MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gxval Structured version   Unicode version

Theorem gxval 21846
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 21845 . . . 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 6098 . . 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 3825 . . . . . . . . 9  |-  ( x  =  A  ->  { x }  =  { A } )
87xpeq2d 4902 . . . . . . . 8  |-  ( x  =  A  ->  ( NN  X.  { x }
)  =  ( NN 
X.  { A }
) )
98seqeq3d 11331 . . . . . . 7  |-  ( x  =  A  ->  seq  1 ( G , 
( NN  X.  {
x } ) )  =  seq  1 ( G ,  ( NN 
X.  { A }
) ) )
109fveq1d 5730 . . . . . 6  |-  ( x  =  A  ->  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 y )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  y
) )
119fveq1d 5730 . . . . . . 7  |-  ( x  =  A  ->  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y )  =  (  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  -u y
) )
1211fveq2d 5732 . . . . . 6  |-  ( x  =  A  ->  ( N `  (  seq  1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) )  =  ( N `  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 -u y ) ) )
1310, 12ifeq12d 3755 . . . . 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 3754 . . . 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 2442 . . . . 5  |-  ( y  =  K  ->  (
y  =  0  <->  K  =  0 ) )
16 breq2 4216 . . . . . 6  |-  ( y  =  K  ->  (
0  <  y  <->  0  <  K ) )
17 fveq2 5728 . . . . . 6  |-  ( y  =  K  ->  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  y )  =  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 K ) )
18 negeq 9298 . . . . . . . 8  |-  ( y  =  K  ->  -u y  =  -u K )
1918fveq2d 5732 . . . . . . 7  |-  ( y  =  K  ->  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u y )  =  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
 -u K ) )
2019fveq2d 5732 . . . . . 6  |-  ( y  =  K  ->  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u y ) )  =  ( N `  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) )
2116, 17, 20ifbieq12d 3761 . . . . 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 3759 . . . 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 2436 . . . 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 5742 . . . . . 6  |-  (GId `  G )  e.  _V
252, 24eqeltri 2506 . . . . 5  |-  U  e. 
_V
26 fvex 5742 . . . . . 6  |-  (  seq  1 ( G , 
( NN  X.  { A } ) ) `  K )  e.  _V
27 fvex 5742 . . . . . 6  |-  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) )  e. 
_V
2826, 27ifex 3797 . . . . 5  |-  if ( 0  <  K , 
(  seq  1 ( G ,  ( NN 
X.  { A }
) ) `  K
) ,  ( N `
 (  seq  1
( G ,  ( NN  X.  { A } ) ) `  -u K ) ) )  e.  _V
2925, 28ifex 3797 . . . 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 6209 . . 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 2488 . 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 1149 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956   ifcif 3739   {csn 3814   class class class wbr 4212    X. cxp 4876   ran crn 4879   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   0cc0 8990   1c1 8991    < clt 9120   -ucneg 9292   NNcn 10000   ZZcz 10282    seq cseq 11323   GrpOpcgr 21774  GIdcgi 21775   invcgn 21776   ^gcgx 21778
This theorem is referenced by:  gxpval  21847  gxnval  21848  gx0  21849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-neg 9294  df-z 10283  df-seq 11324  df-gx 21783
  Copyright terms: Public domain W3C validator