-- Mathlib/Topology/Category/Compactum.lean
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x

Proof state:
case intro
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
claim6 : ∀ S ∈ T2, S.Nonempty
this : ∀ (F : fsu), ↑F ⊆ T1 → (⋂₀ ι F).Nonempty
G : Ultrafilter (Ultrafilter X.A)
h1 : T1 ⊆ (↑G).sets
c1 : X.join G = F
c2 : Ultrafilter.map X.str G = X.incl x
⊢ X.str F = x

Tactic:
simp [← c1, c2]

Result:
no goals

Proof state:
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
⊢ ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q

Tactic:
intro S hS

Result:
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q

Proof state:
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
⊢ ∀ S ∈ T0, S.Nonempty

Tactic:
rintro S ⟨S, hS, rfl⟩

Result:
case intro.intro
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
S : Set X.A
hS : S ∈ F
⊢ (Compactum.basic S).Nonempty

Proof state:
case cond
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ FiniteInter T0

Tactic:
constructor

Result:
case cond.univ_mem
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ Set.univ ∈ T0

case cond.inter_mem
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ ∀ ⦃s : Set (Ultrafilter X.A)⦄, s ∈ T0 → ∀ ⦃t : Set (Ultrafilter X.A)⦄, t ∈ T0 → s ∩ t ∈ T0

Proof state:
case intro.intro.intro.intro
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
S : Set X.A
hS : S ∈ F
G : Ultrafilter X.A
hG : G ∈ Compactum.basic S
hG2 : X.str G = x
⊢ (AA ∩ Compactum.basic S).Nonempty

Tactic:
exact ⟨G, hG2, hG⟩

Result:
no goals

Proof state:
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
claim6 : ∀ S ∈ T2, S.Nonempty
this : ∀ (F : fsu), ↑F ⊆ T1 → (⋂₀ ι F).Nonempty
G : Ultrafilter (Ultrafilter X.A)
h1 : T1 ⊆ (↑G).sets
c1 : X.join G = F
⊢ Ultrafilter.map X.str G = X.incl x

Tactic:
refine Ultrafilter.coe_le_coe.1 fun P hP => ?_

Result:
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
claim6 : ∀ S ∈ T2, S.Nonempty
this : ∀ (F : fsu), ↑F ⊆ T1 → (⋂₀ ι F).Nonempty
G : Ultrafilter (Ultrafilter X.A)
h1 : T1 ⊆ (↑G).sets
c1 : X.join G = F
P : Set X.A
hP : P ∈ ↑(X.incl x)
⊢ P ∈ ↑(Ultrafilter.map X.str G)