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Theorem grpvrinv 27430
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 732 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  /\  x  e.  I
)  ->  G  e.  Grp )
2 elmapi 7040 . . . . . 6  |-  ( X  e.  ( B  ^m  I )  ->  X : I --> B )
32adantl 454 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  X : I --> B )
43ffvelrnda 5872 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  /\  x  e.  I
)  ->  ( X `  x )  e.  B
)
5 grpvlinv.b . . . . 5  |-  B  =  ( Base `  G
)
6 grpvlinv.p . . . . 5  |-  .+  =  ( +g  `  G )
7 grpvlinv.z . . . . 5  |-  .0.  =  ( 0g `  G )
8 grpvlinv.n . . . . 5  |-  N  =  ( inv g `  G )
95, 6, 7, 8grprinv 14854 . . . 4  |-  ( ( G  e.  Grp  /\  ( X `  x )  e.  B )  -> 
( ( X `  x )  .+  ( N `  ( X `  x ) ) )  =  .0.  )
101, 4, 9syl2anc 644 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  /\  x  e.  I
)  ->  ( ( X `  x )  .+  ( N `  ( X `  x )
) )  =  .0.  )
1110mpteq2dva 4297 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( x  e.  I  |->  ( ( X `  x )  .+  ( N `  ( X `  x ) ) ) )  =  ( x  e.  I  |->  .0.  )
)
12 elmapex 7039 . . . . 5  |-  ( X  e.  ( B  ^m  I )  ->  ( B  e.  _V  /\  I  e.  _V ) )
1312simprd 451 . . . 4  |-  ( X  e.  ( B  ^m  I )  ->  I  e.  _V )
1413adantl 454 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  I  e.  _V )
15 fvex 5744 . . . 4  |-  ( N `
 ( X `  x ) )  e. 
_V
1615a1i 11 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  /\  x  e.  I
)  ->  ( N `  ( X `  x
) )  e.  _V )
173feqmptd 5781 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  ->  X  =  ( x  e.  I  |->  ( X `
 x ) ) )
185, 8grpinvf 14851 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
19 fcompt 5906 . . . 4  |-  ( ( N : B --> B  /\  X : I --> B )  ->  ( N  o.  X )  =  ( x  e.  I  |->  ( N `  ( X `
 x ) ) ) )
2018, 2, 19syl2an 465 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( N  o.  X
)  =  ( x  e.  I  |->  ( N `
 ( X `  x ) ) ) )
2114, 4, 16, 17, 20offval2 6324 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( X  o F 
.+  ( N  o.  X ) )  =  ( x  e.  I  |->  ( ( X `  x )  .+  ( N `  ( X `  x ) ) ) ) )
22 fconstmpt 4923 . . 3  |-  ( I  X.  {  .0.  }
)  =  ( x  e.  I  |->  .0.  )
2322a1i 11 . 2  |-  ( ( G  e.  Grp  /\  X  e.  ( B  ^m  I ) )  -> 
( I  X.  {  .0.  } )  =  ( x  e.  I  |->  .0.  ) )
2411, 21, 233eqtr4d 2480 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    e. cmpt 4268    X. cxp 4878    o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083    o Fcof 6305    ^m cmap 7020   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Grpcgrp 14687   inv gcminusg 14688
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  ax-un 4703
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-pw 3803  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-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-map 7022  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815
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