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Theorem gxfval 21838
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 5124 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2520 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 zex 10284 . . . 4  |-  ZZ  e.  _V
6 mpt2exga 6417 . . . 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 644 . . 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 5088 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
98, 2syl6eqr 2486 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
10 eqidd 2437 . . . . 5  |-  ( g  =  G  ->  ZZ  =  ZZ )
11 fveq2 5721 . . . . . . 7  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
12 gxfval.2 . . . . . . 7  |-  U  =  (GId `  G )
1311, 12syl6eqr 2486 . . . . . 6  |-  ( g  =  G  ->  (GId `  g )  =  U )
14 seqeq2 11320 . . . . . . . 8  |-  ( g  =  G  ->  seq  1 ( g ,  ( NN  X.  {
x } ) )  =  seq  1 ( G ,  ( NN 
X.  { x }
) ) )
1514fveq1d 5723 . . . . . . 7  |-  ( g  =  G  ->  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 y )  =  (  seq  1 ( G ,  ( NN 
X.  { x }
) ) `  y
) )
16 fveq2 5721 . . . . . . . . 9  |-  ( g  =  G  ->  ( inv `  g )  =  ( inv `  G
) )
17 gxfval.3 . . . . . . . . 9  |-  N  =  ( inv `  G
)
1816, 17syl6eqr 2486 . . . . . . . 8  |-  ( g  =  G  ->  ( inv `  g )  =  N )
1914fveq1d 5723 . . . . . . . 8  |-  ( g  =  G  ->  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y )  =  (  seq  1 ( G ,  ( NN 
X.  { x }
) ) `  -u y
) )
2018, 19fveq12d 5727 . . . . . . 7  |-  ( g  =  G  ->  (
( inv `  g
) `  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y ) )  =  ( N `  (  seq  1 ( G ,  ( NN  X.  { x } ) ) `  -u y
) ) )
2115, 20ifeq12d 3748 . . . . . 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 3748 . . . . 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 6129 . . . 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 21776 . . . 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 5797 . . 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 650 . 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 2480 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 1652    e. wcel 1725   _Vcvv 2949   ifcif 3732   {csn 3807   class class class wbr 4205    X. cxp 4869   ran crn 4872   ` cfv 5447    e. cmpt2 6076   0cc0 8983   1c1 8984    < clt 9113   -ucneg 9285   NNcn 9993   ZZcz 10275    seq cseq 11316   GrpOpcgr 21767  GIdcgi 21768   invcgn 21769   ^gcgx 21771
This theorem is referenced by:  gxval  21839
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 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-recs 6626  df-rdg 6661  df-neg 9287  df-z 10276  df-seq 11317  df-gx 21776
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