mirror of
https://github.com/Smaug123/agdaproofs
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29 lines
1.7 KiB
Agda
29 lines
1.7 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Groups
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Groups.Abelian.Definition
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open import Numbers.Naturals.Naturals
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open import Setoids.Orders
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open import Setoids.Setoids
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open import Functions
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open import Sets.EquivalenceRelations
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open import Rings.Definition
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open import Modules.Definition
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Modules.DirectSum {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+R_ : A → A → A} {_*R_ : A → A → A} (R : Ring S _+R_ _*R_) {m n o p : _} {M : Set m} {T : Setoid {m} {n} M} {_+_ : M → M → M} {G' : Group T _+_} {G : AbelianGroup G'} {_·1_ : A → M → M} {N : Set o} {U : Setoid {o} {p} N} {_+'_ : N → N → N} {H' : Group U _+'_} {H : AbelianGroup H'} {_·2_ : A → N → N} (M1 : Module R G _·1_) (M2 : Module R H _·2_) where
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open import Groups.Abelian.DirectSum G H
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open import Setoids.DirectSum T U
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directSumModule : Module R directSumAbGroup λ r mn → ((r ·1 (_&&_.fst mn)) ,, (r ·2 (_&&_.snd mn)))
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Module.dotWellDefined directSumModule r=s t=u = directSumLift (Module.dotWellDefined M1 r=s (_&&_.fst t=u)) (Module.dotWellDefined M2 r=s (_&&_.snd t=u))
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Module.dotDistributesLeft directSumModule = directSumLift (Module.dotDistributesLeft M1) (Module.dotDistributesLeft M2)
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Module.dotDistributesRight directSumModule = directSumLift (Module.dotDistributesRight M1) (Module.dotDistributesRight M2)
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Module.dotAssociative directSumModule = directSumLift (Module.dotAssociative M1) (Module.dotAssociative M2)
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Module.dotIdentity directSumModule = directSumLift (Module.dotIdentity M1) (Module.dotIdentity M2)
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