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Theorem grpid 15892
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 2476 . 2  |-  (  .0.  =  X  <->  X  =  .0.  )
2 grpinveu.b . . . . . . 7  |-  B  =  ( Base `  G
)
3 grpinveu.o . . . . . . 7  |-  .0.  =  ( 0g `  G )
42, 3grpidcl 15885 . . . . . 6  |-  ( G  e.  Grp  ->  .0.  e.  B )
5 grpinveu.p . . . . . . . 8  |-  .+  =  ( +g  `  G )
62, 5grprcan 15890 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  .0.  e.  B  /\  X  e.  B )
)  ->  ( ( X  .+  X )  =  (  .0.  .+  X
)  <->  X  =  .0.  ) )
763exp2 1214 . . . . . 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 15887 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  X
)  =  X )
1211eqeq2d 2481 . . 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 1379    e. wcel 1767   ` cfv 5587  (class class class)co 6283   Basecbs 14489   +g cplusg 14554   0gc0g 14694   Grpcgrp 15726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5550  df-fun 5589  df-fv 5595  df-riota 6244  df-ov 6286  df-0g 14696  df-mnd 15731  df-grp 15864
This theorem is referenced by:  isgrpid2  15893  grpidd2  15894  subg0  16009  divs0  16061  ghmid  16075  symgid  16228  isdrng2  17201  lmod0vid  17339  psr0  17839  cnfld0  18229  ldual0v  33956  erng0g  35799
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