Mathlib.Algebra.Group.SecondIsomorphism.Map

import Std.Logic.Eq
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Subgroup
import Mathlib.Algebra.Group.Quotient
import Mathlib.Algebra.Group.Quotient.Mul
import Mathlib.Algebra.Group.Quotient.Group

def SecondIsoPhi.{u} :
  forall (G : Sort succ u), forall (one : G), forall (mul : forall (a : G), forall (b : G), G), forall (inv : forall (a : G), G), forall (N : forall (x : G), Prop), forall (group_args : @GroupLawArgs.{succ u} G one mul inv), forall (normal_args : @NormalSubgroupLawArgs.{succ u} G one mul inv N), forall (Hpred : forall (x : G), Prop), forall (h : G), forall (hh : Hpred h), @NormalQuot.{u} G one mul inv N group_args normal_args :=
  fun G => fun one => fun mul => fun inv => fun N => fun group_args => fun normal_args => fun Hpred => fun h => fun hh => @NormalQuotMk.{u} G one mul inv N group_args normal_args h

theorem second_iso_phi_mk.{u} :
  forall (G : Sort succ u), forall (one : G), forall (mul : forall (a : G), forall (b : G), G), forall (inv : forall (a : G), G), forall (N : forall (x : G), Prop), forall (group_args : @GroupLawArgs.{succ u} G one mul inv), forall (normal_args : @NormalSubgroupLawArgs.{succ u} G one mul inv N), forall (Hpred : forall (x : G), Prop), forall (h : G), forall (hh : Hpred h), @Eq.{succ u} (@NormalQuot.{u} G one mul inv N group_args normal_args) (@SecondIsoPhi.{u} G one mul inv N group_args normal_args Hpred h hh) (@NormalQuotMk.{u} G one mul inv N group_args normal_args h) :=
  fun G => fun one => fun mul => fun inv => fun N => fun group_args => fun normal_args => fun Hpred => fun h => fun hh => @Eq.refl.{succ u} (@NormalQuot.{u} G one mul inv N group_args normal_args) (@NormalQuotMk.{u} G one mul inv N group_args normal_args h)

theorem second_iso_phi_mul.{u} :
  forall (G : Sort succ u), forall (one : G), forall (mul : forall (a : G), forall (b : G), G), forall (inv : forall (a : G), G), forall (N : forall (x : G), Prop), forall (group_args : @GroupLawArgs.{succ u} G one mul inv), forall (normal_args : @NormalSubgroupLawArgs.{succ u} G one mul inv N), forall (Hpred : forall (x : G), Prop), forall (h_args : @SubgroupLawArgs.{succ u} G one mul inv Hpred), forall (a : G), forall (b : G), forall (ha : Hpred a), forall (hb : Hpred b), @Eq.{succ u} (@NormalQuot.{u} G one mul inv N group_args normal_args) (@NormalQuotMul.{u} G one mul inv N group_args normal_args (@SecondIsoPhi.{u} G one mul inv N group_args normal_args Hpred a ha) (@SecondIsoPhi.{u} G one mul inv N group_args normal_args Hpred b hb)) (@SecondIsoPhi.{u} G one mul inv N group_args normal_args Hpred (mul a b) (@subgroup_mul_closed.{succ u} G one mul inv Hpred h_args a b ha hb)) :=
  fun G => fun one => fun mul => fun inv => fun N => fun group_args => fun normal_args => fun Hpred => fun h_args => fun a => fun b => fun ha => fun hb => @Eq.refl.{succ u} (@NormalQuot.{u} G one mul inv N group_args normal_args) (@NormalQuotMk.{u} G one mul inv N group_args normal_args (mul a b))

theorem second_iso_phi_one.{u} :
  forall (G : Sort succ u), forall (one : G), forall (mul : forall (a : G), forall (b : G), G), forall (inv : forall (a : G), G), forall (N : forall (x : G), Prop), forall (group_args : @GroupLawArgs.{succ u} G one mul inv), forall (normal_args : @NormalSubgroupLawArgs.{succ u} G one mul inv N), forall (Hpred : forall (x : G), Prop), forall (h_args : @SubgroupLawArgs.{succ u} G one mul inv Hpred), @Eq.{succ u} (@NormalQuot.{u} G one mul inv N group_args normal_args) (@SecondIsoPhi.{u} G one mul inv N group_args normal_args Hpred one (@subgroup_one.{succ u} G one mul inv Hpred h_args)) (@NormalQuotOne.{u} G one mul inv N group_args normal_args) :=
  fun G => fun one => fun mul => fun inv => fun N => fun group_args => fun normal_args => fun Hpred => fun h_args => @Eq.refl.{succ u} (@NormalQuot.{u} G one mul inv N group_args normal_args) (@NormalQuotMk.{u} G one mul inv N group_args normal_args one)

theorem second_iso_phi_inv.{u} :
  forall (G : Sort succ u), forall (one : G), forall (mul : forall (a : G), forall (b : G), G), forall (inv : forall (a : G), G), forall (N : forall (x : G), Prop), forall (group_args : @GroupLawArgs.{succ u} G one mul inv), forall (normal_args : @NormalSubgroupLawArgs.{succ u} G one mul inv N), forall (Hpred : forall (x : G), Prop), forall (h_args : @SubgroupLawArgs.{succ u} G one mul inv Hpred), forall (h : G), forall (hh : Hpred h), @Eq.{succ u} (@NormalQuot.{u} G one mul inv N group_args normal_args) (@NormalQuotInv.{u} G one mul inv N group_args normal_args (@SecondIsoPhi.{u} G one mul inv N group_args normal_args Hpred h hh)) (@SecondIsoPhi.{u} G one mul inv N group_args normal_args Hpred (inv h) (@subgroup_inv_closed.{succ u} G one mul inv Hpred h_args h hh)) :=
  fun G => fun one => fun mul => fun inv => fun N => fun group_args => fun normal_args => fun Hpred => fun h_args => fun h => fun hh => @Eq.refl.{succ u} (@NormalQuot.{u} G one mul inv N group_args normal_args) (@NormalQuotMk.{u} G one mul inv N group_args normal_args (inv h))