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	|- .0. = ( 0g ` W )
	lspsn0.n	|- N = ( LSpan ` W )
Assertion	lspsn0	|- ( W e. LMod -> ( N ` { .0. } ) = { .0. } )

Proof

Step	Hyp	Ref	Expression
1		lspsn0.z	|- .0. = ( 0g ` W )
2		lspsn0.n	|- N = ( LSpan ` W )
3		eqid	|- ( LSubSp ` W ) = ( LSubSp ` W )
4	1 3	lsssn0	|- ( W e. LMod -> { .0. } e. ( LSubSp ` W ) )
5	3 2	lspid	|- ( ( W e. LMod /\ { .0. } e. ( LSubSp ` W ) ) -> ( N ` { .0. } ) = { .0. } )
6	4 5	mpdan	|- ( W e. LMod -> ( N ` { .0. } ) = { .0. } )