MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpsubeq0 Unicode version

Theorem grpsubeq0 14803
Description: If the difference between two group elements is zero, they are equal. (subeq0 9260 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 2388 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2388 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
4 grpsubid.m . . . . 5  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubval 14776 . . . 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 2396 . 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 14778 . . . 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 14782 . . 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 14786 . . . . 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 2396 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( inv g `  G ) `
 ( ( inv g `  G ) `
 Y ) )  =  X  <->  Y  =  X ) )
18 eqcom 2390 . . 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 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   0gc0g 13651   Grpcgrp 14613   inv gcminusg 14614   -gcsg 14616
This theorem is referenced by:  ghmeqker  14960  ghmf1  14962  odcong  15115  subgdisj1  15251  dprdf11  15509  lmodsubeq0  15931  lvecvscan2  16112  ip2eq  16808  tgphaus  18068  nrmmetd  18494  ply1divmo  19926  dvdsq1p  19951  dvdsr1p  19952  ply1remlem  19953  ig1peu  19962  dchr2sum  20925  kerf1hrm  24079  idomrootle  27181  eqlkr  29215  hdmap11  31967  hdmapinvlem4  32040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-sbg 14742
  Copyright terms: Public domain W3C validator