Theorem spansn

Description: The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 4-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	spansn	$$⊢A\in \mathscr{H}\to \mathrm{span}\left(\left\{A\right\}\right)=\perp \left(\perp \left(\left\{A\right\}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sneq	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \left\{A\right\}=\left\{if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right\}$$
2	1	fveq2d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \mathrm{span}\left(\left\{A\right\}\right)=\mathrm{span}\left(\left\{if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right\}\right)$$
3	1	fveq2d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \perp \left(\left\{A\right\}\right)=\perp \left(\left\{if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right\}\right)$$
4	3	fveq2d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \perp \left(\perp \left(\left\{A\right\}\right)\right)=\perp \left(\perp \left(\left\{if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right\}\right)\right)$$
5	2 4	eqeq12d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \left(\mathrm{span}\left(\left\{A\right\}\right)=\perp \left(\perp \left(\left\{A\right\}\right)\right)\leftrightarrow \mathrm{span}\left(\left\{if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right\}\right)=\perp \left(\perp \left(\left\{if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right\}\right)\right)\right)$$
6		ifhvhv0	$$⊢if\left(A\in \mathscr{H},A,0_{ℎ}\right)\in \mathscr{H}$$
7	6	spansni	$$⊢\mathrm{span}\left(\left\{if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right\}\right)=\perp \left(\perp \left(\left\{if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right\}\right)\right)$$
8	5 7	dedth	$$⊢A\in \mathscr{H}\to \mathrm{span}\left(\left\{A\right\}\right)=\perp \left(\perp \left(\left\{A\right\}\right)\right)$$