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

Theorem grpinvinv 14551
Description: Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
grpinvinv.b  |-  B  =  ( Base `  G
)
grpinvinv.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
grpinvinv  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  ( N `  X )
)  =  X )

Proof of Theorem grpinvinv
StepHypRef Expression
1 grpinvinv.b . . . . 5  |-  B  =  ( Base `  G
)
2 grpinvinv.n . . . . 5  |-  N  =  ( inv g `  G )
31, 2grpinvcl 14543 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
4 eqid 2296 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
5 eqid 2296 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
61, 4, 5, 2grprinv 14545 . . . 4  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
( ( N `  X ) ( +g  `  G ) ( N `
 ( N `  X ) ) )  =  ( 0g `  G ) )
73, 6syldan 456 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X ) ( +g  `  G ) ( N `
 ( N `  X ) ) )  =  ( 0g `  G ) )
81, 4, 5, 2grplinv 14544 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X ) ( +g  `  G ) X )  =  ( 0g `  G ) )
97, 8eqtr4d 2331 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X ) ( +g  `  G ) ( N `
 ( N `  X ) ) )  =  ( ( N `
 X ) ( +g  `  G ) X ) )
10 simpl 443 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  G  e.  Grp )
111, 2grpinvcl 14543 . . . 4  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
( N `  ( N `  X )
)  e.  B )
123, 11syldan 456 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  ( N `  X )
)  e.  B )
13 simpr 447 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  X  e.  B )
141, 4grplcan 14550 . . 3  |-  ( ( G  e.  Grp  /\  ( ( N `  ( N `  X ) )  e.  B  /\  X  e.  B  /\  ( N `  X )  e.  B ) )  ->  ( ( ( N `  X ) ( +g  `  G
) ( N `  ( N `  X ) ) )  =  ( ( N `  X
) ( +g  `  G
) X )  <->  ( N `  ( N `  X
) )  =  X ) )
1510, 12, 13, 3, 14syl13anc 1184 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( ( N `
 X ) ( +g  `  G ) ( N `  ( N `  X )
) )  =  ( ( N `  X
) ( +g  `  G
) X )  <->  ( N `  ( N `  X
) )  =  X ) )
169, 15mpbid 201 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  ( N `  X )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379
This theorem is referenced by:  grpinv11  14553  grpinvnz  14555  grpsubinv  14557  grpinvsub  14564  grpsubeq0  14568  grpnpcan  14573  mulgneg  14601  mulgdir  14608  mulgass  14613  eqger  14683  frgpuptinv  15096  ablsub2inv  15128  mulgdi  15142  invghm  15146  rngm2neg  15400  unitinvinv  15473  unitnegcl  15479  irrednegb  15509  abvneg  15615  lspsnneg  15779  tgpconcomp  17811  islindf4  27411  baerlem5amN  32528  baerlem5bmN  32529  baerlem5abmN  32530
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