-- Mathlib/CategoryTheory/GradedObject/Trifunctor.lean
@[reassoc (attr := simp)]
lemma ι_mapBifunctorComp₂₃MapObjIso_hom (i₁ : I₁) (i₂ : I₂) (i₃ : I₃) (j : J)
    (h : r (i₁, i₂, i₃) = j) :
    ιMapTrifunctorMapObj (bifunctorComp₂₃ F G₂₃) r X₁ X₂ X₃ i₁ i₂ i₃ j h ≫
      (mapBifunctorComp₂₃MapObjIso F G₂₃ ρ₂₃ X₁ X₂ X₃).hom j =
      ιMapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ i₁ i₂ i₃ j h

Proof state:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₂₃ : Type u_6
inst✝⁷ : Category.{u_14, u_1} C₁
inst✝⁶ : Category.{u_13, u_2} C₂
inst✝⁵ : Category.{u_12, u_3} C₃
inst✝⁴ : Category.{u_11, u_4} C₄
inst✝³ : Category.{u_15, u_6} C₂₃
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₂₃ : BifunctorComp₂₃IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝² : (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).HasMap ρ₂₃.p
inst✝¹ : (((mapBifunctor F I₁ ρ₂₃.I₂₃).obj X₁).obj (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃)).HasMap ρ₂₃.q
H : HasGoodTrifunctor₂₃Obj F G₂₃ ρ₂₃ X₁ X₂ X₃
inst✝ : ((((mapTrifunctor (bifunctorComp₂₃ F G₂₃) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap r
i₁ : I₁
i₂ : I₂
i₃ : I₃
j : J
h : r (i₁, i₂, i₃) = j
⊢ ιMapTrifunctorMapObj (bifunctorComp₂₃ F G₂₃) r X₁ X₂ X₃ i₁ i₂ i₃ j h ≫
      (CofanMapObjFun.iso (isColimitCofan₃MapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ j)).inv =
    ιMapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ i₁ i₂ i₃ j h

Tactic:
apply CofanMapObjFun.ιMapObj_iso_inv

Result:
no goals

Proof state:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₂₃ : Type u_6
inst✝⁷ : Category.{u_14, u_1} C₁
inst✝⁶ : Category.{u_13, u_2} C₂
inst✝⁵ : Category.{u_12, u_3} C₃
inst✝⁴ : Category.{u_11, u_4} C₄
inst✝³ : Category.{u_15, u_6} C₂₃
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₂₃ : BifunctorComp₂₃IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝² : (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).HasMap ρ₂₃.p
inst✝¹ : (((mapBifunctor F I₁ ρ₂₃.I₂₃).obj X₁).obj (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃)).HasMap ρ₂₃.q
H : HasGoodTrifunctor₂₃Obj F G₂₃ ρ₂₃ X₁ X₂ X₃
inst✝ : ((((mapTrifunctor (bifunctorComp₂₃ F G₂₃) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap r
i₁ : I₁
i₂ : I₂
i₃ : I₃
j : J
h : r (i₁, i₂, i₃) = j
⊢ ιMapTrifunctorMapObj (bifunctorComp₂₃ F G₂₃) r X₁ X₂ X₃ i₁ i₂ i₃ j h ≫
      (mapBifunctorComp₂₃MapObjIso F G₂₃ ρ₂₃ X₁ X₂ X₃).hom j =
    ιMapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ i₁ i₂ i₃ j h

Tactic:
dsimp [mapBifunctorComp₂₃MapObjIso]

Result:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₂₃ : Type u_6
inst✝⁷ : Category.{u_14, u_1} C₁
inst✝⁶ : Category.{u_13, u_2} C₂
inst✝⁵ : Category.{u_12, u_3} C₃
inst✝⁴ : Category.{u_11, u_4} C₄
inst✝³ : Category.{u_15, u_6} C₂₃
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₂₃ : BifunctorComp₂₃IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝² : (((mapBifunctor G₂₃ I₂ I₃).obj X₂).obj X₃).HasMap ρ₂₃.p
inst✝¹ : (((mapBifunctor F I₁ ρ₂₃.I₂₃).obj X₁).obj (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃)).HasMap ρ₂₃.q
H : HasGoodTrifunctor₂₃Obj F G₂₃ ρ₂₃ X₁ X₂ X₃
inst✝ : ((((mapTrifunctor (bifunctorComp₂₃ F G₂₃) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap r
i₁ : I₁
i₂ : I₂
i₃ : I₃
j : J
h : r (i₁, i₂, i₃) = j
⊢ ιMapTrifunctorMapObj (bifunctorComp₂₃ F G₂₃) r X₁ X₂ X₃ i₁ i₂ i₃ j h ≫
      (CofanMapObjFun.iso (isColimitCofan₃MapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ j)).inv =
    ιMapBifunctorBifunctor₂₃MapObj F G₂₃ ρ₂₃ X₁ X₂ X₃ i₁ i₂ i₃ j h