Theorem lspsnsub

Description: Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015)

	Ref	Expression
Hypotheses	lspsnsub.v	|- V = ( Base ` W )
	lspsnsub.s	|- .- = ( -g ` W )
	lspsnsub.n	|- N = ( LSpan ` W )
	lspsnsub.w	|- ( ph -> W e. LMod )
	lspsnsub.x	|- ( ph -> X e. V )
	lspsnsub.y	|- ( ph -> Y e. V )
Assertion	lspsnsub	|- ( ph -> ( N ` { ( X .- Y ) } ) = ( N ` { ( Y .- X ) } ) )

Proof

Step	Hyp	Ref	Expression
1		lspsnsub.v	|- V = ( Base ` W )
2		lspsnsub.s	|- .- = ( -g ` W )
3		lspsnsub.n	|- N = ( LSpan ` W )
4		lspsnsub.w	|- ( ph -> W e. LMod )
5		lspsnsub.x	|- ( ph -> X e. V )
6		lspsnsub.y	|- ( ph -> Y e. V )
7	1 2	lmodvsubcl	|- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .- Y ) e. V )
8	4 5 6 7	syl3anc	|- ( ph -> ( X .- Y ) e. V )
9		eqid	|- ( invg ` W ) = ( invg ` W )
10	1 9 3	lspsnneg	|- ( ( W e. LMod /\ ( X .- Y ) e. V ) -> ( N ` { ( ( invg ` W ) ` ( X .- Y ) ) } ) = ( N ` { ( X .- Y ) } ) )
11	4 8 10	syl2anc	|- ( ph -> ( N ` { ( ( invg ` W ) ` ( X .- Y ) ) } ) = ( N ` { ( X .- Y ) } ) )
12		lmodgrp	|- ( W e. LMod -> W e. Grp )
13	4 12	syl	|- ( ph -> W e. Grp )
14	1 2 9	grpinvsub	|- ( ( W e. Grp /\ X e. V /\ Y e. V ) -> ( ( invg ` W ) ` ( X .- Y ) ) = ( Y .- X ) )
15	13 5 6 14	syl3anc	|- ( ph -> ( ( invg ` W ) ` ( X .- Y ) ) = ( Y .- X ) )
16	15	sneqd	|- ( ph -> { ( ( invg ` W ) ` ( X .- Y ) ) } = { ( Y .- X ) } )
17	16	fveq2d	|- ( ph -> ( N ` { ( ( invg ` W ) ` ( X .- Y ) ) } ) = ( N ` { ( Y .- X ) } ) )
18	11 17	eqtr3d	|- ( ph -> ( N ` { ( X .- Y ) } ) = ( N ` { ( Y .- X ) } ) )