Theorem lsatlspsn

Description: The span of a nonzero singleton is an atom. (Contributed by NM, 16-Jan-2015)

	Ref	Expression
Hypotheses	lsatset.v	|- V = ( Base ` W )
	lsatset.n	|- N = ( LSpan ` W )
	lsatset.z	|- .0. = ( 0g ` W )
	lsatset.a	|- A = ( LSAtoms ` W )
	lsatlspsn.w	|- ( ph -> W e. LMod )
	lsatlspsn.x	|- ( ph -> X e. ( V \ { .0. } ) )
Assertion	lsatlspsn	|- ( ph -> ( N ` { X } ) e. A )

Proof

Step	Hyp	Ref	Expression
1		lsatset.v	|- V = ( Base ` W )
2		lsatset.n	|- N = ( LSpan ` W )
3		lsatset.z	|- .0. = ( 0g ` W )
4		lsatset.a	|- A = ( LSAtoms ` W )
5		lsatlspsn.w	|- ( ph -> W e. LMod )
6		lsatlspsn.x	|- ( ph -> X e. ( V \ { .0. } ) )
7		eqid	|- ( N ` { X } ) = ( N ` { X } )
8		sneq	|- ( v = X -> { v } = { X } )
9	8	fveq2d	|- ( v = X -> ( N ` { v } ) = ( N ` { X } ) )
10	9	rspceeqv	|- ( ( X e. ( V \ { .0. } ) /\ ( N ` { X } ) = ( N ` { X } ) ) -> E. v e. ( V \ { .0. } ) ( N ` { X } ) = ( N ` { v } ) )
11	6 7 10	sylancl	|- ( ph -> E. v e. ( V \ { .0. } ) ( N ` { X } ) = ( N ` { v } ) )
12	1 2 3 4	islsat	|- ( W e. LMod -> ( ( N ` { X } ) e. A <-> E. v e. ( V \ { .0. } ) ( N ` { X } ) = ( N ` { v } ) ) )
13	5 12	syl	|- ( ph -> ( ( N ` { X } ) e. A <-> E. v e. ( V \ { .0. } ) ( N ` { X } ) = ( N ` { v } ) ) )
14	11 13	mpbird	|- ( ph -> ( N ` { X } ) e. A )