Theorem: CategoryTheory.GradedObject.mapBifunctor₁₂BifunctorMapObj_ext
File: Mathlib/CategoryTheory/GradedObject/Trifunctor.lean

Proof steps:
state_before: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁸ : Category.{u_15, u_1} C₁
inst✝⁷ : Category.{u_16, u_2} C₂
inst✝⁶ : Category.{u_14, u_3} C₃
inst✝⁵ : Category.{u_11, u_4} C₄
inst✝⁴ : Category.{u_13, u_5} C₁₂
inst✝³ : Category.{?u.204416, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₁₂ : BifunctorComp₁₂IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝² : (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p
inst✝¹ : (((mapBifunctor G ρ₁₂.I₁₂ I₃).obj (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj X₃).HasMap ρ₁₂.q
H : HasGoodTrifunctor₁₂Obj F₁₂ G ρ₁₂ X₁ X₂ X₃
inst✝ : ((((mapTrifunctor (bifunctorComp₁₂ F₁₂ G) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap r
j : J
A : C₄
f g : mapBifunctorMapObj G ρ₁₂.q (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂) X₃ j ⟶ A
h :
  ∀ (i₁ : I₁) (i₂ : I₂) (i₃ : I₃) (h : r (i₁, i₂, i₃) = j),
    ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ f =
      ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ g
⊢ f = g
state_after: case h
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁸ : Category.{u_15, u_1} C₁
inst✝⁷ : Category.{u_16, u_2} C₂
inst✝⁶ : Category.{u_14, u_3} C₃
inst✝⁵ : Category.{u_11, u_4} C₄
inst✝⁴ : Category.{u_13, u_5} C₁₂
inst✝³ : Category.{?u.204416, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₁₂ : BifunctorComp₁₂IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝² : (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p
inst✝¹ : (((mapBifunctor G ρ₁₂.I₁₂ I₃).obj (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj X₃).HasMap ρ₁₂.q
H : HasGoodTrifunctor₁₂Obj F₁₂ G ρ₁₂ X₁ X₂ X₃
inst✝ : ((((mapTrifunctor (bifunctorComp₁₂ F₁₂ G) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap r
j : J
A : C₄
f g : mapBifunctorMapObj G ρ₁₂.q (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂) X₃ j ⟶ A
h :
  ∀ (i₁ : I₁) (i₂ : I₂) (i₃ : I₃) (h : r (i₁, i₂, i₃) = j),
    ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ f =
      ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ g
⊢ ∀ (i : ↑(r ⁻¹' {j})),
    Cofan.inj (cofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j) i ≫ f =
      Cofan.inj (cofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j) i ≫ g
tactic: apply Cofan.IsColimit.hom_ext (isColimitCofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j)
state_before: case h
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁸ : Category.{u_15, u_1} C₁
inst✝⁷ : Category.{u_16, u_2} C₂
inst✝⁶ : Category.{u_14, u_3} C₃
inst✝⁵ : Category.{u_11, u_4} C₄
inst✝⁴ : Category.{u_13, u_5} C₁₂
inst✝³ : Category.{?u.204416, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₁₂ : BifunctorComp₁₂IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝² : (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p
inst✝¹ : (((mapBifunctor G ρ₁₂.I₁₂ I₃).obj (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj X₃).HasMap ρ₁₂.q
H : HasGoodTrifunctor₁₂Obj F₁₂ G ρ₁₂ X₁ X₂ X₃
inst✝ : ((((mapTrifunctor (bifunctorComp₁₂ F₁₂ G) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap r
j : J
A : C₄
f g : mapBifunctorMapObj G ρ₁₂.q (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂) X₃ j ⟶ A
h :
  ∀ (i₁ : I₁) (i₂ : I₂) (i₃ : I₃) (h : r (i₁, i₂, i₃) = j),
    ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ f =
      ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ g
⊢ ∀ (i : ↑(r ⁻¹' {j})),
    Cofan.inj (cofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j) i ≫ f =
      Cofan.inj (cofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j) i ≫ g
state_after: case h.mk
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁸ : Category.{u_15, u_1} C₁
inst✝⁷ : Category.{u_16, u_2} C₂
inst✝⁶ : Category.{u_14, u_3} C₃
inst✝⁵ : Category.{u_11, u_4} C₄
inst✝⁴ : Category.{u_13, u_5} C₁₂
inst✝³ : Category.{?u.204416, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₁₂ : BifunctorComp₁₂IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝² : (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p
inst✝¹ : (((mapBifunctor G ρ₁₂.I₁₂ I₃).obj (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj X₃).HasMap ρ₁₂.q
H : HasGoodTrifunctor₁₂Obj F₁₂ G ρ₁₂ X₁ X₂ X₃
inst✝ : ((((mapTrifunctor (bifunctorComp₁₂ F₁₂ G) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap r
j : J
A : C₄
f g : mapBifunctorMapObj G ρ₁₂.q (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂) X₃ j ⟶ A
h :
  ∀ (i₁ : I₁) (i₂ : I₂) (i₃ : I₃) (h : r (i₁, i₂, i₃) = j),
    ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ f =
      ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ g
i : I₁ × I₂ × I₃
hi : i ∈ r ⁻¹' {j}
⊢ Cofan.inj (cofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j) ⟨i, hi⟩ ≫ f =
    Cofan.inj (cofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j) ⟨i, hi⟩ ≫ g
tactic: rintro ⟨i, hi⟩
state_before: case h.mk
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁸ : Category.{u_15, u_1} C₁
inst✝⁷ : Category.{u_16, u_2} C₂
inst✝⁶ : Category.{u_14, u_3} C₃
inst✝⁵ : Category.{u_11, u_4} C₄
inst✝⁴ : Category.{u_13, u_5} C₁₂
inst✝³ : Category.{?u.204416, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₁₂ : BifunctorComp₁₂IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝² : (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p
inst✝¹ : (((mapBifunctor G ρ₁₂.I₁₂ I₃).obj (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj X₃).HasMap ρ₁₂.q
H : HasGoodTrifunctor₁₂Obj F₁₂ G ρ₁₂ X₁ X₂ X₃
inst✝ : ((((mapTrifunctor (bifunctorComp₁₂ F₁₂ G) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap r
j : J
A : C₄
f g : mapBifunctorMapObj G ρ₁₂.q (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂) X₃ j ⟶ A
h :
  ∀ (i₁ : I₁) (i₂ : I₂) (i₃ : I₃) (h : r (i₁, i₂, i₃) = j),
    ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ f =
      ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ g
i : I₁ × I₂ × I₃
hi : i ∈ r ⁻¹' {j}
⊢ Cofan.inj (cofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j) ⟨i, hi⟩ ≫ f =
    Cofan.inj (cofan₃MapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ j) ⟨i, hi⟩ ≫ g
state_after: no goals
tactic: exact h _ _ _ hi