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Theorem grplinv 14528
Description: The left 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
grplinv  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )

Proof of Theorem grplinv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 grpinv.b . . . . 5  |-  B  =  ( Base `  G
)
2 grpinv.p . . . . 5  |-  .+  =  ( +g  `  G )
3 grpinv.u . . . . 5  |-  .0.  =  ( 0g `  G )
4 grpinv.n . . . . 5  |-  N  =  ( inv g `  G )
51, 2, 3, 4grpinvval 14521 . . . 4  |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B ( y  .+  X )  =  .0.  ) )
65adantl 452 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  =  ( iota_ y  e.  B ( y 
.+  X )  =  .0.  ) )
71, 2, 3grpinveu 14516 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E! y  e.  B  ( y  .+  X
)  =  .0.  )
8 riotacl2 6318 . . . 4  |-  ( E! y  e.  B  ( y  .+  X )  =  .0.  ->  ( iota_ y  e.  B ( y  .+  X )  =  .0.  )  e. 
{ y  e.  B  |  ( y  .+  X )  =  .0. 
} )
97, 8syl 15 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( iota_ y  e.  B
( y  .+  X
)  =  .0.  )  e.  { y  e.  B  |  ( y  .+  X )  =  .0. 
} )
106, 9eqeltrd 2357 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  { y  e.  B  |  ( y  .+  X )  =  .0.  } )
11 oveq1 5865 . . . . 5  |-  ( y  =  ( N `  X )  ->  (
y  .+  X )  =  ( ( N `
 X )  .+  X ) )
1211eqeq1d 2291 . . . 4  |-  ( y  =  ( N `  X )  ->  (
( y  .+  X
)  =  .0.  <->  ( ( N `  X )  .+  X )  =  .0.  ) )
1312elrab 2923 . . 3  |-  ( ( N `  X )  e.  { y  e.  B  |  ( y 
.+  X )  =  .0.  }  <->  ( ( N `  X )  e.  B  /\  (
( N `  X
)  .+  X )  =  .0.  ) )
1413simprbi 450 . 2  |-  ( ( N `  X )  e.  { y  e.  B  |  ( y 
.+  X )  =  .0.  }  ->  (
( N `  X
)  .+  X )  =  .0.  )
1510, 14syl 15 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E!wreu 2545   {crab 2547   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363
This theorem is referenced by:  grprinv  14529  grpinvid1  14530  grpinvid2  14531  isgrpinv  14532  grplcan  14534  grpinvinv  14535  grpsubadd  14553  grplactcnv  14564  mulgdirlem  14591  prdsinvlem  14603  imasgrp  14611  issubg2  14636  isnsg3  14651  nmzsubg  14658  ssnmz  14659  eqger  14667  divsgrp  14672  conjghm  14713  galcan  14758  cntzsubg  14812  lsmmod  14984  lsmdisj2  14991  rngnegr  15381  unitlinv  15459  isdrng2  15522  lmodvneg1  15667  psrlinv  16142  tgpconcompeqg  17794  divstgpopn  17802  grpvlinv  27450  lflnegl  29266  dvhgrp  31297
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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