Theorem shoccl

Description: Closure of complement of Hilbert subspace. Part of Remark 3.12 of Beran p. 107. (Contributed by NM, 13-Oct-1999) (New usage is discouraged.)

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

Proof

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