Theorem: isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis
File: Mathlib/Topology/Compactness/Compact.lean

tactic: constructor
state_before: X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
U : Set X
⊢ IsCompact U ∧ IsOpen U ↔ ∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i
state_after: case mp
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
U : Set X
⊢ IsCompact U ∧ IsOpen U → ∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i

case mpr
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
U : Set X
⊢ (∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i) → IsCompact U ∧ IsOpen U

tactic: exact fun ⟨h₁, h₂⟩ ↦ eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open _ hb U h₁ h₂
state_before: case mp
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
U : Set X
⊢ IsCompact U ∧ IsOpen U → ∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i
state_after: no goals

tactic: rintro ⟨s, hs, rfl⟩
state_before: case mpr
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
U : Set X
⊢ (∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i) → IsCompact U ∧ IsOpen U
state_after: case mpr.intro.intro
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s✝ t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
s : Set ι
hs : s.Finite
⊢ IsCompact (⋃ i ∈ s, b i) ∧ IsOpen (⋃ i ∈ s, b i)

tactic: constructor
state_before: case mpr.intro.intro
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s✝ t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
s : Set ι
hs : s.Finite
⊢ IsCompact (⋃ i ∈ s, b i) ∧ IsOpen (⋃ i ∈ s, b i)
state_after: case mpr.intro.intro.left
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s✝ t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
s : Set ι
hs : s.Finite
⊢ IsCompact (⋃ i ∈ s, b i)

case mpr.intro.intro.right
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s✝ t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
s : Set ι
hs : s.Finite
⊢ IsOpen (⋃ i ∈ s, b i)

tactic: exact hs.isCompact_biUnion fun i _ => hb' i
state_before: case mpr.intro.intro.left
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s✝ t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
s : Set ι
hs : s.Finite
⊢ IsCompact (⋃ i ∈ s, b i)
state_after: no goals

tactic: exact isOpen_biUnion fun i _ => hb.isOpen (Set.mem_range_self _)
state_before: case mpr.intro.intro.right
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s✝ t : Set X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsCompact (b i)
s : Set ι
hs : s.Finite
⊢ IsOpen (⋃ i ∈ s, b i)
state_after: no goals