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Theorem grpid 15566
Description: Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)
Hypotheses
Ref Expression
grpinveu.b  |-  B  =  ( Base `  G
)
grpinveu.p  |-  .+  =  ( +g  `  G )
grpinveu.o  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
grpid  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( X  .+  X )  =  X  <-> 
.0.  =  X ) )

Proof of Theorem grpid
StepHypRef Expression
1 eqcom 2443 . 2  |-  (  .0.  =  X  <->  X  =  .0.  )
2 grpinveu.b . . . . . . 7  |-  B  =  ( Base `  G
)
3 grpinveu.o . . . . . . 7  |-  .0.  =  ( 0g `  G )
42, 3grpidcl 15559 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  B )
5 grpinveu.p . . . . . . . 8  |-  .+  =  ( +g  `  G )
62, 5grprcan 15564 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  .0.  e.  B  /\  X  e.  B )
)  ->  ( ( X  .+  X )  =  (  .0.  .+  X
)  <->  X  =  .0.  ) )
763exp2 1200 . . . . . 6  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  (  .0.  e.  B  -> 
( X  e.  B  ->  ( ( X  .+  X )  =  (  .0.  .+  X )  <->  X  =  .0.  ) ) ) ) )
84, 7mpid 41 . . . . 5  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  ( X  e.  B  -> 
( ( X  .+  X )  =  (  .0.  .+  X )  <->  X  =  .0.  ) ) ) )
98pm2.43d 48 . . . 4  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  ( ( X  .+  X
)  =  (  .0.  .+  X )  <->  X  =  .0.  ) ) )
109imp 429 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( X  .+  X )  =  (  .0.  .+  X )  <->  X  =  .0.  ) )
112, 5, 3grplid 15561 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  X
)  =  X )
1211eqeq2d 2452 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( X  .+  X )  =  (  .0.  .+  X )  <->  ( X  .+  X )  =  X ) )
1310, 12bitr3d 255 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  =  .0.  <->  ( X  .+  X )  =  X ) )
141, 13syl5rbb 258 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( X  .+  X )  =  X  <-> 
.0.  =  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   0gc0g 14374   Grpcgrp 15406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-riota 6049  df-ov 6093  df-0g 14376  df-mnd 15411  df-grp 15538
This theorem is referenced by:  isgrpid2  15567  grpidd2  15568  subg0  15680  divs0  15732  ghmid  15746  symgid  15899  isdrng2  16822  lmod0vid  16960  psr0  17448  cnfld0  17799  ldual0v  32517  erng0g  34360
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