-- Mathlib/Algebra/Homology/Monoidal.lean
lemma leftUnitor'_inv (i : I) :
    (leftUnitor' K).inv i = (λ_ (K.X i)).inv ≫ ((singleObjXSelf c 0 (𝟙_ C)).inv ▷ (K.X i)) ≫
      ιTensorObj (tensorUnit C c) K 0 i i (zero_add i)

Proof state:
case e_a.e_f
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ (GradedObject.singleObjApplyIsoOfEq 0 (𝟙_ C) 0 ⋯ ≪≫ (singleObjXIsoOfEq c 0 (𝟙_ C) 0 ⋯).symm).hom =
    GradedObject.Monoidal.tensorUnit₀.hom ≫ (singleObjXSelf c 0 (𝟙_ C)).inv

Tactic:
rfl

Result:
no goals

Proof state:
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ (GradedObject.Monoidal.leftUnitor K.X).inv i ≫ GradedObject.Monoidal.tensorHom (tensorUnitIso C c).hom (𝟙 K.X) i =
    (λ_ (K.X i)).inv ≫ (singleObjXSelf c 0 (𝟙_ C)).inv ▷ K.X i ≫ (tensorUnit C c).ιTensorObj K 0 i i ⋯

Tactic:
rw [GradedObject.Monoidal.leftUnitor_inv_apply, assoc, assoc, Iso.cancel_iso_inv_left,
  GradedObject.Monoidal.ι_tensorHom]

Result:
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ GradedObject.Monoidal.tensorUnit₀.inv ▷ K.X i ≫
      ((tensorUnitIso C c).hom 0 ⊗ 𝟙 K.X i) ≫ GradedObject.Monoidal.ιTensorObj (tensorUnit C c).X K.X 0 i i ⋯ =
    (singleObjXSelf c 0 (𝟙_ C)).inv ▷ K.X i ≫ (tensorUnit C c).ιTensorObj K 0 i i ⋯

Proof state:
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ GradedObject.Monoidal.tensorUnit₀.inv ▷ K.X i ≫
      ((tensorUnitIso C c).hom 0 ⊗ 𝟙 (K.X i)) ≫ GradedObject.Monoidal.ιTensorObj (tensorUnit C c).X K.X 0 i i ⋯ =
    (singleObjXSelf c 0 (𝟙_ C)).inv ▷ K.X i ≫ (tensorUnit C c).ιTensorObj K 0 i i ⋯

Tactic:
rw [tensorHom_id, ← comp_whiskerRight_assoc]

Result:
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ (GradedObject.Monoidal.tensorUnit₀.inv ≫ (tensorUnitIso C c).hom 0) ▷ K.X i ≫
      GradedObject.Monoidal.ιTensorObj (tensorUnit C c).X K.X 0 i i ⋯ =
    (singleObjXSelf c 0 (𝟙_ C)).inv ▷ K.X i ≫ (tensorUnit C c).ιTensorObj K 0 i i ⋯

Proof state:
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ GradedObject.Monoidal.tensorUnit₀.inv ▷ K.X i ≫
      ((tensorUnitIso C c).hom 0 ⊗ 𝟙 K.X i) ≫ GradedObject.Monoidal.ιTensorObj (tensorUnit C c).X K.X 0 i i ⋯ =
    (singleObjXSelf c 0 (𝟙_ C)).inv ▷ K.X i ≫ (tensorUnit C c).ιTensorObj K 0 i i ⋯

Tactic:
dsimp

Result:
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ GradedObject.Monoidal.tensorUnit₀.inv ▷ K.X i ≫
      ((tensorUnitIso C c).hom 0 ⊗ 𝟙 (K.X i)) ≫ GradedObject.Monoidal.ιTensorObj (tensorUnit C c).X K.X 0 i i ⋯ =
    (singleObjXSelf c 0 (𝟙_ C)).inv ▷ K.X i ≫ (tensorUnit C c).ιTensorObj K 0 i i ⋯

Proof state:
case e_a.e_f
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ (if hi : 0 = 0 then GradedObject.singleObjApplyIsoOfEq 0 (𝟙_ C) 0 hi ≪≫ (singleObjXIsoOfEq c 0 (𝟙_ C) 0 hi).symm
      else { hom := 0, inv := 0, hom_inv_id := ⋯, inv_hom_id := ⋯ }).hom =
    GradedObject.Monoidal.tensorUnit₀.hom ≫ (singleObjXSelf c 0 (𝟙_ C)).inv

Tactic:
rw [dif_pos rfl]

Result:
case e_a.e_f
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ (GradedObject.singleObjApplyIsoOfEq 0 (𝟙_ C) 0 ⋯ ≪≫ (singleObjXIsoOfEq c 0 (𝟙_ C) 0 ⋯).symm).hom =
    GradedObject.Monoidal.tensorUnit₀.hom ≫ (singleObjXSelf c 0 (𝟙_ C)).inv

Proof state:
case e_a.e_f
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ (tensorUnitIso C c).hom 0 = GradedObject.Monoidal.tensorUnit₀.hom ≫ (singleObjXSelf c 0 (𝟙_ C)).inv

Tactic:
dsimp [tensorUnitIso]

Result:
case e_a.e_f
C : Type u_1
inst✝⁹ : Category.{u_2, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
i : I
⊢ (if hi : 0 = 0 then GradedObject.singleObjApplyIsoOfEq 0 (𝟙_ C) 0 hi ≪≫ (singleObjXIsoOfEq c 0 (𝟙_ C) 0 hi).symm
      else { hom := 0, inv := 0, hom_inv_id := ⋯, inv_hom_id := ⋯ }).hom =
    GradedObject.Monoidal.tensorUnit₀.hom ≫ (singleObjXSelf c 0 (𝟙_ C)).inv