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

Theorem grpinv11 14553
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 5541 . . . . 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 14551 . . . . . 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 14551 . . . . . 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 2336 . . 3  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  X  =  Y )
1514ex 423 . 2  |-  ( ph  ->  ( ( N `  X )  =  ( N `  Y )  ->  X  =  Y ) )
16 fveq2 5541 . 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 1632    e. wcel 1696   ` cfv 5271   Basecbs 13164   Grpcgrp 14378   inv gcminusg 14379
This theorem is referenced by:  gexdvds  14911  dchrisum0re  20678  mapdpglem30  32514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506
  Copyright terms: Public domain W3C validator