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Theorem mulgfval 14846
Description: Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
Hypotheses
Ref Expression
mulgval.b  |-  B  =  ( Base `  G
)
mulgval.p  |-  .+  =  ( +g  `  G )
mulgval.o  |-  .0.  =  ( 0g `  G )
mulgval.i  |-  I  =  ( inv g `  G )
mulgval.t  |-  .x.  =  (.g
`  G )
Assertion
Ref Expression
mulgfval  |-  .x.  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )
Distinct variable groups:    x, n,  .0.    n, G, x    n, I, x    B, n, x
Allowed substitution hints:    .+ ( x, n)    .x. ( x, n)

Proof of Theorem mulgfval
Dummy variables  w  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgval.t . 2  |-  .x.  =  (.g
`  G )
2 eqidd 2405 . . . . 5  |-  ( w  =  G  ->  ZZ  =  ZZ )
3 fveq2 5687 . . . . . 6  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
4 mulgval.b . . . . . 6  |-  B  =  ( Base `  G
)
53, 4syl6eqr 2454 . . . . 5  |-  ( w  =  G  ->  ( Base `  w )  =  B )
6 fveq2 5687 . . . . . . 7  |-  ( w  =  G  ->  ( 0g `  w )  =  ( 0g `  G
) )
7 mulgval.o . . . . . . 7  |-  .0.  =  ( 0g `  G )
86, 7syl6eqr 2454 . . . . . 6  |-  ( w  =  G  ->  ( 0g `  w )  =  .0.  )
9 seqex 11280 . . . . . . . 8  |-  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) )  e.  _V
109a1i 11 . . . . . . 7  |-  ( w  =  G  ->  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  e. 
_V )
11 id 20 . . . . . . . . . 10  |-  ( s  =  seq  1 ( ( +g  `  w
) ,  ( NN 
X.  { x }
) )  ->  s  =  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) ) )
12 fveq2 5687 . . . . . . . . . . . 12  |-  ( w  =  G  ->  ( +g  `  w )  =  ( +g  `  G
) )
13 mulgval.p . . . . . . . . . . . 12  |-  .+  =  ( +g  `  G )
1412, 13syl6eqr 2454 . . . . . . . . . . 11  |-  ( w  =  G  ->  ( +g  `  w )  = 
.+  )
1514seqeq2d 11285 . . . . . . . . . 10  |-  ( w  =  G  ->  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  =  seq  1 (  .+  ,  ( NN  X.  { x } ) ) )
1611, 15sylan9eqr 2458 . . . . . . . . 9  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
s  =  seq  1
(  .+  ,  ( NN  X.  { x }
) ) )
1716fveq1d 5689 . . . . . . . 8  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( s `  n
)  =  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 n ) )
18 simpl 444 . . . . . . . . . . 11  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  ->  w  =  G )
1918fveq2d 5691 . . . . . . . . . 10  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( inv g `  w )  =  ( inv g `  G
) )
20 mulgval.i . . . . . . . . . 10  |-  I  =  ( inv g `  G )
2119, 20syl6eqr 2454 . . . . . . . . 9  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( inv g `  w )  =  I )
2216fveq1d 5689 . . . . . . . . 9  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( s `  -u n
)  =  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) )
2321, 22fveq12d 5693 . . . . . . . 8  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( ( inv g `  w ) `  (
s `  -u n ) )  =  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) )
2417, 23ifeq12d 3715 . . . . . . 7  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  ->  if ( 0  <  n ,  ( s `  n ) ,  ( ( inv g `  w ) `  (
s `  -u n ) ) )  =  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) )
2510, 24csbied 3253 . . . . . 6  |-  ( w  =  G  ->  [_  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  / 
s ]_ if ( 0  <  n ,  ( s `  n ) ,  ( ( inv g `  w ) `
 ( s `  -u n ) ) )  =  if ( 0  <  n ,  (  seq  1 (  .+  ,  ( NN  X.  { x } ) ) `  n ) ,  ( I `  (  seq  1 (  .+  ,  ( NN  X.  { x } ) ) `  -u n
) ) ) )
268, 25ifeq12d 3715 . . . . 5  |-  ( w  =  G  ->  if ( n  =  0 ,  ( 0g `  w ) ,  [_  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  / 
s ]_ if ( 0  <  n ,  ( s `  n ) ,  ( ( inv g `  w ) `
 ( s `  -u n ) ) ) )  =  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )
272, 5, 26mpt2eq123dv 6095 . . . 4  |-  ( w  =  G  ->  (
n  e.  ZZ ,  x  e.  ( Base `  w )  |->  if ( n  =  0 ,  ( 0g `  w
) ,  [_  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  / 
s ]_ if ( 0  <  n ,  ( s `  n ) ,  ( ( inv g `  w ) `
 ( s `  -u n ) ) ) ) )  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) ) )
28 df-mulg 14770 . . . 4  |- .g  =  (
w  e.  _V  |->  ( n  e.  ZZ ,  x  e.  ( Base `  w )  |->  if ( n  =  0 ,  ( 0g `  w
) ,  [_  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  / 
s ]_ if ( 0  <  n ,  ( s `  n ) ,  ( ( inv g `  w ) `
 ( s `  -u n ) ) ) ) ) )
29 zex 10247 . . . . 5  |-  ZZ  e.  _V
30 fvex 5701 . . . . . 6  |-  ( Base `  G )  e.  _V
314, 30eqeltri 2474 . . . . 5  |-  B  e. 
_V
3229, 31mpt2ex 6384 . . . 4  |-  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )  e.  _V
3327, 28, 32fvmpt 5765 . . 3  |-  ( G  e.  _V  ->  (.g `  G )  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) ) )
34 fvprc 5681 . . . 4  |-  ( -.  G  e.  _V  ->  (.g `  G )  =  (/) )
35 eqid 2404 . . . . . . 7  |-  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )
36 fvex 5701 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
377, 36eqeltri 2474 . . . . . . . 8  |-  .0.  e.  _V
38 fvex 5701 . . . . . . . . 9  |-  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 n )  e. 
_V
39 fvex 5701 . . . . . . . . 9  |-  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) )  e.  _V
4038, 39ifex 3757 . . . . . . . 8  |-  if ( 0  <  n ,  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) )  e. 
_V
4137, 40ifex 3757 . . . . . . 7  |-  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) )  e.  _V
4235, 41fnmpt2i 6379 . . . . . 6  |-  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )  Fn  ( ZZ 
X.  B )
43 fvprc 5681 . . . . . . . . . 10  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
444, 43syl5eq 2448 . . . . . . . . 9  |-  ( -.  G  e.  _V  ->  B  =  (/) )
4544xpeq2d 4861 . . . . . . . 8  |-  ( -.  G  e.  _V  ->  ( ZZ  X.  B )  =  ( ZZ  X.  (/) ) )
46 xp0 5250 . . . . . . . 8  |-  ( ZZ 
X.  (/) )  =  (/)
4745, 46syl6eq 2452 . . . . . . 7  |-  ( -.  G  e.  _V  ->  ( ZZ  X.  B )  =  (/) )
4847fneq2d 5496 . . . . . 6  |-  ( -.  G  e.  _V  ->  ( ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )  Fn  ( ZZ 
X.  B )  <->  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  < 
n ,  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 n ) ,  ( I `  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) ) ) ) )  Fn  (/) ) )
4942, 48mpbii 203 . . . . 5  |-  ( -.  G  e.  _V  ->  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )  Fn  (/) )
50 fn0 5523 . . . . 5  |-  ( ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )  Fn  (/)  <->  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  < 
n ,  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 n ) ,  ( I `  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) ) ) ) )  =  (/) )
5149, 50sylib 189 . . . 4  |-  ( -.  G  e.  _V  ->  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )  =  (/) )
5234, 51eqtr4d 2439 . . 3  |-  ( -.  G  e.  _V  ->  (.g `  G )  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) ) )
5333, 52pm2.61i 158 . 2  |-  (.g `  G
)  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )
541, 53eqtri 2424 1  |-  .x.  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   [_csb 3211   (/)c0 3588   ifcif 3699   {csn 3774   class class class wbr 4172    X. cxp 4835    Fn wfn 5408   ` cfv 5413    e. cmpt2 6042   0cc0 8946   1c1 8947    < clt 9076   -ucneg 9248   NNcn 9956   ZZcz 10238    seq cseq 11278   Basecbs 13424   +g cplusg 13484   0gc0g 13678   inv gcminusg 14641  .gcmg 14644
This theorem is referenced by:  mulgval  14847  mulgfn  14848  mulgpropd  14878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-neg 9250  df-z 10239  df-seq 11279  df-mulg 14770
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