{-# OPTIONS --warning=error --safe --without-K #-}

open import Functions
open import LogicalFormulae
open import Groups.Definition
open import Groups.Groups
open import Groups.FiniteGroups.Definition
open import Groups.Homomorphisms.Definition
open import Groups.Abelian.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Groups.Lemmas
open import Groups.QuotientGroup.Definition
open import Groups.Homomorphisms.Lemmas

module Groups.QuotientGroup.Lemmas {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} (G : Group S _+A_) (H : Group T _+B_) {f : A → B} (fHom : GroupHom G H f) where

quotientGroupLemma : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → {x y : A} → Setoid._∼_ T (underf x) (underf y) → Setoid._∼_ (quotientGroupSetoid G f) x y
quotientGroupLemma {S = S} {T = T} G {H = H} fHom {x} {y} fx=fy = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H (Equivalence.reflexive (Setoid.eq T)) (homRespectsInverse fHom)) (transferToRight'' H fx=fy))
  where
    open Group G
    open Setoid T
    open Equivalence eq

quotientGroupLemma' : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → {x y : A} → Setoid._∼_ (quotientGroupSetoid G f) x y → Setoid._∼_ T (underf x) (underf y)
quotientGroupLemma' {S = S} {T = T} G {H = H} f fx=fy = transferToRight H (transitive (transitive (Group.+WellDefined H (Equivalence.reflexive (Setoid.eq T)) (symmetric (homRespectsInverse f))) (symmetric (GroupHom.groupHom f))) fx=fy)
  where
    open Group G
    open Setoid T
    open Equivalence eq


projectionMapIsGroupHom : GroupHom G (quotientGroupByHom G fHom) id
GroupHom.groupHom projectionMapIsGroupHom {x} {y} = quotientGroupLemma G fHom (Equivalence.reflexive (Setoid.eq T))
GroupHom.wellDefined projectionMapIsGroupHom x=y = quotientGroupLemma G fHom (GroupHom.wellDefined fHom x=y)