Theorem ellspsn6

Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014) (Revised by Mario Carneiro, 8-Jan-2015)

	Ref	Expression
Hypotheses	ellspsn5b.v	|- V = ( Base ` W )
	ellspsn5b.s	|- S = ( LSubSp ` W )
	ellspsn5b.n	|- N = ( LSpan ` W )
	ellspsn5b.w	|- ( ph -> W e. LMod )
	ellspsn5b.a	|- ( ph -> U e. S )
Assertion	ellspsn6	|- ( ph -> ( X e. U <-> ( X e. V /\ ( N ` { X } ) C_ U ) ) )

Proof

Step	Hyp	Ref	Expression
1		ellspsn5b.v	|- V = ( Base ` W )
2		ellspsn5b.s	|- S = ( LSubSp ` W )
3		ellspsn5b.n	|- N = ( LSpan ` W )
4		ellspsn5b.w	|- ( ph -> W e. LMod )
5		ellspsn5b.a	|- ( ph -> U e. S )
6	1 2	lssel	|- ( ( U e. S /\ X e. U ) -> X e. V )
7	5 6	sylan	|- ( ( ph /\ X e. U ) -> X e. V )
8	4	adantr	|- ( ( ph /\ X e. U ) -> W e. LMod )
9	5	adantr	|- ( ( ph /\ X e. U ) -> U e. S )
10		simpr	|- ( ( ph /\ X e. U ) -> X e. U )
11	2 3	lspsnss	|- ( ( W e. LMod /\ U e. S /\ X e. U ) -> ( N ` { X } ) C_ U )
12	8 9 10 11	syl3anc	|- ( ( ph /\ X e. U ) -> ( N ` { X } ) C_ U )
13	7 12	jca	|- ( ( ph /\ X e. U ) -> ( X e. V /\ ( N ` { X } ) C_ U ) )
14	1 3	lspsnid	|- ( ( W e. LMod /\ X e. V ) -> X e. ( N ` { X } ) )
15	4 14	sylan	|- ( ( ph /\ X e. V ) -> X e. ( N ` { X } ) )
16		ssel	|- ( ( N ` { X } ) C_ U -> ( X e. ( N ` { X } ) -> X e. U ) )
17	15 16	syl5com	|- ( ( ph /\ X e. V ) -> ( ( N ` { X } ) C_ U -> X e. U ) )
18	17	impr	|- ( ( ph /\ ( X e. V /\ ( N ` { X } ) C_ U ) ) -> X e. U )
19	13 18	impbida	|- ( ph -> ( X e. U <-> ( X e. V /\ ( N ` { X } ) C_ U ) ) )