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Theorem grprinv 14854
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 14820 . 2  |-  ( ( G  e.  Grp  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
4 grpinv.u . . 3  |-  .0.  =  ( 0g `  G )
51, 4grpidcl 14835 . 2  |-  ( G  e.  Grp  ->  .0.  e.  B )
61, 2, 4grplid 14837 . 2  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
71, 2grpass 14821 . 2  |-  ( ( G  e.  Grp  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )  ->  (
( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )
81, 2, 4grpinvex 14822 . 2  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x
)  =  .0.  )
9 simpr 449 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  X  e.  B )
10 grpinv.n . . 3  |-  N  =  ( inv g `  G )
111, 10grpinvcl 14852 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
121, 2, 4, 10grplinv 14853 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
133, 5, 6, 7, 8, 9, 11, 12grprinvd 6288 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Grpcgrp 14687   inv gcminusg 14688
This theorem is referenced by:  grpinvid1  14855  grpinvid2  14856  grpinvinv  14860  grplmulf1o  14867  grpinvadd  14869  grpsubid  14875  mulgdirlem  14916  subginv  14953  nmzsubg  14983  eqger  14992  divsinv  15001  ghminv  15015  conjnmz  15041  gacan  15084  cntzsubg  15137  oppggrp  15155  oppginv  15157  sylow2blem3  15258  frgpuplem  15406  rngnegl  15705  unitrinv  15785  isdrng2  15847  lmodvnegid  15988  lmodvsinv2  16115  lspsolvlem  16216  ghmcnp  18146  divstgpopn  18151  isngp4  18660  ofldsqr  24242  psgnuni  27401  grpvrinv  27430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815
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