-- Module: Mathlib.RepresentationTheory.Homological.FiniteCyclic
Goal:
k G : Type u
inst : CommRing k
inst_1 : CommGroup G
inst_2 : Fintype G
g : G
hg : ∀ (x : G), Membership.mem (γ := Subgroup G) (Subgroup.zpowers g) x
⊢ ShortComplex.Exact
    (ShortComplex.map
      (ShortComplex.mk (X₁ :=
        ShortComplex.X₁
          (Functor.obj (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
            (Functor.obj (chainComplexFunctor k g) (leftRegular k G))))
        (X₃ :=
        ShortComplex.X₂
          (Functor.obj (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
            (Functor.obj (ChainComplex.single₀ (Rep k G)) (trivial k G k))))
        (ShortComplex.f
          (Functor.obj (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
            (Functor.obj (chainComplexFunctor k g) (leftRegular k G))))
        (ShortComplex.Hom.τ₂
          (CategoryTheory.Functor.map (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
            ((Equiv.symm
                  (ChainComplex.toSingle₀Equiv (Functor.obj (chainComplexFunctor k g) (leftRegular k G))
                    (trivial k G k)) :
                (a :
                    { f : ((chainComplexFunctor k g).obj (leftRegular k G)).X 0 ⟶ trivial k G k //
                      ((chainComplexFunctor k g).obj (leftRegular k G)).d 1 0 ≫ f = 0 }) →
                  (chainComplexFunctor k g).obj (leftRegular k G) ⟶
                    (ChainComplex.single₀ (Rep k G)).obj (trivial k G k))
              (Subtype.mk (p :=
                  fun
                    (f :
                      Quiver.Hom (V := Rep k G)
                        (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 0)
                        (trivial k G k)) ↦
                  Eq (α :=
                    Quiver.Hom (V := Rep k G)
                      (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1)
                      (trivial k G k))
                    (HomologicalComplex.d (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1 0 ≫ f) 0)
                  (leftRegularHom (trivial k G k) 1) ⋯ :
                { f :
                  Quiver.Hom (V := Rep k G)
                    (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 0)
                    (trivial k G k) //
                  Eq (α :=
                    Quiver.Hom (V := Rep k G)
                      (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1)
                      (trivial k G k))
                    (HomologicalComplex.d (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1 0 ≫ f) 0 }))))
        ⋯)
      (forget₂ (Rep k G) (FC := fun (A B : Rep k G) ↦ A.ρ.IntertwiningMap B.ρ) (CC := V) (ModuleCat k) (FD :=
        fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CD := ModuleCat.carrier)))

Tactic:
simpa [ShortComplex.moduleCat_exact_iff_range_eq_ker,
          HomologicalComplex.alternatingConst, ChainComplex.toSingle₀Equiv] using
          leftRegular.range_applyAsHom_sub_eq_ker_linearCombination k g hg

Premises used: HomologicalComplex.alternatingConst, ChainComplex.toSingle₀Equiv, CategoryTheory.ShortComplex.moduleCat_exact_iff_range_eq_ker, Rep.FiniteCyclicGroup.leftRegular.range_applyAsHom_sub_eq_ker_linearCombination