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Theorem mulgfval 14896
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 2439 . . . . 5  |-  ( w  =  G  ->  ZZ  =  ZZ )
3 fveq2 5731 . . . . . 6  |-  ( w  =  G  ->  ( Base `  w )  =  ( Base `  G
) )
4 mulgval.b . . . . . 6  |-  B  =  ( Base `  G
)
53, 4syl6eqr 2488 . . . . 5  |-  ( w  =  G  ->  ( Base `  w )  =  B )
6 fveq2 5731 . . . . . . 7  |-  ( w  =  G  ->  ( 0g `  w )  =  ( 0g `  G
) )
7 mulgval.o . . . . . . 7  |-  .0.  =  ( 0g `  G )
86, 7syl6eqr 2488 . . . . . 6  |-  ( w  =  G  ->  ( 0g `  w )  =  .0.  )
9 seqex 11330 . . . . . . . 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 21 . . . . . . . . . 10  |-  ( s  =  seq  1 ( ( +g  `  w
) ,  ( NN 
X.  { x }
) )  ->  s  =  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) ) )
12 fveq2 5731 . . . . . . . . . . . 12  |-  ( w  =  G  ->  ( +g  `  w )  =  ( +g  `  G
) )
13 mulgval.p . . . . . . . . . . . 12  |-  .+  =  ( +g  `  G )
1412, 13syl6eqr 2488 . . . . . . . . . . 11  |-  ( w  =  G  ->  ( +g  `  w )  = 
.+  )
1514seqeq2d 11335 . . . . . . . . . 10  |-  ( w  =  G  ->  seq  1 ( ( +g  `  w ) ,  ( NN  X.  { x } ) )  =  seq  1 (  .+  ,  ( NN  X.  { x } ) ) )
1611, 15sylan9eqr 2492 . . . . . . . . 9  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
s  =  seq  1
(  .+  ,  ( NN  X.  { x }
) ) )
1716fveq1d 5733 . . . . . . . 8  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( s `  n
)  =  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 n ) )
18 simpl 445 . . . . . . . . . . 11  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  ->  w  =  G )
1918fveq2d 5735 . . . . . . . . . 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 2488 . . . . . . . . 9  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( inv g `  w )  =  I )
2216fveq1d 5733 . . . . . . . . 9  |-  ( ( w  =  G  /\  s  =  seq  1
( ( +g  `  w
) ,  ( NN 
X.  { x }
) ) )  -> 
( s `  -u n
)  =  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) )
2321, 22fveq12d 5737 . . . . . . . 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 3757 . . . . . . 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 3295 . . . . . 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 3757 . . . . 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 6139 . . . 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 14820 . . . 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 10296 . . . . 5  |-  ZZ  e.  _V
30 fvex 5745 . . . . . 6  |-  ( Base `  G )  e.  _V
314, 30eqeltri 2508 . . . . 5  |-  B  e. 
_V
3229, 31mpt2ex 6428 . . . 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 5809 . . 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 5725 . . . 4  |-  ( -.  G  e.  _V  ->  (.g `  G )  =  (/) )
35 eqid 2438 . . . . . . 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 5745 . . . . . . . . 9  |-  ( 0g
`  G )  e. 
_V
377, 36eqeltri 2508 . . . . . . . 8  |-  .0.  e.  _V
38 fvex 5745 . . . . . . . . 9  |-  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 n )  e. 
_V
39 fvex 5745 . . . . . . . . 9  |-  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) )  e.  _V
4038, 39ifex 3799 . . . . . . . 8  |-  if ( 0  <  n ,  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) )  e. 
_V
4137, 40ifex 3799 . . . . . . 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 6423 . . . . . 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 5725 . . . . . . . . . 10  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
444, 43syl5eq 2482 . . . . . . . . 9  |-  ( -.  G  e.  _V  ->  B  =  (/) )
4544xpeq2d 4905 . . . . . . . 8  |-  ( -.  G  e.  _V  ->  ( ZZ  X.  B )  =  ( ZZ  X.  (/) ) )
46 xp0 5294 . . . . . . . 8  |-  ( ZZ 
X.  (/) )  =  (/)
4745, 46syl6eq 2486 . . . . . . 7  |-  ( -.  G  e.  _V  ->  ( ZZ  X.  B )  =  (/) )
4847fneq2d 5540 . . . . . 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 204 . . . . 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 5567 . . . . 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 190 . . . 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 2473 . . 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 159 . 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 2458 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   [_csb 3253   (/)c0 3630   ifcif 3741   {csn 3816   class class class wbr 4215    X. cxp 4879    Fn wfn 5452   ` cfv 5457    e. cmpt2 6086   0cc0 8995   1c1 8996    < clt 9125   -ucneg 9297   NNcn 10005   ZZcz 10287    seq cseq 11328   Basecbs 13474   +g cplusg 13534   0gc0g 13728   inv gcminusg 14691  .gcmg 14694
This theorem is referenced by:  mulgval  14897  mulgfn  14898  mulgpropd  14928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-neg 9299  df-z 10288  df-seq 11329  df-mulg 14820
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