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Theorem grpvlinv 27427
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 7037 . . . 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 7038 . . 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 14833 . . 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 14849 . . 3  |-  ( G  e.  Grp  ->  N : B --> B )
1211adantr 452 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  N : B --> B )
13 fcompt 5904 . . 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 14851 . . 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 6331 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 1652    e. wcel 1725   _Vcvv 2956   {csn 3814    e. cmpt 4266    X. cxp 4876    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303    ^m cmap 7018   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   inv gcminusg 14686
This theorem is referenced by:  mendrng  27477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-map 7020  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813
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