-- Mathlib/Condensed/Light/Epi.lean
lemma epi_π_app_zero_of_epi : Epi (c.π.app ⟨0⟩)

Proof state:
case hF
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ ∀ (n : ℕ), Epi ((F ⋙ forget R).map (homOfLE ⋯).op)

Tactic:
exact fun _ ↦ (forget R).map_epi _

Result:
no goals

Proof state:
case hc
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
this : (forget R).IsRightAdjoint
⊢ IsLimit ((forget R).mapCone c)

Tactic:
exact isLimitOfPreserves _ hc

Result:
no goals

Proof state:
case hc
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ IsLimit ((forget R).mapCone c)

Tactic:
have := (freeForgetAdjunction R).isRightAdjoint

Result:
case hc
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
this : (forget R).IsRightAdjoint
⊢ IsLimit ((forget R).mapCone c)

Proof state:
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ Epi ((forget R).map (c.π.app (Opposite.op 0)))

Tactic:
change Epi (((forget R).mapCone c).π.app ⟨0⟩)

Result:
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ Epi (((forget R).mapCone c).π.app (Opposite.op 0))

Proof state:
case h
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ ∀ (G : ℕᵒᵖ ⥤ LightProfinite),
    (∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (G.map (homOfLE ⋯).op))) →
      Function.Surjective ⇑(ConcreteCategory.hom (limit.π G (Opposite.op 0)))

Tactic:
exact fun x h ↦ Concrete.surjective_π_app_zero_of_surjective_map (limit.isLimit x) h

Result:
no goals

Proof state:
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ Epi (c.π.app (Opposite.op 0))

Tactic:
apply Functor.epi_of_epi_map (forget R)

Result:
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ Epi ((forget R).map (c.π.app (Opposite.op 0)))

Proof state:
case h
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ ∀ (G : ℕᵒᵖ ⥤ LightProfinite),
    (∀ (n : ℕ), EffectiveEpi (G.map (homOfLE ⋯).op)) → EffectiveEpi (limit.π G (Opposite.op 0))

Tactic:
simp only [LightProfinite.effectiveEpi_iff_surjective]

Result:
case h
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ ∀ (G : ℕᵒᵖ ⥤ LightProfinite),
    (∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (G.map (homOfLE ⋯).op))) →
      Function.Surjective ⇑(ConcreteCategory.hom (limit.π G (Opposite.op 0)))

Proof state:
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ Epi (((forget R).mapCone c).π.app (Opposite.op 0))

Tactic:
apply coherentTopology.epi_π_app_zero_of_epi

Result:
case h
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ ∀ (G : ℕᵒᵖ ⥤ LightProfinite),
    (∀ (n : ℕ), EffectiveEpi (G.map (homOfLE ⋯).op)) → EffectiveEpi (limit.π G (Opposite.op 0))

case hc
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ IsLimit ((forget R).mapCone c)

case hF
R : Type u_1
inst✝ : Ring R
F : ℕᵒᵖ ⥤ LightCondMod R
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Epi (F.map (homOfLE ⋯).op)
⊢ ∀ (n : ℕ), Epi ((F ⋙ forget R).map (homOfLE ⋯).op)