internlm/Lean-Github huggingface dataset:
Theorem: Profinite.EffectiveEpiFamily.π'_comp_ι_hom
Proof state before:
case w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
x✝ : (forget Profinite).obj (X a)
⊢ ↑(π' π surj a ≫ (ι₂Iso π surj).hom) x✝ = ↑(π a) x✝
Tactic:
rfl
Proof state after:
no goals
Theorem: Profinite.EffectiveEpiFamily.π_comp_ι_inv
Proof state before:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ π a ≫ (ι₂Iso π surj).inv = π' π surj a
Tactic:
rw [Iso.comp_inv_eq]
Proof state after:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ π a = π' π surj a ≫ (ι₂Iso π surj).hom
Theorem: Profinite.EffectiveEpiFamily.π_comp_ι_inv
Proof state before:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ π a = π' π surj a ≫ (ι₂Iso π surj).hom
Tactic:
exact π'_comp_ι_hom _ surj _
Proof state after:
no goals
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
Tactic:
tfae_have 1 → 2
Proof state after:
case tfae_1_to_2
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π)

X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
Tactic:
tfae_have 2 → 3
Proof state after:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b

X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
Tactic:
tfae_have 3 → 1
Proof state after:
case tfae_3_to_1
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π

X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
tfae_3_to_1 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
tfae_3_to_1 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
Tactic:
tfae_finish
Proof state after:
no goals
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_1_to_2
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π)
Tactic:
intro
Proof state after:
case tfae_1_to_2
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π)
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_1_to_2
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π)
Tactic:
infer_instance
Proof state after:
no goals
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
Tactic:
intro e
Proof state after:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Epi (Sigma.desc π)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Epi (Sigma.desc π)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
Tactic:
rw [epi_iff_surjective] at e
Proof state after:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
Tactic:
let i : ∐ X ≅ finiteCoproduct X :=
  (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
Proof state after:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
Tactic:
intro b
Proof state after:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
b : ↑B.toCompHaus.toTop
⊢ ∃ a x, ↑(π a) x = b
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
b : ↑B.toCompHaus.toTop
⊢ ∃ a x, ↑(π a) x = b
Tactic:
obtain ⟨t,rfl⟩ := e b
Proof state after:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t
Tactic:
let q := i.hom t
Proof state after:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t
Tactic:
refine ⟨q.1,q.2,?_⟩
Proof state after:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t
Tactic:
have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id] ; rfl
Proof state after:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t
Tactic:
rw [this]
Proof state after:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) (↑i.inv (↑i.hom t))
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) (↑i.inv (↑i.hom t))
Tactic:
show _ = (i.inv ≫ Sigma.desc π) (i.hom t)
Proof state after:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t)
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t)
Tactic:
suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
  rw [this] ; rfl
Proof state after:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π
Tactic:
rw [Iso.inv_comp_eq]
Proof state after:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π
Tactic:
apply colimit.hom_ext
Proof state after:
case tfae_2_to_3.intro.w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ∀ (j : Discrete α),
    colimit.ι (Discrete.functor X) j ≫ Sigma.desc π =
      colimit.ι (Discrete.functor X) j ≫ i.hom ≫ finiteCoproduct.desc X π
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro.w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ∀ (j : Discrete α),
    colimit.ι (Discrete.functor X) j ≫ Sigma.desc π =
      colimit.ι (Discrete.functor X) j ≫ i.hom ≫ finiteCoproduct.desc X π
Tactic:
rintro ⟨a⟩
Proof state after:
case tfae_2_to_3.intro.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
⊢ colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π =
    colimit.ι (Discrete.functor X) { as := a } ≫ i.hom ≫ finiteCoproduct.desc X π
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
⊢ colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π =
    colimit.ι (Discrete.functor X) { as := a } ≫ i.hom ≫ finiteCoproduct.desc X π
Tactic:
simp only [Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app,
  colimit.comp_coconePointUniqueUpToIso_hom_assoc]
Proof state after:
case tfae_2_to_3.intro.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
⊢ π a = (finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π
Theorem: Profinite.effectiveEpiFamily_tfae
Proof state before:
case tfae_2_to_3.intro.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
⊢ π a = (finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π
Tactic:
ext
Proof state after:
case tfae_2_to_3.intro.w.mk.w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
  IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
x✝ : (forget Profinite).obj (X a)
⊢ ↑(π a) x✝ = ↑((finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝