Theorem lspsnsubn0 19140

Description: Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)

Hypotheses

Ref	Expression
lspsnsubn0.v	|- V = ( Base ` W )
lspsnsubn0.o	|- .0. = ( 0g ` W )
lspsnsubn0.m	|- .- = ( -g ` W )
lspsnsubn0.w	|- ( ph -> W e. LMod )
lspsnsubn0.x	|- ( ph -> X e. V )
lspsnsubn0.y	|- ( ph -> Y e. V )
lspsnsubn0.e	|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )

Assertion

Ref	Expression
lspsnsubn0	|- ( ph -> ( X .- Y ) =/= .0. )

Proof of Theorem lspsnsubn0

Step	Hyp	Ref	Expression
1		lspsnsubn0.e	. 2 |- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
2		lspsnsubn0.w	. . . . 5 |- ( ph -> W e. LMod )
3		lspsnsubn0.x	. . . . 5 |- ( ph -> X e. V )
4		lspsnsubn0.y	. . . . 5 |- ( ph -> Y e. V )
5		lspsnsubn0.v	. . . . . 6 |- V = ( Base ` W )
6		lspsnsubn0.o	. . . . . 6 |- .0. = ( 0g ` W )
7		lspsnsubn0.m	. . . . . 6 |- .- = ( -g ` W )
8	5, 6, 7	lmodsubeq0 18922	. . . . 5 |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( ( X .- Y ) = .0. <-> X = Y ) )
9	2, 3, 4, 8	syl3anc 1326	. . . 4 |- ( ph -> ( ( X .- Y ) = .0. <-> X = Y ) )
10		sneq 4187	. . . . 5 |- ( X = Y -> { X } = { Y } )
11	10	fveq2d 6195	. . . 4 |- ( X = Y -> ( N ` { X } ) = ( N ` { Y } ) )
12	9, 11	syl6bi 243	. . 3 |- ( ph -> ( ( X .- Y ) = .0. -> ( N ` { X } ) = ( N ` { Y } ) ) )
13	12	necon3d 2815	. 2 |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) -> ( X .- Y ) =/= .0. ) )
14	1, 13	mpd 15	1 |- ( ph -> ( X .- Y ) =/= .0. )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   0gc0g 16100   -gcsg 17424   LModclmod 18863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-lmod 18865

This theorem is referenced by:  mapdpglem4N  36965