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Theorem mulgfval 14568
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 2284 . . . . 5  |-  ( w  =  G  ->  ZZ  =  ZZ )
3 fveq2 5525 . . . . . 6  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
4 mulgval.b . . . . . 6  |-  B  =  ( Base `  G
)
53, 4syl6eqr 2333 . . . . 5  |-  ( w  =  G  ->  ( Base `  w )  =  B )
6 fveq2 5525 . . . . . . 7  |-  ( w  =  G  ->  ( 0g `  w )  =  ( 0g `  G
) )
7 mulgval.o . . . . . . 7  |-  .0.  =  ( 0g `  G )
86, 7syl6eqr 2333 . . . . . 6  |-  ( w  =  G  ->  ( 0g `  w )  =  .0.  )
9 seqex 11048 . . . . . . . 8  |-  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) )  e.  _V
109a1i 10 . . . . . . 7  |-  ( w  =  G  ->  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  e. 
_V )
11 id 19 . . . . . . . . . 10  |-  ( s  =  seq  1 ( ( +g  `  w
) ,  ( NN 
X.  { x }
) )  ->  s  =  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) ) )
12 fveq2 5525 . . . . . . . . . . . 12  |-  ( w  =  G  ->  ( +g  `  w )  =  ( +g  `  G
) )
13 mulgval.p . . . . . . . . . . . 12  |-  .+  =  ( +g  `  G )
1412, 13syl6eqr 2333 . . . . . . . . . . 11  |-  ( w  =  G  ->  ( +g  `  w )  = 
.+  )
1514seqeq2d 11053 . . . . . . . . . 10  |-  ( w  =  G  ->  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  =  seq  1 (  .+  ,  ( NN  X.  { x } ) ) )
1611, 15sylan9eqr 2337 . . . . . . . . 9  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
s  =  seq  1
(  .+  ,  ( NN  X.  { x }
) ) )
1716fveq1d 5527 . . . . . . . 8  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( s `  n
)  =  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 n ) )
18 simpl 443 . . . . . . . . . . 11  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  ->  w  =  G )
1918fveq2d 5529 . . . . . . . . . 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 2333 . . . . . . . . 9  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( inv g `  w )  =  I )
2216fveq1d 5527 . . . . . . . . 9  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( s `  -u n
)  =  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) )
2321, 22fveq12d 5531 . . . . . . . 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 3581 . . . . . . 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 3123 . . . . . 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 3581 . . . . 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 5910 . . . 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 14492 . . . 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 10033 . . . . 5  |-  ZZ  e.  _V
30 fvex 5539 . . . . . 6  |-  ( Base `  G )  e.  _V
314, 30eqeltri 2353 . . . . 5  |-  B  e. 
_V
3229, 31mpt2ex 6198 . . . 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 5602 . . 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 5519 . . . 4  |-  ( -.  G  e.  _V  ->  (.g `  G )  =  (/) )
35 eqid 2283 . . . . . . 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 5539 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
377, 36eqeltri 2353 . . . . . . . 8  |-  .0.  e.  _V
38 fvex 5539 . . . . . . . . 9  |-  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 n )  e. 
_V
39 fvex 5539 . . . . . . . . 9  |-  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) )  e.  _V
4038, 39ifex 3623 . . . . . . . 8  |-  if ( 0  <  n ,  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) )  e. 
_V
4137, 40ifex 3623 . . . . . . 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 6193 . . . . . 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 5519 . . . . . . . . . 10  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
444, 43syl5eq 2327 . . . . . . . . 9  |-  ( -.  G  e.  _V  ->  B  =  (/) )
4544xpeq2d 4713 . . . . . . . 8  |-  ( -.  G  e.  _V  ->  ( ZZ  X.  B )  =  ( ZZ  X.  (/) ) )
46 xp0 5098 . . . . . . . 8  |-  ( ZZ 
X.  (/) )  =  (/)
4745, 46syl6eq 2331 . . . . . . 7  |-  ( -.  G  e.  _V  ->  ( ZZ  X.  B )  =  (/) )
4847fneq2d 5336 . . . . . 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 202 . . . . 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 5363 . . . . 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 188 . . . 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 2318 . . 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 156 . 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 2303 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081   (/)c0 3455   ifcif 3565   {csn 3640   class class class wbr 4023    X. cxp 4687    Fn wfn 5250   ` cfv 5255    e. cmpt2 5860   0cc0 8737   1c1 8738    < clt 8867   -ucneg 9038   NNcn 9746   ZZcz 10024    seq cseq 11046   Basecbs 13148   +g cplusg 13208   0gc0g 13400   inv gcminusg 14363  .gcmg 14366
This theorem is referenced by:  mulgval  14569  mulgfn  14570  mulgpropd  14600
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-neg 9040  df-z 10025  df-seq 11047  df-mulg 14492
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