Theorem ocss

Description: An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000) (New usage is discouraged.)

	Ref	Expression
Assertion	ocss	$$⊢A\subseteq \mathscr{H}\to \perp \left(A\right)\subseteq \mathscr{H}$$

Proof

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