open import Algebra
module Algebra.Properties.AbelianGroup
{g₁ g₂} (G : AbelianGroup g₁ g₂) where
import Algebra.Properties.Group as GP
open import Function
import Relation.Binary.EqReasoning as EqR
open AbelianGroup G
open EqR setoid
open GP group public
private
lemma₁ : ∀ x y → x ∙ y ∙ x ⁻¹ ≈ y
lemma₁ x y = begin
x ∙ y ∙ x ⁻¹ ≈⟨ comm _ _ ⟨ ∙-cong ⟩ refl ⟩
y ∙ x ∙ x ⁻¹ ≈⟨ assoc _ _ _ ⟩
y ∙ (x ∙ x ⁻¹) ≈⟨ refl ⟨ ∙-cong ⟩ inverseʳ _ ⟩
y ∙ ε ≈⟨ identityʳ _ ⟩
y ∎
lemma₂ : ∀ x y → x ∙ (y ∙ (x ∙ y) ⁻¹ ∙ y ⁻¹) ≈ y ⁻¹
lemma₂ x y = 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 ≈⟨ inverseʳ _ ⟨ ∙-cong ⟩ refl ⟩
ε ∙ y ⁻¹ ≈⟨ identityˡ _ ⟩
y ⁻¹ ∎
⁻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 $ (lemma₂ x y) ⟨ ∙-cong ⟩ refl ⟩
x ∙ (y ∙ (x ∙ y) ⁻¹ ∙ y ⁻¹) ∙ x ⁻¹ ≈⟨ lemma₁ _ _ ⟩
y ∙ (x ∙ y) ⁻¹ ∙ y ⁻¹ ≈⟨ lemma₁ _ _ ⟩
(x ∙ y) ⁻¹ ∎
where