Theorem shsupcl

Description: Closure of the subspace supremum of set of subsets of Hilbert space. (Contributed by NM, 26-Nov-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shsupcl	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ( span ‘ ∪ 𝐴 ) ∈ Sℋ )

Proof

Step	Hyp	Ref	Expression
1		uniss	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ∪ 𝐴 ⊆ ∪ 𝒫 ℋ )
2		unipw	⊢ ∪ 𝒫 ℋ = ℋ
3	1 2	sseqtrdi	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ∪ 𝐴 ⊆ ℋ )
4		spancl	⊢ ( ∪ 𝐴 ⊆ ℋ → ( span ‘ ∪ 𝐴 ) ∈ Sℋ )
5	3 4	syl	⊢ ( 𝐴 ⊆ 𝒫 ℋ → ( span ‘ ∪ 𝐴 ) ∈ Sℋ )