MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpinv11 Unicode version

Theorem grpinv11 14537
Description: The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)
Hypotheses
Ref Expression
grpinvinv.b  |-  B  =  ( Base `  G
)
grpinvinv.n  |-  N  =  ( inv g `  G )
grpinv11.g  |-  ( ph  ->  G  e.  Grp )
grpinv11.x  |-  ( ph  ->  X  e.  B )
grpinv11.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
grpinv11  |-  ( ph  ->  ( ( N `  X )  =  ( N `  Y )  <-> 
X  =  Y ) )

Proof of Theorem grpinv11
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( ( N `  X )  =  ( N `  Y )  ->  ( N `  ( N `  X ) )  =  ( N `  ( N `  Y )
) )
21adantl 452 . . . 4  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  ( N `  ( N `  X
) )  =  ( N `  ( N `
 Y ) ) )
3 grpinv11.g . . . . . 6  |-  ( ph  ->  G  e.  Grp )
4 grpinv11.x . . . . . 6  |-  ( ph  ->  X  e.  B )
5 grpinvinv.b . . . . . . 7  |-  B  =  ( Base `  G
)
6 grpinvinv.n . . . . . . 7  |-  N  =  ( inv g `  G )
75, 6grpinvinv 14535 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  ( N `  X )
)  =  X )
83, 4, 7syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  ( N `  X )
)  =  X )
98adantr 451 . . . 4  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  ( N `  ( N `  X
) )  =  X )
10 grpinv11.y . . . . . 6  |-  ( ph  ->  Y  e.  B )
115, 6grpinvinv 14535 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
123, 10, 11syl2anc 642 . . . . 5  |-  ( ph  ->  ( N `  ( N `  Y )
)  =  Y )
1312adantr 451 . . . 4  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  ( N `  ( N `  Y
) )  =  Y )
142, 9, 133eqtr3d 2323 . . 3  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  X  =  Y )
1514ex 423 . 2  |-  ( ph  ->  ( ( N `  X )  =  ( N `  Y )  ->  X  =  Y ) )
16 fveq2 5525 . 2  |-  ( X  =  Y  ->  ( N `  X )  =  ( N `  Y ) )
1715, 16impbid1 194 1  |-  ( ph  ->  ( ( N `  X )  =  ( N `  Y )  <-> 
X  =  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255   Basecbs 13148   Grpcgrp 14362   inv gcminusg 14363
This theorem is referenced by:  gexdvds  14895  dchrisum0re  20662  mapdpglem30  31892
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
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-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-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490
  Copyright terms: Public domain W3C validator