Theorem lspssv

Description: A span is a set of vectors. (Contributed by NM, 22-Feb-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lspss.v	$$⊢V={\mathrm{Base}}_W$$
	lspss.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
Assertion	lspssv	$$⊢\left(W\in \mathrm{LMod}\wedge U\subseteq V\right)\to N\left(U\right)\subseteq V$$

Proof

Step	Hyp	Ref	Expression
1		lspss.v	$$⊢V={\mathrm{Base}}_W$$
2		lspss.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
3		eqid	$$⊢\mathrm{LSubSp}\left(W\right)=\mathrm{LSubSp}\left(W\right)$$
4	1 3 2	lspcl	$$⊢\left(W\in \mathrm{LMod}\wedge U\subseteq V\right)\to N\left(U\right)\in \mathrm{LSubSp}\left(W\right)$$
5	1 3	lssss	$$⊢N\left(U\right)\in \mathrm{LSubSp}\left(W\right)\to N\left(U\right)\subseteq V$$
6	4 5	syl	$$⊢\left(W\in \mathrm{LMod}\wedge U\subseteq V\right)\to N\left(U\right)\subseteq V$$