Theorem chel

Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chel	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in H\right)\to A\in \mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		chss	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to H\subseteq \mathscr{H}$$
2	1	sselda	$$⊢\left(H\in {\mathbf{C}}_{\mathscr{H}}\wedge A\in H\right)\to A\in \mathscr{H}$$