-- Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality
Goal:
k G : Type u
inst : CommRing k
inst_1 : Group G
A : Rep k G
S : Subgroup G
inst_2 : S.Normal
inst_3 : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ (↑A : Type u)
hxc : x ∈ cycles₁ A
hx :
  Eq (α := (↑(H1 (A.ofQuotient S)) : Type u))
    (((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (X := ModuleCat.of k ↥(cycles₁ (A.ofQuotient S))) (Y :=
              groupHomology (A.ofQuotient S) 1) (H1π (A.ofQuotient S)) :
            (↑(ModuleCat.of k ↥(cycles₁ (A.ofQuotient S))) : Type u) →ₗ[k]
              (↑(groupHomology (A.ofQuotient S) 1) : Type u)) :
          (a : ↥(cycles₁ (A.ofQuotient S))) → (↑(groupHomology (A.ofQuotient S) 1) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
                ModuleCat.carrier) (Y := ModuleCat.of k ↥(cycles₁ (A.ofQuotient S)))
                (mapCycles₁ (QuotientGroup.mk' S) (Iso.inv (A.resOfQuotientIso S))) :
              (a : ↥(cycles₁ A)) → ↥(cycles₁ (A.ofQuotient S)))
            (Subtype.mk (p := fun (x : G →₀ (↑A : Type u)) ↦ x ∈ cycles₁ A) x hxc) :
          ↥(cycles₁ (A.ofQuotient S))) :
      (↑(groupHomology (A.ofQuotient S) 1) : Type u))
    (0 : (↑(H1 (A.ofQuotient S)) : Type u))
y : (↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ (↑(A.ofQuotient S) : Type u))) : Type u)
hy :
  Eq (α := G ⧸ S →₀ (↑(A.ofQuotient S) : Type u))
    ((ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S)) :
          (a : (↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ (↑(A.ofQuotient S) : Type u))) : Type u)) →
            G ⧸ S →₀ (↑(A.ofQuotient S) : Type u))
        y :
      G ⧸ S →₀ (↑(A.ofQuotient S) : Type u))
    ((↑((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
                ModuleCat.carrier) (Y := ModuleCat.of k ↥(cycles₁ (A.ofQuotient S)))
                (mapCycles₁ (QuotientGroup.mk' S) (Iso.inv (A.resOfQuotientIso S))) :
              (a : ↥(cycles₁ A)) → ↥(cycles₁ (A.ofQuotient S)))
            (Subtype.mk (p := fun (x : G →₀ (↑A : Type u)) ↦ x ∈ cycles₁ A) x hxc)) :
        G ⧸ S →₀ (↑(A.ofQuotient S) : Type u)) :
      G ⧸ S →₀ (↑(A.ofQuotient S) : Type u))
s : (b : G ⧸ S) → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S : (a : G) → G ⧸ S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ (↑A : Type u) :=
  (lmapDomain (↑A : Type u) k (Prod.map s s) : (a : (G ⧸ S) × G ⧸ S →₀ (↑A : Type u)) → G × G →₀ (↑A : Type u)) y
hz :
  Eq (α := G ⧸ S →₀ (↑A : Type u))
    ((lmapDomain (↑A : Type u) k (⇑(QuotientGroup.mk' S) : (a : G) → G ⧸ S) :
          (a : G →₀ (↑A : Type u)) → G ⧸ S →₀ (↑A : Type u))
        ((ModuleCat.Hom.hom (d₂₁ A) :
            (a : (↑(ModuleCat.of k (G × G →₀ (↑A : Type u))) : Type u)) →
              (↑(ModuleCat.of k (G →₀ (↑A : Type u))) : Type u))
          z) :
      G ⧸ S →₀ (↑A : Type u))
    ((ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S)) :
          (a : (↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ (↑(A.ofQuotient S) : Type u))) : Type u)) →
            (↑(ModuleCat.of k (G ⧸ S →₀ (↑(A.ofQuotient S) : Type u))) : Type u))
        y :
      (↑(ModuleCat.of k (G ⧸ S →₀ (↑(A.ofQuotient S) : Type u))) : Type u))
v : G →₀ (↑A : Type u) :=
  HSub.hSub (β := (↑(ModuleCat.of k (G →₀ (↑A : Type u))) : Type u)) x
    ((ModuleCat.Hom.hom (d₂₁ A) :
          (a : (↑(ModuleCat.of k (G × G →₀ (↑A : Type u))) : Type u)) →
            (↑(ModuleCat.of k (G →₀ (↑A : Type u))) : Type u))
        z :
      (↑(ModuleCat.of k (G →₀ (↑A : Type u))) : Type u))
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := fun (g : G) ↦ (s (↑g : G ⧸ S), (s (↑g : G ⧸ S))⁻¹ * g)
he : Function.Injective e
ve : G × G →₀ (↑A : Type u) := mapDomain e v
⊢ (↑(HAdd.hAdd (β := (↑(ModuleCat.of k (G →₀ (↑A : Type u))) : Type u)) v
            ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
                    ModuleCat.carrier) (d₂₁ A) :
                  (a : (↑(ModuleCat.of k (G × G →₀ (↑A : Type u))) : Type u)) →
                    (↑(ModuleCat.of k (G →₀ (↑A : Type u))) : Type u))
                ve :
              (↑(ModuleCat.of k (G →₀ (↑A : Type u))) : Type u))).support :
      Set G) ⊆
    (↑S : Set G)

Tactic:
have hS : ((v + d₂₁ _ ve).support : Set G) ⊆ S := by
  /- We have `d(ve) = ∑ ρ(s(π(g))⁻¹)(v(g))·s(π(g))⁻¹g - ∑ v(g)·g + ∑ v(g)·s(π(g))`.
    The second sum is `v`, so cancels: -/
    simp only [d₂₁, ve, ModuleCat.hom_ofHom, coe_lsum, sum_mapDomain_index_inj he, sum_single,
      LinearMap.add_apply, LinearMap.sub_apply, LinearMap.coe_comp, Function.comp_apply,
      lsingle_apply, sum_add, sum_sub, mul_inv_cancel_left, ← add_assoc, add_sub_cancel, e]
    intro w hw
    · obtain (hl | hr) := Finset.mem_union.1 (support_add hw)
    /- The first sum clearly has support in `S`: -/
      · obtain ⟨t, _, ht⟩ := Finset.mem_biUnion.1 (support_sum hl)
        apply support_single_subset at ht
        simp_all [← QuotientGroup.eq]
    /- The third sum is 0, by `hv`. -/
      · simp_all [mapDomain]