Theorem lspsncmp

Description: Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015)

	Ref	Expression
Hypotheses	lspsncmp.v	|- V = ( Base ` W )
	lspsncmp.o	|- .0. = ( 0g ` W )
	lspsncmp.n	|- N = ( LSpan ` W )
	lspsncmp.w	|- ( ph -> W e. LVec )
	lspsncmp.x	|- ( ph -> X e. ( V \ { .0. } ) )
	lspsncmp.y	|- ( ph -> Y e. V )
Assertion	lspsncmp	|- ( ph -> ( ( N ` { X } ) C_ ( N ` { Y } ) <-> ( N ` { X } ) = ( N ` { Y } ) ) )

Proof

Step	Hyp	Ref	Expression
1		lspsncmp.v	|- V = ( Base ` W )
2		lspsncmp.o	|- .0. = ( 0g ` W )
3		lspsncmp.n	|- N = ( LSpan ` W )
4		lspsncmp.w	|- ( ph -> W e. LVec )
5		lspsncmp.x	|- ( ph -> X e. ( V \ { .0. } ) )
6		lspsncmp.y	|- ( ph -> Y e. V )
7	4	adantr	|- ( ( ph /\ ( N ` { X } ) C_ ( N ` { Y } ) ) -> W e. LVec )
8	6	adantr	|- ( ( ph /\ ( N ` { X } ) C_ ( N ` { Y } ) ) -> Y e. V )
9		eqid	|- ( LSubSp ` W ) = ( LSubSp ` W )
10		lveclmod	|- ( W e. LVec -> W e. LMod )
11	4 10	syl	|- ( ph -> W e. LMod )
12	1 9 3	lspsncl	|- ( ( W e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` W ) )
13	11 6 12	syl2anc	|- ( ph -> ( N ` { Y } ) e. ( LSubSp ` W ) )
14	5	eldifad	|- ( ph -> X e. V )
15	1 9 3 11 13 14	ellspsn5b	|- ( ph -> ( X e. ( N ` { Y } ) <-> ( N ` { X } ) C_ ( N ` { Y } ) ) )
16	15	biimpar	|- ( ( ph /\ ( N ` { X } ) C_ ( N ` { Y } ) ) -> X e. ( N ` { Y } ) )
17		eldifsni	|- ( X e. ( V \ { .0. } ) -> X =/= .0. )
18	5 17	syl	|- ( ph -> X =/= .0. )
19	18	adantr	|- ( ( ph /\ ( N ` { X } ) C_ ( N ` { Y } ) ) -> X =/= .0. )
20	1 2 3 7 8 16 19	lspsneleq	|- ( ( ph /\ ( N ` { X } ) C_ ( N ` { Y } ) ) -> ( N ` { X } ) = ( N ` { Y } ) )
21	20	ex	|- ( ph -> ( ( N ` { X } ) C_ ( N ` { Y } ) -> ( N ` { X } ) = ( N ` { Y } ) ) )
22		eqimss	|- ( ( N ` { X } ) = ( N ` { Y } ) -> ( N ` { X } ) C_ ( N ` { Y } ) )
23	21 22	impbid1	|- ( ph -> ( ( N ` { X } ) C_ ( N ` { Y } ) <-> ( N ` { X } ) = ( N ` { Y } ) ) )