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Theorem gxfval 21686
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 5064 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2464 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 zex 10216 . . . 4  |-  ZZ  e.  _V
6 mpt2exga 6356 . . . 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 5028 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
98, 2syl6eqr 2430 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
10 eqidd 2381 . . . . 5  |-  ( g  =  G  ->  ZZ  =  ZZ )
11 fveq2 5661 . . . . . . 7  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
12 gxfval.2 . . . . . . 7  |-  U  =  (GId `  G )
1311, 12syl6eqr 2430 . . . . . 6  |-  ( g  =  G  ->  (GId `  g )  =  U )
14 seqeq2 11247 . . . . . . . 8  |-  ( g  =  G  ->  seq  1 ( g ,  ( NN  X.  {
x } ) )  =  seq  1 ( G ,  ( NN 
X.  { x }
) ) )
1514fveq1d 5663 . . . . . . 7  |-  ( g  =  G  ->  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 y )  =  (  seq  1 ( G ,  ( NN 
X.  { x }
) ) `  y
) )
16 fveq2 5661 . . . . . . . . 9  |-  ( g  =  G  ->  ( inv `  g )  =  ( inv `  G
) )
17 gxfval.3 . . . . . . . . 9  |-  N  =  ( inv `  G
)
1816, 17syl6eqr 2430 . . . . . . . 8  |-  ( g  =  G  ->  ( inv `  g )  =  N )
1914fveq1d 5663 . . . . . . . 8  |-  ( g  =  G  ->  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y )  =  (  seq  1 ( G ,  ( NN 
X.  { x }
) ) `  -u y
) )
2018, 19fveq12d 5667 . . . . . . 7  |-  ( g  =  G  ->  (
( inv `  g
) `  (  seq  1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y ) )  =  ( N `  (  seq  1 ( G ,  ( NN  X.  { x } ) ) `  -u y
) ) )
2115, 20ifeq12d 3691 . . . . . 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 3691 . . . . 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 6068 . . . 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 21624 . . . 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 5736 . . 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 2424 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 1649    e. wcel 1717   _Vcvv 2892   ifcif 3675   {csn 3750   class class class wbr 4146    X. cxp 4809   ran crn 4812   ` cfv 5387    e. cmpt2 6015   0cc0 8916   1c1 8917    < clt 9046   -ucneg 9217   NNcn 9925   ZZcz 10207    seq cseq 11243   GrpOpcgr 21615  GIdcgi 21616   invcgn 21617   ^gcgx 21619
This theorem is referenced by:  gxval  21687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-neg 9219  df-z 10208  df-seq 11244  df-gx 21624
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