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Theorem grpvlinv 27318
Description: Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
grpvlinv.b  |-  B  =  ( Base `  G
)
grpvlinv.p  |-  .+  =  ( +g  `  G )
grpvlinv.n  |-  N  =  ( inv g `  G )
grpvlinv.z  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
grpvlinv  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( ( N  o.  X )  o F 
.+  X )  =  ( I  X.  {  .0.  } ) )

Proof of Theorem grpvlinv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapex 6996 . . . 4  |-  ( X  e.  ( B  ^m  I )  ->  ( B  e.  _V  /\  I  e.  _V ) )
21simprd 450 . . 3  |-  ( X  e.  ( B  ^m  I )  ->  I  e.  _V )
32adantl 453 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  I  e.  _V )
4 elmapi 6997 . . 3  |-  ( X  e.  ( B  ^m  I )  ->  X : I --> B )
54adantl 453 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  X : I --> B )
6 grpvlinv.b . . . 4  |-  B  =  ( Base `  G
)
7 grpvlinv.z . . . 4  |-  .0.  =  ( 0g `  G )
86, 7grpidcl 14788 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  B )
98adantr 452 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  .0.  e.  B )
10 grpvlinv.n . . . 4  |-  N  =  ( inv g `  G )
116, 10grpinvf 14804 . . 3  |-  ( G  e.  Grp  ->  N : B --> B )
1211adantr 452 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  N : B --> B )
13 fcompt 5863 . . 3  |-  ( ( N : B --> B  /\  X : I --> B )  ->  ( N  o.  X )  =  ( x  e.  I  |->  ( N `  ( X `
 x ) ) ) )
1411, 4, 13syl2an 464 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( N  o.  X
)  =  ( x  e.  I  |->  ( N `
 ( X `  x ) ) ) )
15 grpvlinv.p . . . 4  |-  .+  =  ( +g  `  G )
166, 15, 7, 10grplinv 14806 . . 3  |-  ( ( G  e.  Grp  /\  y  e.  B )  ->  ( ( N `  y )  .+  y
)  =  .0.  )
1716adantlr 696 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  /\  y  e.  B
)  ->  ( ( N `  y )  .+  y )  =  .0.  )
183, 5, 9, 12, 14, 17caofinvl 6290 1  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( ( N  o.  X )  o F 
.+  X )  =  ( I  X.  {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   {csn 3774    e. cmpt 4226    X. cxp 4835    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262    ^m cmap 6977   Basecbs 13424   +g cplusg 13484   0gc0g 13678   Grpcgrp 14640   inv gcminusg 14641
This theorem is referenced by:  mendrng  27368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-map 6979  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768
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