Theorem shel

Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004) (New usage is discouraged.)

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

Proof

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