Theorem grpsubeq0 17501

Description: If the difference between two group elements is zero, they are equal. (subeq0 10307 analog.) (Contributed by NM, 31-Mar-2014.)

Hypotheses

Ref	Expression
grpsubid.b	|- B = ( Base ` G )
grpsubid.o	|- .0. = ( 0g ` G )
grpsubid.m	|- .- = ( -g ` G )

Assertion

Ref	Expression
grpsubeq0	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .- Y ) = .0. <-> X = Y ) )

Proof of Theorem grpsubeq0

Step	Hyp	Ref	Expression
1		grpsubid.b	. . . . 5 |- B = ( Base ` G )
2		eqid 2622	. . . . 5 |- ( +g ` G ) = ( +g ` G )
3		eqid 2622	. . . . 5 |- ( invg ` G ) = ( invg ` G )
4		grpsubid.m	. . . . 5 |- .- = ( -g ` G )
5	1, 2, 3, 4	grpsubval 17465	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
6	5	3adant1 1079	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
7	6	eqeq1d 2624	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .- Y ) = .0. <-> ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) = .0. ) )
8		simp1 1061	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> G e. Grp )
9	1, 3	grpinvcl 17467	. . . 4 |- ( ( G e. Grp /\ Y e. B ) -> ( ( invg ` G ) ` Y ) e. B )
10	9	3adant2 1080	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( invg ` G ) ` Y ) e. B )
11		simp2 1062	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> X e. B )
12		grpsubid.o	. . . 4 |- .0. = ( 0g ` G )
13	1, 2, 12, 3	grpinvid2 17471	. . 3 |- ( ( G e. Grp /\ ( ( invg ` G ) ` Y ) e. B /\ X e. B ) -> ( ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = X <-> ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) = .0. ) )
14	8, 10, 11, 13	syl3anc 1326	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = X <-> ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) = .0. ) )
15	1, 3	grpinvinv 17482	. . . . 5 |- ( ( G e. Grp /\ Y e. B ) -> ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = Y )
16	15	3adant2 1080	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = Y )
17	16	eqeq1d 2624	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = X <-> Y = X ) )
18		eqcom 2629	. . 3 |- ( Y = X <-> X = Y )
19	17, 18	syl6bb 276	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = X <-> X = Y ) )
20	7, 14, 19	3bitr2d 296	1 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .- Y ) = .0. <-> X = Y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424

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

This theorem is referenced by:  ghmeqker  17687  ghmf1  17689  odcong  17968  subgdisj1  18104  dprdf11  18422  kerf1hrm  18743  lmodsubeq0  18922  lvecvscan2  19112  ip2eq  19998  mdetuni0  20427  tgphaus  21920  nrmmetd  22379  ply1divmo  23895  dvdsq1p  23920  dvdsr1p  23921  ply1remlem  23922  ig1peu  23931  dchr2sum  24998  eqlkr  34386  hdmap11  37140  hdmapinvlem4  37213  idomrootle  37773  lidldomn1  41921