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Theorem grpoid 21659
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 21653 . . . . 5  |-  ( G  e.  GrpOp  ->  U  e.  X )
41grporcan 21657 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  U  e.  X  /\  A  e.  X )
)  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U
) )
543exp2 1171 . . . . 5  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( U  e.  X  ->  ( A  e.  X  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U ) ) ) ) )
63, 5mpid 39 . . . 4  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( A  e.  X  ->  (
( A G A )  =  ( U G A )  <->  A  =  U ) ) ) )
76pm2.43d 46 . . 3  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U ) ) )
87imp 419 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A G A )  =  ( U G A )  <->  A  =  U ) )
91, 2grpolid 21655 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( U G A )  =  A )
109eqeq2d 2398 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A G A )  =  ( U G A )  <->  ( A G A )  =  A ) )
118, 10bitr3d 247 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A  =  U  <->  ( A G A )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   ran crn 4819   ` cfv 5394  (class class class)co 6020   GrpOpcgr 21622  GIdcgi 21623
This theorem is referenced by:  subgoid  21743  ghomid  21801  hhssnv  22612  ghomgrpilem2  24876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-ov 6023  df-riota 6485  df-grpo 21627  df-gid 21628
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