-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : J) ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this_1 : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
j : J
rw :
  a ≫
        (productTriangle T).mor₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) -
      a ≫
        (productTriangle T).mor₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) φ'.hom₁ ≫
            Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) =
    0
rw₁ :
  a ≫
      (productTriangle T).mor₃ ≫
        Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) =
    a ≫
      (productTriangle T).mor₃ ≫
        Functor.map (shiftFunctor C (A := ℤ) 1) φ'.hom₁ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j))
rw₂ :
  a ≫
      (productTriangle T).mor₃ ≫
        Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) =
    a ≫
      (productTriangle T).mor₃ ≫
        Functor.map (shiftFunctor C (A := ℤ) 1) (𝟙 T'.obj₁) ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j))
rw₃ :
  a ≫
      (productTriangle T).mor₃ ≫
        Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) =
    a ≫
      (productTriangle T).mor₃ ≫
        𝟙 (Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) T'.obj₁) ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j))
rw₄ rw₅ :
  a ≫
      (productTriangle T).mor₃ ≫
        Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) =
    a ≫
      (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j))
⊢ a ≫
        (productTriangle T).mor₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) -
      a' ≫
        T'.mor₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) φ'.hom₁ ≫
            Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) =
    0

Tactic:
rw [reassoc_of% ha', sub_eq_zero, h₁, Functor.map_id, id_comp]

Premises used: sub_eq_zero, CategoryTheory.Functor.map_id, CategoryTheory.Category.id_comp