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Theorem gxval 20925
Description: The result of the group power operator. (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
gxval  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  =  if ( K  =  0 ,  U ,  if ( 0  <  K ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) `  K ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) ) ) )

Proof of Theorem gxval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gxfval.1 . . . . 5  |-  X  =  ran  G
2 gxfval.2 . . . . 5  |-  U  =  (GId `  G )
3 gxfval.3 . . . . 5  |-  N  =  ( inv `  G
)
4 gxfval.4 . . . . 5  |-  P  =  ( ^g `  G
)
51, 2, 3, 4gxfval 20924 . . . 4  |-  ( 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 ) ) ) ) ) )
65oveqd 5875 . . 3  |-  ( G  e.  GrpOp  ->  ( A P K )  =  ( A ( 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 ) ) ) ) ) K ) )
7 sneq 3651 . . . . . . . . 9  |-  ( x  =  A  ->  { x }  =  { A } )
87xpeq2d 4713 . . . . . . . 8  |-  ( x  =  A  ->  ( NN  X.  { x }
)  =  ( NN 
X.  { A }
) )
98seqeq3d 11054 . . . . . . 7  |-  ( x  =  A  ->  seq  1 ( G , 
( NN  X.  {
x } ) )  =  seq  1 ( G ,  ( NN 
X.  { A }
) ) )
109fveq1d 5527 . . . . . 6  |-  ( x  =  A  ->  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 y )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  y
) )
119fveq1d 5527 . . . . . . 7  |-  ( x  =  A  ->  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  -u y
) )
1211fveq2d 5529 . . . . . 6  |-  ( x  =  A  ->  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) )  =  ( N `  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 -u y ) ) )
1310, 12ifeq12d 3581 . . . . 5  |-  ( x  =  A  ->  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { x } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) ) )  =  if ( 0  <  y ,  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  y
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u y ) ) ) )
1413ifeq2d 3580 . . . 4  |-  ( x  =  A  ->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { x } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) ) ) )  =  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u y ) ) ) ) )
15 eqeq1 2289 . . . . 5  |-  ( y  =  K  ->  (
y  =  0  <->  K  =  0 ) )
16 breq2 4027 . . . . . 6  |-  ( y  =  K  ->  (
0  <  y  <->  0  <  K ) )
17 fveq2 5525 . . . . . 6  |-  ( y  =  K  ->  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  y )  =  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 K ) )
18 negeq 9044 . . . . . . . 8  |-  ( y  =  K  ->  -u y  =  -u K )
1918fveq2d 5529 . . . . . . 7  |-  ( y  =  K  ->  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u y )  =  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 -u K ) )
2019fveq2d 5529 . . . . . 6  |-  ( y  =  K  ->  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u y ) )  =  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) )
2116, 17, 20ifbieq12d 3587 . . . . 5  |-  ( y  =  K  ->  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u y ) ) )  =  if ( 0  <  K ,  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 K ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) ) )
2215, 21ifbieq2d 3585 . . . 4  |-  ( y  =  K  ->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) `  y ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u y ) ) ) )  =  if ( K  =  0 ,  U ,  if ( 0  <  K , 
(  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) ) ) )
23 eqid 2283 . . . 4  |-  ( 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 ) ) ) ) )  =  ( 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 fvex 5539 . . . . . 6  |-  (GId `  G )  e.  _V
252, 24eqeltri 2353 . . . . 5  |-  U  e. 
_V
26 fvex 5539 . . . . . 6  |-  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  K )  e.  _V
27 fvex 5539 . . . . . 6  |-  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) )  e. 
_V
2826, 27ifex 3623 . . . . 5  |-  if ( 0  <  K , 
(  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) )  e.  _V
2925, 28ifex 3623 . . . 4  |-  if ( K  =  0 ,  U ,  if ( 0  <  K , 
(  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) ) )  e.  _V
3014, 22, 23, 29ovmpt2 5983 . . 3  |-  ( ( A  e.  X  /\  K  e.  ZZ )  ->  ( A ( 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 ) ) ) ) ) K )  =  if ( K  =  0 ,  U ,  if ( 0  < 
K ,  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  K ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) ) ) )
316, 30sylan9eq 2335 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  K  e.  ZZ )
)  ->  ( A P K )  =  if ( K  =  0 ,  U ,  if ( 0  <  K ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) `  K ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) ) ) )
32313impb 1147 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  =  if ( K  =  0 ,  U ,  if ( 0  <  K ,  (  seq  1
( G ,  ( NN  X.  { A } ) ) `  K ) ,  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   ifcif 3565   {csn 3640   class class class wbr 4023    X. cxp 4687   ran crn 4690   ` cfv 5255  (class class class)co 5858    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:  gxpval  20926  gxnval  20927  gx0  20928
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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