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Theorem gxfval 20924
Description: The value of the group power operator function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
gxfval.1  |-  X  =  ran  G
gxfval.2  |-  U  =  (GId `  G )
gxfval.3  |-  N  =  ( inv `  G
)
gxfval.4  |-  P  =  ( ^g `  G
)
Assertion
Ref Expression
gxfval  |-  ( G  e.  GrpOp  ->  P  =  ( x  e.  X ,  y  e.  ZZ  |->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { x } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) ) ) )
Distinct variable groups:    x, G, y    x, U, y    x, X, y    x, N, y
Allowed substitution hints:    P( x, y)

Proof of Theorem gxfval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 gxfval.4 . 2  |-  P  =  ( ^g `  G
)
2 gxfval.1 . . . . 5  |-  X  =  ran  G
3 rnexg 4940 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2367 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 zex 10033 . . . 4  |-  ZZ  e.  _V
6 mpt2exga 6197 . . . 4  |-  ( ( X  e.  _V  /\  ZZ  e.  _V )  -> 
( x  e.  X ,  y  e.  ZZ  |->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { x } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) ) )  e. 
_V )
74, 5, 6sylancl 643 . . 3  |-  ( G  e.  GrpOp  ->  ( x  e.  X ,  y  e.  ZZ  |->  if ( y  =  0 ,  U ,  if ( 0  < 
y ,  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) ) )  e. 
_V )
8 rneq 4904 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
98, 2syl6eqr 2333 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
10 eqidd 2284 . . . . 5  |-  ( g  =  G  ->  ZZ  =  ZZ )
11 fveq2 5525 . . . . . . 7  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
12 gxfval.2 . . . . . . 7  |-  U  =  (GId `  G )
1311, 12syl6eqr 2333 . . . . . 6  |-  ( g  =  G  ->  (GId `  g )  =  U )
14 seqeq2 11050 . . . . . . . 8  |-  ( g  =  G  ->  seq  1 ( g ,  ( NN  X.  {
x } ) )  =  seq  1 ( G ,  ( NN 
X.  { x }
) ) )
1514fveq1d 5527 . . . . . . 7  |-  ( g  =  G  ->  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 y )  =  (  seq  1 ( G ,  ( NN 
X.  { x }
) ) `  y
) )
16 fveq2 5525 . . . . . . . . 9  |-  ( g  =  G  ->  ( inv `  g )  =  ( inv `  G
) )
17 gxfval.3 . . . . . . . . 9  |-  N  =  ( inv `  G
)
1816, 17syl6eqr 2333 . . . . . . . 8  |-  ( g  =  G  ->  ( inv `  g )  =  N )
1914fveq1d 5527 . . . . . . . 8  |-  ( g  =  G  ->  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y )  =  (  seq  1 ( G ,  ( NN 
X.  { x }
) ) `  -u y
) )
2018, 19fveq12d 5531 . . . . . . 7  |-  ( g  =  G  ->  (
( inv `  g
) `  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y ) )  =  ( N `  (  seq  1 ( G ,  ( NN  X.  { x } ) ) `  -u y
) ) )
2115, 20ifeq12d 3581 . . . . . 6  |-  ( g  =  G  ->  if ( 0  <  y ,  (  seq  1
( g ,  ( NN  X.  { x } ) ) `  y ) ,  ( ( inv `  g
) `  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y ) ) )  =  if ( 0  <  y ,  (  seq  1 ( G ,  ( NN 
X.  { x }
) ) `  y
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { x } ) ) `  -u y ) ) ) )
2213, 21ifeq12d 3581 . . . . 5  |-  ( g  =  G  ->  if ( y  =  0 ,  (GId `  g
) ,  if ( 0  <  y ,  (  seq  1 ( g ,  ( NN 
X.  { x }
) ) `  y
) ,  ( ( inv `  g ) `
 (  seq  1
( g ,  ( NN  X.  { x } ) ) `  -u y ) ) ) )  =  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1 ( G ,  ( NN 
X.  { x }
) ) `  y
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { x } ) ) `  -u y ) ) ) ) )
239, 10, 22mpt2eq123dv 5910 . . . 4  |-  ( g  =  G  ->  (
x  e.  ran  g ,  y  e.  ZZ  |->  if ( y  =  0 ,  (GId `  g
) ,  if ( 0  <  y ,  (  seq  1 ( g ,  ( NN 
X.  { x }
) ) `  y
) ,  ( ( inv `  g ) `
 (  seq  1
( g ,  ( NN  X.  { x } ) ) `  -u y ) ) ) ) )  =  ( x  e.  X , 
y  e.  ZZ  |->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { x } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) ) ) )
24 df-gx 20862 . . . 4  |-  ^g  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g ,  y  e.  ZZ  |->  if ( y  =  0 ,  (GId
`  g ) ,  if ( 0  < 
y ,  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 y ) ,  ( ( inv `  g
) `  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y ) ) ) ) ) )
2523, 24fvmptg 5600 . . 3  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X , 
y  e.  ZZ  |->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { x } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) ) )  e. 
_V )  ->  ( ^g `  G )  =  ( x  e.  X ,  y  e.  ZZ  |->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { x } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) ) ) )
267, 25mpdan 649 . 2  |-  ( G  e.  GrpOp  ->  ( ^g `  G )  =  ( x  e.  X , 
y  e.  ZZ  |->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { x } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) ) ) )
271, 26syl5eq 2327 1  |-  ( G  e.  GrpOp  ->  P  =  ( x  e.  X ,  y  e.  ZZ  |->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { x } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   ifcif 3565   {csn 3640   class class class wbr 4023    X. cxp 4687   ran crn 4690   ` cfv 5255    e. cmpt2 5860   0cc0 8737   1c1 8738    < clt 8867   -ucneg 9038   NNcn 9746   ZZcz 10024    seq cseq 11046   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855   ^gcgx 20857
This theorem is referenced by:  gxval  20925
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  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-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-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-neg 9040  df-z 10025  df-seq 11047  df-gx 20862
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