Theorem: CategoryTheory.Pretriangulated.productTriangle_distinguished
File: Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean

Proof steps:
state_before: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
⊢ productTriangle T ∈ distinguishedTriangles
state_after: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
⊢ productTriangle T ∈ distinguishedTriangles
tactic: let f₁ := Pi.map (fun j => (T j).mor₁)
state_before: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
⊢ productTriangle T ∈ distinguishedTriangles
state_after: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
hT' : Triangle.mk f₁ f₂ f₃ ∈ distinguishedTriangles
⊢ productTriangle T ∈ distinguishedTriangles
tactic: obtain ⟨Z, f₂, f₃, hT'⟩ := distinguished_cocone_triangle f₁
state_before: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
hT' : Triangle.mk f₁ f₂ f₃ ∈ distinguishedTriangles
⊢ productTriangle T ∈ distinguishedTriangles
state_after: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
hT' : Triangle.mk f₁ f₂ f₃ ∈ distinguishedTriangles
T' : Triangle C := Triangle.mk f₁ f₂ f₃
⊢ productTriangle T ∈ distinguishedTriangles
tactic: let T' := Triangle.mk f₁ f₂ f₃
state_before: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
hT' : Triangle.mk f₁ f₂ f₃ ∈ distinguishedTriangles
T' : Triangle C := Triangle.mk f₁ f₂ f₃
⊢ productTriangle T ∈ distinguishedTriangles
state_after: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
⊢ productTriangle T ∈ distinguishedTriangles
tactic: change T' ∈ distTriang C at hT'
state_before: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
⊢ productTriangle T ∈ distinguishedTriangles
state_after: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun j => (T j).obj₂) j) ⋯
⊢ productTriangle T ∈ distinguishedTriangles
tactic: let φ : ∀ j, T' ⟶ T j := fun j => completeDistinguishedTriangleMorphism _ _
    hT' (hT j) (Pi.π _ j) (Pi.π _ j) (by simp [f₁, T'])
state_before: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun j => (T j).obj₂) j) ⋯
⊢ productTriangle T ∈ distinguishedTriangles
state_after: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun j => (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
⊢ productTriangle T ∈ distinguishedTriangles
tactic: let φ' := productTriangle.lift _ φ
state_before: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun j => (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
⊢ productTriangle T ∈ distinguishedTriangles
state_after: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun j => (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
⊢ productTriangle T ∈ distinguishedTriangles
tactic: have h₁ : φ'.hom₁ = 𝟙 _ := by aesop_cat
state_before: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun j => (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
⊢ productTriangle T ∈ distinguishedTriangles
state_after: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun 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
⊢ productTriangle T ∈ distinguishedTriangles
tactic: have h₂ : φ'.hom₂ = 𝟙 _ := by aesop_cat
state_before: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun 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
⊢ productTriangle T ∈ distinguishedTriangles
state_after: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun 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 : IsIso φ'.hom₁
⊢ productTriangle T ∈ distinguishedTriangles
tactic: have : IsIso φ'.hom₁ := by rw [h₁]; infer_instance
state_before: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun 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 : IsIso φ'.hom₁
⊢ productTriangle T ∈ distinguishedTriangles
state_after: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun 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✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
⊢ productTriangle T ∈ distinguishedTriangles
tactic: have : IsIso φ'.hom₂ := by rw [h₂]; infer_instance
state_before: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun 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✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
⊢ productTriangle T ∈ distinguishedTriangles
state_after: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun 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✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
⊢ IsIso φ'.hom₃
tactic: suffices IsIso φ'.hom₃ by
    have : IsIso φ' := by
      apply Triangle.isIso_of_isIsos
      all_goals infer_instance
    exact isomorphic_distinguished _ hT' _ (asIso φ').symm
state_before: case intro.intro.intro
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun 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✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
⊢ IsIso φ'.hom₃
state_after: case intro.intro.intro.refine_1
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun 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✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
⊢ Function.Injective fun x => x ≫ φ'.hom₃

case intro.intro.intro.refine_2
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j => (T j).obj₁
inst✝² : HasProduct fun j => (T j).obj₂
inst✝¹ : HasProduct fun j => (T j).obj₃
inst✝ : HasProduct fun j => (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j => (T j).obj₁) ⟶ ∏ᶜ fun j => (T j).obj₂ := Pi.map fun j => (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j => (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j => (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j =>
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j => (T j).obj₁) j) (Pi.π (fun 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✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
⊢ Function.Surjective fun x => x ≫ φ'.hom₃
tactic: refine isIso_of_yoneda_map_bijective _ (fun A => ⟨?_, ?_⟩)