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Theorem grprinv 14529
Description: The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grpinv.b  |-  B  =  ( Base `  G
)
grpinv.p  |-  .+  =  ( +g  `  G )
grpinv.u  |-  .0.  =  ( 0g `  G )
grpinv.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
grprinv  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )

Proof of Theorem grprinv
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpinv.b . . 3  |-  B  =  ( Base `  G
)
2 grpinv.p . . 3  |-  .+  =  ( +g  `  G )
31, 2grpcl 14495 . 2  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
4 grpinv.u . . 3  |-  .0.  =  ( 0g `  G )
51, 4grpidcl 14510 . 2  |-  ( G  e.  Grp  ->  .0.  e.  B )
61, 2, 4grplid 14512 . 2  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
71, 2grpass 14496 . 2  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )  ->  (
( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )
81, 2, 4grpinvex 14497 . 2  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x
)  =  .0.  )
9 simpr 447 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  X  e.  B )
10 grpinv.n . . 3  |-  N  =  ( inv g `  G )
111, 10grpinvcl 14527 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
121, 2, 4, 10grplinv 14528 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
133, 5, 6, 7, 8, 9, 11, 12grprinvd 6059 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363
This theorem is referenced by:  grpinvid1  14530  grpinvid2  14531  grpinvinv  14535  grplmulf1o  14542  grpinvadd  14544  grpsubid  14550  mulgdirlem  14591  subginv  14628  nmzsubg  14658  eqger  14667  divsinv  14676  ghminv  14690  conjnmz  14716  gacan  14759  cntzsubg  14812  oppggrp  14830  oppginv  14832  sylow2blem3  14933  frgpuplem  15081  rngnegl  15380  unitrinv  15460  isdrng2  15522  lmodvnegid  15666  lmodvsinv2  15794  lspsolvlem  15895  ghmcnp  17797  divstgpopn  17802  isngp4  18133  psgnuni  27422  grpvrinv  27451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490
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