Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: case h.h
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
x : ↥(cycles₁ A)
hx :
  (ConcreteCategory.hom (H1π (A.quotientToCoinvariants S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.toCoinvariantsMkQ S))) x) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.quotientToCoinvariants S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.quotientToCoinvariants S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.toCoinvariantsMkQ S))) x)
z : ↥(cycles₁ (res S.subtype (A.toCoinvariants S)))
hz :
  (ConcreteCategory.hom (H1π (A.toCoinvariants S)))
      ((ConcreteCategory.hom (mapCycles₁ S.subtype (𝟙 (res S.subtype (A.toCoinvariants S))))) z) =
    (ConcreteCategory.hom (H1π (A.toCoinvariants S)))
      ((ConcreteCategory.hom (mapCycles₁ (MonoidHom.id G) (A.toCoinvariantsMkQ S))) x)
w : ↑(ModuleCat.of k (G × G →₀ ↑(A.toCoinvariants S)))
hzw :
  (ModuleCat.Hom.hom (d₂₁ (A.toCoinvariants S))) w =
    ↑((ConcreteCategory.hom (mapCycles₁ S.subtype (𝟙 (res S.subtype (A.toCoinvariants S))))) z) -
      ↑((ConcreteCategory.hom (mapCycles₁ (MonoidHom.id G) (A.toCoinvariantsMkQ S))) x)
Z : ↥S →₀ ↑A
hZ : mapRange ⇑(Coinvariants.mk (MonoidHom.comp A.ρ S.subtype)) ⋯ Z = ↑z
W : G × G →₀ ↑A
hW : mapRange ⇑(Coinvariants.mk (MonoidHom.comp A.ρ S.subtype)) ⋯ W = w
b : G →₀ ↑A := ↑x + (ConcreteCategory.hom (d₂₁ A)) W - (lmapDomain (↑A) k ⇑S.subtype) Z
⊢ (ConcreteCategory.hom (H1π A)) x ∈ (ModuleCat.Hom.hom (H1CoresCoinf A S).f).range

Tactic:
have hb : ∀ g, b g ∈ Coinvariants.ker (A.ρ.comp S.subtype) :=
    fun g => (Coinvariants.mk_eq_iff _).1 <| by
      have := Finsupp.ext_iff.1 (congr($((mapShortComplexH1 (B := toCoinvariants A S)
        (MonoidHom.id G) (toCoinvariantsMkQ A S)).comm₁₂.symm) W)) g
      simp only [shortComplexH1, mapShortComplexH1_τ₂, ModuleCat.ofHom_comp, MonoidHom.coe_id,
        lmapDomain_id, ModuleCat.ofHom_id, res_obj_ρ, hom_ofHom, Category.id_comp,
        mapShortComplexH1_τ₁, Prod.map_id, ModuleCat.hom_comp, ModuleCat.hom_ofHom,
        LinearMap.coe_comp, Function.comp_apply, mapRange.linearMap_apply, mapRange_apply] at this
      simp only [← mapRange_apply (f := Coinvariants.mk <| A.ρ.comp S.subtype)
        (hf := map_zero _) (a := g), ← mapRange.linearMap_apply (R := k)]
      simp [← mapDomain_mapRange, hZ, this, hW, hzw, coe_mapCycles₁ S.subtype,
        coe_mapCycles₁ (MonoidHom.id G)]

Goal state after:
case h.h
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
x : ↥(cycles₁ A)
hx :
  (ConcreteCategory.hom (H1π (A.quotientToCoinvariants S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.toCoinvariantsMkQ S))) x) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.quotientToCoinvariants S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.quotientToCoinvariants S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.toCoinvariantsMkQ S))) x)
z : ↥(cycles₁ (res S.subtype (A.toCoinvariants S)))
hz :
  (ConcreteCategory.hom (H1π (A.toCoinvariants S)))
      ((ConcreteCategory.hom (mapCycles₁ S.subtype (𝟙 (res S.subtype (A.toCoinvariants S))))) z) =
    (ConcreteCategory.hom (H1π (A.toCoinvariants S)))
      ((ConcreteCategory.hom (mapCycles₁ (MonoidHom.id G) (A.toCoinvariantsMkQ S))) x)
w : ↑(ModuleCat.of k (G × G →₀ ↑(A.toCoinvariants S)))
hzw :
  (ModuleCat.Hom.hom (d₂₁ (A.toCoinvariants S))) w =
    ↑((ConcreteCategory.hom (mapCycles₁ S.subtype (𝟙 (res S.subtype (A.toCoinvariants S))))) z) -
      ↑((ConcreteCategory.hom (mapCycles₁ (MonoidHom.id G) (A.toCoinvariantsMkQ S))) x)
Z : ↥S →₀ ↑A
hZ : mapRange ⇑(Coinvariants.mk (MonoidHom.comp A.ρ S.subtype)) ⋯ Z = ↑z
W : G × G →₀ ↑A
hW : mapRange ⇑(Coinvariants.mk (MonoidHom.comp A.ρ S.subtype)) ⋯ W = w
b : G →₀ ↑A := ↑x + (ConcreteCategory.hom (d₂₁ A)) W - (lmapDomain (↑A) k ⇑S.subtype) Z
hb : ∀ (g : G), b g ∈ Coinvariants.ker (MonoidHom.comp A.ρ S.subtype)
⊢ (ConcreteCategory.hom (H1π A)) x ∈ (ModuleCat.Hom.hom (H1CoresCoinf A S).f).range