Theorem spansnch

Description: The span of a Hilbert space singleton belongs to the Hilbert lattice. (Contributed by NM, 9-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	spansnch	$$⊢A\in \mathscr{H}\to \mathrm{span}\left(\left\{A\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		spansn	$$⊢A\in \mathscr{H}\to \mathrm{span}\left(\left\{A\right\}\right)=\perp \left(\perp \left(\left\{A\right\}\right)\right)$$
2		snssi	$$⊢A\in \mathscr{H}\to \left\{A\right\}\subseteq \mathscr{H}$$
3		occl	$$⊢\left\{A\right\}\subseteq \mathscr{H}\to \perp \left(\left\{A\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
4		choccl	$$⊢\perp \left(\left\{A\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}\to \perp \left(\perp \left(\left\{A\right\}\right)\right)\in {\mathbf{C}}_{\mathscr{H}}$$
5	2 3 4	3syl	$$⊢A\in \mathscr{H}\to \perp \left(\perp \left(\left\{A\right\}\right)\right)\in {\mathbf{C}}_{\mathscr{H}}$$
6	1 5	eqeltrd	$$⊢A\in \mathscr{H}\to \mathrm{span}\left(\left\{A\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$