Theorem: CategoryTheory.Pretriangulated.Triangle.shiftFunctorAdd'_inv_app_hom₁
File: Mathlib/CategoryTheory/Triangulated/TriangleShift.lean

Proof steps:
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b n : ℤ
h : a + b = n
T : Triangle C
⊢ ((shiftFunctor C n).obj T).mor₁ ≫ ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₂).hom =
    ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₁).hom ≫ ((shiftFunctor C a ⋙ shiftFunctor C b).obj T).mor₁
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((shiftFunctor C (a + b)).obj T).mor₁ ≫ ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₂).hom =
    ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₁).hom ≫
      ((shiftFunctor C a ⋙ shiftFunctor C b).obj T).mor₁
tactic: subst h
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((shiftFunctor C (a + b)).obj T).mor₁ ≫ ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₂).hom =
    ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₁).hom ≫
      ((shiftFunctor C a ⋙ shiftFunctor C b).obj T).mor₁
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((a + b).negOnePow • (CategoryTheory.shiftFunctor C (a + b)).map T.mor₁) ≫
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ =
    (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁ ≫
      (b.negOnePow • (CategoryTheory.shiftFunctor C b).map (a.negOnePow • (CategoryTheory.shiftFunctor C a).map T.mor₁))
tactic: dsimp
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((a + b).negOnePow • (CategoryTheory.shiftFunctor C (a + b)).map T.mor₁) ≫
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ =
    (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁ ≫
      (b.negOnePow • (CategoryTheory.shiftFunctor C b).map (a.negOnePow • (CategoryTheory.shiftFunctor C a).map T.mor₁))
state_after: no goals
tactic: rw [Linear.units_smul_comp, NatTrans.naturality, Linear.comp_units_smul, Functor.comp_map,
          Functor.map_units_smul, Linear.comp_units_smul, smul_smul, Int.negOnePow_add, mul_comm]
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b n : ℤ
h : a + b = n
T : Triangle C
⊢ ((shiftFunctor C n).obj T).mor₂ ≫ ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₃).hom =
    ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₂).hom ≫ ((shiftFunctor C a ⋙ shiftFunctor C b).obj T).mor₂
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((shiftFunctor C (a + b)).obj T).mor₂ ≫ ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₃).hom =
    ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₂).hom ≫
      ((shiftFunctor C a ⋙ shiftFunctor C b).obj T).mor₂
tactic: subst h
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((shiftFunctor C (a + b)).obj T).mor₂ ≫ ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₃).hom =
    ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₂).hom ≫
      ((shiftFunctor C a ⋙ shiftFunctor C b).obj T).mor₂
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((a + b).negOnePow • (CategoryTheory.shiftFunctor C (a + b)).map T.mor₂) ≫
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₃ =
    (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ ≫
      (b.negOnePow • (CategoryTheory.shiftFunctor C b).map (a.negOnePow • (CategoryTheory.shiftFunctor C a).map T.mor₂))
tactic: dsimp
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((a + b).negOnePow • (CategoryTheory.shiftFunctor C (a + b)).map T.mor₂) ≫
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₃ =
    (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₂ ≫
      (b.negOnePow • (CategoryTheory.shiftFunctor C b).map (a.negOnePow • (CategoryTheory.shiftFunctor C a).map T.mor₂))
state_after: no goals
tactic: rw [Linear.units_smul_comp, NatTrans.naturality, Linear.comp_units_smul, Functor.comp_map,
          Functor.map_units_smul, Linear.comp_units_smul, smul_smul, Int.negOnePow_add, mul_comm]
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b n : ℤ
h : a + b = n
T : Triangle C
⊢ ((shiftFunctor C n).obj T).mor₃ ≫
      (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₁).hom =
    ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₃).hom ≫ ((shiftFunctor C a ⋙ shiftFunctor C b).obj T).mor₃
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((shiftFunctor C (a + b)).obj T).mor₃ ≫
      (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₁).hom =
    ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₃).hom ≫
      ((shiftFunctor C a ⋙ shiftFunctor C b).obj T).mor₃
tactic: subst h
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((shiftFunctor C (a + b)).obj T).mor₃ ≫
      (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₁).hom =
    ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).app T.obj₃).hom ≫
      ((shiftFunctor C a ⋙ shiftFunctor C b).obj T).mor₃
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((a + b).negOnePow •
        (CategoryTheory.shiftFunctor C (a + b)).map T.mor₃ ≫ (shiftFunctorComm C 1 (a + b)).hom.app T.obj₁) ≫
      (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁) =
    (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₃ ≫
      (b.negOnePow •
        (CategoryTheory.shiftFunctor C b).map
            (a.negOnePow • (CategoryTheory.shiftFunctor C a).map T.mor₃ ≫ (shiftFunctorComm C 1 a).hom.app T.obj₁) ≫
          (shiftFunctorComm C 1 b).hom.app ((CategoryTheory.shiftFunctor C a).obj T.obj₁))
tactic: dsimp
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ ((a + b).negOnePow •
        (CategoryTheory.shiftFunctor C (a + b)).map T.mor₃ ≫ (shiftFunctorComm C 1 (a + b)).hom.app T.obj₁) ≫
      (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁) =
    (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₃ ≫
      (b.negOnePow •
        (CategoryTheory.shiftFunctor C b).map
            (a.negOnePow • (CategoryTheory.shiftFunctor C a).map T.mor₃ ≫ (shiftFunctorComm C 1 a).hom.app T.obj₁) ≫
          (shiftFunctorComm C 1 b).hom.app ((CategoryTheory.shiftFunctor C a).obj T.obj₁))
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ (a + b).negOnePow •
      (CategoryTheory.shiftFunctor C (a + b)).map T.mor₃ ≫
        (shiftFunctorComm C 1 (a + b)).hom.app T.obj₁ ≫
          (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁) =
    (b.negOnePow * a.negOnePow) •
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₃ ≫
        (CategoryTheory.shiftFunctor C b).map ((CategoryTheory.shiftFunctor C a).map T.mor₃) ≫
          (CategoryTheory.shiftFunctor C b).map ((shiftFunctorComm C 1 a).hom.app T.obj₁) ≫
            (shiftFunctorComm C 1 b).hom.app ((CategoryTheory.shiftFunctor C a).obj T.obj₁)
tactic: rw [Linear.units_smul_comp, Linear.comp_units_smul, Functor.map_units_smul,
          Linear.units_smul_comp, Linear.comp_units_smul, smul_smul, assoc,
          Functor.map_comp, assoc]
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ (a + b).negOnePow •
      (CategoryTheory.shiftFunctor C (a + b)).map T.mor₃ ≫
        (shiftFunctorComm C 1 (a + b)).hom.app T.obj₁ ≫
          (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁) =
    (b.negOnePow * a.negOnePow) •
      (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₃ ≫
        (CategoryTheory.shiftFunctor C b).map ((CategoryTheory.shiftFunctor C a).map T.mor₃) ≫
          (CategoryTheory.shiftFunctor C b).map ((shiftFunctorComm C 1 a).hom.app T.obj₁) ≫
            (shiftFunctorComm C 1 b).hom.app ((CategoryTheory.shiftFunctor C a).obj T.obj₁)
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ (a + b).negOnePow •
      (CategoryTheory.shiftFunctor C (a + b)).map T.mor₃ ≫
        (shiftFunctorComm C 1 (a + b)).hom.app T.obj₁ ≫
          (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁) =
    (b.negOnePow * a.negOnePow) •
      (CategoryTheory.shiftFunctor C (a + b)).map T.mor₃ ≫
        (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app ((CategoryTheory.shiftFunctor C 1).obj T.obj₁) ≫
          (CategoryTheory.shiftFunctor C b).map ((shiftFunctorComm C 1 a).hom.app T.obj₁) ≫
            (shiftFunctorComm C 1 b).hom.app ((CategoryTheory.shiftFunctor C a).obj T.obj₁)
tactic: erw [← NatTrans.naturality_assoc]
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b : ℤ
T : Triangle C
⊢ (a + b).negOnePow •
      (CategoryTheory.shiftFunctor C (a + b)).map T.mor₃ ≫
        (shiftFunctorComm C 1 (a + b)).hom.app T.obj₁ ≫
          (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app T.obj₁) =
    (b.negOnePow * a.negOnePow) •
      (CategoryTheory.shiftFunctor C (a + b)).map T.mor₃ ≫
        (CategoryTheory.shiftFunctorAdd' C a b (a + b) ⋯).hom.app ((CategoryTheory.shiftFunctor C 1).obj T.obj₁) ≫
          (CategoryTheory.shiftFunctor C b).map ((shiftFunctorComm C 1 a).hom.app T.obj₁) ≫
            (shiftFunctorComm C 1 b).hom.app ((CategoryTheory.shiftFunctor C a).obj T.obj₁)
state_after: no goals
tactic: simp only [shiftFunctorAdd'_eq_shiftFunctorAdd, Int.negOnePow_add,
          shiftFunctorComm_hom_app_comp_shift_shiftFunctorAdd_hom_app, add_comm a]
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
a b n : ℤ
h : a + b = n
⊢ ∀ {X Y : Triangle C} (f : X ⟶ Y),
    (shiftFunctor C n).map f ≫
        ((fun T =>
              ((shiftFunctor C n).obj T).isoMk ((shiftFunctor C a ⋙ shiftFunctor C b).obj T)
                ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₁)
                ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₂)
                ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₃) ⋯ ⋯ ⋯)
            Y).hom =
      ((fun T =>
              ((shiftFunctor C n).obj T).isoMk ((shiftFunctor C a ⋙ shiftFunctor C b).obj T)
                ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₁)
                ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₂)
                ((CategoryTheory.shiftFunctorAdd' C a b n h).app T.obj₃) ⋯ ⋯ ⋯)
            X).hom ≫
        (shiftFunctor C a ⋙ shiftFunctor C b).map f
state_after: no goals
tactic: aesop_cat