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https://github.com/Smaug123/agdaproofs
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25 lines
1.8 KiB
Agda
25 lines
1.8 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Setoids.Setoids
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open import Groups.Definition
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open import Sets.EquivalenceRelations
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open import Groups.Abelian.Definition
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open import Groups.Homomorphisms.Definition
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open import Groups.DirectSum.Definition
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open import Groups.Isomorphisms.Definition
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module Groups.Abelian.Lemmas where
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directSumAbelianGroup : {m n o p : _} → {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {underG : Group S _·A_} {underH : Group T _·B_} (G : AbelianGroup underG) (h : AbelianGroup underH) → (AbelianGroup (directSumGroup underG underH))
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AbelianGroup.commutative (directSumAbelianGroup {A = A} {B} G H) = AbelianGroup.commutative G ,, AbelianGroup.commutative H
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subgroupOfAbelianIsAbelian : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} → SetoidInjection T S f → AbelianGroup G → AbelianGroup H
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AbelianGroup.commutative (subgroupOfAbelianIsAbelian {S = S} {_+B_ = _+B_} {fHom = fHom} fInj record { commutative = commutative }) {x} {y} = SetoidInjection.injective fInj (transitive (GroupHom.groupHom fHom) (transitive commutative (symmetric (GroupHom.groupHom fHom))))
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where
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open Setoid S
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open Equivalence eq
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abelianIsGroupProperty : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} → GroupsIsomorphic G H → AbelianGroup H → AbelianGroup G
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abelianIsGroupProperty iso abH = subgroupOfAbelianIsAbelian {fHom = GroupIso.groupHom (GroupsIsomorphic.proof iso)} (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof iso))) abH
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