-- 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
zero :
  Function.Surjective.{u + 1, u + 1} (α :=
    (↑(ShortComplex.X₂
          (Functor.obj (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
            (Functor.obj (chainComplexFunctor k g) (leftRegular k G)))) :
      Type u))
    (β :=
    (↑(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)))) :
      Type u))
    (⇑(Hom.hom
          (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 }))))) :
      (a :
          (↑((HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0).obj
                  ((chainComplexFunctor k g).obj (leftRegular k G))).X₂ :
            Type u)) →
        (↑((HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0).obj
                ((ChainComplex.single₀ (Rep k G)).obj (trivial k G k))).X₂ :
          Type u))
⊢ Epi.{u, u + 1} (C := Rep k G) (X :=
    ShortComplex.X₂
      (Functor.obj (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
        (Functor.obj (chainComplexFunctor k g) (leftRegular k G))))
    (Y :=
    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.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 }))))

Tactic:
rw [Rep.epi_iff_surjective]

Premises used: Rep.epi_iff_surjective