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Theorem grpid 14799
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 2410 . 2  |-  (  .0.  =  X  <->  X  =  .0.  )
2 grpinveu.b . . . . . . 7  |-  B  =  ( Base `  G
)
3 grpinveu.o . . . . . . 7  |-  .0.  =  ( 0g `  G )
42, 3grpidcl 14792 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  B )
5 grpinveu.p . . . . . . . 8  |-  .+  =  ( +g  `  G )
62, 5grprcan 14797 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  .0.  e.  B  /\  X  e.  B )
)  ->  ( ( X  .+  X )  =  (  .0.  .+  X
)  <->  X  =  .0.  ) )
763exp2 1171 . . . . . 6  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  (  .0.  e.  B  -> 
( X  e.  B  ->  ( ( X  .+  X )  =  (  .0.  .+  X )  <->  X  =  .0.  ) ) ) ) )
84, 7mpid 39 . . . . 5  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  ( X  e.  B  -> 
( ( X  .+  X )  =  (  .0.  .+  X )  <->  X  =  .0.  ) ) ) )
98pm2.43d 46 . . . 4  |-  ( G  e.  Grp  ->  ( X  e.  B  ->  ( ( X  .+  X
)  =  (  .0.  .+  X )  <->  X  =  .0.  ) ) )
109imp 419 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( X  .+  X )  =  (  .0.  .+  X )  <->  X  =  .0.  ) )
112, 5, 3grplid 14794 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  X
)  =  X )
1211eqeq2d 2419 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( X  .+  X )  =  (  .0.  .+  X )  <->  ( X  .+  X )  =  X ) )
1310, 12bitr3d 247 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  =  .0.  <->  ( X  .+  X )  =  X ) )
141, 13syl5rbb 250 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( X  .+  X )  =  X  <-> 
.0.  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5417  (class class class)co 6044   Basecbs 13428   +g cplusg 13488   0gc0g 13682   Grpcgrp 14644
This theorem is referenced by:  isgrpid2  14800  grpidd2  14801  subg0  14909  divs0  14957  ghmid  14971  symgid  15063  isdrng2  15804  lmod0vid  15941  psr0  16422  cnfld0  16684  ldual0v  29637  erng0g  31480
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-riota 6512  df-0g 13686  df-mnd 14649  df-grp 14771
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