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Theorem gxfval 23749
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 6515 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2527 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 zex 10660 . . . 4  |-  ZZ  e.  _V
6 mpt2exga 6654 . . . 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 662 . . 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 5070 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
98, 2syl6eqr 2493 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
10 eqidd 2444 . . . . 5  |-  ( g  =  G  ->  ZZ  =  ZZ )
11 fveq2 5696 . . . . . . 7  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
12 gxfval.2 . . . . . . 7  |-  U  =  (GId `  G )
1311, 12syl6eqr 2493 . . . . . 6  |-  ( g  =  G  ->  (GId `  g )  =  U )
14 seqeq2 11815 . . . . . . . 8  |-  ( g  =  G  ->  seq 1 ( g ,  ( NN  X.  {
x } ) )  =  seq 1 ( G ,  ( NN 
X.  { x }
) ) )
1514fveq1d 5698 . . . . . . 7  |-  ( g  =  G  ->  (  seq 1 ( g ,  ( NN  X.  {
x } ) ) `
 y )  =  (  seq 1 ( G ,  ( NN 
X.  { x }
) ) `  y
) )
16 fveq2 5696 . . . . . . . . 9  |-  ( g  =  G  ->  ( inv `  g )  =  ( inv `  G
) )
17 gxfval.3 . . . . . . . . 9  |-  N  =  ( inv `  G
)
1816, 17syl6eqr 2493 . . . . . . . 8  |-  ( g  =  G  ->  ( inv `  g )  =  N )
1914fveq1d 5698 . . . . . . . 8  |-  ( g  =  G  ->  (  seq 1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y )  =  (  seq 1 ( G ,  ( NN 
X.  { x }
) ) `  -u y
) )
2018, 19fveq12d 5702 . . . . . . 7  |-  ( g  =  G  ->  (
( inv `  g
) `  (  seq 1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y ) )  =  ( N `  (  seq 1 ( G ,  ( NN  X.  { x } ) ) `  -u y
) ) )
2115, 20ifeq12d 3814 . . . . . 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 3814 . . . . 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 6153 . . . 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 23687 . . . 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 5777 . . 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 668 . 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 2487 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2977   ifcif 3796   {csn 3882   class class class wbr 4297    X. cxp 4843   ran crn 4846   ` cfv 5423    e. cmpt2 6098   0cc0 9287   1c1 9288    < clt 9423   -ucneg 9601   NNcn 10327   ZZcz 10651    seqcseq 11811   GrpOpcgr 23678  GIdcgi 23679   invcgn 23680   ^gcgx 23682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-neg 9603  df-z 10652  df-seq 11812  df-gx 23687
This theorem is referenced by:  gxval  23750
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