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Theorem grpsubeq0 14552
Description: If the difference between two group elements is zero, they are equal. (subeq0 9073 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 2283 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2283 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
4 grpsubid.m . . . . 5  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubval 14525 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) ) )
653adant1 973 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) ) )
76eqeq1d 2291 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  =  .0.  <->  ( X ( +g  `  G
) ( ( inv g `  G ) `
 Y ) )  =  .0.  ) )
8 simp1 955 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  G  e.  Grp )
91, 3grpinvcl 14527 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  Y
)  e.  B )
1093adant2 974 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  Y
)  e.  B )
11 simp2 956 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
12 grpsubid.o . . . 4  |-  .0.  =  ( 0g `  G )
131, 2, 12, 3grpinvid2 14531 . . 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 1182 . 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 14535 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  (
( inv g `  G ) `  Y
) )  =  Y )
16153adant2 974 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( inv g `  G ) `  (
( inv g `  G ) `  Y
) )  =  Y )
1716eqeq1d 2291 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( inv g `  G ) `
 ( ( inv g `  G ) `
 Y ) )  =  X  <->  Y  =  X ) )
18 eqcom 2285 . . 3  |-  ( Y  =  X  <->  X  =  Y )
1917, 18syl6bb 252 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( inv g `  G ) `
 ( ( inv g `  G ) `
 Y ) )  =  X  <->  X  =  Y ) )
207, 14, 193bitr2d 272 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 176    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365
This theorem is referenced by:  ghmeqker  14709  ghmf1  14711  odcong  14864  subgdisj1  15000  dprdf11  15258  lmodsubeq0  15684  lvecvscan2  15865  ip2eq  16557  tgphaus  17799  nrmmetd  18097  ply1divmo  19521  dvdsq1p  19546  dvdsr1p  19547  ply1remlem  19548  ig1peu  19557  dchr2sum  20512  idomrootle  26923  eqlkr  28662  hdmap11  31414  hdmapinvlem4  31487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491
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