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

Theorem grpoinvid 20915
Description: The inverse of the identity element of a group. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvid.1  |-  U  =  (GId `  G )
grpinvid.2  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoinvid  |-  ( G  e.  GrpOp  ->  ( N `  U )  =  U )

Proof of Theorem grpoinvid
StepHypRef Expression
1 eqid 2296 . . . 4  |-  ran  G  =  ran  G
2 grpinvid.1 . . . 4  |-  U  =  (GId `  G )
31, 2grpoidcl 20900 . . 3  |-  ( G  e.  GrpOp  ->  U  e.  ran  G )
41, 2grpolid 20902 . . 3  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G )  -> 
( U G U )  =  U )
53, 4mpdan 649 . 2  |-  ( G  e.  GrpOp  ->  ( U G U )  =  U )
6 grpinvid.2 . . . 4  |-  N  =  ( inv `  G
)
71, 2, 6grpoinvid1 20913 . . 3  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G  /\  U  e.  ran  G )  -> 
( ( N `  U )  =  U  <-> 
( U G U )  =  U ) )
83, 3, 7mpd3an23 1279 . 2  |-  ( G  e.  GrpOp  ->  ( ( N `  U )  =  U  <->  ( U G U )  =  U ) )
95, 8mpbird 223 1  |-  ( G  e.  GrpOp  ->  ( N `  U )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871
This theorem is referenced by:  gxnn0neg  20946  gxinv  20953  gxid  20956  grpodivone  25476
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-pr 4230  ax-un 4528
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-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-grpo 20874  df-gid 20875  df-ginv 20876
  Copyright terms: Public domain W3C validator