-- Mathlib/CategoryTheory/Localization/Triangulated.lean
lemma rotate_essImageDistTriang [Preadditive D] [L.Additive]
    [∀ (n : ℤ), (shiftFunctor D n).Additive] (T : Triangle D) :
  T ∈ L.essImageDistTriang ↔ T.rotate ∈ L.essImageDistTriang

Proof state:
case mp.intro.intro
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
T' : Triangle C
e' : T ≅ L.mapTriangle.obj T'
hT' : T' ∈ distinguishedTriangles
⊢ T.rotate ∈ L.essImageDistTriang

Tactic:
exact ⟨T'.rotate, (rotate D).mapIso e' ≪≫ L.mapTriangleRotateIso.app T',
  rot_of_distTriang T' hT'⟩

Result:
no goals

Proof state:
case mpr
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
⊢ T.rotate ∈ L.essImageDistTriang → T ∈ L.essImageDistTriang

Tactic:
rintro ⟨T', e', hT'⟩

Result:
case mpr.intro.intro
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
T' : Triangle C
e' : T.rotate ≅ L.mapTriangle.obj T'
hT' : T' ∈ distinguishedTriangles
⊢ T ∈ L.essImageDistTriang

Proof state:
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
⊢ T ∈ L.essImageDistTriang ↔ T.rotate ∈ L.essImageDistTriang

Tactic:
constructor

Result:
case mp
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
⊢ T ∈ L.essImageDistTriang → T.rotate ∈ L.essImageDistTriang

case mpr
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
⊢ T.rotate ∈ L.essImageDistTriang → T ∈ L.essImageDistTriang

Proof state:
case mp
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
⊢ T ∈ L.essImageDistTriang → T.rotate ∈ L.essImageDistTriang

Tactic:
rintro ⟨T', e', hT'⟩

Result:
case mp.intro.intro
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
T' : Triangle C
e' : T ≅ L.mapTriangle.obj T'
hT' : T' ∈ distinguishedTriangles
⊢ T.rotate ∈ L.essImageDistTriang

Proof state:
case mpr.intro.intro
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{u_4, u_1} C
inst✝¹⁰ : Category.{u_3, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
T' : Triangle C
e' : T.rotate ≅ L.mapTriangle.obj T'
hT' : T' ∈ distinguishedTriangles
⊢ T ∈ L.essImageDistTriang

Tactic:
exact ⟨T'.invRotate, (triangleRotation D).unitIso.app T ≪≫ (invRotate D).mapIso e' ≪≫
  L.mapTriangleInvRotateIso.app T', inv_rot_of_distTriang T' hT'⟩

Result:
no goals