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Theorem grpid 14845
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 2440 . 2  |-  (  .0.  =  X  <->  X  =  .0.  )
2 grpinveu.b . . . . . . 7  |-  B  =  ( Base `  G
)
3 grpinveu.o . . . . . . 7  |-  .0.  =  ( 0g `  G )
42, 3grpidcl 14838 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  B )
5 grpinveu.p . . . . . . . 8  |-  .+  =  ( +g  `  G )
62, 5grprcan 14843 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  .0.  e.  B  /\  X  e.  B )
)  ->  ( ( X  .+  X )  =  (  .0.  .+  X
)  <->  X  =  .0.  ) )
763exp2 1172 . . . . . 6  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  (  .0.  e.  B  -> 
( X  e.  B  ->  ( ( X  .+  X )  =  (  .0.  .+  X )  <->  X  =  .0.  ) ) ) ) )
84, 7mpid 40 . . . . 5  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  ( X  e.  B  -> 
( ( X  .+  X )  =  (  .0.  .+  X )  <->  X  =  .0.  ) ) ) )
98pm2.43d 47 . . . 4  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  ( ( X  .+  X
)  =  (  .0.  .+  X )  <->  X  =  .0.  ) ) )
109imp 420 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( X  .+  X )  =  (  .0.  .+  X )  <->  X  =  .0.  ) )
112, 5, 3grplid 14840 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  X
)  =  X )
1211eqeq2d 2449 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( X  .+  X )  =  (  .0.  .+  X )  <->  ( X  .+  X )  =  X ) )
1310, 12bitr3d 248 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  =  .0.  <->  ( X  .+  X )  =  X ) )
141, 13syl5rbb 251 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( X  .+  X )  =  X  <-> 
.0.  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   Basecbs 13474   +g cplusg 13534   0gc0g 13728   Grpcgrp 14690
This theorem is referenced by:  isgrpid2  14846  grpidd2  14847  subg0  14955  divs0  15003  ghmid  15017  symgid  15109  isdrng2  15850  lmod0vid  15987  psr0  16468  cnfld0  16730  ldual0v  30022  erng0g  31865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-riota 6552  df-0g 13732  df-mnd 14695  df-grp 14817
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