Theorem chelii

Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chssi.1	⊢ 𝐻 ∈ Cℋ
	cheli.1	⊢ 𝐴 ∈ 𝐻
Assertion	chelii	⊢ 𝐴 ∈ ℋ

Proof

Step	Hyp	Ref	Expression
1		chssi.1	⊢ 𝐻 ∈ Cℋ
2		cheli.1	⊢ 𝐴 ∈ 𝐻
3	1	chssii	⊢ 𝐻 ⊆ ℋ
4	3 2	sselii	⊢ 𝐴 ∈ ℋ