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

Theorem grpsubeq0 15925
Description: If the difference between two group elements is zero, they are equal. (subeq0 9841 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 2467 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2467 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
4 grpsubid.m . . . . 5  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubval 15894 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) )
653adant1 1014 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) ) )
76eqeq1d 2469 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  =  .0.  <->  ( X ( +g  `  G
) ( ( invg `  G ) `
 Y ) )  =  .0.  ) )
8 simp1 996 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  G  e.  Grp )
91, 3grpinvcl 15896 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
1093adant2 1015 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
11 simp2 997 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
12 grpsubid.o . . . 4  |-  .0.  =  ( 0g `  G )
131, 2, 12, 3grpinvid2 15900 . . 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 1228 . 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 15906 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
16153adant2 1015 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( invg `  G ) `  (
( invg `  G ) `  Y
) )  =  Y )
1716eqeq1d 2469 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( invg `  G ) `
 ( ( invg `  G ) `
 Y ) )  =  X  <->  Y  =  X ) )
18 eqcom 2476 . . 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 973    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   0gc0g 14691   Grpcgrp 15723   invgcminusg 15724   -gcsg 15726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-0g 14693  df-mnd 15728  df-grp 15858  df-minusg 15859  df-sbg 15860
This theorem is referenced by:  ghmeqker  16088  ghmf1  16090  odcong  16369  subgdisj1  16505  dprdf11  16853  dprdf11OLD  16860  kerf1hrm  17175  lmodsubeq0  17352  lvecvscan2  17541  ip2eq  18455  mdetuni0  18890  tgphaus  20350  nrmmetd  20830  ply1divmo  22271  dvdsq1p  22296  dvdsr1p  22297  ply1remlem  22298  ig1peu  22307  dchr2sum  23276  idomrootle  30757  eqlkr  33896  hdmap11  36648  hdmapinvlem4  36721
  Copyright terms: Public domain W3C validator