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Theorem mulgval 14819
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 )
mulgval.s  |-  S  =  seq  1 (  .+  ,  ( NN  X.  { X } ) )
Assertion
Ref Expression
mulgval  |-  ( ( N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X
)  =  if ( N  =  0 ,  .0.  ,  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) ) )

Proof of Theorem mulgval
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  n  =  N )
21eqeq1d 2395 . . 3  |-  ( ( n  =  N  /\  x  =  X )  ->  ( n  =  0  <-> 
N  =  0 ) )
31breq2d 4165 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  ( 0  <  n  <->  0  <  N ) )
4 simpr 448 . . . . . . . . 9  |-  ( ( n  =  N  /\  x  =  X )  ->  x  =  X )
54sneqd 3770 . . . . . . . 8  |-  ( ( n  =  N  /\  x  =  X )  ->  { x }  =  { X } )
65xpeq2d 4842 . . . . . . 7  |-  ( ( n  =  N  /\  x  =  X )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { X } ) )
76seqeq3d 11258 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  ->  seq  1 (  .+  ,  ( NN  X.  { x } ) )  =  seq  1
(  .+  ,  ( NN  X.  { X }
) ) )
8 mulgval.s . . . . . 6  |-  S  =  seq  1 (  .+  ,  ( NN  X.  { X } ) )
97, 8syl6eqr 2437 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  seq  1 (  .+  ,  ( NN  X.  { x } ) )  =  S )
109, 1fveq12d 5674 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
)  =  ( S `
 N ) )
111negeqd 9232 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  -> 
-u n  =  -u N )
129, 11fveq12d 5674 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  -u n
)  =  ( S `
 -u N ) )
1312fveq2d 5672 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  ( I `  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) )  =  ( I `  ( S `  -u N
) ) )
143, 10, 13ifbieq12d 3704 . . 3  |-  ( ( n  =  N  /\  x  =  X )  ->  if ( 0  < 
n ,  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 n ) ,  ( I `  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) ) )  =  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) )
152, 14ifbieq2d 3702 . 2  |-  ( ( n  =  N  /\  x  =  X )  ->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  n
) ,  ( I `
 (  seq  1
(  .+  ,  ( NN  X.  { x }
) ) `  -u n
) ) ) )  =  if ( N  =  0 ,  .0.  ,  if ( 0  < 
N ,  ( S `
 N ) ,  ( I `  ( S `  -u N ) ) ) ) )
16 mulgval.b . . 3  |-  B  =  ( Base `  G
)
17 mulgval.p . . 3  |-  .+  =  ( +g  `  G )
18 mulgval.o . . 3  |-  .0.  =  ( 0g `  G )
19 mulgval.i . . 3  |-  I  =  ( inv g `  G )
20 mulgval.t . . 3  |-  .x.  =  (.g
`  G )
2116, 17, 18, 19, 20mulgfval 14818 . 2  |-  .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
) ) ) ) )
22 fvex 5682 . . . 4  |-  ( 0g
`  G )  e. 
_V
2318, 22eqeltri 2457 . . 3  |-  .0.  e.  _V
24 fvex 5682 . . . 4  |-  ( S `
 N )  e. 
_V
25 fvex 5682 . . . 4  |-  ( I `
 ( S `  -u N ) )  e. 
_V
2624, 25ifex 3740 . . 3  |-  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) )  e.  _V
2723, 26ifex 3740 . 2  |-  if ( N  =  0 ,  .0.  ,  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) )  e.  _V
2815, 21, 27ovmpt2a 6143 1  |-  ( ( N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X
)  =  if ( N  =  0 ,  .0.  ,  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   ifcif 3682   {csn 3757   class class class wbr 4153    X. cxp 4816   ` cfv 5394  (class class class)co 6020   0cc0 8923   1c1 8924    < clt 9053   -ucneg 9224   NNcn 9932   ZZcz 10214    seq cseq 11250   Basecbs 13396   +g cplusg 13456   0gc0g 13650   inv gcminusg 14613  .gcmg 14616
This theorem is referenced by:  mulg0  14822  mulgnn  14823  mulgnegnn  14827  subgmulg  14885
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-recs 6569  df-rdg 6604  df-neg 9226  df-z 10215  df-seq 11251  df-mulg 14742
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