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Theorem gxfval 24963
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 6716 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2559 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 zex 10873 . . . 4  |-  ZZ  e.  _V
6 mpt2exga 6859 . . . 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 5228 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
98, 2syl6eqr 2526 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
10 eqidd 2468 . . . . 5  |-  ( g  =  G  ->  ZZ  =  ZZ )
11 fveq2 5866 . . . . . . 7  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
12 gxfval.2 . . . . . . 7  |-  U  =  (GId `  G )
1311, 12syl6eqr 2526 . . . . . 6  |-  ( g  =  G  ->  (GId `  g )  =  U )
14 seqeq2 12079 . . . . . . . 8  |-  ( g  =  G  ->  seq 1 ( g ,  ( NN  X.  {
x } ) )  =  seq 1 ( G ,  ( NN 
X.  { x }
) ) )
1514fveq1d 5868 . . . . . . 7  |-  ( g  =  G  ->  (  seq 1 ( g ,  ( NN  X.  {
x } ) ) `
 y )  =  (  seq 1 ( G ,  ( NN 
X.  { x }
) ) `  y
) )
16 fveq2 5866 . . . . . . . . 9  |-  ( g  =  G  ->  ( inv `  g )  =  ( inv `  G
) )
17 gxfval.3 . . . . . . . . 9  |-  N  =  ( inv `  G
)
1816, 17syl6eqr 2526 . . . . . . . 8  |-  ( g  =  G  ->  ( inv `  g )  =  N )
1914fveq1d 5868 . . . . . . . 8  |-  ( g  =  G  ->  (  seq 1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y )  =  (  seq 1 ( G ,  ( NN 
X.  { x }
) ) `  -u y
) )
2018, 19fveq12d 5872 . . . . . . 7  |-  ( g  =  G  ->  (
( inv `  g
) `  (  seq 1 ( g ,  ( NN  X.  {
x } ) ) `
 -u y ) )  =  ( N `  (  seq 1 ( G ,  ( NN  X.  { x } ) ) `  -u y
) ) )
2115, 20ifeq12d 3959 . . . . . 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 3959 . . . . 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 6343 . . . 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 24901 . . . 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 5948 . . 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 2520 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 1379    e. wcel 1767   _Vcvv 3113   ifcif 3939   {csn 4027   class class class wbr 4447    X. cxp 4997   ran crn 5000   ` cfv 5588    |-> cmpt2 6286   0cc0 9492   1c1 9493    < clt 9628   -ucneg 9806   NNcn 10536   ZZcz 10864    seqcseq 12075   GrpOpcgr 24892  GIdcgi 24893   invcgn 24894   ^gcgx 24896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-neg 9808  df-z 10865  df-seq 12076  df-gx 24901
This theorem is referenced by:  gxval  24964
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