mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-11 14:48:42 +00:00
38 lines
1.7 KiB
Agda
38 lines
1.7 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Functions
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open import Groups.Groups
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open import Groups.Definition
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open import Groups.Lemmas
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open import Rings.Definition
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open import Setoids.Setoids
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open import Setoids.Orders
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open import Sets.EquivalenceRelations
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module Rings.Lemmas where
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ringMinusExtracts : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) → {x y : A} → Setoid._∼_ S (x * Group.inverse (Ring.additiveGroup R) y) (Group.inverse (Ring.additiveGroup R) (x * y))
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ringMinusExtracts {S = S} {_+_ = _+_} {_*_ = _*_} R {x = x} {y} = transferToRight' additiveGroup (transitive (symmetric *DistributesOver+) (transitive (*WellDefined reflexive invLeft) (Ring.timesZero R)))
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where
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open Setoid S
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open Equivalence eq
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open Ring R
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open Group additiveGroup
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groupLemmaMove0G : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → (Setoid._∼_ S (Group.0G G) (Group.inverse G x)) → Setoid._∼_ S x (Group.0G G)
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groupLemmaMove0G {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitive (symmetric p) (invIdent G))
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where
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open Group G
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open Setoid S
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open Equivalence eq
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p : inverse 0G ∼ inverse (inverse x)
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p = inverseWellDefined G pr
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groupLemmaMove0G' : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → Setoid._∼_ S x (Group.0G G) → (Setoid._∼_ S (Group.0G G) (Group.inverse G x))
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groupLemmaMove0G' {S = S} G {x} pr = transferToRight' G (transitive identLeft pr)
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where
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open Group G
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open Setoid S
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open Equivalence eq
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