open import Algebra
module Algebra.Properties.AbelianGroup
{g₁ g₂} (G : AbelianGroup g₁ g₂) where
import Algebra.Properties.Group as GP
open import Data.Product
open import Function
import Relation.Binary.EqReasoning as EqR
open AbelianGroup G
open EqR setoid
open GP group public
private
lemma : ∀ x y → x ∙ y ∙ x ⁻¹ ≈ y
lemma x y = begin
x ∙ y ∙ x ⁻¹ ≈⟨ comm _ _ ⟨ ∙-cong ⟩ refl ⟩
y ∙ x ∙ x ⁻¹ ≈⟨ assoc _ _ _ ⟩
y ∙ (x ∙ x ⁻¹) ≈⟨ refl ⟨ ∙-cong ⟩ proj₂ inverse _ ⟩
y ∙ ε ≈⟨ proj₂ identity _ ⟩
y ∎
⁻¹-∙-comm : ∀ x y → x ⁻¹ ∙ y ⁻¹ ≈ (x ∙ y) ⁻¹
⁻¹-∙-comm x y = begin
x ⁻¹ ∙ y ⁻¹ ≈⟨ comm _ _ ⟩
y ⁻¹ ∙ x ⁻¹ ≈⟨ sym $ lem ⟨ ∙-cong ⟩ refl ⟩
x ∙ (y ∙ (x ∙ y) ⁻¹ ∙ y ⁻¹) ∙ x ⁻¹ ≈⟨ lemma _ _ ⟩
y ∙ (x ∙ y) ⁻¹ ∙ y ⁻¹ ≈⟨ lemma _ _ ⟩
(x ∙ y) ⁻¹ ∎
where
lem = begin
x ∙ (y ∙ (x ∙ y) ⁻¹ ∙ y ⁻¹) ≈⟨ sym $ assoc _ _ _ ⟩
x ∙ (y ∙ (x ∙ y) ⁻¹) ∙ y ⁻¹ ≈⟨ sym $ assoc _ _ _ ⟨ ∙-cong ⟩ refl ⟩
x ∙ y ∙ (x ∙ y) ⁻¹ ∙ y ⁻¹ ≈⟨ proj₂ inverse _ ⟨ ∙-cong ⟩ refl ⟩
ε ∙ y ⁻¹ ≈⟨ proj₁ identity _ ⟩
y ⁻¹ ∎