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Theorem grpoid 25342
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 25336 . . . . 5  |-  ( G  e.  GrpOp  ->  U  e.  X )
41grporcan 25340 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  U  e.  X  /\  A  e.  X )
)  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U
) )
543exp2 1212 . . . . 5  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( U  e.  X  ->  ( A  e.  X  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U ) ) ) ) )
63, 5mpid 41 . . . 4  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( A  e.  X  ->  (
( A G A )  =  ( U G A )  <->  A  =  U ) ) ) )
76pm2.43d 48 . . 3  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( ( A G A )  =  ( U G A )  <->  A  =  U ) ) )
87imp 427 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A G A )  =  ( U G A )  <->  A  =  U ) )
91, 2grpolid 25338 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( U G A )  =  A )
109eqeq2d 2396 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A G A )  =  ( U G A )  <->  ( A G A )  =  A ) )
118, 10bitr3d 255 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A  =  U  <->  ( A G A )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   ran crn 4914   ` cfv 5496  (class class class)co 6196   GrpOpcgr 25305  GIdcgi 25306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fo 5502  df-fv 5504  df-riota 6158  df-ov 6199  df-grpo 25310  df-gid 25311
This theorem is referenced by:  subgoid  25426  ghomidOLD  25484  hhssnv  26297  ghomgrpilem2  29215
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