-- Module: Mathlib.CategoryTheory.Abelian.Injective.Resolution
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
inst_2 : EnoughInjectives.{v, u} C
Z : C
zero :
  ShortComplex.Exact.{v, u}
      (ShortComplex.mk (X₁ :=
        ShortComplex.X₂.{v, u}
          (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
            (Functor.obj (CochainComplex.single₀ C) Z)))
        (X₂ :=
        ShortComplex.X₂.{v, u}
          (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)))
        (ShortComplex.Hom.τ₂
          (Functor.map (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
            ((Equiv.symm (CochainComplex.fromSingle₀Equiv (ofCocomplex Z) Z) :
                (a : { f : Z ⟶ (ofCocomplex Z).X 0 // f ≫ (ofCocomplex Z).d 0 1 = 0 }) →
                  (CochainComplex.single₀ C).obj Z ⟶ ofCocomplex Z)
              (Subtype.mk (p := fun (f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0) ↦
                  Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1)
                    (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1) 0)
                  (Injective.ι Z) ⋯ :
                { f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0 //
                  Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1)
                    (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1) 0 }))))
        (ShortComplex.g
          (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)))
        ⋯) ∧
    Mono.{v, u} (X :=
      ShortComplex.X₂.{v, u}
        (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
          (Functor.obj (CochainComplex.single₀ C) Z)))
      (Y :=
      ShortComplex.X₂.{v, u}
        (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)))
      (ShortComplex.Hom.τ₂
        (Functor.map (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
          ((Equiv.symm (CochainComplex.fromSingle₀Equiv (ofCocomplex Z) Z) :
              (a : { f : Z ⟶ (ofCocomplex Z).X 0 // f ≫ (ofCocomplex Z).d 0 1 = 0 }) →
                (CochainComplex.single₀ C).obj Z ⟶ ofCocomplex Z)
            (Subtype.mk (p := fun (f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0) ↦
                Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1)
                  (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1) 0)
                (Injective.ι Z) ⋯ :
              { f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0 //
                Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1)
                  (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1) 0 }))))
zero₁ :
  Eq.{v + 1} (α :=
    ShortComplex.X₁.{v, u}
        (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
          (Functor.obj (CochainComplex.single₀ C) Z)) ⟶
      ShortComplex.X₂.{v, u}
        (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
          (Functor.obj (CochainComplex.single₀ C) Z)))
    (ShortComplex.f
      (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
        (Functor.obj (CochainComplex.single₀ C) Z)))
    0
zero₂ :
  Eq.{v + 1} (α :=
    ShortComplex.X₂.{v, u}
        (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
          (Functor.obj (CochainComplex.single₀ C) Z)) ⟶
      ShortComplex.X₃.{v, u}
        (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
          (Functor.obj (CochainComplex.single₀ C) Z)))
    (ShortComplex.g
      (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1)
        (Functor.obj (CochainComplex.single₀ C) Z)))
    0
zero₃ :
  Eq.{v + 1} (α :=
    ShortComplex.X₁.{v, u}
        (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)) ⟶
      ShortComplex.X₂.{v, u}
        (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)))
    (ShortComplex.f (Functor.obj (HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1) (ofCocomplex Z)))
    0
⊢ QuasiIsoAt.{v, u, 0} (ι := ℕ) (C := C) (c := ComplexShape.up ℕ) (K := Functor.obj (CochainComplex.single₀ C) Z) (L :=
    ofCocomplex Z)
    ((Equiv.symm (CochainComplex.fromSingle₀Equiv (ofCocomplex Z) Z) :
        (a : { f : Z ⟶ (ofCocomplex Z).X 0 // f ≫ (ofCocomplex Z).d 0 1 = 0 }) →
          (CochainComplex.single₀ C).obj Z ⟶ ofCocomplex Z)
      (Subtype.mk (p := fun (f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0) ↦
          Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1) (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1)
            0)
          (Injective.ι Z) ⋯ :
        { f : Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 0 //
          Eq (α := Z ⟶ HomologicalComplex.X.{v, u, 0} (ofCocomplex Z) 1) (f ≫ HomologicalComplex.d (ofCocomplex Z) 0 1)
            0 }))
    0

Tactic:
rw [CochainComplex.quasiIsoAt₀_iff, ShortComplex.quasiIso_iff_of_zeros]

Premises used: CategoryTheory.ShortComplex.quasiIso_iff_of_zeros, CochainComplex.quasiIsoAt₀_iff