Definition. We define struct_b_abelian_group to be λX ⇒ struct_b X ∧ unpack_b_o X (λX' op ⇒ explicit_Group X' op ∧ explicit_abelian X' op) of type set → prop.

In Proofgold the corresponding term root is 68631c... and object id is a355bd...

Theorem. (MetaCat_struct_b_abelian_group)

MetaCat struct_b_abelian_group Hom_struct_b struct_id struct_comp

In Proofgold the corresponding term root is 631eb7... and proposition id is da44ba...

Proof:
Proof not loaded.

L12

Theorem. (MetaCat_struct_b_abelian_group_Forgetful)

MetaFunctor struct_b_abelian_group Hom_struct_b 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 1e15e3... and proposition id is 7ae19b...

Proof:
Proof not loaded.

L22

Proposition. (MetaCat_struct_b_abelian_group_initial)

∃Y : set, ∃uniqa : set → set, initial_p struct_b_abelian_group Hom_struct_b struct_id struct_comp Y uniqa

In Proofgold the corresponding term root is 954d8d... and proposition id is 750bb9...

Proof:
Proof not loaded.

L26

Proposition. (MetaCat_struct_b_abelian_group_terminal)

∃Y : set, ∃uniqa : set → set, terminal_p struct_b_abelian_group Hom_struct_b struct_id struct_comp Y uniqa

In Proofgold the corresponding term root is c5474d... and proposition id is ad3864...

Proof:
Proof not loaded.

L30

Proposition. (MetaCat_struct_b_abelian_group_coproduct_constr)

∃coprod : set → set → set, ∃i0 i1 : set → set → set, ∃copair : set → set → set → set → set → set, coproduct_constr_p struct_b_abelian_group Hom_struct_b struct_id struct_comp coprod i0 i1 copair

In Proofgold the corresponding term root is 3245f0... and proposition id is d1f2d8...

Proof:
Proof not loaded.

L37

Proposition. (MetaCat_struct_b_abelian_group_product_constr)

∃prod : set → set → set, ∃pi0 pi1 : set → set → set, ∃pair : set → set → set → set → set → set, product_constr_p struct_b_abelian_group Hom_struct_b struct_id struct_comp prod pi0 pi1 pair

In Proofgold the corresponding term root is 1790b1... and proposition id is 760b6a...

Proof:
Proof not loaded.

L44

Proposition. (MetaCat_struct_b_abelian_group_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_abelian_group Hom_struct_b struct_id struct_comp quot canonmap fac

In Proofgold the corresponding term root is aeae63... and proposition id is c94cfe...

Proof:
Proof not loaded.

L51

Proposition. (MetaCat_struct_b_abelian_group_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_abelian_group Hom_struct_b struct_id struct_comp quot canonmap fac

In Proofgold the corresponding term root is b5511c... and proposition id is 994bf4...

Proof:
Proof not loaded.

L58

Proposition. (MetaCat_struct_b_abelian_group_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_abelian_group Hom_struct_b struct_id struct_comp po i0 i1 copair

In Proofgold the corresponding term root is 1fb0dd... and proposition id is 066718...

Proof:
Proof not loaded.

L66

Proposition. (MetaCat_struct_b_abelian_group_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_abelian_group Hom_struct_b struct_id struct_comp pb pi0 pi1 pair

In Proofgold the corresponding term root is 43013d... and proposition id is e443a4...

Proof:
Proof not loaded.

L74

Proposition. (MetaCat_struct_b_abelian_group_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_abelian_group Hom_struct_b struct_id struct_comp prod pi0 pi1 pair exp a lm

In Proofgold the corresponding term root is e2e65e... and proposition id is f8b926...

Proof:
Proof not loaded.

L84

Proposition. (MetaCat_struct_b_abelian_group_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_abelian_group Hom_struct_b struct_id struct_comp one uniqa Omega tru ch constr

In Proofgold the corresponding term root is 237288... and proposition id is 78047c...

Proof:
Proof not loaded.

L92

Proposition. (MetaCat_struct_b_abelian_group_nno)

∃one : set, ∃uniqa : set → set, ∃N : set, ∃zer suc : set, ∃rec : set → set → set → set, nno_p struct_b_abelian_group Hom_struct_b struct_id struct_comp one uniqa N zer suc rec

In Proofgold the corresponding term root is c62cc4... and proposition id is 8006d8...

Proof:
Proof not loaded.

L101

Proposition. (MetaCat_struct_b_abelian_group_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_abelian_group Hom_struct_b struct_id struct_comp F0 F1 (λX ⇒ X 0) (λX Y f ⇒ f) eta eps

In Proofgold the corresponding term root is 6ebdd6... and proposition id is 3295db...

Proof:
Proof not loaded.