Theorem lspidm

Description: The span of a set of vectors is idempotent. (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	lspidm	$$⊢\left(W\in \mathrm{LMod}\wedge U\subseteq V\right)\to N\left(N\left(U\right)\right)=N\left(U\right)$$

Proof

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