Theorem lspsnel5a

Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015)

	Ref	Expression
Hypotheses	lspsnel5a.s	|- S = ( LSubSp ` W )
	lspsnel5a.n	|- N = ( LSpan ` W )
	lspsnel5a.w	|- ( ph -> W e. LMod )
	lspsnel5a.a	|- ( ph -> U e. S )
	lspsnel5a.x	|- ( ph -> X e. U )
Assertion	lspsnel5a	|- ( ph -> ( N ` { X } ) C_ U )

Proof

Step	Hyp	Ref	Expression
1		lspsnel5a.s	|- S = ( LSubSp ` W )
2		lspsnel5a.n	|- N = ( LSpan ` W )
3		lspsnel5a.w	|- ( ph -> W e. LMod )
4		lspsnel5a.a	|- ( ph -> U e. S )
5		lspsnel5a.x	|- ( ph -> X e. U )
6		eqid	|- ( Base ` W ) = ( Base ` W )
7	6 1	lssel	|- ( ( U e. S /\ X e. U ) -> X e. ( Base ` W ) )
8	4 5 7	syl2anc	|- ( ph -> X e. ( Base ` W ) )
9	6 1 2 3 4 8	lspsnel5	|- ( ph -> ( X e. U <-> ( N ` { X } ) C_ U ) )
10	5 9	mpbid	|- ( ph -> ( N ` { X } ) C_ U )