l3lab/ntp-mathlib huggingface dataset:
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    

Proof state:
case tfae_2_to_3.intro
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := i.hom t
⊢ (π q.fst) q.snd = (Sigma.desc π) t

Next tactic:
have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by 

Proof state:
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := i.hom t
⊢ t = (i.hom ≫ i.inv) t

Next tactic:
simp only [i.hom_inv_id]
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; 

Proof state:
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := i.hom t
⊢ t = (𝟙 (∐ X)) t

Next tactic:
rfl
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
    

Proof state:
case tfae_2_to_3.intro
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := i.hom t
this : t = i.inv (i.hom t)
⊢ (π q.fst) q.snd = (Sigma.desc π) t

Next tactic:
rw [this]
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
    rw [this]
    

Proof state:
case tfae_2_to_3.intro
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := i.hom t
this : t = i.inv (i.hom t)
⊢ (π q.fst) q.snd = (Sigma.desc π) (i.inv (i.hom t))

Next tactic:
show _ = (i.inv ≫ Sigma.desc π) (i.hom t)
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
    rw [this]
    show _ = (i.inv ≫ Sigma.desc π) (i.hom t)
    

Proof state:
case tfae_2_to_3.intro
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := i.hom t
this : t = i.inv (i.hom t)
⊢ (π q.fst) q.snd = (i.inv ≫ Sigma.desc π) (i.hom t)

Next tactic:
suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
      rw [this]; rfl
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
    rw [this]
    show _ = (i.inv ≫ Sigma.desc π) (i.hom t)
    suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
      

Proof state:
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := i.hom t
this✝ : t = i.inv (i.hom t)
this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π
⊢ (π q.fst) q.snd = (i.inv ≫ Sigma.desc π) (i.hom t)

Next tactic:
rw [this]
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
    rw [this]
    show _ = (i.inv ≫ Sigma.desc π) (i.hom t)
    suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
      rw [this]; 

Proof state:
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := i.hom t
this✝ : t = i.inv (i.hom t)
this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π
⊢ (π q.fst) q.snd = (finiteCoproduct.desc X π) (i.hom t)

Next tactic:
rfl
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
    rw [this]
    show _ = (i.inv ≫ Sigma.desc π) (i.hom t)
    suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
      rw [this]; rfl
    

Proof state:
case tfae_2_to_3.intro
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := i.hom t
this : t = i.inv (i.hom t)
⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π

Next tactic:
rw [Iso.inv_comp_eq]
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
    rw [this]
    show _ = (i.inv ≫ Sigma.desc π) (i.hom t)
    suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
      rw [this]; rfl
    rw [Iso.inv_comp_eq]
    

Proof state:
case tfae_2_to_3.intro
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := i.hom t
this : t = i.inv (i.hom t)
⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π

Next tactic:
apply colimit.hom_ext
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
    rw [this]
    show _ = (i.inv ≫ Sigma.desc π) (i.hom t)
    suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
      rw [this]; rfl
    rw [Iso.inv_comp_eq]
    apply colimit.hom_ext
    

Proof state:
case tfae_2_to_3.intro.w
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := 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

Next tactic:
rintro ⟨a⟩
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
    rw [this]
    show _ = (i.inv ≫ Sigma.desc π) (i.hom t)
    suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
      rw [this]; rfl
    rw [Iso.inv_comp_eq]
    apply colimit.hom_ext
    rintro ⟨a⟩
    

Proof state:
case tfae_2_to_3.intro.w.mk
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := 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

Next tactic:
simp only [Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app,
      colimit.comp_coconePointUniqueUpToIso_hom_assoc]
open List in
theorem effectiveEpiFamily_tfae
    {α : Type} [Fintype α] {B : Profinite.{u}}
    (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ] := by
  tfae_have 2 → 1
  · intro
    simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
  tfae_have 1 → 2
  · intro; infer_instance
  tfae_have 3 → 2
  · intro e
    rw [epi_iff_surjective]
    intro b
    obtain ⟨t, x, h⟩ := e b
    refine ⟨Sigma.ι X t x, ?_⟩
    change (Sigma.ι X t ≫ Sigma.desc π) x = _
    simpa using h
  tfae_have 2 → 3
  · intro e; rw [epi_iff_surjective] at e
    let i : ∐ X ≅ finiteCoproduct X :=
      (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
    intro b
    obtain ⟨t, rfl⟩ := e b
    let q := i.hom t
    refine ⟨q.1,q.2,?_⟩
    have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id]; rfl
    rw [this]
    show _ = (i.inv ≫ Sigma.desc π) (i.hom t)
    suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
      rw [this]; rfl
    rw [Iso.inv_comp_eq]
    apply colimit.hom_ext
    rintro ⟨a⟩
    simp only [Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app,
      colimit.comp_coconePointUniqueUpToIso_hom_assoc]
    

Proof state:
case tfae_2_to_3.intro.w.mk
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, (π a) x = b) → 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 : (forget Profinite).obj (finiteCoproduct X) := 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

Next tactic:
ext