Theorem sheli

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

	Ref	Expression
Hypothesis	shssi.1	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}$$
Assertion	sheli	$$⊢A\in H\to A\in \mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		shssi.1	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}$$
2	1	shssii	$$⊢H\subseteq \mathscr{H}$$
3	2	sseli	$$⊢A\in H\to A\in \mathscr{H}$$