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Theorem grpsubeq0 15612
Description: If the difference between two group elements is zero, they are equal. (subeq0 9635 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 2443 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2443 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
4 grpsubid.m . . . . 5  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubval 15581 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) )
653adant1 1006 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) )
76eqeq1d 2451 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  =  .0.  <->  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) )  =  .0.  ) )
8 simp1 988 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  G  e.  Grp )
91, 3grpinvcl 15583 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
1093adant2 1007 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
11 simp2 989 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
12 grpsubid.o . . . 4  |-  .0.  =  ( 0g `  G )
131, 2, 12, 3grpinvid2 15587 . . 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.  ) )
148, 10, 11, 13syl3anc 1218 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( invg `  G ) `
 ( ( invg `  G ) `
 Y ) )  =  X  <->  ( X
( +g  `  G ) ( ( invg `  G ) `  Y
) )  =  .0.  ) )
151, 3grpinvinv 15593 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
16153adant2 1007 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
1716eqeq1d 2451 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( invg `  G ) `
 ( ( invg `  G ) `
 Y ) )  =  X  <->  Y  =  X ) )
18 eqcom 2445 . . 3  |-  ( Y  =  X  <->  X  =  Y )
1917, 18syl6bb 261 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( invg `  G ) `
 ( ( invg `  G ) `
 Y ) )  =  X  <->  X  =  Y ) )
207, 14, 193bitr2d 281 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  =  .0.  <->  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   0gc0g 14378   Grpcgrp 15410   invgcminusg 15411   -gcsg 15413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547
This theorem is referenced by:  ghmeqker  15773  ghmf1  15775  odcong  16052  subgdisj1  16188  dprdf11  16513  dprdf11OLD  16520  kerf1hrm  16831  lmodsubeq0  17004  lvecvscan2  17193  ip2eq  18082  mdetuni0  18427  tgphaus  19687  nrmmetd  20167  ply1divmo  21607  dvdsq1p  21632  dvdsr1p  21633  ply1remlem  21634  ig1peu  21643  dchr2sum  22612  idomrootle  29560  eqlkr  32744  hdmap11  35496  hdmapinvlem4  35569
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