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Theorem gxval 20941
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 20940 . . . 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 5891 . . 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 3664 . . . . . . . . 9  |-  ( x  =  A  ->  { x }  =  { A } )
87xpeq2d 4729 . . . . . . . 8  |-  ( x  =  A  ->  ( NN  X.  { x }
)  =  ( NN 
X.  { A }
) )
98seqeq3d 11070 . . . . . . 7  |-  ( x  =  A  ->  seq  1 ( G , 
( NN  X.  {
x } ) )  =  seq  1 ( G ,  ( NN 
X.  { A }
) ) )
109fveq1d 5543 . . . . . 6  |-  ( x  =  A  ->  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 y )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  y
) )
119fveq1d 5543 . . . . . . 7  |-  ( x  =  A  ->  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  -u y
) )
1211fveq2d 5545 . . . . . 6  |-  ( x  =  A  ->  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) )  =  ( N `  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 -u y ) ) )
1310, 12ifeq12d 3594 . . . . 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 3593 . . . 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 2302 . . . . 5  |-  ( y  =  K  ->  (
y  =  0  <->  K  =  0 ) )
16 breq2 4043 . . . . . 6  |-  ( y  =  K  ->  (
0  <  y  <->  0  <  K ) )
17 fveq2 5541 . . . . . 6  |-  ( y  =  K  ->  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  y )  =  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 K ) )
18 negeq 9060 . . . . . . . 8  |-  ( y  =  K  ->  -u y  =  -u K )
1918fveq2d 5545 . . . . . . 7  |-  ( y  =  K  ->  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u y )  =  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 -u K ) )
2019fveq2d 5545 . . . . . 6  |-  ( y  =  K  ->  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u y ) )  =  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) )
2116, 17, 20ifbieq12d 3600 . . . . 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 3598 . . . 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 2296 . . . 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 5555 . . . . . 6  |-  (GId `  G )  e.  _V
252, 24eqeltri 2366 . . . . 5  |-  U  e. 
_V
26 fvex 5555 . . . . . 6  |-  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  K )  e.  _V
27 fvex 5555 . . . . . 6  |-  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) )  e. 
_V
2826, 27ifex 3636 . . . . 5  |-  if ( 0  <  K , 
(  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) )  e.  _V
2925, 28ifex 3636 . . . 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 5999 . . 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 2348 . 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 1632    e. wcel 1696   _Vcvv 2801   ifcif 3578   {csn 3653   class class class wbr 4039    X. cxp 4703   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   0cc0 8753   1c1 8754    < clt 8883   -ucneg 9054   NNcn 9762   ZZcz 10040    seq cseq 11062   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871   ^gcgx 20873
This theorem is referenced by:  gxpval  20942  gxnval  20943  gx0  20944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-neg 9056  df-z 10041  df-seq 11063  df-gx 20878
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