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

Proof steps:
state_before: 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
X₁ X₂ : C
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles
state_after: case 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
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles
tactic: obtain ⟨Y, g, h, mem⟩ := distinguished_cocone_triangle₂ (0 : X₂ ⟶ X₁⟦(1 : ℤ)⟧)
state_before: case 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
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles
state_after: case intro.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
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : (Triangle.mk g h 0).mor₁ ≫ e.hom = biprod.inl
he₂ : (Triangle.mk g h 0).mor₂ = e.hom ≫ biprod.snd
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles
tactic: obtain ⟨e, ⟨he₁, he₂⟩⟩ := exists_iso_binaryBiproduct_of_distTriang _ mem rfl
state_before: case intro.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
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : (Triangle.mk g h 0).mor₁ ≫ e.hom = biprod.inl
he₂ : (Triangle.mk g h 0).mor₂ = e.hom ≫ biprod.snd
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles
state_after: case intro.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
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles
tactic: dsimp at he₁ he₂
state_before: case intro.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
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ binaryBiproductTriangle X₁ X₂ ∈ distinguishedTriangles
state_after: case intro.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
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ Triangle.mk g h 0 ≅ binaryBiproductTriangle X₁ X₂
tactic: refine isomorphic_distinguished _ mem _ (Iso.symm ?_)
state_before: case intro.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
X₁ X₂ Y : C
g : X₁ ⟶ Y
h : Y ⟶ X₂
mem : Triangle.mk g h 0 ∈ distinguishedTriangles
e : (Triangle.mk g h 0).obj₂ ≅ (Triangle.mk g h 0).obj₁ ⊞ (Triangle.mk g h 0).obj₃
he₁ : g ≫ e.hom = biprod.inl
he₂ : h = e.hom ≫ biprod.snd
⊢ Triangle.mk g h 0 ≅ binaryBiproductTriangle X₁ X₂
state_after: no goals
tactic: refine Triangle.isoMk _ _ (Iso.refl _) e (Iso.refl _)
    (by aesop_cat) (by aesop_cat) (by aesop_cat)