Theorem lspcl

Description: The span of a set of vectors is a subspace. ( spancl analog.) (Contributed by NM, 9-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

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

Proof

Step	Hyp	Ref	Expression
1		lspval.v	$$⊢V={\mathrm{Base}}_W$$
2		lspval.s	$$⊢S=\mathrm{LSubSp}\left(W\right)$$
3		lspval.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
4	1 2 3	lspf	$$⊢W\in \mathrm{LMod}\to N:𝒫V⟶S$$
5	1	fvexi	$$⊢V\in \mathrm{V}$$
6	5	elpw2	$$⊢U\in 𝒫V\leftrightarrow U\subseteq V$$
7	6	biimpri	$$⊢U\subseteq V\to U\in 𝒫V$$
8		ffvelrn	$$⊢\left(N:𝒫V⟶S\wedge U\in 𝒫V\right)\to N\left(U\right)\in S$$
9	4 7 8	syl2an	$$⊢\left(W\in \mathrm{LMod}\wedge U\subseteq V\right)\to N\left(U\right)\in S$$