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Theorem grpoid 20890
Description: Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinveu.1  |-  X  =  ran  G
grpinveu.2  |-  U  =  (GId `  G )
Assertion
Ref Expression
grpoid  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A  =  U  <->  ( A G A )  =  A ) )

Proof of Theorem grpoid
StepHypRef Expression
1 grpinveu.1 . . . . . 6  |-  X  =  ran  G
2 grpinveu.2 . . . . . 6  |-  U  =  (GId `  G )
31, 2grpoidcl 20884 . . . . 5  |-  ( G  e.  GrpOp  ->  U  e.  X )
41grporcan 20888 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  U  e.  X  /\  A  e.  X )
)  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U
) )
543exp2 1169 . . . . 5  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( U  e.  X  ->  ( A  e.  X  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U ) ) ) ) )
63, 5mpid 37 . . . 4  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( A  e.  X  ->  (
( A G A )  =  ( U G A )  <->  A  =  U ) ) ) )
76pm2.43d 44 . . 3  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U ) ) )
87imp 418 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A G A )  =  ( U G A )  <->  A  =  U ) )
91, 2grpolid 20886 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( U G A )  =  A )
109eqeq2d 2294 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A G A )  =  ( U G A )  <->  ( A G A )  =  A ) )
118, 10bitr3d 246 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A  =  U  <->  ( A G A )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854
This theorem is referenced by:  subgoid  20974  ghomid  21032  hhssnv  21841  ghomgrpilem2  23993
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-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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-riota 6304  df-grpo 20858  df-gid 20859
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