Theorem: IsLindelof.elim_countable_subfamily_closed
File: Mathlib/Topology/Compactness/Lindelof.lean

Proof steps:
state_before: X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
⊢ ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
state_after: X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
⊢ ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
tactic: let U := tᶜ
state_before: X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
⊢ ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
state_after: X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
⊢ ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
tactic: have hUo : ∀ i, IsOpen (U i) := by simp only [U, Pi.compl_apply, isOpen_compl_iff]; exact htc
state_before: X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
⊢ ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
state_after: X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
⊢ ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
tactic: have hsU : s ⊆ ⋃ i, U i := by
    simp only [U, Pi.compl_apply]
    rw [← compl_iInter]
    apply disjoint_compl_left_iff_subset.mp
    simp only [compl_iInter, compl_iUnion, compl_compl]
    apply Disjoint.symm
    exact disjoint_iff_inter_eq_empty.mpr hst
state_before: X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
⊢ ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
state_after: case intro.intro
X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
u : Set ι
hucount : u.Countable
husub : s ⊆ ⋃ i ∈ u, U i
⊢ ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
tactic: rcases hs.elim_countable_subcover U hUo hsU with ⟨u, ⟨hucount, husub⟩⟩
state_before: case intro.intro
X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
u : Set ι
hucount : u.Countable
husub : s ⊆ ⋃ i ∈ u, U i
⊢ ∃ u, u.Countable ∧ s ∩ ⋂ i ∈ u, t i = ∅
state_after: case right
X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
u : Set ι
hucount : u.Countable
husub : s ⊆ ⋃ i ∈ u, U i
⊢ s ∩ ⋂ i ∈ u, t i = ∅
tactic: use u, hucount
state_before: case right
X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
u : Set ι
hucount : u.Countable
husub : s ⊆ ⋃ i ∈ u, U i
⊢ s ∩ ⋂ i ∈ u, t i = ∅
state_after: case right
X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
u : Set ι
hucount : u.Countable
husub : Disjoint (⋃ i ∈ u, U i)ᶜ s
⊢ s ∩ ⋂ i ∈ u, t i = ∅
tactic: rw [← disjoint_compl_left_iff_subset] at husub
state_before: case right
X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
u : Set ι
hucount : u.Countable
husub : Disjoint (⋃ i ∈ u, U i)ᶜ s
⊢ s ∩ ⋂ i ∈ u, t i = ∅
state_after: case right
X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
u : Set ι
hucount : u.Countable
husub : Disjoint (⋂ i ∈ u, t i) s
⊢ s ∩ ⋂ i ∈ u, t i = ∅
tactic: simp only [U, Pi.compl_apply, compl_iUnion, compl_compl] at husub
state_before: case right
X : Type u
Y : Type v
ι✝ : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t✝ : Set X
ι : Type v
hs : IsLindelof s
t : ι → Set X
htc : ∀ (i : ι), IsClosed (t i)
hst : s ∩ ⋂ i, t i = ∅
U : ι → Set X := tᶜ
hUo : ∀ (i : ι), IsOpen (U i)
hsU : s ⊆ ⋃ i, U i
u : Set ι
hucount : u.Countable
husub : Disjoint (⋂ i ∈ u, t i) s
⊢ s ∩ ⋂ i ∈ u, t i = ∅
state_after: no goals
tactic: exact disjoint_iff_inter_eq_empty.mp (Disjoint.symm husub)