Theorem elspancl

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

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

Proof

Step	Hyp	Ref	Expression
1		spancl	$$⊢A\subseteq \mathscr{H}\to \mathrm{span}\left(A\right)\in {\mathbf{S}}_{\mathscr{H}}$$
2		shel	$$⊢\left(\mathrm{span}\left(A\right)\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in \mathrm{span}\left(A\right)\right)\to B\in \mathscr{H}$$
3	1 2	sylan	$$⊢\left(A\subseteq \mathscr{H}\wedge B\in \mathrm{span}\left(A\right)\right)\to B\in \mathscr{H}$$