Theorem: TopologicalSpace.IsSeparable.span
File: Mathlib/Topology/Algebra/Module/Basic.lean

tactic: rw [span_eq_iUnion_nat]
state_before: R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
⊢ IsSeparable ↑(Submodule.span R s)
state_after: R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
⊢ IsSeparable (⋃ n, (fun f => ∑ i : Fin n, (f i).1 • (f i).2) '' {f | ∀ (i : Fin n), (f i).2 ∈ s})

tactic: refine .iUnion fun n ↦ .image ?_ ?_
state_before: R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
⊢ IsSeparable (⋃ n, (fun f => ∑ i : Fin n, (f i).1 • (f i).2) '' {f | ∀ (i : Fin n), (f i).2 ∈ s})
state_after: case refine_1
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
⊢ IsSeparable {f | ∀ (i : Fin n), (f i).2 ∈ s}

case refine_2
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
⊢ Continuous fun f => ∑ i : Fin n, (f i).1 • (f i).2

tactic: have : IsSeparable {f : Fin n → R × M | ∀ (i : Fin n), f i ∈ Set.univ ×ˢ s} := by
  apply isSeparable_pi (fun i ↦ .prod (.of_separableSpace Set.univ) hs)
state_before: case refine_1
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
⊢ IsSeparable {f | ∀ (i : Fin n), (f i).2 ∈ s}
state_after: case refine_1
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
this : IsSeparable {f | ∀ (i : Fin n), f i ∈ Set.univ ×ˢ s}
⊢ IsSeparable {f | ∀ (i : Fin n), (f i).2 ∈ s}

tactic: rwa [Set.univ_prod] at this
state_before: case refine_1
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
this : IsSeparable {f | ∀ (i : Fin n), f i ∈ Set.univ ×ˢ s}
⊢ IsSeparable {f | ∀ (i : Fin n), (f i).2 ∈ s}
state_after: no goals

tactic: apply isSeparable_pi (fun i ↦ .prod (.of_separableSpace Set.univ) hs)
state_before: R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
⊢ IsSeparable {f | ∀ (i : Fin n), f i ∈ Set.univ ×ˢ s}
state_after: no goals

tactic: apply continuous_finset_sum _ (fun i _ ↦ ?_)
state_before: case refine_2
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
⊢ Continuous fun f => ∑ i : Fin n, (f i).1 • (f i).2
state_after: R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
i : Fin n
x✝ : i ∈ Finset.univ
⊢ Continuous fun a => (a i).1 • (a i).2

tactic: exact (continuous_fst.comp (continuous_apply i)).smul (continuous_snd.comp (continuous_apply i))
state_before: R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
i : Fin n
x✝ : i ∈ Finset.univ
⊢ Continuous fun a => (a i).1 • (a i).2
state_after: no goals