Theorem elspansncl

Description: A member of a span of a singleton is a vector. (Contributed by NM, 17-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	elspansncl	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathrm{span}\left(\left\{A\right\}\right)\right)\to B\in \mathscr{H}$$

Proof

Step	Hyp	Ref	Expression
1		snssi	$$⊢A\in \mathscr{H}\to \left\{A\right\}\subseteq \mathscr{H}$$
2		elspancl	$$⊢\left(\left\{A\right\}\subseteq \mathscr{H}\wedge B\in \mathrm{span}\left(\left\{A\right\}\right)\right)\to B\in \mathscr{H}$$
3	1 2	sylan	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathrm{span}\left(\left\{A\right\}\right)\right)\to B\in \mathscr{H}$$