Theorem: isSigmaCompact_iUnion
File: Mathlib/Topology/Compactness/SigmaCompact.lean

tactic: choose K hcomp hcov using fun i ↦ hcomp i
state_before: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s✝ t : Set X
inst✝ : Countable ι
s : ι → Set X
hcomp : ∀ (i : ι), IsSigmaCompact (s i)
⊢ IsSigmaCompact (⋃ i, s i)
state_after: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s✝ t : Set X
inst✝ : Countable ι
s : ι → Set X
hcomp✝ : ∀ (i : ι), IsSigmaCompact (s i)
K : ι → ℕ → Set X
hcomp : ∀ (i : ι) (n : ℕ), IsCompact (K i n)
hcov : ∀ (i : ι), ⋃ n, K i n = s i
⊢ IsSigmaCompact (⋃ i, s i)

tactic: have := calc
  ⋃ i, s i
  _ = ⋃ i, ⋃ n, (K i n) := by simp_rw [hcov]
  _ = ⋃ (i) (n : ℕ), (K.uncurry ⟨i, n⟩) := by rw [Function.uncurry_def]
  _ = ⋃ x, K.uncurry x := by rw [← iUnion_prod']
state_before: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s✝ t : Set X
inst✝ : Countable ι
s : ι → Set X
hcomp✝ : ∀ (i : ι), IsSigmaCompact (s i)
K : ι → ℕ → Set X
hcomp : ∀ (i : ι) (n : ℕ), IsCompact (K i n)
hcov : ∀ (i : ι), ⋃ n, K i n = s i
⊢ IsSigmaCompact (⋃ i, s i)
state_after: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s✝ t : Set X
inst✝ : Countable ι
s : ι → Set X
hcomp✝ : ∀ (i : ι), IsSigmaCompact (s i)
K : ι → ℕ → Set X
hcomp : ∀ (i : ι) (n : ℕ), IsCompact (K i n)
hcov : ∀ (i : ι), ⋃ n, K i n = s i
this : ⋃ i, s i = ⋃ x, Function.uncurry K x
⊢ IsSigmaCompact (⋃ i, s i)

tactic: rw [this]
state_before: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s✝ t : Set X
inst✝ : Countable ι
s : ι → Set X
hcomp✝ : ∀ (i : ι), IsSigmaCompact (s i)
K : ι → ℕ → Set X
hcomp : ∀ (i : ι) (n : ℕ), IsCompact (K i n)
hcov : ∀ (i : ι), ⋃ n, K i n = s i
this : ⋃ i, s i = ⋃ x, Function.uncurry K x
⊢ IsSigmaCompact (⋃ i, s i)
state_after: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s✝ t : Set X
inst✝ : Countable ι
s : ι → Set X
hcomp✝ : ∀ (i : ι), IsSigmaCompact (s i)
K : ι → ℕ → Set X
hcomp : ∀ (i : ι) (n : ℕ), IsCompact (K i n)
hcov : ∀ (i : ι), ⋃ n, K i n = s i
this : ⋃ i, s i = ⋃ x, Function.uncurry K x
⊢ IsSigmaCompact (⋃ x, Function.uncurry K x)

tactic: exact isSigmaCompact_iUnion_of_isCompact K.uncurry fun x ↦ (hcomp x.1 x.2)
state_before: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s✝ t : Set X
inst✝ : Countable ι
s : ι → Set X
hcomp✝ : ∀ (i : ι), IsSigmaCompact (s i)
K : ι → ℕ → Set X
hcomp : ∀ (i : ι) (n : ℕ), IsCompact (K i n)
hcov : ∀ (i : ι), ⋃ n, K i n = s i
this : ⋃ i, s i = ⋃ x, Function.uncurry K x
⊢ IsSigmaCompact (⋃ x, Function.uncurry K x)
state_after: no goals

tactic: simp_rw [hcov]
state_before: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s✝ t : Set X
inst✝ : Countable ι
s : ι → Set X
hcomp✝ : ∀ (i : ι), IsSigmaCompact (s i)
K : ι → ℕ → Set X
hcomp : ∀ (i : ι) (n : ℕ), IsCompact (K i n)
hcov : ∀ (i : ι), ⋃ n, K i n = s i
⊢ ⋃ i, s i = ⋃ i, ⋃ n, K i n
state_after: no goals

tactic: rw [Function.uncurry_def]
state_before: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s✝ t : Set X
inst✝ : Countable ι
s : ι → Set X
hcomp✝ : ∀ (i : ι), IsSigmaCompact (s i)
K : ι → ℕ → Set X
hcomp : ∀ (i : ι) (n : ℕ), IsCompact (K i n)
hcov : ∀ (i : ι), ⋃ n, K i n = s i
⊢ ⋃ i, ⋃ n, K i n = ⋃ i, ⋃ n, Function.uncurry K (i, n)
state_after: no goals

tactic: rw [← iUnion_prod']
state_before: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s✝ t : Set X
inst✝ : Countable ι
s : ι → Set X
hcomp✝ : ∀ (i : ι), IsSigmaCompact (s i)
K : ι → ℕ → Set X
hcomp : ∀ (i : ι) (n : ℕ), IsCompact (K i n)
hcov : ∀ (i : ι), ⋃ n, K i n = s i
⊢ ⋃ i, ⋃ n, Function.uncurry K (i, n) = ⋃ x, Function.uncurry K x
state_after: no goals