Theorem: SimplicialObject.Split.Hom.ext
File: Mathlib/AlgebraicTopology/SplitSimplicialObject.lean

Proof steps:
state_before: C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
Φ₁ Φ₂ : S₁.Hom S₂
h : ∀ (n : ℕ), Φ₁.f n = Φ₂.f n
⊢ Φ₁ = Φ₂
state_after: case mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
Φ₂ : S₁.Hom S₂
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = Φ₂.f n
⊢ { F := F₁, f := f₁, comm := c₁ } = Φ₂
tactic: rcases Φ₁ with ⟨F₁, f₁, c₁⟩
state_before: case mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
Φ₂ : S₁.Hom S₂
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = Φ₂.f n
⊢ { F := F₁, f := f₁, comm := c₁ } = Φ₂
state_after: case mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
f₂ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₂ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₂, comm := c₂ }.f n
⊢ { F := F₁, f := f₁, comm := c₁ } = { F := F₂, f := f₂, comm := c₂ }
tactic: rcases Φ₂ with ⟨F₂, f₂, c₂⟩
state_before: case mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
f₂ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₂ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₂, comm := c₂ }.f n
⊢ { F := F₁, f := f₁, comm := c₁ } = { F := F₂, f := f₂, comm := c₂ }
state_after: case mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
f₂ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₂ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₂, comm := c₂ }.f n
h' : f₁ = f₂
⊢ { F := F₁, f := f₁, comm := c₁ } = { F := F₂, f := f₂, comm := c₂ }
tactic: have h' : f₁ = f₂ := by
    ext
    apply h
state_before: case mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
f₂ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₂ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₂, comm := c₂ }.f n
h' : f₁ = f₂
⊢ { F := F₁, f := f₁, comm := c₁ } = { F := F₂, f := f₂, comm := c₂ }
state_after: case mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
⊢ { F := F₁, f := f₁, comm := c₁ } = { F := F₂, f := f₁, comm := c₂ }
tactic: subst h'
state_before: case mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
⊢ { F := F₁, f := f₁, comm := c₁ } = { F := F₂, f := f₁, comm := c₂ }
state_after: case mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
⊢ F₁ = F₂
tactic: simp only [mk.injEq, and_true]
state_before: case mk.mk
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
⊢ F₁ = F₂
state_after: case mk.mk.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
⊢ ∀ (n : ℕ), S₁.s.φ F₁ n = S₁.s.φ F₂ n
tactic: apply S₁.s.hom_ext
state_before: case mk.mk.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
⊢ ∀ (n : ℕ), S₁.s.φ F₁ n = S₁.s.φ F₂ n
state_after: case mk.mk.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
n : ℕ
⊢ S₁.s.φ F₁ n = S₁.s.φ F₂ n
tactic: intro n
state_before: case mk.mk.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
n : ℕ
⊢ S₁.s.φ F₁ n = S₁.s.φ F₂ n
state_after: case mk.mk.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
n : ℕ
⊢ S₁.s.ι n ≫ F₁.app (op [n]) = S₁.s.ι n ≫ F₂.app (op [n])
tactic: dsimp
state_before: case mk.mk.h
C : Type u_1
inst✝ : Category.{u_2, u_1} C
S₁ S₂ : Split C
F₁ : S₁.X ⟶ S₂.X
f₁ : (n : ℕ) → S₁.s.N n ⟶ S₂.s.N n
c₁ : ∀ (n : ℕ), S₁.s.ι n ≫ F₁.app (op [n]) = f₁ n ≫ S₂.s.ι n
F₂ : S₁.X ⟶ S₂.X
c₂ : ∀ (n : ℕ), S₁.s.ι n ≫ F₂.app (op [n]) = f₁ n ≫ S₂.s.ι n
h : ∀ (n : ℕ), { F := F₁, f := f₁, comm := c₁ }.f n = { F := F₂, f := f₁, comm := c₂ }.f n
n : ℕ
⊢ S₁.s.ι n ≫ F₁.app (op [n]) = S₁.s.ι n ≫ F₂.app (op [n])
state_after: no goals
tactic: rw [c₁, c₂]