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Theorem grpvrinv 27451
Description: Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-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
grpvrinv  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( X  o F 
.+  ( N  o.  X ) )  =  ( I  X.  {  .0.  } ) )

Proof of Theorem grpvrinv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 730 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  /\  x  e.  I
)  ->  G  e.  Grp )
2 elmapi 6792 . . . . . 6  |-  ( X  e.  ( B  ^m  I )  ->  X : I --> B )
32adantl 452 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  X : I --> B )
4 ffvelrn 5663 . . . . 5  |-  ( ( X : I --> B  /\  x  e.  I )  ->  ( X `  x
)  e.  B )
53, 4sylan 457 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  /\  x  e.  I
)  ->  ( X `  x )  e.  B
)
6 grpvlinv.b . . . . 5  |-  B  =  ( Base `  G
)
7 grpvlinv.p . . . . 5  |-  .+  =  ( +g  `  G )
8 grpvlinv.z . . . . 5  |-  .0.  =  ( 0g `  G )
9 grpvlinv.n . . . . 5  |-  N  =  ( inv g `  G )
106, 7, 8, 9grprinv 14529 . . . 4  |-  ( ( G  e.  Grp  /\  ( X `  x )  e.  B )  -> 
( ( X `  x )  .+  ( N `  ( X `  x ) ) )  =  .0.  )
111, 5, 10syl2anc 642 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  /\  x  e.  I
)  ->  ( ( X `  x )  .+  ( N `  ( X `  x )
) )  =  .0.  )
1211mpteq2dva 4106 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( x  e.  I  |->  ( ( X `  x )  .+  ( N `  ( X `  x ) ) ) )  =  ( x  e.  I  |->  .0.  )
)
13 elmapex 6791 . . . . 5  |-  ( X  e.  ( B  ^m  I )  ->  ( B  e.  _V  /\  I  e.  _V ) )
1413simprd 449 . . . 4  |-  ( X  e.  ( B  ^m  I )  ->  I  e.  _V )
1514adantl 452 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  I  e.  _V )
16 fvex 5539 . . . 4  |-  ( N `
 ( X `  x ) )  e. 
_V
1716a1i 10 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  /\  x  e.  I
)  ->  ( N `  ( X `  x
) )  e.  _V )
183feqmptd 5575 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  X  =  ( x  e.  I  |->  ( X `
 x ) ) )
196, 9grpinvf 14526 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
20 fcompt 5694 . . . 4  |-  ( ( N : B --> B  /\  X : I --> B )  ->  ( N  o.  X )  =  ( x  e.  I  |->  ( N `  ( X `
 x ) ) ) )
2119, 2, 20syl2an 463 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( N  o.  X
)  =  ( x  e.  I  |->  ( N `
 ( X `  x ) ) ) )
2215, 5, 17, 18, 21offval2 6095 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( X  o F 
.+  ( N  o.  X ) )  =  ( x  e.  I  |->  ( ( X `  x )  .+  ( N `  ( X `  x ) ) ) ) )
23 fconstmpt 4732 . . 3  |-  ( I  X.  {  .0.  }
)  =  ( x  e.  I  |->  .0.  )
2423a1i 10 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( I  X.  {  .0.  } )  =  ( x  e.  I  |->  .0.  ) )
2512, 22, 243eqtr4d 2325 1  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( X  o F 
.+  ( N  o.  X ) )  =  ( I  X.  {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076    ^m cmap 6772   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363
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  ax-un 4512
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-pw 3627  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-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490
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