Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: case w
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := (lmapDomain (↑A) k (Prod.map s s)) y
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := x - (ModuleCat.Hom.hom (d₂₁ A)) z
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := fun g ↦ (s ↑g, (s ↑g)⁻¹ * g)
he : Function.Injective e
ve : G × G →₀ ↑A := mapDomain e v
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this✝¹ :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
this✝ :
  (comapDomain ?w.f✝ (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun g a ↦ (A.ρ g⁻¹) a) ∘ ?w.f✝) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
this :
  (comapDomain ?w.f✝ (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun g a ↦ (A.ρ g⁻¹) a) ∘ ?w.f✝) =
    (comapDomain ?w.f (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun x a ↦ a) ∘ ?w.f)
⊢ ((comapDomain Subtype.val (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum fun g a ↦ ((res S.subtype A).ρ g⁻¹) a) =
    (comapDomain Subtype.val (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum fun x a ↦ a

pp: case w.f
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := ⋯
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := ⋯
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := ⋯
he : Function.Injective e
ve : G × G →₀ ↑A := ⋯
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this✝ :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
this :
  (comapDomain ?w.f✝ (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun g a ↦ (A.ρ g⁻¹) a) ∘ ?w.f✝) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ ?m.632 → G

pp: case w.hf
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := (lmapDomain (↑A) k (Prod.map s s)) y
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := x - (ModuleCat.Hom.hom (d₂₁ A)) z
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := fun g ↦ (s ↑g, (s ↑g)⁻¹ * g)
he : Function.Injective e
ve : G × G →₀ ↑A := mapDomain e v
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this✝ :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
this :
  (comapDomain ?w.f✝ (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun g a ↦ (A.ρ g⁻¹) a) ∘ ?w.f✝) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ Set.BijOn ?w.f (?w.f ⁻¹' ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support)
    ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support

pp: k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := ⋯
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := ⋯
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := ⋯
he : Function.Injective e
ve : G × G →₀ ↑A := ⋯
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this✝ :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
this :
  (comapDomain ?w.f✝ (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun g a ↦ (A.ρ g⁻¹) a) ∘ ?w.f✝) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ Type ?u.425

pp: case w.f
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := ⋯
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := ⋯
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := ⋯
he : Function.Injective e
ve : G × G →₀ ↑A := ⋯
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ ?m.619 → G

pp: case w.hf
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := (lmapDomain (↑A) k (Prod.map s s)) y
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := x - (ModuleCat.Hom.hom (d₂₁ A)) z
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := fun g ↦ (s ↑g, (s ↑g)⁻¹ * g)
he : Function.Injective e
ve : G × G →₀ ↑A := mapDomain e v
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ Set.BijOn ?w.f✝ (?w.f✝ ⁻¹' ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support)
    ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support

pp: k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := ⋯
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := ⋯
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := ⋯
he : Function.Injective e
ve : G × G →₀ ↑A := ⋯
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ Type ?u.421

Tactic:
]

Goal state after:
case w
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := (lmapDomain (↑A) k (Prod.map s s)) y
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := x - (ModuleCat.Hom.hom (d₂₁ A)) z
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := fun g ↦ (s ↑g, (s ↑g)⁻¹ * g)
he : Function.Injective e
ve : G × G →₀ ↑A := mapDomain e v
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this✝¹ :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
this✝ :
  (comapDomain ?w.f✝ (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun g a ↦ (A.ρ g⁻¹) a) ∘ ?w.f✝) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
this :
  (comapDomain ?w.f✝ (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun g a ↦ (A.ρ g⁻¹) a) ∘ ?w.f✝) =
    (comapDomain ?w.f (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun x a ↦ a) ∘ ?w.f)
⊢ ((comapDomain Subtype.val (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum fun g a ↦ ((res S.subtype A).ρ g⁻¹) a) =
    (comapDomain Subtype.val (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum fun x a ↦ a

case w.f
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := ⋯
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := ⋯
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := ⋯
he : Function.Injective e
ve : G × G →₀ ↑A := ⋯
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this✝ :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
this :
  (comapDomain ?w.f✝ (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun g a ↦ (A.ρ g⁻¹) a) ∘ ?w.f✝) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ ?m.632 → G

case w.hf
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := (lmapDomain (↑A) k (Prod.map s s)) y
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := x - (ModuleCat.Hom.hom (d₂₁ A)) z
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := fun g ↦ (s ↑g, (s ↑g)⁻¹ * g)
he : Function.Injective e
ve : G × G →₀ ↑A := mapDomain e v
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this✝ :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
this :
  (comapDomain ?w.f✝ (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun g a ↦ (A.ρ g⁻¹) a) ∘ ?w.f✝) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ Set.BijOn ?w.f (?w.f ⁻¹' ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support)
    ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support

k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := ⋯
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := ⋯
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := ⋯
he : Function.Injective e
ve : G × G →₀ ↑A := ⋯
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this✝ :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
this :
  (comapDomain ?w.f✝ (v + (ConcreteCategory.hom (d₂₁ A)) ve) ⋯).sum ((fun g a ↦ (A.ρ g⁻¹) a) ∘ ?w.f✝) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ Type ?u.425

case w.f
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := ⋯
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := ⋯
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := ⋯
he : Function.Injective e
ve : G × G →₀ ↑A := ⋯
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ ?m.619 → G

case w.hf
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := (lmapDomain (↑A) k (Prod.map s s)) y
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := x - (ModuleCat.Hom.hom (d₂₁ A)) z
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := fun g ↦ (s ↑g, (s ↑g)⁻¹ * g)
he : Function.Injective e
ve : G × G →₀ ↑A := mapDomain e v
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ Set.BijOn ?w.f✝ (?w.f✝ ⁻¹' ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support)
    ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support

k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := ⋯
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := ⋯
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := ⋯
he : Function.Injective e
ve : G × G →₀ ↑A := ⋯
hS : ↑(v + (ConcreteCategory.hom (d₂₁ A)) ve).support ⊆ ↑S
this :
  ((v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun g a ↦ (A.ρ g⁻¹) a) =
    (v + (ConcreteCategory.hom (d₂₁ A)) ve).sum fun x a ↦ a
⊢ Type ?u.421