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Theorem grpsubeq0 14867
Description: If the difference between two group elements is zero, they are equal. (subeq0 9319 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
StepHypRef Expression
1 grpsubid.b . . . . 5  |-  B  =  ( Base `  G
)
2 eqid 2435 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2435 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
4 grpsubid.m . . . . 5  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubval 14840 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) ) )
653adant1 975 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) ) )
76eqeq1d 2443 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  =  .0.  <->  ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) )  =  .0.  ) )
8 simp1 957 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  G  e.  Grp )
91, 3grpinvcl 14842 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  Y
)  e.  B )
1093adant2 976 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  Y
)  e.  B )
11 simp2 958 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
12 grpsubid.o . . . 4  |-  .0.  =  ( 0g `  G )
131, 2, 12, 3grpinvid2 14846 . . 3  |-  ( ( G  e.  Grp  /\  ( ( inv g `  G ) `  Y
)  e.  B  /\  X  e.  B )  ->  ( ( ( inv g `  G ) `
 ( ( inv g `  G ) `
 Y ) )  =  X  <->  ( X
( +g  `  G ) ( ( inv g `  G ) `  Y
) )  =  .0.  ) )
148, 10, 11, 13syl3anc 1184 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( inv g `  G ) `
 ( ( inv g `  G ) `
 Y ) )  =  X  <->  ( X
( +g  `  G ) ( ( inv g `  G ) `  Y
) )  =  .0.  ) )
151, 3grpinvinv 14850 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  (
( inv g `  G ) `  Y
) )  =  Y )
16153adant2 976 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  (
( inv g `  G ) `  Y
) )  =  Y )
1716eqeq1d 2443 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( inv g `  G ) `
 ( ( inv g `  G ) `
 Y ) )  =  X  <->  Y  =  X ) )
18 eqcom 2437 . . 3  |-  ( Y  =  X  <->  X  =  Y )
1917, 18syl6bb 253 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( inv g `  G ) `
 ( ( inv g `  G ) `
 Y ) )  =  X  <->  X  =  Y ) )
207, 14, 193bitr2d 273 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  =  .0.  <->  X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   0gc0g 13715   Grpcgrp 14677   inv gcminusg 14678   -gcsg 14680
This theorem is referenced by:  ghmeqker  15024  ghmf1  15026  odcong  15179  subgdisj1  15315  dprdf11  15573  lmodsubeq0  15995  lvecvscan2  16176  ip2eq  16876  tgphaus  18138  nrmmetd  18614  ply1divmo  20050  dvdsq1p  20075  dvdsr1p  20076  ply1remlem  20077  ig1peu  20086  dchr2sum  21049  kerf1hrm  24254  idomrootle  27479  eqlkr  29834  hdmap11  32586  hdmapinvlem4  32659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806
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