Theorem lspsnel5

Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014)

	Ref	Expression
Hypotheses	lspsnel5.v	|- V = ( Base ` W )
	lspsnel5.s	|- S = ( LSubSp ` W )
	lspsnel5.n	|- N = ( LSpan ` W )
	lspsnel5.w	|- ( ph -> W e. LMod )
	lspsnel5.a	|- ( ph -> U e. S )
	lspsnel5.x	|- ( ph -> X e. V )
Assertion	lspsnel5	|- ( ph -> ( X e. U <-> ( N ` { X } ) C_ U ) )

Proof

Step	Hyp	Ref	Expression
1		lspsnel5.v	|- V = ( Base ` W )
2		lspsnel5.s	|- S = ( LSubSp ` W )
3		lspsnel5.n	|- N = ( LSpan ` W )
4		lspsnel5.w	|- ( ph -> W e. LMod )
5		lspsnel5.a	|- ( ph -> U e. S )
6		lspsnel5.x	|- ( ph -> X e. V )
7	1 2 3 4 5	lspsnel6	|- ( ph -> ( X e. U <-> ( X e. V /\ ( N ` { X } ) C_ U ) ) )
8	6 7	mpbirand	|- ( ph -> ( X e. U <-> ( N ` { X } ) C_ U ) )