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Theorem grpoinvid 20899
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 2283 . . . 4  |-  ran  G  =  ran  G
2 grpinvid.1 . . . 4  |-  U  =  (GId `  G )
31, 2grpoidcl 20884 . . 3  |-  ( G  e.  GrpOp  ->  U  e.  ran  G )
41, 2grpolid 20886 . . 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 20897 . . 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 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855
This theorem is referenced by:  gxnn0neg  20930  gxinv  20937  gxid  20940  grpodivone  25373
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-pr 4214  ax-un 4512
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-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-grpo 20858  df-gid 20859  df-ginv 20860
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