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Theorem grpinvid 16303
Description: The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.)
Hypotheses
Ref Expression
grpinvid.u  |-  .0.  =  ( 0g `  G )
grpinvid.n  |-  N  =  ( invg `  G )
Assertion
Ref Expression
grpinvid  |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )

Proof of Theorem grpinvid
StepHypRef Expression
1 eqid 2454 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
2 grpinvid.u . . . 4  |-  .0.  =  ( 0g `  G )
31, 2grpidcl 16280 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
4 eqid 2454 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
51, 4, 2grplid 16282 . . 3  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
63, 5mpdan 666 . 2  |-  ( G  e.  Grp  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
7 grpinvid.n . . . 4  |-  N  =  ( invg `  G )
81, 4, 2, 7grpinvid1 16300 . . 3  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
)  /\  .0.  e.  ( Base `  G )
)  ->  ( ( N `  .0.  )  =  .0.  <->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  )
)
93, 3, 8mpd3an23 1324 . 2  |-  ( G  e.  Grp  ->  (
( N `  .0.  )  =  .0.  <->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  ) )
106, 9mpbird 232 1  |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   0gc0g 14932   Grpcgrp 16255   invgcminusg 16256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-minusg 16260
This theorem is referenced by:  grpinvnz  16311  grpsubid1  16325  mulgneg  16362  mulgz  16365  0subg  16428  eqgid  16455  odnncl  16771  gexdvds  16806  gsumzinv  17170  gsumzinvOLD  17171  gsumsub  17174  dprdfinv  17257  dprdfinvOLD  17264  mplsubglem  18291  mplsubglemOLD  18293  dsmmsubg  18950  dchrisum0re  23899  baerlem3lem1  37850
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