Module: Mathlib.Topology.Category.LightProfinite.Injective

Goal state before:
pp: S : LightProfinite
inst✝ : Nonempty ↑S.toTop
X Y : Profinite
g : X ⟶ lightToProfinite.obj S
f : X ⟶ Y
h : Mono f
this✝¹ : ∀ (n : ℕ), Finite ↑(lightToProfinite.obj (S.component n)).toTop
this✝ : Nonempty ↑(lightToProfinite.obj (S.component 0)).toTop
this : ∀ (n : ℕ), Epi (S.transitionMap n)
k0 : Y ⟶ lightToProfinite.obj (S.diagram.obj (Opposite.op 0))
h_down0 : f ≫ k0 = g ≫ lightToProfinite.map (S.proj 0)
h_comm :
  ∀ (n : ℕ) (k : Y ⟶ lightToProfinite.obj (S.component n)),
    f ≫ k = g ≫ lightToProfinite.map (S.proj n) →
      f ≫ k = g ≫ lightToProfinite.map (S.proj (n + 1)) ≫ lightToProfinite.map (S.transitionMap n)
h_step :
  ∀ (n : ℕ) (k : Y ⟶ lightToProfinite.obj (S.component n)),
    f ≫ k = g ≫ lightToProfinite.map (S.proj n) →
      ∃ k', k' ≫ lightToProfinite.map (S.transitionMap n) = k ∧ f ≫ k' = g ≫ lightToProfinite.map (S.proj (n + 1))
lifts : ℕ → Type (max 0 u) := fun n ↦ { k // f ≫ k = g ≫ lightToProfinite.map (S.proj n) }
next : (n : ℕ) → lifts n → lifts (n + 1) := fun n k ↦ ⟨⋯.choose, ⋯⟩
k_seq : (n : ℕ) → lifts n := fun n ↦ Nat.rec ⟨k0, h_down0⟩ next n
h_down : ∀ (n : ℕ), f ≫ ↑(k_seq n) = g ≫ lightToProfinite.map (S.proj n)
h_up : ∀ (n : ℕ), ↑(k_seq (n + 1)) ≫ lightToProfinite.map (S.transitionMap n) = ↑(k_seq n)
k_cone : Cone (S.diagram ⋙ lightToProfinite) := { pt := Y, π := NatTrans.ofOpSequence (fun n ↦ ↑(k_seq n)) ⋯ }
hg : g = (isLimitOfPreserves lightToProfinite S.asLimit).lift (k_cone.extend f)
⊢ f ≫ (isLimitOfPreserves lightToProfinite S.asLimit).lift k_cone = g

Tactic:
rw [hg]

Goal state after:
S : LightProfinite
inst✝ : Nonempty ↑S.toTop
X Y : Profinite
g : X ⟶ lightToProfinite.obj S
f : X ⟶ Y
h : Mono f
this✝¹ : ∀ (n : ℕ), Finite ↑(lightToProfinite.obj (S.component n)).toTop
this✝ : Nonempty ↑(lightToProfinite.obj (S.component 0)).toTop
this : ∀ (n : ℕ), Epi (S.transitionMap n)
k0 : Y ⟶ lightToProfinite.obj (S.diagram.obj (Opposite.op 0))
h_down0 : f ≫ k0 = g ≫ lightToProfinite.map (S.proj 0)
h_comm :
  ∀ (n : ℕ) (k : Y ⟶ lightToProfinite.obj (S.component n)),
    f ≫ k = g ≫ lightToProfinite.map (S.proj n) →
      f ≫ k = g ≫ lightToProfinite.map (S.proj (n + 1)) ≫ lightToProfinite.map (S.transitionMap n)
h_step :
  ∀ (n : ℕ) (k : Y ⟶ lightToProfinite.obj (S.component n)),
    f ≫ k = g ≫ lightToProfinite.map (S.proj n) →
      ∃ k', k' ≫ lightToProfinite.map (S.transitionMap n) = k ∧ f ≫ k' = g ≫ lightToProfinite.map (S.proj (n + 1))
lifts : ℕ → Type (max 0 u) := fun n ↦ { k // f ≫ k = g ≫ lightToProfinite.map (S.proj n) }
next : (n : ℕ) → lifts n → lifts (n + 1) := fun n k ↦ ⟨⋯.choose, ⋯⟩
k_seq : (n : ℕ) → lifts n := fun n ↦ Nat.rec ⟨k0, h_down0⟩ next n
h_down : ∀ (n : ℕ), f ≫ ↑(k_seq n) = g ≫ lightToProfinite.map (S.proj n)
h_up : ∀ (n : ℕ), ↑(k_seq (n + 1)) ≫ lightToProfinite.map (S.transitionMap n) = ↑(k_seq n)
k_cone : Cone (S.diagram ⋙ lightToProfinite) := { pt := Y, π := NatTrans.ofOpSequence (fun n ↦ ↑(k_seq n)) ⋯ }
hg : g = (isLimitOfPreserves lightToProfinite S.asLimit).lift (k_cone.extend f)
⊢ f ≫ (isLimitOfPreserves lightToProfinite S.asLimit).lift k_cone =
    (isLimitOfPreserves lightToProfinite S.asLimit).lift (k_cone.extend f)