Theorem lspssid

Description: A set of vectors is a subset of its span. ( spanss2 analog.) (Contributed by NM, 6-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	lspssid	$$⊢\left(W\in \mathrm{LMod}\wedge U\subseteq V\right)\to U\subseteq N\left(U\right)$$

Proof

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