Theorem ellspsn5

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

	Ref	Expression
Hypotheses	ellspsn5.s	|- S = ( LSubSp ` W )
	ellspsn5.n	|- N = ( LSpan ` W )
	ellspsn5.w	|- ( ph -> W e. LMod )
	ellspsn5.a	|- ( ph -> U e. S )
	ellspsn5.x	|- ( ph -> X e. U )
Assertion	ellspsn5	|- ( ph -> ( N ` { X } ) C_ U )

Proof

Step	Hyp	Ref	Expression
1		ellspsn5.s	|- S = ( LSubSp ` W )
2		ellspsn5.n	|- N = ( LSpan ` W )
3		ellspsn5.w	|- ( ph -> W e. LMod )
4		ellspsn5.a	|- ( ph -> U e. S )
5		ellspsn5.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	ellspsn5b	|- ( ph -> ( X e. U <-> ( N ` { X } ) C_ U ) )
10	5 9	mpbid	|- ( ph -> ( N ` { X } ) C_ U )