mirror of
https://github.com/Smaug123/agdaproofs
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33 lines
1.5 KiB
Agda
33 lines
1.5 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import Groups.Definition
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open import Groups.Lemmas
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open import Groups.Abelian.Definition
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open import Setoids.Setoids
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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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module Modules.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+R_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+R_ _*_} {m n : _} {M : Set m} {T : Setoid {m} {n} M} {_+_ : M → M → M} {G' : Group T _+_} {G : AbelianGroup G'} {_·_ : A → M → M} (mod : Module R G _·_) where
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open Group G'
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open Ring R
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open Setoid T
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open Equivalence eq
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open Module mod
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moduleTimesZero : {x : M} → (0R · x) ∼ 0G
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moduleTimesZero {x} = equalsDoubleImpliesZero G' (symmetric x=2x)
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where
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x=2x : (0R · x) ∼ (0R · x) + (0R · x)
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x=2x = transitive (dotWellDefined (Equivalence.symmetric (Setoid.eq S) (Group.identLeft additiveGroup)) reflexive) dotDistributesRight
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moduleTimes-1 : {x : M} → ((Group.inverse additiveGroup 1R) · x) ∼ inverse x
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moduleTimes-1 {x} = transitive (transferToRight' G' j) (inverseWellDefined G' dotIdentity)
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where
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i : ((1R · x) + ((Group.inverse additiveGroup 1R) · x)) ∼ 0G
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i = transitive (symmetric (transitive (dotWellDefined (Equivalence.symmetric (Setoid.eq S) (Group.invRight additiveGroup {1R})) reflexive) dotDistributesRight)) (moduleTimesZero)
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j : (((Group.inverse additiveGroup 1R) · x) + (1R · x)) ∼ 0G
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j = transitive (AbelianGroup.commutative G) i
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