Theorem chss

Description: A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	chss	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to H\subseteq \mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		chsh	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to H\in {\mathbf{S}}_{\mathscr{H}}$$
2		shss	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to H\subseteq \mathscr{H}$$
3	1 2	syl	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}\to H\subseteq \mathscr{H}$$