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Theorem grpinvid 15709
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 15686 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
4 eqid 2454 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
51, 4, 2grplid 15688 . . 3  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
63, 5mpdan 668 . 2  |-  ( G  e.  Grp  ->  (  .0.  ( +g  `  G
)  .0.  )  =  .0.  )
7 grpinvid.n . . . 4  |-  N  =  ( invg `  G )
81, 4, 2, 7grpinvid1 15706 . . 3  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
)  /\  .0.  e.  ( Base `  G )
)  ->  ( ( N `  .0.  )  =  .0.  <->  (  .0.  ( +g  `  G )  .0.  )  =  .0.  )
)
93, 3, 8mpd3an23 1317 . 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 1370    e. wcel 1758   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   0gc0g 14498   Grpcgrp 15530   invgcminusg 15531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-0g 14500  df-mnd 15535  df-grp 15665  df-minusg 15666
This theorem is referenced by:  grpinvnz  15717  grpsubid1  15731  mulgneg  15765  mulgz  15768  0subg  15826  eqgid  15853  odnncl  16170  gexdvds  16205  gsumzinv  16565  gsumzinvOLD  16566  gsumsub  16570  gsumsubOLD  16571  dprdfinv  16632  dprdfinvOLD  16639  mplsubglem  17635  mplsubglemOLD  17637  dsmmsubg  18294  dchrisum0re  22896  baerlem3lem1  35691
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