Theorem shocss

Description: An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		shss	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to A\subseteq \mathscr{H}$$
2		ocss	$$⊢A\subseteq \mathscr{H}\to \perp \left(A\right)\subseteq \mathscr{H}$$
3	1 2	syl	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to \perp \left(A\right)\subseteq \mathscr{H}$$