Theorem lspsn0

Description: Span of the singleton of the zero vector. ( spansn0 analog.) (Contributed by NM, 15-Jan-2014) (Proof shortened by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lspsn0.z	$$⊢\underset{\dot{}}{0}=0_W$$
	lspsn0.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
Assertion	lspsn0	$$⊢W\in \mathrm{LMod}\to N\left(\left\{\underset{\dot{}}{0}\right\}\right)=\left\{\underset{\dot{}}{0}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		lspsn0.z	$$⊢\underset{\dot{}}{0}=0_W$$
2		lspsn0.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
3		eqid	$$⊢\mathrm{LSubSp}\left(W\right)=\mathrm{LSubSp}\left(W\right)$$
4	1 3	lsssn0	$$⊢W\in \mathrm{LMod}\to \left\{\underset{\dot{}}{0}\right\}\in \mathrm{LSubSp}\left(W\right)$$
5	3 2	lspid	$$⊢\left(W\in \mathrm{LMod}\wedge \left\{\underset{\dot{}}{0}\right\}\in \mathrm{LSubSp}\left(W\right)\right)\to N\left(\left\{\underset{\dot{}}{0}\right\}\right)=\left\{\underset{\dot{}}{0}\right\}$$
6	4 5	mpdan	$$⊢W\in \mathrm{LMod}\to N\left(\left\{\underset{\dot{}}{0}\right\}\right)=\left\{\underset{\dot{}}{0}\right\}$$