Theorem: CategoryTheory.Pretriangulated.complete_distinguished_triangle_morphism₂
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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a = c ≫ T₂.mor₃
⊢ ∃ b, T₁.mor₁ ≫ b = a ≫ T₂.mor₁ ∧ T₁.mor₂ ≫ c = b ≫ T₂.mor₂
state_after: case 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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a✝ : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a✝ = c ≫ T₂.mor₃
a : T₁.invRotate.obj₃ ⟶ T₂.invRotate.obj₃
ha₁ : T₁.invRotate.mor₂ ≫ a = a✝ ≫ T₂.invRotate.mor₂
ha₂ : T₁.invRotate.mor₃ ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c) = a ≫ T₂.invRotate.mor₃
⊢ ∃ b, T₁.mor₁ ≫ b = a✝ ≫ T₂.mor₁ ∧ T₁.mor₂ ≫ c = b ≫ T₂.mor₂
tactic: obtain ⟨a, ⟨ha₁, ha₂⟩⟩ := complete_distinguished_triangle_morphism _ _
    (inv_rot_of_distTriang _ hT₁) (inv_rot_of_distTriang _ hT₂) (c⟦(-1 : ℤ)⟧') a (by
    dsimp
    simp only [neg_comp, comp_neg, ← Functor.map_comp_assoc, ← comm,
      Functor.map_comp, shift_shift_neg', Functor.id_obj, assoc, Iso.inv_hom_id_app, comp_id])
state_before: case 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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a✝ : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a✝ = c ≫ T₂.mor₃
a : T₁.invRotate.obj₃ ⟶ T₂.invRotate.obj₃
ha₁ : T₁.invRotate.mor₂ ≫ a = a✝ ≫ T₂.invRotate.mor₂
ha₂ : T₁.invRotate.mor₃ ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c) = a ≫ T₂.invRotate.mor₃
⊢ ∃ b, T₁.mor₁ ≫ b = a✝ ≫ T₂.mor₁ ∧ T₁.mor₂ ≫ c = b ≫ T₂.mor₂
state_after: case 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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a✝ : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a✝ = c ≫ T₂.mor₃
a : T₁.invRotate.obj₃ ⟶ T₂.invRotate.obj₃
ha₁ : T₁.invRotate.mor₂ ≫ a = a✝ ≫ T₂.invRotate.mor₂
ha₂ : T₁.invRotate.mor₃ ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c) = a ≫ T₂.invRotate.mor₃
⊢ T₁.mor₂ ≫ c = a ≫ T₂.mor₂
tactic: refine ⟨a, ⟨ha₁, ?_⟩⟩
state_before: case 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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a✝ : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a✝ = c ≫ T₂.mor₃
a : T₁.invRotate.obj₃ ⟶ T₂.invRotate.obj₃
ha₁ : T₁.invRotate.mor₂ ≫ a = a✝ ≫ T₂.invRotate.mor₂
ha₂ : T₁.invRotate.mor₃ ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c) = a ≫ T₂.invRotate.mor₃
⊢ T₁.mor₂ ≫ c = a ≫ T₂.mor₂
state_after: case 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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a✝ : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a✝ = c ≫ T₂.mor₃
a : T₁.invRotate.obj₃ ⟶ T₂.invRotate.obj₃
ha₁ : T₁.invRotate.mor₂ ≫ a = a✝ ≫ T₂.invRotate.mor₂
ha₂ :
  (T₁.mor₂ ≫ (shiftEquiv C 1).counitIso.inv.app T₁.3) ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c) =
    a ≫ T₂.mor₂ ≫ (shiftEquiv C 1).counitIso.inv.app T₂.3
⊢ T₁.mor₂ ≫ c = a ≫ T₂.mor₂
tactic: dsimp only [Triangle.invRotate, Triangle.mk] at ha₂
state_before: case 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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a✝ : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a✝ = c ≫ T₂.mor₃
a : T₁.invRotate.obj₃ ⟶ T₂.invRotate.obj₃
ha₁ : T₁.invRotate.mor₂ ≫ a = a✝ ≫ T₂.invRotate.mor₂
ha₂ :
  (T₁.mor₂ ≫ (shiftEquiv C 1).counitIso.inv.app T₁.3) ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c) =
    a ≫ T₂.mor₂ ≫ (shiftEquiv C 1).counitIso.inv.app T₂.3
⊢ T₁.mor₂ ≫ c = a ≫ T₂.mor₂
state_after: case 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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a✝ : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a✝ = c ≫ T₂.mor₃
a : T₁.invRotate.obj₃ ⟶ T₂.invRotate.obj₃
ha₁ : T₁.invRotate.mor₂ ≫ a = a✝ ≫ T₂.invRotate.mor₂
ha₂ :
  (T₁.mor₂ ≫ (shiftEquiv C 1).counitIso.inv.app T₁.3) ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c) =
    a ≫ T₂.mor₂ ≫ (shiftEquiv C 1).counitIso.inv.app T₂.3
⊢ T₁.mor₂ ≫ c ≫ (shiftEquiv C 1).counitIso.inv.app T₂.obj₃ =
    (T₁.mor₂ ≫ (shiftEquiv C 1).counitIso.inv.app T₁.3) ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c)
tactic: rw [← cancel_mono ((shiftEquiv C (1 : ℤ)).counitIso.inv.app T₂.obj₃), assoc, assoc, ← ha₂]
state_before: case 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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a✝ : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a✝ = c ≫ T₂.mor₃
a : T₁.invRotate.obj₃ ⟶ T₂.invRotate.obj₃
ha₁ : T₁.invRotate.mor₂ ≫ a = a✝ ≫ T₂.invRotate.mor₂
ha₂ :
  (T₁.mor₂ ≫ (shiftEquiv C 1).counitIso.inv.app T₁.3) ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c) =
    a ≫ T₂.mor₂ ≫ (shiftEquiv C 1).counitIso.inv.app T₂.3
⊢ T₁.mor₂ ≫ c ≫ (shiftEquiv C 1).counitIso.inv.app T₂.obj₃ =
    (T₁.mor₂ ≫ (shiftEquiv C 1).counitIso.inv.app T₁.3) ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c)
state_after: no goals
tactic: simp only [shiftEquiv'_counitIso, shift_neg_shift', assoc, Iso.inv_hom_id_app_assoc]