Theorem lsat2el

Description: Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015)

	Ref	Expression
Hypotheses	lsat2el.o	|- .0. = ( 0g ` W )
	lsat2el.a	|- A = ( LSAtoms ` W )
	lsat2el.w	|- ( ph -> W e. LVec )
	lsat2el.p	|- ( ph -> P e. A )
	lsat2el.q	|- ( ph -> Q e. A )
	lsat2el.x	|- ( ph -> X =/= .0. )
	lsat2el.x1	|- ( ph -> X e. P )
	lsat2el.x2	|- ( ph -> X e. Q )
Assertion	lsat2el	|- ( ph -> P = Q )

Proof

Step	Hyp	Ref	Expression
1		lsat2el.o	|- .0. = ( 0g ` W )
2		lsat2el.a	|- A = ( LSAtoms ` W )
3		lsat2el.w	|- ( ph -> W e. LVec )
4		lsat2el.p	|- ( ph -> P e. A )
5		lsat2el.q	|- ( ph -> Q e. A )
6		lsat2el.x	|- ( ph -> X =/= .0. )
7		lsat2el.x1	|- ( ph -> X e. P )
8		lsat2el.x2	|- ( ph -> X e. Q )
9		eqid	|- ( LSpan ` W ) = ( LSpan ` W )
10	1 9 2 3 4 7 6	lsatel	|- ( ph -> P = ( ( LSpan ` W ) ` { X } ) )
11	1 9 2 3 5 8 6	lsatel	|- ( ph -> Q = ( ( LSpan ` W ) ` { X } ) )
12	10 11	eqtr4d	|- ( ph -> P = Q )