Definition. We define struct_b_b_e_crng to be λR ⇒ struct_b_b_e R ∧ unpack_b_b_e_o R (λR plus mult zero ⇒ explicit_Rng R zero plus mult ∧ (∀x y ∈ R, mult x y = mult y x)) of type set → prop.
In Proofgold the corresponding term root is a27cd8... and object id is 72966a...
Theorem. (MetaCat_struct_b_b_e_crng)
MetaCat struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp
In Proofgold the corresponding term root is 8ccfb9... and proposition id is 2eee93...
Proof:
We prove the intermediate claim L1: ∀X, struct_b_b_e_crng X → struct_b_b_e X.
Let X be given.
Assume HX.
Apply HX to the current goal.
Assume H _.
An exact proof term for the current goal is H.
An exact proof term for the current goal is MetaCat_struct_b_b_e_gen struct_b_b_e_crng L1.
∎
Theorem. (MetaCat_struct_b_b_e_crng_Forgetful)
MetaFunctor struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp (λ_ ⇒ True) SetHom (λX ⇒ lam_id X) (λX Y Z f g ⇒ (lam_comp X f g)) (λX ⇒ X 0) (λX Y f ⇒ f)
In Proofgold the corresponding term root is 367b3e... and proposition id is d5d286...
Proof:
We prove the intermediate claim L1: ∀X, struct_b_b_e_crng X → struct_b_b_e X.
Let X be given.
Assume HX.
Apply HX to the current goal.
Assume H _.
An exact proof term for the current goal is H.
An exact proof term for the current goal is MetaCat_struct_b_b_e_Forgetful_gen struct_b_b_e_crng L1.
∎
Proposition. (MetaCat_struct_b_b_e_crng_initial)
∃Y : set, ∃uniqa : set → set, initial_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 37f6f7... and proposition id is 0e3a37...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_terminal)
∃Y : set, ∃uniqa : set → set, terminal_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp Y uniqa
In Proofgold the corresponding term root is 68df42... and proposition id is 23816c...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_coproduct_constr)
∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp coprod i0 i1 copair
In Proofgold the corresponding term root is 98f1e6... and proposition id is 5d66b3...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_product_constr)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp prod pi0 pi1 pair
In Proofgold the corresponding term root is 8843e1... and proposition id is 256257...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_coequalizer_constr)
∃quot : set → set → set → set → set, ∃canonmap : set → set → set → set → set, ∃fac : set → set → set → set → set → set → set, coequalizer_constr_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is aa6332... and proposition id is 64e778...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_equalizer_constr)
∃quot : set → set → set → set → set, ∃canonmap : set → set → set → set → set, ∃fac : set → set → set → set → set → set → set, equalizer_constr_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp quot canonmap fac
In Proofgold the corresponding term root is df3c36... and proposition id is 6b4e01...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_pushout_constr)
∃po : set → set → set → set → set → set, ∃i0 : set → set → set → set → set → set, ∃i1 : set → set → set → set → set → set, ∃copair : set → set → set → set → set → set → set → set → set, pushout_constr_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp po i0 i1 copair
In Proofgold the corresponding term root is 721f79... and proposition id is 406b91...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_pullback_constr)
∃pb : set → set → set → set → set → set, ∃pi0 : set → set → set → set → set → set, ∃pi1 : set → set → set → set → set → set, ∃pair : set → set → set → set → set → set → set → set → set, pullback_constr_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp pb pi0 pi1 pair
In Proofgold the corresponding term root is c78a61... and proposition id is de7c82...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_product_exponent)
∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, ∃exp : set → set → set, ∃a : set → set → set, ∃lm : set → set → set → set → set, product_exponent_constr_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp prod pi0 pi1 pair exp a lm
In Proofgold the corresponding term root is d5ef5c... and proposition id is da9ac3...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_subobject_classifier)
∃one : set, ∃uniqa : set → set, ∃Omega : set, ∃tru : set, ∃ch : set → set → set → set, ∃constr : set → set → set → set → set → set → set, subobject_classifier_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp one uniqa Omega tru ch constr
In Proofgold the corresponding term root is ea2298... and proposition id is 60634a...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_nno)
∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp one uniqa N zer suc rec
In Proofgold the corresponding term root is 83b532... and proposition id is dbe72a...
Proof:
The rest of the proof is missing.

Proposition. (MetaCat_struct_b_b_e_crng_left_adjoint_forgetful)
∃F0 : set → set, ∃F1 : set → set → set → set, ∃eta eps : set → set, MetaAdjunction_strict (λ_ ⇒ True) SetHom (λX ⇒ (lam_id X)) (λX Y Z f g ⇒ (lam_comp X f g)) struct_b_b_e_crng Hom_struct_b_b_e struct_id struct_comp F0 F1 (λX ⇒ X 0) (λX Y f ⇒ f) eta eps
In Proofgold the corresponding term root is 3918b6... and proposition id is ea9f0f...
Proof:
The rest of the proof is missing.