-- Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
lemma isIso₂_of_isIso₁₃ {T T' : Triangle C} (φ : T ⟶ T') (hT : T ∈ distTriang C)
    (hT' : T' ∈ distTriang C) (h₁ : IsIso φ.hom₁) (h₃ : IsIso φ.hom₃) : IsIso φ.hom₂

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
g : A ⟶ T.obj₁
hf : g ≫ T.mor₁ ≫ φ.hom₂ = 0
h : A ⟶ T'.invRotate.obj₁
hh : g ≫ φ.hom₁ = h ≫ T'.invRotate.mor₁
⊢ (h ≫ inv ((shiftFunctor C (-1)).map φ.hom₃)) ≫ T.invRotate.mor₁ = g

Tactic:
have eq := ((invRotate C).map φ).comm₁

Result:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
g : A ⟶ T.obj₁
hf : g ≫ T.mor₁ ≫ φ.hom₂ = 0
h : A ⟶ T'.invRotate.obj₁
hh : g ≫ φ.hom₁ = h ≫ T'.invRotate.mor₁
eq : ((invRotate C).obj T).mor₁ ≫ ((invRotate C).map φ).hom₂ = ((invRotate C).map φ).hom₁ ≫ ((invRotate C).obj T').mor₁
⊢ (h ≫ inv ((shiftFunctor C (-1)).map φ.hom₃)) ≫ T.invRotate.mor₁ = g

Proof state:
case refine_2.intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
y₂ : A ⟶ T'.obj₂
x₃ : A ⟶ T.obj₃
hx₃ : x₃ ≫ φ.hom₃ = y₂ ≫ T'.mor₂
x₂ : A ⟶ T.obj₂
hx₂ : x₃ = x₂ ≫ T.mor₂
y₁ : A ⟶ T'.obj₁
hy₁ : y₂ - x₂ ≫ φ.hom₂ = y₁ ≫ T'.mor₁
⊢ ∃ a, (fun x => x ≫ φ.hom₂) a = y₂

Tactic:
obtain ⟨x₁, hx₁⟩ : ∃ (x₁ : A ⟶ T.obj₁), x₁ ≫ φ.hom₁ = y₁ := ⟨y₁ ≫ inv φ.hom₁, by simp⟩

Result:
case refine_2.intro.intro.intro.intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
y₂ : A ⟶ T'.obj₂
x₃ : A ⟶ T.obj₃
hx₃ : x₃ ≫ φ.hom₃ = y₂ ≫ T'.mor₂
x₂ : A ⟶ T.obj₂
hx₂ : x₃ = x₂ ≫ T.mor₂
y₁ : A ⟶ T'.obj₁
hy₁ : y₂ - x₂ ≫ φ.hom₂ = y₁ ≫ T'.mor₁
x₁ : A ⟶ T.obj₁
hx₁ : x₁ ≫ φ.hom₁ = y₁
⊢ ∃ a, (fun x => x ≫ φ.hom₂) a = y₂

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
⊢ IsIso φ.hom₂

Tactic:
have : Mono φ.hom₂ := by
  rw [mono_iff_cancel_zero]
  intro A f hf
  obtain ⟨g, rfl⟩ := Triangle.coyoneda_exact₂ _ hT f
    (by rw [← cancel_mono φ.hom₃, assoc, φ.comm₂, reassoc_of% hf, zero_comp, zero_comp])
  rw [assoc] at hf
  obtain ⟨h, hh⟩ := Triangle.coyoneda_exact₂ T'.invRotate (inv_rot_of_distTriang _ hT')
    (g ≫ φ.hom₁) (by dsimp; rw [assoc, ← φ.comm₁, hf])
  obtain ⟨k, rfl⟩ : ∃ (k : A ⟶ T.invRotate.obj₁), k ≫ T.invRotate.mor₁ = g := by
    refine ⟨h ≫ inv (φ.hom₃⟦(-1 : ℤ)⟧'), ?_⟩
    have eq := ((invRotate C).map φ).comm₁
    dsimp only [invRotate] at eq
    rw [← cancel_mono φ.hom₁, assoc, assoc, eq, IsIso.inv_hom_id_assoc, hh]
  erw [assoc, comp_distTriang_mor_zero₁₂ _ (inv_rot_of_distTriang _ hT), comp_zero]

Result:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
⊢ IsIso φ.hom₂

Proof state:
case refine_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
f₁ f₂ : A ⟶ T.obj₂
h : (fun x => x ≫ φ.hom₂) f₁ = (fun x => x ≫ φ.hom₂) f₂
⊢ f₁ = f₂

Tactic:
simpa only [← cancel_mono φ.hom₂] using h

Result:
no goals

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
⊢ IsIso φ.hom₂

Tactic:
refine isIso_of_yoneda_map_bijective _ (fun A => ⟨?_, ?_⟩)

Result:
case refine_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
⊢ Function.Injective fun x => x ≫ φ.hom₂

case refine_2
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
⊢ Function.Surjective fun x => x ≫ φ.hom₂