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Theorem gxval 25074
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 25073 . . . 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 6312 . . 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 4043 . . . . . . . . 9  |-  ( x  =  A  ->  { x }  =  { A } )
87xpeq2d 5029 . . . . . . . 8  |-  ( x  =  A  ->  ( NN  X.  { x }
)  =  ( NN 
X.  { A }
) )
98seqeq3d 12095 . . . . . . 7  |-  ( x  =  A  ->  seq 1 ( G , 
( NN  X.  {
x } ) )  =  seq 1 ( G ,  ( NN 
X.  { A }
) ) )
109fveq1d 5874 . . . . . 6  |-  ( x  =  A  ->  (  seq 1 ( G , 
( NN  X.  {
x } ) ) `
 y )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  y
) )
119fveq1d 5874 . . . . . . 7  |-  ( x  =  A  ->  (  seq 1 ( G , 
( NN  X.  {
x } ) ) `
 -u y )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  -u y
) )
1211fveq2d 5876 . . . . . 6  |-  ( x  =  A  ->  ( N `  (  seq 1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) )  =  ( N `  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 -u y ) ) )
1310, 12ifeq12d 3965 . . . . 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 3964 . . . 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 2471 . . . . 5  |-  ( y  =  K  ->  (
y  =  0  <->  K  =  0 ) )
16 breq2 4457 . . . . . 6  |-  ( y  =  K  ->  (
0  <  y  <->  0  <  K ) )
17 fveq2 5872 . . . . . 6  |-  ( y  =  K  ->  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  y )  =  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 K ) )
18 negeq 9824 . . . . . . . 8  |-  ( y  =  K  ->  -u y  =  -u K )
1918fveq2d 5876 . . . . . . 7  |-  ( y  =  K  ->  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  -u y )  =  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 -u K ) )
2019fveq2d 5876 . . . . . 6  |-  ( y  =  K  ->  ( N `  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  -u y ) )  =  ( N `  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) )
2116, 17, 20ifbieq12d 3972 . . . . 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 3970 . . . 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 2467 . . . 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 5882 . . . . . 6  |-  (GId `  G )  e.  _V
252, 24eqeltri 2551 . . . . 5  |-  U  e. 
_V
26 fvex 5882 . . . . . 6  |-  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  K )  e.  _V
27 fvex 5882 . . . . . 6  |-  ( N `
 (  seq 1
( G ,  ( NN  X.  { A } ) ) `  -u K ) )  e. 
_V
2826, 27ifex 4014 . . . . 5  |-  if ( 0  <  K , 
(  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 K ) ,  ( N `  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) )  e.  _V
2925, 28ifex 4014 . . . 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 6433 . . 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 2528 . 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 1192 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118   ifcif 3945   {csn 4033   class class class wbr 4453    X. cxp 5003   ran crn 5006   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   0cc0 9504   1c1 9505    < clt 9640   -ucneg 9818   NNcn 10548   ZZcz 10876    seqcseq 12087   GrpOpcgr 25002  GIdcgi 25003   invcgn 25004   ^gcgx 25006
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-neg 9820  df-z 10877  df-seq 12088  df-gx 25011
This theorem is referenced by:  gxpval  25075  gxnval  25076  gx0  25077
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