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

Theorem gxval 23917
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 23916 . . . 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 6220 . . 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 3998 . . . . . . . . 9  |-  ( x  =  A  ->  { x }  =  { A } )
87xpeq2d 4975 . . . . . . . 8  |-  ( x  =  A  ->  ( NN  X.  { x }
)  =  ( NN 
X.  { A }
) )
98seqeq3d 11934 . . . . . . 7  |-  ( x  =  A  ->  seq 1 ( G , 
( NN  X.  {
x } ) )  =  seq 1 ( G ,  ( NN 
X.  { A }
) ) )
109fveq1d 5804 . . . . . 6  |-  ( x  =  A  ->  (  seq 1 ( G , 
( NN  X.  {
x } ) ) `
 y )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  y
) )
119fveq1d 5804 . . . . . . 7  |-  ( x  =  A  ->  (  seq 1 ( G , 
( NN  X.  {
x } ) ) `
 -u y )  =  (  seq 1 ( G ,  ( NN 
X.  { A }
) ) `  -u y
) )
1211fveq2d 5806 . . . . . 6  |-  ( x  =  A  ->  ( N `  (  seq 1 ( G , 
( NN  X.  {
x } ) ) `
 -u y ) )  =  ( N `  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 -u y ) ) )
1310, 12ifeq12d 3920 . . . . 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 3919 . . . 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 2458 . . . . 5  |-  ( y  =  K  ->  (
y  =  0  <->  K  =  0 ) )
16 breq2 4407 . . . . . 6  |-  ( y  =  K  ->  (
0  <  y  <->  0  <  K ) )
17 fveq2 5802 . . . . . 6  |-  ( y  =  K  ->  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  y )  =  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 K ) )
18 negeq 9716 . . . . . . . 8  |-  ( y  =  K  ->  -u y  =  -u K )
1918fveq2d 5806 . . . . . . 7  |-  ( y  =  K  ->  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  -u y )  =  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 -u K ) )
2019fveq2d 5806 . . . . . 6  |-  ( y  =  K  ->  ( N `  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  -u y ) )  =  ( N `  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) )
2116, 17, 20ifbieq12d 3927 . . . . 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 3925 . . . 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 2454 . . . 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 5812 . . . . . 6  |-  (GId `  G )  e.  _V
252, 24eqeltri 2538 . . . . 5  |-  U  e. 
_V
26 fvex 5812 . . . . . 6  |-  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  K )  e.  _V
27 fvex 5812 . . . . . 6  |-  ( N `
 (  seq 1
( G ,  ( NN  X.  { A } ) ) `  -u K ) )  e. 
_V
2826, 27ifex 3969 . . . . 5  |-  if ( 0  <  K , 
(  seq 1 ( G ,  ( NN  X.  { A } ) ) `
 K ) ,  ( N `  (  seq 1 ( G , 
( NN  X.  { A } ) ) `  -u K ) ) )  e.  _V
2925, 28ifex 3969 . . . 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 6339 . . 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 2515 . 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 1184 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 965    = wceq 1370    e. wcel 1758   _Vcvv 3078   ifcif 3902   {csn 3988   class class class wbr 4403    X. cxp 4949   ran crn 4952   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   0cc0 9396   1c1 9397    < clt 9532   -ucneg 9710   NNcn 10436   ZZcz 10760    seqcseq 11926   GrpOpcgr 23845  GIdcgi 23846   invcgn 23847   ^gcgx 23849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-neg 9712  df-z 10761  df-seq 11927  df-gx 23854
This theorem is referenced by:  gxpval  23918  gxnval  23919  gx0  23920
  Copyright terms: Public domain W3C validator