Theorem shocsh

Description: The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of Beran p. 107. (Contributed by NM, 10-Oct-1999) (New usage is discouraged.)

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

Proof

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