ≤ y
In Proofgold the corresponding term root is 26eb53... and proposition id is 655b0c...
Proof:
Proof not loaded.
L25072
Theorem. (pos_mul_SNo_Lt2)
∀x y z w, SNo x → SNo y → SNo z → SNo w → 0 < x → 0 < y → x < z → y < w → x * y < z * w
In Proofgold the corresponding term root is f53e1e... and proposition id is d23a11...
Proof:
Proof not loaded.
L25087
Theorem. (nonneg_mul_SNo_Le2)
∀x y z w, SNo x → SNo y → SNo z → SNo w → 0 ≤ x → 0 ≤ y → x ≤ z → y ≤ w → x * y ≤ z * w
In Proofgold the corresponding term root is 37f3f7... and proposition id is e53b91...
Proof:
Proof not loaded.
L25102
Theorem. (mul_SNo_pos_pos)
∀x y, SNo x → SNo y → 0 < x → 0 < y → 0 < x * y
In Proofgold the corresponding term root is fda250... and proposition id is d984dc...
Proof:
Proof not loaded.
L25118
Theorem. (mul_SNo_pos_neg)
∀x y, SNo x → SNo y → 0 < x → y < 0 → x * y < 0
In Proofgold the corresponding term root is 4dec1e... and proposition id is 697e77...
Proof:
Proof not loaded.
L25132
Theorem. (mul_SNo_neg_pos)
∀x y, SNo x → SNo y → x < 0 → 0 < y → x * y < 0
In Proofgold the corresponding term root is 6f7ff5... and proposition id is 6e7adc...
Proof:
Proof not loaded.
L25146
Theorem. (mul_SNo_neg_neg)
∀x y, SNo x → SNo y → x < 0 → y < 0 → 0 < x * y
In Proofgold the corresponding term root is 86d604... and proposition id is 03517a...
Proof:
Proof not loaded.
L25162
Theorem. (mul_SNo_nonneg_nonneg)
∀x y, SNo x → SNo y → 0 ≤ x → 0 ≤ y → 0 ≤ x * y
In Proofgold the corresponding term root is 53d610... and proposition id is fc765b...
Proof:
Proof not loaded.
L25178
Theorem. (mul_SNo_nonpos_pos)
∀x y, SNo x → SNo y → x ≤ 0 → 0 < y → x * y ≤ 0
In Proofgold the corresponding term root is c597ec... and proposition id is fcd532...
Proof:
Proof not loaded.
L25195
Theorem. (mul_SNo_nonpos_neg)
∀x y, SNo x → SNo y → x ≤ 0 → y < 0 → 0 ≤ x * y
In Proofgold the corresponding term root is 54e1f5... and proposition id is 906128...
Proof:
Proof not loaded.
L25212
Theorem. (nonpos_mul_SNo_Le)
∀x y z, SNo x → x ≤ 0 → SNo y → SNo z → z ≤ y → x * y ≤ x * z
In Proofgold the corresponding term root is ebad85... and proposition id is c0938c...
Proof:
Proof not loaded.
L25240
Theorem. (SNo_zero_or_sqr_pos)
∀x, SNo x → x = 0 ∨ 0 < x * x
In Proofgold the corresponding term root is 4929cf... and proposition id is 489689...
Proof:
Proof not loaded.
L25258
Theorem. (SNo_pos_sqr_uniq)
∀x y, SNo x → SNo y → 0 < x → 0 < y → x * x = y * y → x = y
In Proofgold the corresponding term root is f41831... and proposition id is 31cc9e...
Proof:
Proof not loaded.
L25277
Theorem. (SNo_nonneg_sqr_uniq)
∀x y, SNo x → SNo y → 0 ≤ x → 0 ≤ y → x * x = y * y → x = y
In Proofgold the corresponding term root is 68d62c... and proposition id is d45b76...
Proof:
Proof not loaded.
L25302
Theorem. (SNo_foil)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + y) * (z + w) = x * z + x * w + y * z + y * w
In Proofgold the corresponding term root is d65f34... and proposition id is ab6bae...
Proof:
Proof not loaded.
L25328
Theorem. (mul_SNo_minus_minus)
∀x y, SNo x → SNo y → (- x) * (- y) = x * y
In Proofgold the corresponding term root is cc94f8... and proposition id is 2674e5...
Proof:
Proof not loaded.
L25337
Theorem. (mul_SNo_com_3_0_1)
∀x y z, SNo x → SNo y → SNo z → x * y * z = y * x * z
In Proofgold the corresponding term root is 00a644... and proposition id is cda6e7...
Proof:
Proof not loaded.
L25347
Theorem. (mul_SNo_com_3b_1_2)
∀x y z, SNo x → SNo y → SNo z → (x * y) * z = (x * z) * y
In Proofgold the corresponding term root is 91c3e7... and proposition id is e48f00...
Proof:
Proof not loaded.
L25357
Theorem. (mul_SNo_com_4_inner_mid)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x * y) * (z * w) = (x * z) * (y * w)
In Proofgold the corresponding term root is 0bafcb... and proposition id is e23f15...
Proof:
Proof not loaded.
L25369
Theorem. (SNo_foil_mm)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x + - y) * (z + - w) = x * z + - x * w + - y * z + y * w
In Proofgold the corresponding term root is c5dda6... and proposition id is 6b5f54...
Proof:
Proof not loaded.
L25384
Theorem. (mul_SNo_nonzero_cancel)
∀x y z, SNo x → x ≠ 0 → SNo y → SNo z → x * y = x * z → y = z
In Proofgold the corresponding term root is 67c26c... and proposition id is 035c5c...
Proof:
Proof not loaded.
L25420
Theorem. (mul_SNoCutP_lem)
∀Lx Rx Ly Ry x y, SNoCutP Lx Rx → SNoCutP Ly Ry → x = SNoCut Lx Rx → y = SNoCut Ly Ry → SNoCutP ({w 0 * y + x * w 1 + - w 0 * w 1|w ∈ Lx ⨯ Ly} ∪ {z 0 * y + x * z 1 + - z 0 * z 1|z ∈ Rx ⨯ Ry}) ({w 0 * y + x * w 1 + - w 0 * w 1|w ∈ Lx ⨯ Ry} ∪ {z 0 * y + x * z 1 + - z 0 * z 1|z ∈ Rx ⨯ Ly}) ∧ x * y = SNoCut ({w 0 * y + x * w 1 + - w 0 * w 1|w ∈ Lx ⨯ Ly} ∪ {z 0 * y + x * z 1 + - z 0 * z 1|z ∈ Rx ⨯ Ry}) ({w 0 * y + x * w 1 + - w 0 * w 1|w ∈ Lx ⨯ Ry} ∪ {z 0 * y + x * z 1 + - z 0 * z 1|z ∈ Rx ⨯ Ly}) ∧ ∀q : prop, (∀LxLy' RxRy' LxRy' RxLy', (∀u ∈ LxLy', ∀p : prop, (∀w ∈ Lx, ∀w' ∈ Ly, SNo w → SNo w' → w < x → w' < y → u = w * y + x * w' + - w * w' → p) → p) → (∀w ∈ Lx, ∀w' ∈ Ly, w * y + x * w' + - w * w' ∈ LxLy') → (∀u ∈ RxRy', ∀p : prop, (∀z ∈ Rx, ∀z' ∈ Ry, SNo z → SNo z' → x < z → y < z' → u = z * y + x * z' + - z * z' → p) → p) → (∀z ∈ Rx, ∀z' ∈ Ry, z * y + x * z' + - z * z' ∈ RxRy') → (∀u ∈ LxRy', ∀p : prop, (∀w ∈ Lx, ∀z ∈ Ry, SNo w → SNo z → w < x → y < z → u = w * y + x * z + - w * z → p) → p) → (∀w ∈ Lx, ∀z ∈ Ry, w * y + x * z + - w * z ∈ LxRy') → (∀u ∈ RxLy', ∀p : prop, (∀z ∈ Rx, ∀w ∈ Ly, SNo z → SNo w → x < z → w < y → u = z * y + x * w + - z * w → p) → p) → (∀z ∈ Rx, ∀w ∈ Ly, z * y + x * w + - z * w ∈ RxLy') → SNoCutP (LxLy' ∪ RxRy') (LxRy' ∪ RxLy') → x * y = SNoCut (LxLy' ∪ RxRy') (LxRy' ∪ RxLy') → q) → q
In Proofgold the corresponding term root is 912d2d... and proposition id is 4f23eb...
Proof:
Proof not loaded.
L26713
Theorem. (mul_SNoCut_abs)
∀Lx Rx Ly Ry x y, SNoCutP Lx Rx → SNoCutP Ly Ry → x = SNoCut Lx Rx → y = SNoCut Ly Ry → ∀q : prop, (∀LxLy' RxRy' LxRy' RxLy', (∀u ∈ LxLy', ∀p : prop, (∀w ∈ Lx, ∀w' ∈ Ly, SNo w → SNo w' → w < x → w' < y → u = w * y + x * w' + - w * w' → p) → p) → (∀w ∈ Lx, ∀w' ∈ Ly, w * y + x * w' + - w * w' ∈ LxLy') → (∀u ∈ RxRy', ∀p : prop, (∀z ∈ Rx, ∀z' ∈ Ry, SNo z → SNo z' → x < z → y < z' → u = z * y + x * z' + - z * z' → p) → p) → (∀z ∈ Rx, ∀z' ∈ Ry, z * y + x * z' + - z * z' ∈ RxRy') → (∀u ∈ LxRy', ∀p : prop, (∀w ∈ Lx, ∀z ∈ Ry, SNo w → SNo z → w < x → y < z → u = w * y + x * z + - w * z → p) → p) → (∀w ∈ Lx, ∀z ∈ Ry, w * y + x * z + - w * z ∈ LxRy') → (∀u ∈ RxLy', ∀p : prop, (∀z ∈ Rx, ∀w ∈ Ly, SNo z → SNo w → x < z → w < y → u = z * y + x * w + - z * w → p) → p) → (∀z ∈ Rx, ∀w ∈ Ly, z * y + x * w + - z * w ∈ RxLy') → SNoCutP (LxLy' ∪ RxRy') (LxRy' ∪ RxLy') → x * y = SNoCut (LxLy' ∪ RxRy') (LxRy' ∪ RxLy') → q) → q
In Proofgold the corresponding term root is 0c7af3... and proposition id is 32f888...
Proof:
Proof not loaded.
L26738
Theorem. (mul_SNo_SNoCut_SNoL_interpolate)
∀Lx Rx Ly Ry, SNoCutP Lx Rx → SNoCutP Ly Ry → ∀x y, x = SNoCut Lx Rx → y = SNoCut Ly Ry → ∀u ∈ SNoL (x * y), (∃v ∈ Lx, ∃w ∈ Ly, u + v * w ≤ v * y + x * w) ∨ (∃v ∈ Rx, ∃w ∈ Ry, u + v * w ≤ v * y + x * w)
In Proofgold the corresponding term root is a1d92b... and proposition id is cc79b7...
Proof:
Proof not loaded.
L26978
Theorem. (mul_SNo_SNoCut_SNoL_interpolate_impred)
∀Lx Rx Ly Ry, SNoCutP Lx Rx → SNoCutP Ly Ry → ∀x y, x = SNoCut Lx Rx → y = SNoCut Ly Ry → ∀u ∈ SNoL (x * y), ∀p : prop, (∀v ∈ Lx, ∀w ∈ Ly, u + v * w ≤ v * y + x * w → p) → (∀v ∈ Rx, ∀w ∈ Ry, u + v * w ≤ v * y + x * w → p) → p
In Proofgold the corresponding term root is 600e8b... and proposition id is f55f47...
Proof:
Proof not loaded.
L27004
Theorem. (mul_SNo_SNoCut_SNoR_interpolate)
∀Lx Rx Ly Ry, SNoCutP Lx Rx → SNoCutP Ly Ry → ∀x y, x = SNoCut Lx Rx → y = SNoCut Ly Ry → ∀u ∈ SNoR (x * y), (∃v ∈ Lx, ∃w ∈ Ry, v * y + x * w ≤ u + v * w) ∨ (∃v ∈ Rx, ∃w ∈ Ly, v * y + x * w ≤ u + v * w)
In Proofgold the corresponding term root is 1f215f... and proposition id is a2fa89...
Proof:
Proof not loaded.
L27244
Theorem. (mul_SNo_SNoCut_SNoR_interpolate_impred)
∀Lx Rx Ly Ry, SNoCutP Lx Rx → SNoCutP Ly Ry → ∀x y, x = SNoCut Lx Rx → y = SNoCut Ly Ry → ∀u ∈ SNoR (x * y), ∀p : prop, (∀v ∈ Lx, ∀w ∈ Ry, v * y + x * w ≤ u + v * w → p) → (∀v ∈ Rx, ∀w ∈ Ly, v * y + x * w ≤ u + v * w → p) → p
In Proofgold the corresponding term root is 544226... and proposition id is a2323a...
Proof:
Proof not loaded.
L27270
Theorem. (nonpos_nonneg_0)
∀m n ∈ ω, m = - n → m = 0 ∧ n = 0
In Proofgold the corresponding term root is 94b9ef... and proposition id is bac05d...
Proof:
Proof not loaded.
L27302
Theorem. (mul_minus_SNo_distrR)
∀x y, SNo x → SNo y → x * (- y) = - (x * y)
In Proofgold the corresponding term root is ab775a... and proposition id is 2e7475...
Proof:
Proof not loaded.
End of Section SurrealMul
Beginning of Section Int
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
L27322
Definition. We define int to be ω ∪ {- n|n ∈ ω} of type set.
In Proofgold the corresponding term root is 90ee85... and object id is 331669...
L27324
Theorem. (int_SNo_cases)
∀p : set → prop, (∀n ∈ ω, p n) → (∀n ∈ ω, p (- n)) → ∀x ∈ int, p x
In Proofgold the corresponding term root is 60107a... and proposition id is 14278d...
Proof:
Proof not loaded.
L27338
Theorem. (int_3_cases)
∀n ∈ int, ∀p : prop, (∀m ∈ ω, n = - ordsucc m → p) → (n = 0 → p) → (∀m ∈ ω, n = ordsucc m → p) → p
In Proofgold the corresponding term root is 67b16d... and proposition id is 5c24db...
Proof:
Proof not loaded.
L27367
Theorem. (int_SNo)
∀x ∈ int, SNo x
In Proofgold the corresponding term root is 19fdd2... and proposition id is 6089ec...
Proof:
Proof not loaded.
L27375
Theorem. (Subq_omega_int)
ω ⊆ int
In Proofgold the corresponding term root is 405dda... and proposition id is b1bb2b...
Proof:
Proof not loaded.
L27382
Theorem. (int_minus_SNo_omega)
∀n ∈ ω, - n ∈ int
In Proofgold the corresponding term root is 869503... and proposition id is 3a2593...
Proof:
Proof not loaded.
L27390
Theorem. (int_add_SNo_lem)
∀n ∈ ω, ∀m, nat_p m → - n + m ∈ int
In Proofgold the corresponding term root is 8f0047... and proposition id is aab9d5...
Proof:
Proof not loaded.
L27446
Theorem. (int_add_SNo)
∀x y ∈ int, x + y ∈ int
In Proofgold the corresponding term root is d66975... and proposition id is 1036d9...
Proof:
Proof not loaded.
L27475
Theorem. (int_minus_SNo)
∀x ∈ int, - x ∈ int
In Proofgold the corresponding term root is 00e714... and proposition id is b2c591...
Proof:
Proof not loaded.
L27484
Theorem. (int_mul_SNo)
∀x y ∈ int, x * y ∈ int
In Proofgold the corresponding term root is a7bc92... and proposition id is f40564...
Proof:
Proof not loaded.
L27553
Theorem. (nonneg_int_nat_p)
∀n ∈ int, 0 ≤ n → nat_p n
In Proofgold the corresponding term root is 17e526... and proposition id is aa7e89...
Proof:
Proof not loaded.
End of Section Int
Beginning of Section BezoutAndGCD
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
L27591
Theorem. (quotient_remainder_nat)
∀n ∈ ω ∖ {0}, ∀m, nat_p m → ∃q ∈ ω, ∃r ∈ n, m = q * n + r
In Proofgold the corresponding term root is 9215c7... and proposition id is b1d852...
Proof:
Proof not loaded.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
L27672
Theorem. (mul_SNo_nonpos_nonneg)
∀x y, SNo x → SNo y → x ≤ 0 → 0 ≤ y → x * y ≤ 0
In Proofgold the corresponding term root is be40cc... and proposition id is 07ca23...
Proof:
Proof not loaded.
L27683
Theorem. (ordinal_0_In_ordsucc)
∀alpha, ordinal alpha → 0 ∈ ordsucc alpha
In Proofgold the corresponding term root is 62fdc5... and proposition id is a0377c...
Proof:
Proof not loaded.
L27692
Theorem. (ordinal_ordsucc_pos)
∀alpha, ordinal alpha → 0 < ordsucc alpha
In Proofgold the corresponding term root is 9af5d3... and proposition id is 7e3b01...
Proof:
Proof not loaded.
L27699
Theorem. (quotient_remainder_int)
∀n ∈ ω ∖ {0}, ∀m ∈ int, ∃q ∈ int, ∃r ∈ n, m = q * n + r
In Proofgold the corresponding term root is a18a83... and proposition id is 60c3e2...
Proof:
Proof not loaded.
L27842
Definition. We define divides_int to be λm n ⇒ m ∈ int ∧ n ∈ int ∧ ∃k ∈ int, m * k = n of type set → set → prop.
In Proofgold the corresponding term root is c1f22f... and object id is ed2a3f...
L27844
Theorem. (divides_int_ref)
∀n ∈ int, divides_int n n
In Proofgold the corresponding term root is 6f5bb9... and proposition id is 5a3a8d...
Proof:
Proof not loaded.
L27859
Theorem. (divides_int_0)
∀n ∈ int, divides_int n 0
In Proofgold the corresponding term root is f1256b... and proposition id is 0ec1b5...
Proof:
Proof not loaded.
L27876
Theorem. (divides_int_add_SNo)
∀m n k, divides_int m n → divides_int m k → divides_int m (n + k)
In Proofgold the corresponding term root is 8466f3... and proposition id is 8cb5db...
Proof:
Proof not loaded.
L27913
Theorem. (divides_int_mul_SNo)
∀m n m' n', divides_int m m' → divides_int n n' → divides_int (m * n) (m' * n')
In Proofgold the corresponding term root is 6a57ce... and proposition id is b51f19...
Proof:
Proof not loaded.
L27951
Theorem. (divides_nat_divides_int)
∀m n, divides_nat m n → divides_int m n
In Proofgold the corresponding term root is c2b8d5... and proposition id is 5cfa58...
Proof:
Proof not loaded.
L27973
Theorem. (divides_int_divides_nat)
∀m n ∈ ω, divides_int m n → divides_nat m n
In Proofgold the corresponding term root is 57df20... and proposition id is 6a6a50...
Proof:
Proof not loaded.
L28027
Theorem. (divides_int_minus_SNo)
∀m n, divides_int m n → divides_int m (- n)
In Proofgold the corresponding term root is 28e67d... and proposition id is dcc59b...
Proof:
Proof not loaded.
L28051
Theorem. (divides_int_mul_SNo_L)
∀m n, ∀k ∈ int, divides_int m n → divides_int m (n * k)
In Proofgold the corresponding term root is 202890... and proposition id is 1af430...
Proof:
Proof not loaded.
L28081
Theorem. (divides_int_mul_SNo_R)
∀m n, ∀k ∈ int, divides_int m n → divides_int m (k * n)
In Proofgold the corresponding term root is 08581b... and proposition id is 573697...
Proof:
Proof not loaded.
L28094
Theorem. (divides_int_1)
∀n ∈ int, divides_int 1 n
In Proofgold the corresponding term root is c86e00... and proposition id is cb6ed6...
Proof:
Proof not loaded.
L28106
Theorem. (divides_int_pos_Le)
∀m n, divides_int m n → 0 < n → m ≤ n
In Proofgold the corresponding term root is 310556... and proposition id is 586793...
Proof:
Proof not loaded.
L28189
Definition. We define gcd_reln to be λm n d ⇒ divides_int d m ∧ divides_int d n ∧ ∀d', divides_int d' m → divides_int d' n → d' ≤ d of type set → set → set → prop.
In Proofgold the corresponding term root is 9ed1e3... and object id is 5f6921...
L28191
Theorem. (gcd_reln_uniq)
∀a b c d, gcd_reln a b c → gcd_reln a b d → c = d
In Proofgold the corresponding term root is 77eeb9... and proposition id is 2e39ac...
Proof:
Proof not loaded.
L28218
Definition. We define int_lin_comb to be λa b c ⇒ a ∈ int ∧ b ∈ int ∧ c ∈ int ∧ ∃m n ∈ int, m * a + n * b = c of type set → set → set → prop.
In Proofgold the corresponding term root is 5ca407... and object id is 46c13c...
L28220
Theorem. (int_lin_comb_I)
∀a b c ∈ int, (∃m n ∈ int, m * a + n * b = c) → int_lin_comb a b c
In Proofgold the corresponding term root is ee0af1... and proposition id is c74b8d...
Proof:
Proof not loaded.
L28231
Theorem. (int_lin_comb_E)
∀a b c, int_lin_comb a b c → ∀p : prop, (a ∈ int → b ∈ int → c ∈ int → ∀m n ∈ int, m * a + n * b = c → p) → p
In Proofgold the corresponding term root is c38ec0... and proposition id is f9a364...
Proof:
Proof not loaded.
L28247
Theorem. (int_lin_comb_E1)
∀a b c, int_lin_comb a b c → a ∈ int
In Proofgold the corresponding term root is 8760f2... and proposition id is 607ef2...
Proof:
Proof not loaded.
L28253
Theorem. (int_lin_comb_E2)
∀a b c, int_lin_comb a b c → b ∈ int
In Proofgold the corresponding term root is d61e74... and proposition id is 239909...
Proof:
Proof not loaded.
L28259
Theorem. (int_lin_comb_E3)
∀a b c, int_lin_comb a b c → c ∈ int
In Proofgold the corresponding term root is 2d6f43... and proposition id is 4114d5...
Proof:
Proof not loaded.
L28265
Theorem. (int_lin_comb_E4)
∀a b c, int_lin_comb a b c → ∀p : prop, (∀m n ∈ int, m * a + n * b = c → p) → p
In Proofgold the corresponding term root is a419d4... and proposition id is 7b482f...
Proof:
Proof not loaded.
L28274
Theorem. (least_pos_int_lin_comb_ex)
∀a b ∈ int, ¬ (a = 0 ∧ b = 0) → ∃c, int_lin_comb a b c ∧ 0 < c ∧ ∀c', int_lin_comb a b c' → 0 < c' → c ≤ c'
In Proofgold the corresponding term root is adb37d... and proposition id is 3821bd...
Proof:
Proof not loaded.
L28474
Theorem. (int_lin_comb_sym)
∀a b d, int_lin_comb a b d → int_lin_comb b a d
In Proofgold the corresponding term root is 78fc6f... and proposition id is 75991b...
Proof:
Proof not loaded.
L28495
Theorem. (least_pos_int_lin_comb_divides_int)
∀a b d, int_lin_comb a b d → 0 < d → (∀c, int_lin_comb a b c → 0 < c → d ≤ c) → divides_int d a
In Proofgold the corresponding term root is f354de... and proposition id is 68c498...
Proof:
Proof not loaded.
L28653
Theorem. (least_pos_int_lin_comb_gcd)
∀a b d, int_lin_comb a b d → 0 < d → (∀c, int_lin_comb a b c → 0 < c → d ≤ c) → gcd_reln a b d
In Proofgold the corresponding term root is 9c320d... and proposition id is f5a534...
Proof:
Proof not loaded.
L28733
Theorem. (BezoutThm)
∀a b ∈ int, ¬ (a = 0 ∧ b = 0) → ∀d, gcd_reln a b d ↔ int_lin_comb a b d ∧ 0 < d ∧ ∀d', int_lin_comb a b d' → 0 < d' → d ≤ d'
In Proofgold the corresponding term root is 01513f... and proposition id is ed22be...
Proof:
Proof not loaded.
L28764
Theorem. (gcd_id)
∀m ∈ ω ∖ {0}, gcd_reln m m m
In Proofgold the corresponding term root is 97c37f... and proposition id is 4d0b43...
Proof:
Proof not loaded.
L28863
Theorem. (gcd_0)
∀m ∈ ω ∖ {0}, gcd_reln 0 m m
In Proofgold the corresponding term root is a22353... and proposition id is f93fc3...
Proof:
Proof not loaded.
L28890
Theorem. (gcd_sym)
∀m n d, gcd_reln m n d → gcd_reln n m d
In Proofgold the corresponding term root is 0a399e... and proposition id is ac8b69...
Proof:
Proof not loaded.
L28905
Theorem. (gcd_minus)
∀m n d, gcd_reln m n d → gcd_reln m (- n) d
In Proofgold the corresponding term root is 261ac3... and proposition id is d054b8...
Proof:
Proof not loaded.
L28928
Theorem. (euclidean_algorithm_prop_1)
∀m n d, n ∈ int → gcd_reln m (n + - m) d → gcd_reln m n d
In Proofgold the corresponding term root is a6b8d4... and proposition id is 29c3b8...
Proof:
Proof not loaded.
L28964
Theorem. (euclidean_algorithm)
(∀m ∈ ω ∖ {0}, gcd_reln m m m) ∧ (∀m ∈ ω ∖ {0}, gcd_reln 0 m m) ∧ (∀m ∈ ω ∖ {0}, gcd_reln m 0 m) ∧ (∀m n ∈ ω, m < n → ∀d, gcd_reln m (n + - m) d → gcd_reln m n d) ∧ (∀m n ∈ ω, n < m → ∀d, gcd_reln n m d → gcd_reln m n d) ∧ (∀m ∈ ω, ∀n ∈ int, n < 0 → ∀d, gcd_reln m (- n) d → gcd_reln m n d) ∧ (∀m n ∈ int, m < 0 → ∀d, gcd_reln (- m) n d → gcd_reln m n d)
In Proofgold the corresponding term root is d87f07... and proposition id is 24974e...
Proof:
Proof not loaded.
L29011
Theorem. (Euclid_lemma)
∀p, prime_nat p → ∀a b ∈ int, divides_int p (a * b) → divides_int p a ∨ divides_int p b
In Proofgold the corresponding term root is 786cc5... and proposition id is 2d187a...
Proof:
Proof not loaded.
End of Section BezoutAndGCD
Beginning of Section PrimeFactorization
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
L29133
Theorem. (prime_not_divides_int_1)
∀p, prime_nat p → ¬ divides_int p 1
In Proofgold the corresponding term root is b9f761... and proposition id is 10be61...
Proof:
Proof not loaded.
L29157
Definition. We define Pi_SNo to be λf n ⇒ nat_primrec 1 (λi r ⇒ r * f i) n of type (set → set) → set → set.
In Proofgold the corresponding term root is 3e6f6f... and object id is 05ecb8...
L29160
Theorem. (Pi_SNo_0)
∀f : set → set, Pi_SNo f 0 = 1
In Proofgold the corresponding term root is 89e731... and proposition id is b6e187...
Proof:
Proof not loaded.
L29165
Theorem. (Pi_SNo_S)
∀f : set → set, ∀n, nat_p n → Pi_SNo f (ordsucc n) = Pi_SNo f n * f n
In Proofgold the corresponding term root is bfe386... and proposition id is 33475e...
Proof:
Proof not loaded.
L29171
Theorem. (Pi_SNo_In_omega)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, f i ∈ ω) → Pi_SNo f n ∈ ω
In Proofgold the corresponding term root is b7aa6a... and proposition id is 63bfef...
Proof:
Proof not loaded.
L29196
Theorem. (Pi_SNo_In_int)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, f i ∈ int) → Pi_SNo f n ∈ int
In Proofgold the corresponding term root is 011f99... and proposition id is 22c84f...
Proof:
Proof not loaded.
L29221
Theorem. (divides_int_prime_nat_eq)
∀p q, prime_nat p → prime_nat q → divides_int p q → p = q
In Proofgold the corresponding term root is 5e30a7... and proposition id is d07df4...
Proof:
Proof not loaded.
L29246
Theorem. (Euclid_lemma_Pi_SNo)
∀f : set → set, ∀p, prime_nat p → ∀n, nat_p n → (∀i ∈ n, f i ∈ int) → divides_int p (Pi_SNo f n) → ∃i ∈ n, divides_int p (f i)
In Proofgold the corresponding term root is 50a15b... and proposition id is 2326d1...
Proof:
Proof not loaded.
L29287
Theorem. (divides_nat_mul_SNo_R)
∀m n ∈ ω, divides_nat m (m * n)
In Proofgold the corresponding term root is 68b6fa... and proposition id is a297e7...
Proof:
Proof not loaded.
L29301
Theorem. (divides_nat_mul_SNo_L)
∀m n ∈ ω, divides_nat n (m * n)
In Proofgold the corresponding term root is 1f5742... and proposition id is a0cc9d...
Proof:
Proof not loaded.
L29308
Theorem. (Pi_SNo_divides)
∀f : set → set, ∀n, nat_p n → (∀i ∈ n, f i ∈ ω) → (∀i ∈ n, divides_nat (f i) (Pi_SNo f n))
In Proofgold the corresponding term root is dd587f... and proposition id is 3899cc...
Proof:
Proof not loaded.
L29340
Definition. We define nonincrfinseq to be λA n f ⇒ ∀i ∈ n, A (f i) ∧ ∀j ∈ i, f i ≤ f j of type (set → prop) → set → (set → set) → prop.
In Proofgold the corresponding term root is f77d44... and object id is 168aa7...
L29342
Theorem. (Pi_SNo_eq)
∀f g : set → set, ∀m, nat_p m → (∀i ∈ m, f i = g i) → Pi_SNo f m = Pi_SNo g m
In Proofgold the corresponding term root is c1ba17... and proposition id is 9a5d69...
Proof:
Proof not loaded.
L29365
Theorem. (prime_factorization_ex_uniq)
∀n, nat_p n → 0 ∈ n → ∃k ∈ ω, ∃f : set → set, nonincrfinseq prime_nat k f ∧ Pi_SNo f k = n ∧ ∀k' ∈ ω, ∀f' : set → set, nonincrfinseq prime_nat k' f' → Pi_SNo f' k' = n → k' = k ∧ ∀i ∈ k, f' i = f i
In Proofgold the corresponding term root is ae2a40... and proposition id is a96cd8...
Proof:
Proof not loaded.
End of Section PrimeFactorization
Beginning of Section SurrealExp
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
L29865
Definition. We define exp_SNo_nat to be λn m : set ⇒ nat_primrec 1 (λ_ r ⇒ n * r) m of type set → set → set.
In Proofgold the corresponding term root is cc5143... and object id is ea8104...
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
L29867
Theorem. (exp_SNo_nat_0)
∀x, SNo x → x ^ 0 = 1
In Proofgold the corresponding term root is c266f2... and proposition id is fe30fb...
Proof:
Proof not loaded.
L29872
Theorem. (exp_SNo_nat_S)
∀x, SNo x → ∀n, nat_p n → x ^ (ordsucc n) = x * x ^ n
In Proofgold the corresponding term root is 6b5ef9... and proposition id is ae55b7...
Proof:
Proof not loaded.
L29877
Theorem. (exp_SNo_nat_1)
∀x, SNo x → x ^ 1 = x
In Proofgold the corresponding term root is 9a6103... and proposition id is d52b56...
Proof:
Proof not loaded.
L29886
Theorem. (SNo_exp_SNo_nat)
∀x, SNo x → ∀n, nat_p n → SNo (x ^ n)
In Proofgold the corresponding term root is 7a77b4... and proposition id is f317bf...
Proof:
Proof not loaded.
L29896
Theorem. (nat_exp_SNo_nat)
∀x, nat_p x → ∀n, nat_p n → nat_p (x ^ n)
In Proofgold the corresponding term root is 14de73... and proposition id is 39327e...
Proof:
Proof not loaded.
L29911
Theorem. (eps_ordsucc_half_add)
∀n, nat_p n → eps_ (ordsucc n) + eps_ (ordsucc n) = eps_ n
In Proofgold the corresponding term root is 6f06cd... and proposition id is 1a029d...
Proof:
Proof not loaded.
L30071
Theorem. (eps_1_half_eq1)
eps_ 1 + eps_ 1 = 1
In Proofgold the corresponding term root is 356e75... and proposition id is 2163a7...
Proof:
Proof not loaded.
L30077
Theorem. (eps_1_half_eq2)
2 * eps_ 1 = 1
In Proofgold the corresponding term root is 69f0c6... and proposition id is 95d3cb...
Proof:
Proof not loaded.
L30086
Theorem. (double_eps_1)
∀x y z, SNo x → SNo y → SNo z → x + x = y + z → x = eps_ 1 * (y + z)
In Proofgold the corresponding term root is 42def0... and proposition id is 896286...
Proof:
Proof not loaded.
L30107
Theorem. (exp_SNo_1_bd)
∀x, SNo x → 1 ≤ x → ∀n, nat_p n → 1 ≤ x ^ n
In Proofgold the corresponding term root is a5c549... and proposition id is 002e5b...
Proof:
Proof not loaded.
L30127
Theorem. (exp_SNo_2_bd)
∀n, nat_p n → n < 2 ^ n
In Proofgold the corresponding term root is 1297d1... and proposition id is 88dd94...
Proof:
Proof not loaded.
L30161
Theorem. (mul_SNo_eps_power_2)
∀n, nat_p n → eps_ n * 2 ^ n = 1
In Proofgold the corresponding term root is 6921ff... and proposition id is e69e44...
Proof:
Proof not loaded.
L30186
Theorem. (eps_bd_1)
∀n ∈ ω, eps_ n ≤ 1
In Proofgold the corresponding term root is b781ab... and proposition id is 1165c4...
Proof:
Proof not loaded.
L30204
Theorem. (mul_SNo_eps_power_2')
∀n, nat_p n → 2 ^ n * eps_ n = 1
In Proofgold the corresponding term root is 721eb5... and proposition id is ce9559...
Proof:
Proof not loaded.
L30213
Theorem. (exp_SNo_nat_mul_add)
∀x, SNo x → ∀m, nat_p m → ∀n, nat_p n → x ^ m * x ^ n = x ^ (m + n)
In Proofgold the corresponding term root is 3333a9... and proposition id is f04547...
Proof:
Proof not loaded.
L30245
Theorem. (exp_SNo_nat_mul_add')
∀x, SNo x → ∀m n ∈ ω, x ^ m * x ^ n = x ^ (m + n)
In Proofgold the corresponding term root is 364123... and proposition id is bde099...
Proof:
Proof not loaded.
L30250
Theorem. (exp_SNo_nat_pos)
∀x, SNo x → 0 < x → ∀n, nat_p n → 0 < x ^ n
In Proofgold the corresponding term root is 795806... and proposition id is 6538ad...
Proof:
Proof not loaded.
L30264
Theorem. (mul_SNo_eps_eps_add_SNo)
∀m n ∈ ω, eps_ m * eps_ n = eps_ (m + n)
In Proofgold the corresponding term root is 857b85... and proposition id is ea6877...
Proof:
Proof not loaded.
L30314
Theorem. (SNoS_omega_Lev_equip)
∀n, nat_p n → equip {x ∈ SNoS_ ω|SNoLev x = n} (2 ^ n)
In Proofgold the corresponding term root is 2712ea... and proposition id is 81ef79...
Proof:
Proof not loaded.
L30711
Theorem. (SNoS_finite)
∀n ∈ ω, finite (SNoS_ n)
In Proofgold the corresponding term root is b18610... and proposition id is c5538e...
Proof:
Proof not loaded.
L30763
Theorem. (SNoS_omega_SNoL_finite)
∀x ∈ SNoS_ ω, finite (SNoL x)
In Proofgold the corresponding term root is bf4159... and proposition id is 473468...
Proof:
Proof not loaded.
L30783
Theorem. (SNoS_omega_SNoR_finite)
∀x ∈ SNoS_ ω, finite (SNoR x)
In Proofgold the corresponding term root is c352d0... and proposition id is eafcc9...
Proof:
Proof not loaded.
End of Section SurrealExp
Beginning of Section SNoMaxMin
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
L30816
Definition. We define SNo_max_of to be λX x ⇒ x ∈ X ∧ SNo x ∧ ∀y ∈ X, SNo y → y ≤ x of type set → set → prop.
In Proofgold the corresponding term root is 9570d4... and object id is 0e9f0e...
L30817
Definition. We define SNo_min_of to be λX x ⇒ x ∈ X ∧ SNo x ∧ ∀y ∈ X, SNo y → x ≤ y of type set → set → prop.
In Proofgold the corresponding term root is 5eb8d7... and object id is 144946...
L30818
Theorem. (minus_SNo_max_min)
∀X y, (∀x ∈ X, SNo x) → SNo_max_of X y → SNo_min_of {- x|x ∈ X} (- y)
In Proofgold the corresponding term root is 2403cf... and proposition id is f6c519...
Proof:
Proof not loaded.
L30839
Theorem. (minus_SNo_max_min')
∀X y, (∀x ∈ X, SNo x) → SNo_max_of {- x|x ∈ X} y → SNo_min_of X (- y)
In Proofgold the corresponding term root is 544581... and proposition id is b07737...
Proof:
Proof not loaded.
L30858
Theorem. (minus_SNo_min_max)
∀X y, (∀x ∈ X, SNo x) → SNo_min_of X y → SNo_max_of {- x|x ∈ X} (- y)
In Proofgold the corresponding term root is d435c2... and proposition id is fbf599...
Proof:
Proof not loaded.
L30879
Theorem. (double_SNo_max_1)
∀x y, SNo x → SNo_max_of (SNoL x) y → ∀z, SNo z → x < z → y + z < x + x → ∃w ∈ SNoR z, y + w = x + x
In Proofgold the corresponding term root is 3155e8... and proposition id is d66034...
Proof:
Proof not loaded.
L31071
Theorem. (double_SNo_min_1)
∀x y, SNo x → SNo_min_of (SNoR x) y → ∀z, SNo z → z < x → x + x < y + z → ∃w ∈ SNoL z, y + w = x + x
In Proofgold the corresponding term root is fa7d8b... and proposition id is 084f04...
Proof:
Proof not loaded.
L31141
Theorem. (finite_max_exists)
∀X, (∀x ∈ X, SNo x) → finite X → X ≠ 0 → ∃x, SNo_max_of X x
In Proofgold the corresponding term root is 182021... and proposition id is 9baa0b...
Proof:
Proof not loaded.
L31287
Theorem. (finite_min_exists)
∀X, (∀x ∈ X, SNo x) → finite X → X ≠ 0 → ∃x, SNo_min_of X x
In Proofgold the corresponding term root is efc9f1... and proposition id is 25808a...
Proof:
Proof not loaded.
L31355
Theorem. (SNoS_omega_SNoL_max_exists)
∀x ∈ SNoS_ ω, SNoL x = 0 ∨ ∃y, SNo_max_of (SNoL x) y
In Proofgold the corresponding term root is 4539ed... and proposition id is faaae1...
Proof:
Proof not loaded.
L31370
Theorem. (SNoS_omega_SNoR_min_exists)
∀x ∈ SNoS_ ω, SNoR x = 0 ∨ ∃y, SNo_min_of (SNoR x) y
In Proofgold the corresponding term root is b0aa0c... and proposition id is baaae8...
Proof:
Proof not loaded.
End of Section SNoMaxMin
Beginning of Section DiadicRationals
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
L31396
Theorem. (nonneg_diadic_rational_p_SNoS_omega)
∀k ∈ ω, ∀n, nat_p n → eps_ k * n ∈ SNoS_ ω
In Proofgold the corresponding term root is 9323b5... and proposition id is d9b318...
Proof:
Proof not loaded.
L31420
Definition. We define diadic_rational_p to be λx ⇒ ∃k ∈ ω, ∃m ∈ int, x = eps_ k * m of type set → prop.
In Proofgold the corresponding term root is a02ca9... and object id is 25bca1...
L31422
Theorem. (diadic_rational_p_SNoS_omega)
∀x, diadic_rational_p x → x ∈ SNoS_ ω
In Proofgold the corresponding term root is b7e7af... and proposition id is 3dea67...
Proof:
Proof not loaded.
L31447
Theorem. (int_diadic_rational_p)
∀m ∈ int, diadic_rational_p m
In Proofgold the corresponding term root is 88ce9c... and proposition id is 8a4cf7...
Proof:
Proof not loaded.
L31458
Theorem. (omega_diadic_rational_p)
∀m ∈ ω, diadic_rational_p m
In Proofgold the corresponding term root is 40b048... and proposition id is c250b0...
Proof:
Proof not loaded.
L31463
Theorem. (eps_diadic_rational_p)
∀k ∈ ω, diadic_rational_p (eps_ k)
In Proofgold the corresponding term root is d306dc... and proposition id is cc1098...
Proof:
Proof not loaded.
L31475
Theorem. (minus_SNo_diadic_rational_p)
∀x, diadic_rational_p x → diadic_rational_p (- x)
In Proofgold the corresponding term root is bb6f7d... and proposition id is bc4825...
Proof:
Proof not loaded.
L31501
Theorem. (mul_SNo_diadic_rational_p)
∀x y, diadic_rational_p x → diadic_rational_p y → diadic_rational_p (x * y)
In Proofgold the corresponding term root is ed6a37... and proposition id is e4ff6e...
Proof:
Proof not loaded.
L31543
Theorem. (add_SNo_diadic_rational_p)
∀x y, diadic_rational_p x → diadic_rational_p y → diadic_rational_p (x + y)
In Proofgold the corresponding term root is bd7ab2... and proposition id is 4e6254...
Proof:
Proof not loaded.
L31621
Theorem. (SNoS_omega_diadic_rational_p_lem)
∀n, nat_p n → ∀x, SNo x → SNoLev x = n → diadic_rational_p x
In Proofgold the corresponding term root is 85a5ea... and proposition id is bf97d9...
Proof:
Proof not loaded.
L31826
Theorem. (SNoS_omega_diadic_rational_p)
∀x ∈ SNoS_ ω, diadic_rational_p x
In Proofgold the corresponding term root is a9c764... and proposition id is 963bd5...
Proof:
Proof not loaded.
L31839
Theorem. (mul_SNo_SNoS_omega)
∀x y ∈ SNoS_ ω, x * y ∈ SNoS_ ω
In Proofgold the corresponding term root is d17aa1... and proposition id is 768371...
Proof:
Proof not loaded.
End of Section DiadicRationals
Beginning of Section SurrealDiv
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
L31860
Definition. We define SNoL_pos to be λx ⇒ {w ∈ SNoL x|0 < w} of type set → set.
In Proofgold the corresponding term root is 399f06... and object id is 65d5ad...
L31861
Theorem. (SNo_recip_pos_pos)
∀x xi, SNo x → SNo xi → 0 < x → x * xi = 1 → 0 < xi
In Proofgold the corresponding term root is 8f0063... and proposition id is 5b48db...
Proof:
Proof not loaded.
L31883
Theorem. (SNo_recip_lem1)
∀x x' x'i y y', SNo x → 0 < x → x' ∈ SNoL_pos x → SNo x'i → x' * x'i = 1 → SNo y → x * y < 1 → SNo y' → 1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i → 1 < x * y'
In Proofgold the corresponding term root is 761151... and proposition id is b197c8...
Proof:
Proof not loaded.
L31929
Theorem. (SNo_recip_lem2)
∀x x' x'i y y', SNo x → 0 < x → x' ∈ SNoL_pos x → SNo x'i → x' * x'i = 1 → SNo y → 1 < x * y → SNo y' → 1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i → x * y' < 1
In Proofgold the corresponding term root is 423150... and proposition id is 1a06f3...
Proof:
Proof not loaded.
L31975
Theorem. (SNo_recip_lem3)
∀x x' x'i y y', SNo x → 0 < x → x' ∈ SNoR x → SNo x'i → x' * x'i = 1 → SNo y → x * y < 1 → SNo y' → 1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i → x * y' < 1
In Proofgold the corresponding term root is b6e635... and proposition id is 22668c...
Proof:
Proof not loaded.
L32020
Theorem. (SNo_recip_lem4)
∀x x' x'i y y', SNo x → 0 < x → x' ∈ SNoR x → SNo x'i → x' * x'i = 1 → SNo y → 1 < x * y → SNo y' → 1 + - x * y' = (1 + - x * y) * (x' + - x) * x'i → 1 < x * y'
In Proofgold the corresponding term root is 417306... and proposition id is b572bc...
Proof:
Proof not loaded.
L32065
Definition. We define SNo_recipauxset to be λY x X g ⇒ ⋃y ∈ Y{(1 + (x' + - x) * y) * g x'|x' ∈ X} of type set → set → set → (set → set) → set.
In Proofgold the corresponding term root is f6355f... and object id is d40943...
L32067
Theorem. (SNo_recipauxset_I)
∀Y x X, ∀g : set → set, ∀y ∈ Y, ∀x' ∈ X, (1 + (x' + - x) * y) * g x' ∈ SNo_recipauxset Y x X g
In Proofgold the corresponding term root is fb23ea... and proposition id is 3cd34a...
Proof:
Proof not loaded.
L32076
Theorem. (SNo_recipauxset_E)
∀Y x X, ∀g : set → set, ∀z ∈ SNo_recipauxset Y x X g, ∀p : prop, (∀y ∈ Y, ∀x' ∈ X, z = (1 + (x' + - x) * y) * g x' → p) → p
In Proofgold the corresponding term root is ea0c30... and proposition id is 9c174a...
Proof:
Proof not loaded.
L32088
Theorem. (SNo_recipauxset_ext)
∀Y x X, ∀g h : set → set, (∀x' ∈ X, g x' = h x') → SNo_recipauxset Y x X g = SNo_recipauxset Y x X h
In Proofgold the corresponding term root is 7eb20f... and proposition id is a033a6...
Proof:
Proof not loaded.
L32102
Definition. We define SNo_recipaux to be λx g ⇒ nat_primrec ({0},0) (λk p ⇒ (p 0 ∪ SNo_recipauxset (p 0) x (SNoR x) g ∪ SNo_recipauxset (p 1) x (SNoL_pos x) g,p 1 ∪ SNo_recipauxset (p 0) x (SNoL_pos x) g ∪ SNo_recipauxset (p 1) x (SNoR x) g)) of type set → (set → set) → set → set.
In Proofgold the corresponding term root is 8cdbea... and object id is 6fc28a...
L32110
Theorem. (SNo_recipaux_0)
∀x, ∀g : set → set, SNo_recipaux x g 0 = ({0},0)
In Proofgold the corresponding term root is 1a1104... and proposition id is a9fc4d...
Proof:
Proof not loaded.
L32119
Theorem. (SNo_recipaux_S)
∀x, ∀g : set → set, ∀n, nat_p n → SNo_recipaux x g (ordsucc n) = (SNo_recipaux x g n 0 ∪ SNo_recipauxset (SNo_recipaux x g n 0) x (SNoR x) g ∪ SNo_recipauxset (SNo_recipaux x g n 1) x (SNoL_pos x) g,SNo_recipaux x g n 1 ∪ SNo_recipauxset (SNo_recipaux x g n 0) x (SNoL_pos x) g ∪ SNo_recipauxset (SNo_recipaux x g n 1) x (SNoR x) g)
In Proofgold the corresponding term root is 35af7d... and proposition id is 55b106...
Proof:
Proof not loaded.
L32133
Theorem. (SNo_recipaux_lem1)
∀x, SNo x → 0 < x → ∀g : set → set, (∀x' ∈ SNoS_ (SNoLev x), 0 < x' → SNo (g x') ∧ x' * g x' = 1) → ∀k, nat_p k → (∀y ∈ SNo_recipaux x g k 0, SNo y ∧ x * y < 1) ∧ (∀y ∈ SNo_recipaux x g k 1, SNo y ∧ 1 < x * y)
In Proofgold the corresponding term root is af615d... and proposition id is 7c3309...
Proof:
Proof not loaded.
L32376
Theorem. (SNo_recipaux_lem2)
∀x, SNo x → 0 < x → ∀g : set → set, (∀x' ∈ SNoS_ (SNoLev x), 0 < x' → SNo (g x') ∧ x' * g x' = 1) → SNoCutP (⋃k ∈ ωSNo_recipaux x g k 0) (⋃k ∈ ωSNo_recipaux x g k 1)
In Proofgold the corresponding term root is 38996c... and proposition id is c6f6db...
Proof:
Proof not loaded.
L32437
Theorem. (SNo_recipaux_ext)
∀x, SNo x → ∀g h : set → set, (∀x' ∈ SNoS_ (SNoLev x), g x' = h x') → ∀k, nat_p k → SNo_recipaux x g k = SNo_recipaux x h k
In Proofgold the corresponding term root is b2de41... and proposition id is e7b024...
Proof:
Proof not loaded.
Beginning of Section recip_SNo_pos
L32527
Let G : set → (set → set) → set ≝ λx g ⇒ SNoCut (⋃k ∈ ωSNo_recipaux x g k 0) (⋃k ∈ ωSNo_recipaux x g k 1)
L32528
Definition. We define recip_SNo_pos to be SNo_rec_i G of type set → set.
In Proofgold the corresponding term root is 88641a... and object id is 2f4ee1...
L32529
Theorem. (recip_SNo_pos_eq)
∀x, SNo x → recip_SNo_pos x = G x recip_SNo_pos
In Proofgold the corresponding term root is 2e3e0e... and proposition id is ebab77...
Proof:
Proof not loaded.
L32554
Theorem. (recip_SNo_pos_prop1)
∀x, SNo x → 0 < x → SNo (recip_SNo_pos x) ∧ x * recip_SNo_pos x = 1
In Proofgold the corresponding term root is 42138c... and proposition id is 785262...
Proof:
Proof not loaded.
L33177
Theorem. (SNo_recip_SNo_pos)
∀x, SNo x → 0 < x → SNo (recip_SNo_pos x)
In Proofgold the corresponding term root is 90e6e6... and proposition id is 6b9294...
Proof:
Proof not loaded.
L33182
Theorem. (recip_SNo_pos_invR)
∀x, SNo x → 0 < x → x * recip_SNo_pos x = 1
In Proofgold the corresponding term root is 12c412... and proposition id is a4f83c...
Proof:
Proof not loaded.
L33187
Theorem. (recip_SNo_pos_is_pos)
∀x, SNo x → 0 < x → 0 < recip_SNo_pos x
In Proofgold the corresponding term root is c95744... and proposition id is 30d6a5...
Proof:
Proof not loaded.
L33212
Theorem. (recip_SNo_pos_invol)
∀x, SNo x → 0 < x → recip_SNo_pos (recip_SNo_pos x) = x
In Proofgold the corresponding term root is 5e7409... and proposition id is a3c6c5...
Proof:
Proof not loaded.
L33235
Theorem. (recip_SNo_pos_eps_)
∀n, nat_p n → recip_SNo_pos (eps_ n) = 2 ^ n
In Proofgold the corresponding term root is ce7e5d... and proposition id is 3c7219...
Proof:
Proof not loaded.
L33256
Theorem. (recip_SNo_pos_pow_2)
∀n, nat_p n → recip_SNo_pos (2 ^ n) = eps_ n
In Proofgold the corresponding term root is e4f02a... and proposition id is 9ca0e9...
Proof:
Proof not loaded.
L33262
Theorem. (recip_SNo_pos_2)
recip_SNo_pos 2 = eps_ 1
In Proofgold the corresponding term root is 1953c2... and proposition id is 4addd0...
Proof:
Proof not loaded.
End of Section recip_SNo_pos
L33269
Definition. We define recip_SNo to be λx ⇒ if 0 < x then recip_SNo_pos x else if x < 0 then - recip_SNo_pos (- x) else 0 of type set → set.
In Proofgold the corresponding term root is aa96a2... and object id is 0d39b7...
L33270
Theorem. (recip_SNo_poscase)
∀x, 0 < x → recip_SNo x = recip_SNo_pos x
In Proofgold the corresponding term root is 152688... and proposition id is 2f43fd...
Proof:
Proof not loaded.
L33275
Theorem. (recip_SNo_negcase)
∀x, SNo x → x < 0 → recip_SNo x = - recip_SNo_pos (- x)
In Proofgold the corresponding term root is c96182... and proposition id is 126cc9...
Proof:
Proof not loaded.
L33287
Theorem. (recip_SNo_0)
recip_SNo 0 = 0
In Proofgold the corresponding term root is bda1b2... and proposition id is 466dab...
Proof:
Proof not loaded.
L33294
Theorem. (SNo_recip_SNo)
∀x, SNo x → SNo (recip_SNo x)
In Proofgold the corresponding term root is 675572... and proposition id is 9f0369...
Proof:
Proof not loaded.
L33314
Theorem. (recip_SNo_invR)
∀x, SNo x → x ≠ 0 → x * recip_SNo x = 1
In Proofgold the corresponding term root is 33c26e... and proposition id is 446f16...
Proof:
Proof not loaded.
L33336
Theorem. (recip_SNo_invL)
∀x, SNo x → x ≠ 0 → recip_SNo x * x = 1
In Proofgold the corresponding term root is 85d435... and proposition id is e80fdb...
Proof:
Proof not loaded.
L33343
Theorem. (mul_SNo_nonzero_cancel_L)
∀x y z, SNo x → x ≠ 0 → SNo y → SNo z → x * y = x * z → y = z
In Proofgold the corresponding term root is 67c26c... and proposition id is 035c5c...
Proof:
Proof not loaded.
L33359
Theorem. (recip_SNo_pow_2)
∀n, nat_p n → recip_SNo (2 ^ n) = eps_ n
In Proofgold the corresponding term root is 761b63... and proposition id is 80f328...
Proof:
Proof not loaded.
L33367
Theorem. (recip_SNo_of_pos_is_pos)
∀x, SNo x → 0 < x → 0 < recip_SNo x
In Proofgold the corresponding term root is 8f5692... and proposition id is 1015d4...
Proof:
Proof not loaded.
L33373
Definition. We define div_SNo to be λx y ⇒ x * recip_SNo y of type set → set → set.
In Proofgold the corresponding term root is a77cc8... and object id is 8b12fd...
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
L33377
Theorem. (SNo_div_SNo)
∀x y, SNo x → SNo y → SNo (x :/: y)
In Proofgold the corresponding term root is cebe75... and proposition id is c761b5...
Proof:
Proof not loaded.
L33383
Theorem. (div_SNo_0_num)
∀x, SNo x → 0 :/: x = 0
In Proofgold the corresponding term root is 28c3dc... and proposition id is ec5f74...
Proof:
Proof not loaded.
L33388
Theorem. (div_SNo_0_denum)
∀x, SNo x → x :/: 0 = 0
In Proofgold the corresponding term root is d6eb4e... and proposition id is 46b3d2...
Proof:
Proof not loaded.
L33395
Theorem. (mul_div_SNo_invL)
∀x y, SNo x → SNo y → y ≠ 0 → (x :/: y) * y = x
In Proofgold the corresponding term root is 2484b5... and proposition id is 07d80b...
Proof:
Proof not loaded.
L33405
Theorem. (mul_div_SNo_invR)
∀x y, SNo x → SNo y → y ≠ 0 → y * (x :/: y) = x
In Proofgold the corresponding term root is 9b260e... and proposition id is df334a...
Proof:
Proof not loaded.
L33411
Theorem. (mul_div_SNo_R)
∀x y z, SNo x → SNo y → SNo z → (x :/: y) * z = (x * z) :/: y
In Proofgold the corresponding term root is a78538... and proposition id is daa8eb...
Proof:
Proof not loaded.
L33432
Theorem. (mul_div_SNo_L)
∀x y z, SNo x → SNo y → SNo z → z * (x :/: y) = (z * x) :/: y
In Proofgold the corresponding term root is 95a4c6... and proposition id is c10e5c...
Proof:
Proof not loaded.
L33440
Theorem. (div_mul_SNo_invL)
∀x y, SNo x → SNo y → y ≠ 0 → (x * y) :/: y = x
In Proofgold the corresponding term root is c74971... and proposition id is b99cd5...
Proof:
Proof not loaded.
L33447
Theorem. (div_div_SNo)
∀x y z, SNo x → SNo y → SNo z → (x :/: y) :/: z = x :/: (y * z)
In Proofgold the corresponding term root is 3e9827... and proposition id is de0369...
Proof:
Proof not loaded.
L33497
Theorem. (mul_div_SNo_both)
∀x y z w, SNo x → SNo y → SNo z → SNo w → (x :/: y) * (z :/: w) = (x * z) :/: (y * w)
In Proofgold the corresponding term root is 06b28b... and proposition id is e61f18...
Proof:
Proof not loaded.
L33506
Theorem. (recip_SNo_pos_pos)
∀x, SNo x → 0 < x → 0 < recip_SNo_pos x
In Proofgold the corresponding term root is c95744... and proposition id is 30d6a5...
Proof:
Proof not loaded.
L33522
Theorem. (div_SNo_pos_pos)
∀x y, SNo x → SNo y → 0 < x → 0 < y → 0 < x :/: y
In Proofgold the corresponding term root is acf73b... and proposition id is 9935dd...
Proof:
Proof not loaded.
L33530
Theorem. (div_SNo_neg_pos)
∀x y, SNo x → SNo y → x < 0 → 0 < y → x :/: y < 0
In Proofgold the corresponding term root is ff8d4c... and proposition id is df8e4a...
Proof:
Proof not loaded.
L33538
Theorem. (div_SNo_pos_LtL)
∀x y z, SNo x → SNo y → SNo z → 0 < y → x < z * y → x :/: y < z
In Proofgold the corresponding term root is 6ffbf3... and proposition id is 87c1ac...
Proof:
Proof not loaded.
L33559
Theorem. (div_SNo_pos_LtR)
∀x y z, SNo x → SNo y → SNo z → 0 < y → z * y < x → z < x :/: y
In Proofgold the corresponding term root is caaa93... and proposition id is c36da0...
Proof:
Proof not loaded.
L33580
Theorem. (div_SNo_pos_LtL')
∀x y z, SNo x → SNo y → SNo z → 0 < y → x :/: y < z → x < z * y
In Proofgold the corresponding term root is 6d81ed... and proposition id is de6389...
Proof:
Proof not loaded.
L33599
Theorem. (div_SNo_pos_LtR')
∀x y z, SNo x → SNo y → SNo z → 0 < y → z < x :/: y → z * y < x
In Proofgold the corresponding term root is 0c5a47... and proposition id is a7e3d6...
Proof:
Proof not loaded.
L33618
Theorem. (mul_div_SNo_nonzero_eq)
∀x y z, SNo x → SNo y → SNo z → y ≠ 0 → x = y * z → x :/: y = z
In Proofgold the corresponding term root is 4c4da2... and proposition id is bea168...
Proof:
Proof not loaded.
End of Section SurrealDiv
Beginning of Section Reals
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
L33641
Theorem. (SNoS_omega_drat_intvl)
∀x ∈ SNoS_ ω, ∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k
In Proofgold the corresponding term root is d24dbb... and proposition id is eb9878...
Proof:
Proof not loaded.
L33669
Theorem. (SNoS_ordsucc_omega_bdd_above)
∀x ∈ SNoS_ (ordsucc ω), x < ω → ∃N ∈ ω, x < N
In Proofgold the corresponding term root is c51d13... and proposition id is 7fd9f0...
Proof:
Proof not loaded.
L33716
Theorem. (SNoS_ordsucc_omega_bdd_below)
∀x ∈ SNoS_ (ordsucc ω), - ω < x → ∃N ∈ ω, - N < x
In Proofgold the corresponding term root is d7d14e... and proposition id is 09271c...
Proof:
Proof not loaded.
L33737
Theorem. (SNoS_ordsucc_omega_bdd_drat_intvl)
∀x ∈ SNoS_ (ordsucc ω), - ω < x → x < ω → ∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k
In Proofgold the corresponding term root is 57512e... and proposition id is 0ae7ab...
Proof:
Proof not loaded.
L33875
Definition. We define real to be {x ∈ SNoS_ (ordsucc ω)|x ≠ ω ∧ x ≠ - ω ∧ (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x)} of type set.
In Proofgold the corresponding term root is e26ffa... and object id is 5277ef...
L33877
Theorem. (real_I)
∀x ∈ SNoS_ (ordsucc ω), x ≠ ω → x ≠ - ω → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → x ∈ real
In Proofgold the corresponding term root is b05b9f... and proposition id is 9f45f4...
Proof:
Proof not loaded.
L33893
Theorem. (real_E)
∀x ∈ real, ∀p : prop, (SNo x → SNoLev x ∈ ordsucc ω → x ∈ SNoS_ (ordsucc ω) → - ω < x → x < ω → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k) → p) → p
In Proofgold the corresponding term root is ac0c5d... and proposition id is 447bb9...
Proof:
Proof not loaded.
L33936
Theorem. (real_SNo)
∀x ∈ real, SNo x
In Proofgold the corresponding term root is 17c3d6... and proposition id is d4cbb8...
Proof:
Proof not loaded.
L33941
Theorem. (real_SNoS_omega_prop)
∀x ∈ real, ∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x
In Proofgold the corresponding term root is 6da66c... and proposition id is cb7f5e...
Proof:
Proof not loaded.
L33946
Theorem. (SNoS_omega_real)
SNoS_ ω ⊆ real
In Proofgold the corresponding term root is a469c6... and proposition id is d9411b...
Proof:
Proof not loaded.
L34037
Theorem. (real_0)
0 ∈ real
In Proofgold the corresponding term root is 53bbc2... and proposition id is 493130...
Proof:
Proof not loaded.
L34041
Theorem. (real_1)
1 ∈ real
In Proofgold the corresponding term root is bbf720... and proposition id is 790b4d...
Proof:
Proof not loaded.
L34045
Theorem. (SNoLev_In_real_SNoS_omega)
∀x ∈ real, ∀w, SNo w → SNoLev w ∈ SNoLev x → w ∈ SNoS_ ω
In Proofgold the corresponding term root is defc4e... and proposition id is 30df9a...
Proof:
Proof not loaded.
L34062
Theorem. (real_SNoCut_SNoS_omega)
∀L R ⊆ SNoS_ ω, SNoCutP L R → L ≠ 0 → R ≠ 0 → (∀w ∈ L, ∃w' ∈ L, w < w') → (∀z ∈ R, ∃z' ∈ R, z' < z) → SNoCut L R ∈ real
In Proofgold the corresponding term root is 9f1398... and proposition id is 49872c...
Proof:
Proof not loaded.
L34311
Theorem. (real_SNoCut)
∀L R ⊆ real, SNoCutP L R → L ≠ 0 → R ≠ 0 → (∀w ∈ L, ∃w' ∈ L, w < w') → (∀z ∈ R, ∃z' ∈ R, z' < z) → SNoCut L R ∈ real
In Proofgold the corresponding term root is 6c4b7d... and proposition id is 463b8d...
Proof:
Proof not loaded.
L34708
Theorem. (minus_SNo_prereal_1)
∀x, SNo x → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - - x) < eps_ k) → q = - x)
In Proofgold the corresponding term root is 824b89... and proposition id is 1d4018...
Proof:
Proof not loaded.
L34734
Theorem. (minus_SNo_prereal_2)
∀x, SNo x → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k) → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < - x ∧ - x < q + eps_ k)
In Proofgold the corresponding term root is 68fe96... and proposition id is 8ce93f...
Proof:
Proof not loaded.
L34764
Theorem. (SNo_prereal_incr_lower_pos)
∀x, SNo x → 0 < x → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k) → ∀k ∈ ω, ∀p : prop, (∀q ∈ SNoS_ ω, 0 < q → q < x → x < q + eps_ k → p) → p
In Proofgold the corresponding term root is 96a462... and proposition id is 947a7e...
Proof:
Proof not loaded.
L34850
Theorem. (real_minus_SNo)
∀x ∈ real, - x ∈ real
In Proofgold the corresponding term root is e8e8be... and proposition id is 4b8379...
Proof:
Proof not loaded.
L34877
Theorem. (SNo_prereal_incr_lower_approx)
∀x, SNo x → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k) → ∃f ∈ SNoS_ ωω, ∀n ∈ ω, f n < x ∧ x < f n + eps_ n ∧ ∀i ∈ n, f i < f n
In Proofgold the corresponding term root is 29e857... and proposition id is 85c0ef...
Proof:
Proof not loaded.
L35093
Theorem. (SNo_prereal_decr_upper_approx)
∀x, SNo x → (∀q ∈ SNoS_ ω, (∀k ∈ ω, abs_SNo (q + - x) < eps_ k) → q = x) → (∀k ∈ ω, ∃q ∈ SNoS_ ω, q < x ∧ x < q + eps_ k) → ∃g ∈ SNoS_ ωω, ∀n ∈ ω, g n + - eps_ n < x ∧ x < g n ∧ ∀i ∈ n, g n < g i
In Proofgold the corresponding term root is e28afc... and proposition id is 07ed25...
Proof:
Proof not loaded.
L35156
Theorem. (SNoCutP_SNoCut_lim)
∀lambda, ordinal lambda → (∀alpha ∈ lambda, ordsucc alpha ∈ lambda) → ∀L R ⊆ SNoS_ lambda, SNoCutP L R → SNoLev (SNoCut L R) ∈ ordsucc lambda
In Proofgold the corresponding term root is 1a37c9... and proposition id is f1520f...
Proof:
Proof not loaded.
L35222
Theorem. (SNoCutP_SNoCut_omega)
∀L R ⊆ SNoS_ ω, SNoCutP L R → SNoLev (SNoCut L R) ∈ ordsucc ω
In Proofgold the corresponding term root is d11617... and proposition id is a23f8a...
Proof:
Proof not loaded.
L35227
Theorem. (SNo_approx_real_lem)
∀f g ∈ SNoS_ ωω, (∀n m ∈ ω, f n < g m) → ∀p : prop, (SNoCutP {f n|n ∈ ω} {g n|n ∈ ω} → SNo (SNoCut {f n|n ∈ ω} {g n|n ∈ ω}) → SNoLev (SNoCut {f n|n ∈ ω} {g n|n ∈ ω}) ∈ ordsucc ω → SNoCut {f n|n ∈ ω} {g n|n ∈ ω} ∈ SNoS_ (ordsucc ω) → (∀n ∈ ω, f n < SNoCut {f n|n ∈ ω} {g n|n ∈ ω}) → (∀n ∈ ω, SNoCut {f n|n ∈ ω} {g n|n ∈ ω} < g n) → p) → p
In Proofgold the corresponding term root is d275e7... and proposition id is 3ca0b1...
Proof:
Proof not loaded.
L35328
Theorem. (SNo_approx_real)
∀x, SNo x → ∀f g ∈ SNoS_ ωω, (∀n ∈ ω, f n < x) → (∀n ∈ ω, x < f n + eps_ n) → (∀n ∈ ω, ∀i ∈ n, f i < f n) → (∀n ∈ ω, x < g n) → (∀n ∈ ω, ∀i ∈ n, g n < g i) → x = SNoCut {f n|n ∈ ω} {g n|n ∈ ω} → x ∈ real
In Proofgold the corresponding term root is 8b24c5... and proposition id is fca80e...
Proof:
Proof not loaded.
L35545
Theorem. (SNo_approx_real_rep)
∀x ∈ real, ∀p : prop, (∀f g ∈ SNoS_ ωω, (∀n ∈ ω, f n < x) → (∀n ∈ ω, x < f n + eps_ n) → (∀n ∈ ω, ∀i ∈ n, f i < f n) → (∀n ∈ ω, g n + - eps_ n < x) → (∀n ∈ ω, x < g n) → (∀n ∈ ω, ∀i ∈ n, g n < g i) → SNoCutP {f n|n ∈ ω} {g n|n ∈ ω} → x = SNoCut {f n|n ∈ ω} {g n|n ∈ ω} → p) → p
In Proofgold the corresponding term root is 91befd... and proposition id is a38d26...
Proof:
Proof not loaded.
L35728
Theorem. (real_add_SNo)
∀x y ∈ real, x + y ∈ real
In Proofgold the corresponding term root is 695361... and proposition id is 3c841f...
Proof:
Proof not loaded.
L36204
Theorem. (SNoS_ordsucc_omega_bdd_eps_pos)
∀x ∈ SNoS_ (ordsucc ω), 0 < x → x < ω → ∃N ∈ ω, eps_ N * x < 1
In Proofgold the corresponding term root is 064957... and proposition id is 9096d4...
Proof:
Proof not loaded.
L36255
Theorem. (real_mul_SNo_pos)
∀x y ∈ real, 0 < x → 0 < y → x * y ∈ real
In Proofgold the corresponding term root is dfe1b6... and proposition id is 8ba463...
Proof:
Proof not loaded.
L37620
Theorem. (real_mul_SNo)
∀x y ∈ real, x * y ∈ real
In Proofgold the corresponding term root is b48f84... and proposition id is 51dc42...
Proof:
Proof not loaded.
L37678
Theorem. (nonneg_real_nat_interval)
∀x ∈ real, 0 ≤ x → ∃n ∈ ω, n ≤ x ∧ x < ordsucc n
In Proofgold the corresponding term root is 00f333... and proposition id is a6e1f8...
Proof:
Proof not loaded.
L37757
Theorem. (pos_real_left_approx_double)
∀x ∈ real, 0 < x → x ≠ 2 → (∀m ∈ ω, x ≠ eps_ m) → ∃w ∈ SNoL_pos x, x < 2 * w
In Proofgold the corresponding term root is 721fe7... and proposition id is b80017...
Proof:
Proof not loaded.
L38122
Theorem. (real_recip_SNo_lem1)
∀x, SNo x → x ∈ real → 0 < x → recip_SNo_pos x ∈ real ∧ ∀k, nat_p k → (SNo_recipaux x recip_SNo_pos k 0 ⊆ real) ∧ (SNo_recipaux x recip_SNo_pos k 1 ⊆ real)
In Proofgold the corresponding term root is 129489... and proposition id is cfc2ec...
Proof:
Proof not loaded.
L38866
Theorem. (real_recip_SNo_pos)
∀x ∈ real, 0 < x → recip_SNo_pos x ∈ real
In Proofgold the corresponding term root is 8169ab... and proposition id is 406f08...
Proof:
Proof not loaded.
L38872
Theorem. (real_recip_SNo)
∀x ∈ real, recip_SNo x ∈ real
In Proofgold the corresponding term root is 336956... and proposition id is 64deb9...
Proof:
Proof not loaded.
L38898
Theorem. (real_div_SNo)
∀x y ∈ real, x :/: y ∈ real
In Proofgold the corresponding term root is 1b354d... and proposition id is 4ff694...
Proof:
Proof not loaded.
End of Section Reals
Beginning of Section even_odd
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_nat.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_nat.
L38915
Theorem. (nat_le2_cases)
∀m, nat_p m → m ⊆ 2 → m = 0 ∨ m = 1 ∨ m = 2
In Proofgold the corresponding term root is baf753... and proposition id is 166808...
Proof:
Proof not loaded.
L38936
Theorem. (prime_nat_2_lem)
∀m, nat_p m → ∀n, nat_p n → m * n = 2 → m = 1 ∨ m = 2
In Proofgold the corresponding term root is acdedc... and proposition id is 4e2fe6...
Proof:
Proof not loaded.
L38967
Theorem. (prime_nat_2)
prime_nat 2
In Proofgold the corresponding term root is 3729fc... and proposition id is 9dc412...
Proof:
Proof not loaded.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
L38987
Theorem. (not_eq_2m_2n1)
∀m n ∈ int, 2 * m ≠ 2 * n + 1
In Proofgold the corresponding term root is db57c3... and proposition id is af26fb...
Proof:
Proof not loaded.
End of Section even_odd
Beginning of Section form100_22b
L39029
Let tag : set → set ≝ λalpha ⇒ SetAdjoin alpha {1}
Notation. We use ' as a postfix operator with priority 100 corresponding to applying term tag.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
L39036
Theorem. (atleastp_SNoS_ordsucc_omega_Power_omega)
atleastp (SNoS_ (ordsucc ω)) (𝒫 ω)
In Proofgold the corresponding term root is dcf01d... and proposition id is e3c12b...
Proof:
Proof not loaded.
L39203
Theorem. (Repl_finite)
∀f : set → set, ∀X, finite X → finite {f x|x ∈ X}
In Proofgold the corresponding term root is 9e4868... and proposition id is 56bf85...
Proof:
Proof not loaded.
L39256
Theorem. (infinite_bigger)
∀X ⊆ ω, infinite X → ∀m ∈ ω, ∃n ∈ X, m ∈ n
In Proofgold the corresponding term root is 1f57b0... and proposition id is c36de8...
Proof:
Proof not loaded.
L39285
Theorem. (equip_real_Power_omega)
equip real (𝒫 ω)
In Proofgold the corresponding term root is be2eea... and proposition id is 6e3566...
Proof:
Proof not loaded.
L41035
Theorem. (form100_22_real_uncountable_atleastp)
¬ atleastp real ω
In Proofgold the corresponding term root is 5bec50... and proposition id is e8cda3...
Proof:
Proof not loaded.
L41048
Theorem. (form100_22_real_uncountable_equip)
¬ equip real ω
In Proofgold the corresponding term root is bea95a... and proposition id is 5a8d6a...
Proof:
Proof not loaded.
L41056
Theorem. (form100_22_real_uncountable)
atleastp ω real ∧ ¬ equip real ω
In Proofgold the corresponding term root is 296976... and proposition id is c2bf76...
Proof:
Proof not loaded.
End of Section form100_22b
Beginning of Section rational
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
L41082
Definition. We define rational to be {x ∈ real|∃m ∈ int, ∃n ∈ ω ∖ {0}, x = m :/: n} of type set.
In Proofgold the corresponding term root is efaf16... and object id is 6ea078...
End of Section rational
Beginning of Section form100_3
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
(*** The Denumerability of the Rational Numbers ***)
L41096
Theorem. (Subq_int_SNoS_omega)
int ⊆ SNoS_ ω
In Proofgold the corresponding term root is 947247... and proposition id is 412d00...
Proof:
Proof not loaded.
L41105
Theorem. (Subq_SNoS_omega_rational)
SNoS_ ω ⊆ rational
In Proofgold the corresponding term root is e4a867... and proposition id is 694b2d...
Proof:
Proof not loaded.
L41153
Theorem. (Subq_rational_real)
rational ⊆ real
In Proofgold the corresponding term root is 0691ee... and proposition id is 2749c2...
Proof:
Proof not loaded.
L41157
Theorem. (rational_minus_SNo)
∀q ∈ rational, - q ∈ rational
In Proofgold the corresponding term root is 31c42e... and proposition id is 6d2672...
Proof:
Proof not loaded.
L41216
Definition. We define nat_pair to be λm n ⇒ 2 ^ m * (2 * n + 1) of type set → set → set.
In Proofgold the corresponding term root is 4f6fe3... and object id is 89fb54...
L41218
Theorem. (nat_pair_In_omega)
∀m n ∈ ω, nat_pair m n ∈ ω
In Proofgold the corresponding term root is 6914a5... and proposition id is d0bc54...
Proof:
Proof not loaded.
L41230
Theorem. (nat_pair_0)
∀m n m' n' ∈ ω, nat_pair m n = nat_pair m' n' → m = m'
In Proofgold the corresponding term root is 26314d... and proposition id is 1af62d...
Proof:
Proof not loaded.
L41439
Theorem. (nat_pair_1)
∀m n m' n' ∈ ω, nat_pair m n = nat_pair m' n' → n = n'
In Proofgold the corresponding term root is 7f1deb... and proposition id is 7929c6...
Proof:
Proof not loaded.
L41476
Theorem. (form100_3)
equip ω rational
In Proofgold the corresponding term root is a25be9... and proposition id is 680b23...
Proof:
Proof not loaded.
End of Section form100_3
Beginning of Section SurrealSqrt
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
Notation. We use < as an infix operator with priority 490 and no associativity corresponding to applying term SNoLt.
Notation. We use ≤ as an infix operator with priority 490 and no associativity corresponding to applying term SNoLe.
Notation. We use ^ as an infix operator with priority 342 and which associates to the right corresponding to applying term exp_SNo_nat.
L41810
Definition. We define SNoL_nonneg to be λx ⇒ {w ∈ SNoL x|0 ≤ w} of type set → set.
In Proofgold the corresponding term root is 426e35... and object id is 1ec1d7...
L41811
Theorem. (SNoL_nonneg_0)
SNoL_nonneg 0 = 0
In Proofgold the corresponding term root is ab6dce... and proposition id is 67aa58...
Proof:
Proof not loaded.
L41817
Theorem. (SNoL_nonneg_1)
SNoL_nonneg 1 = 1
In Proofgold the corresponding term root is 74d4e8... and proposition id is e3fc28...
Proof:
Proof not loaded.
L41831
Definition. We define SNo_sqrtauxset to be λY Z x ⇒ ⋃y ∈ Y{(x + y * z) :/: (y + z)|z ∈ Z, 0 < y + z} of type set → set → set → set.
In Proofgold the corresponding term root is 211c64... and object id is 3fc62b...
L41833
Theorem. (SNo_sqrtauxset_I)
∀Y Z x, ∀y ∈ Y, ∀z ∈ Z, 0 < y + z → (x + y * z) :/: (y + z) ∈ SNo_sqrtauxset Y Z x
In Proofgold the corresponding term root is 4b16a1... and proposition id is c09ca4...
Proof:
Proof not loaded.
L41843
Theorem. (SNo_sqrtauxset_E)
∀Y Z x, ∀u ∈ SNo_sqrtauxset Y Z x, ∀p : prop, (∀y ∈ Y, ∀z ∈ Z, 0 < y + z → u = (x + y * z) :/: (y + z) → p) → p
In Proofgold the corresponding term root is 0bec9e... and proposition id is b7810c...
Proof:
Proof not loaded.
L41857
Theorem. (SNo_sqrtauxset_0)
∀Z x, SNo_sqrtauxset 0 Z x = 0
In Proofgold the corresponding term root is 2d39ce... and proposition id is db93fb...
Proof:
Proof not loaded.
L41865
Theorem. (SNo_sqrtauxset_0')
∀Y x, SNo_sqrtauxset Y 0 x = 0
In Proofgold the corresponding term root is b0f585... and proposition id is d57918...
Proof:
Proof not loaded.
L41874
Definition. We define SNo_sqrtaux to be λx g ⇒ nat_primrec ({g w|w ∈ SNoL_nonneg x},{g z|z ∈ SNoR x}) (λk p ⇒ (p 0 ∪ SNo_sqrtauxset (p 0) (p 1) x,p 1 ∪ SNo_sqrtauxset (p 0) (p 0) x ∪ SNo_sqrtauxset (p 1) (p 1) x)) of type set → (set → set) → set → set.
In Proofgold the corresponding term root is e233df... and object id is 2ebc89...
L41881
Theorem. (SNo_sqrtaux_0)
∀x, ∀g : set → set, SNo_sqrtaux x g 0 = ({g w|w ∈ SNoL_nonneg x},{g z|z ∈ SNoR x})
In Proofgold the corresponding term root is d46ed0... and proposition id is 182e5e...
Proof:
Proof not loaded.
L41889
Theorem. (SNo_sqrtaux_S)
∀x, ∀g : set → set, ∀n, nat_p n → SNo_sqrtaux x g (ordsucc n) = (SNo_sqrtaux x g n 0 ∪ SNo_sqrtauxset (SNo_sqrtaux x g n 0) (SNo_sqrtaux x g n 1) x,SNo_sqrtaux x g n 1 ∪ SNo_sqrtauxset (SNo_sqrtaux x g n 0) (SNo_sqrtaux x g n 0) x ∪ SNo_sqrtauxset (SNo_sqrtaux x g n 1) (SNo_sqrtaux x g n 1) x)
In Proofgold the corresponding term root is bb0dd8... and proposition id is 7223d6...
Proof:
Proof not loaded.
L41903
Theorem. (SNo_sqrtaux_mon_lem)
∀x, ∀g : set → set, ∀m, nat_p m → ∀n, nat_p n → SNo_sqrtaux x g m 0 ⊆ SNo_sqrtaux x g (add_nat m n) 0 ∧ SNo_sqrtaux x g m 1 ⊆ SNo_sqrtaux x g (add_nat m n) 1
In Proofgold the corresponding term root is be3c0c... and proposition id is 75cbc1...
Proof:
Proof not loaded.
L41945
Theorem. (SNo_sqrtaux_mon)
∀x, ∀g : set → set, ∀m, nat_p m → ∀n, nat_p n → m ⊆ n → SNo_sqrtaux x g m 0 ⊆ SNo_sqrtaux x g n 0 ∧ SNo_sqrtaux x g m 1 ⊆ SNo_sqrtaux x g n 1
In Proofgold the corresponding term root is 53a992... and proposition id is f80eb0...
Proof:
Proof not loaded.
L41959
Theorem. (SNo_sqrtaux_ext)
∀x, SNo x → ∀g h : set → set, (∀x' ∈ SNoS_ (SNoLev x), g x' = h x') → ∀k, nat_p k → SNo_sqrtaux x g k = SNo_sqrtaux x h k
In Proofgold the corresponding term root is 9e623f... and proposition id is 88e0ac...
Proof:
Proof not loaded.
Beginning of Section sqrt_SNo_nonneg
L42000
Let G : set → (set → set) → set ≝ λx g ⇒ SNoCut (⋃k ∈ ωSNo_sqrtaux x g k 0) (⋃k ∈ ωSNo_sqrtaux x g k 1)
L42001
Definition. We define sqrt_SNo_nonneg to be SNo_rec_i G of type set → set.
In Proofgold the corresponding term root is 71d152... and object id is 20211a...
L42002
Theorem. (sqrt_SNo_nonneg_eq)
∀x, SNo x → sqrt_SNo_nonneg x = G x sqrt_SNo_nonneg
In Proofgold the corresponding term root is 63c3e8... and proposition id is edc75b...
Proof:
Proof not loaded.
L42027
Theorem. (sqrt_SNo_nonneg_prop1a)
∀x, SNo x → 0 ≤ x → (∀w ∈ SNoS_ (SNoLev x), 0 ≤ w → SNo (sqrt_SNo_nonneg w) ∧ 0 ≤ sqrt_SNo_nonneg w ∧ sqrt_SNo_nonneg w * sqrt_SNo_nonneg w = w) → ∀k, nat_p k → (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y ∧ 0 ≤ y ∧ y * y < x) ∧ (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y ∧ 0 ≤ y ∧ x < y * y)
In Proofgold the corresponding term root is 111bf5... and proposition id is 44bfa1...
Proof:
Proof not loaded.
L42465
Theorem. (sqrt_SNo_nonneg_prop1b)
∀x, SNo x → 0 ≤ x → (∀k, nat_p k → (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y ∧ 0 ≤ y ∧ y * y < x) ∧ (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y ∧ 0 ≤ y ∧ x < y * y)) → SNoCutP (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1)
In Proofgold the corresponding term root is 79d32b... and proposition id is 3a96af...
Proof:
Proof not loaded.
L42534
Theorem. (sqrt_SNo_nonneg_prop1c)
∀x, SNo x → 0 ≤ x → SNoCutP (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1) → (∀z ∈ (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1), ∀p : prop, (SNo z → 0 ≤ z → x < z * z → p) → p) → 0 ≤ G x sqrt_SNo_nonneg
In Proofgold the corresponding term root is afcc07... and proposition id is abe562...
Proof:
Proof not loaded.
L42570
Theorem. (sqrt_SNo_nonneg_prop1d)
∀x, SNo x → 0 ≤ x → (∀w ∈ SNoS_ (SNoLev x), 0 ≤ w → SNo (sqrt_SNo_nonneg w) ∧ 0 ≤ sqrt_SNo_nonneg w ∧ sqrt_SNo_nonneg w * sqrt_SNo_nonneg w = w) → SNoCutP (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1) → 0 ≤ G x sqrt_SNo_nonneg → G x sqrt_SNo_nonneg * G x sqrt_SNo_nonneg < x → False
In Proofgold the corresponding term root is 9308b8... and proposition id is 9979dd...
Proof:
Proof not loaded.
L42853
Theorem. (sqrt_SNo_nonneg_prop1e)
∀x, SNo x → 0 ≤ x → (∀w ∈ SNoS_ (SNoLev x), 0 ≤ w → SNo (sqrt_SNo_nonneg w) ∧ 0 ≤ sqrt_SNo_nonneg w ∧ sqrt_SNo_nonneg w * sqrt_SNo_nonneg w = w) → SNoCutP (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1) → 0 ≤ G x sqrt_SNo_nonneg → x < G x sqrt_SNo_nonneg * G x sqrt_SNo_nonneg → False
In Proofgold the corresponding term root is bd498d... and proposition id is c07c4e...
Proof:
Proof not loaded.
L43237
Theorem. (sqrt_SNo_nonneg_prop1)
∀x, SNo x → 0 ≤ x → SNo (sqrt_SNo_nonneg x) ∧ 0 ≤ sqrt_SNo_nonneg x ∧ sqrt_SNo_nonneg x * sqrt_SNo_nonneg x = x
In Proofgold the corresponding term root is db6879... and proposition id is b09b96...
Proof:
Proof not loaded.
End of Section sqrt_SNo_nonneg
L43320
Theorem. (SNo_sqrtaux_0_1_prop)
∀x, SNo x → 0 ≤ x → ∀k, nat_p k → (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y ∧ 0 ≤ y ∧ y * y < x) ∧ (∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y ∧ 0 ≤ y ∧ x < y * y)
In Proofgold the corresponding term root is 955f65... and proposition id is a2bb45...
Proof:
Proof not loaded.
L43331
Theorem. (SNo_sqrtaux_0_prop)
∀x, SNo x → 0 ≤ x → ∀k, nat_p k → ∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 0, SNo y ∧ 0 ≤ y ∧ y * y < x
In Proofgold the corresponding term root is abda0d... and proposition id is 8db9d4...
Proof:
Proof not loaded.
L43340
Theorem. (SNo_sqrtaux_1_prop)
∀x, SNo x → 0 ≤ x → ∀k, nat_p k → ∀y ∈ SNo_sqrtaux x sqrt_SNo_nonneg k 1, SNo y ∧ 0 ≤ y ∧ x < y * y
In Proofgold the corresponding term root is d59bfa... and proposition id is 95a41e...
Proof:
Proof not loaded.
L43349
Theorem. (SNo_sqrt_SNo_SNoCutP)
∀x, SNo x → 0 ≤ x → SNoCutP (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1)
In Proofgold the corresponding term root is 33c123... and proposition id is d04ad9...
Proof:
Proof not loaded.
L43357
Theorem. (SNo_sqrt_SNo_nonneg)
∀x, SNo x → 0 ≤ x → SNo (sqrt_SNo_nonneg x)
In Proofgold the corresponding term root is 016da2... and proposition id is c5c4da...
Proof:
Proof not loaded.
L43363
Theorem. (sqrt_SNo_nonneg_nonneg)
∀x, SNo x → 0 ≤ x → 0 ≤ sqrt_SNo_nonneg x
In Proofgold the corresponding term root is 40638f... and proposition id is 6c1cd0...
Proof:
Proof not loaded.
L43369
Theorem. (sqrt_SNo_nonneg_sqr)
∀x, SNo x → 0 ≤ x → sqrt_SNo_nonneg x * sqrt_SNo_nonneg x = x
In Proofgold the corresponding term root is c79d06... and proposition id is 15814c...
Proof:
Proof not loaded.
L43375
Theorem. (sqrt_SNo_nonneg_0)
sqrt_SNo_nonneg 0 = 0
In Proofgold the corresponding term root is 0afb2a... and proposition id is c6b1a7...
Proof:
Proof not loaded.
L43455
Theorem. (sqrt_SNo_nonneg_1)
sqrt_SNo_nonneg 1 = 1
In Proofgold the corresponding term root is 1801e5... and proposition id is d5a6d5...
Proof:
Proof not loaded.
L43579
Theorem. (sqrt_SNo_nonneg_0inL0)
∀x, SNo x → 0 ≤ x → 0 ∈ SNoLev x → 0 ∈ SNo_sqrtaux x sqrt_SNo_nonneg 0 0
In Proofgold the corresponding term root is afb66e... and proposition id is 6bcc63...
Proof:
Proof not loaded.
L43607
Theorem. (sqrt_SNo_nonneg_Lnonempty)
∀x, SNo x → 0 ≤ x → 0 ∈ SNoLev x → (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 0) ≠ 0
In Proofgold the corresponding term root is 1725fa... and proposition id is c1c8c8...
Proof:
Proof not loaded.
L43625
Theorem. (sqrt_SNo_nonneg_Rnonempty)
∀x, SNo x → 0 ≤ x → 1 ∈ SNoLev x → (⋃k ∈ ωSNo_sqrtaux x sqrt_SNo_nonneg k 1) ≠ 0
In Proofgold the corresponding term root is b397c9... and proposition id is 04384e...
Proof:
Proof not loaded.
L43699
Theorem. (SNo_sqrtauxset_real)
∀Y Z x, Y ⊆ real → Z ⊆ real → x ∈ real → SNo_sqrtauxset Y Z x ⊆ real
In Proofgold the corresponding term root is 5d5b3c... and proposition id is d040c4...
Proof:
Proof not loaded.
L43726
Theorem. (SNo_sqrtauxset_real_nonneg)
∀Y Z x, Y ⊆ {w ∈ real|0 ≤ w} → Z ⊆ {z ∈ real|0 ≤ z} → x ∈ real → 0 ≤ x → SNo_sqrtauxset Y Z x ⊆ {w ∈ real|0 ≤ w}
In Proofgold the corresponding term root is 1c421c... and proposition id is 3226b5...
Proof:
Proof not loaded.
L43797
Theorem. (sqrt_SNo_nonneg_SNoS_omega)
∀x ∈ SNoS_ ω, 0 ≤ x → sqrt_SNo_nonneg x ∈ real
In Proofgold the corresponding term root is 6aa091... and proposition id is 85b0b0...
Proof:
Proof not loaded.
L44221
Theorem. (sqrt_SNo_nonneg_real)
∀x ∈ real, 0 ≤ x → sqrt_SNo_nonneg x ∈ real
In Proofgold the corresponding term root is 17e0d9... and proposition id is 511860...
Proof:
Proof not loaded.
End of Section SurrealSqrt
Beginning of Section form100_1
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
L44647
Theorem. (divides_int_div_SNo_int)
∀m n, divides_int m n → n :/: m ∈ int
In Proofgold the corresponding term root is 33dd36... and proposition id is a3726c...
Proof:
Proof not loaded.
L44684
Theorem. (form100_1_lem1)
∀m, nat_p m → ∀n, nat_p n → m * m = 2 * n * n → n = 0
In Proofgold the corresponding term root is 8d3f60... and proposition id is 9bd188...
Proof:
Proof not loaded.
L44847
Theorem. (form100_1_lem2)
∀m ∈ ω, ∀n ∈ ω ∖ 1, m * m ≠ 2 * n * n
In Proofgold the corresponding term root is 1bc378... and proposition id is 1129e9...
Proof:
Proof not loaded.
Notation. We use - as a prefix operator with priority 358 corresponding to applying term minus_SNo.
Notation. We use + as an infix operator with priority 360 and which associates to the right corresponding to applying term add_SNo.
Notation. We use * as an infix operator with priority 355 and which associates to the right corresponding to applying term mul_SNo.
Notation. We use :/: as an infix operator with priority 353 and no associativity corresponding to applying term div_SNo.
L44869
Theorem. (sqrt_2_irrational)
sqrt_SNo_nonneg 2 ∈ real ∖ rational
In Proofgold the corresponding term root is 2c9cbf... and proposition id is 44a1b3...
Proof:
Proof not loaded.
End of Section form100_1