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	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
	cheli.1	$$⊢A\in H$$
Assertion	chelii	$$⊢A\in \mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		chssi.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
2		cheli.1	$$⊢A\in H$$
3	1	chssii	$$⊢H\subseteq \mathscr{H}$$
4	3 2	sselii	$$⊢A\in \mathscr{H}$$