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Theorem mulgval 16016
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  =  ( invg `  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 457 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  n  =  N )
21eqeq1d 2469 . . 3  |-  ( ( n  =  N  /\  x  =  X )  ->  ( n  =  0  <-> 
N  =  0 ) )
31breq2d 4465 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  ( 0  <  n  <->  0  <  N ) )
4 simpr 461 . . . . . . . . 9  |-  ( ( n  =  N  /\  x  =  X )  ->  x  =  X )
54sneqd 4045 . . . . . . . 8  |-  ( ( n  =  N  /\  x  =  X )  ->  { x }  =  { X } )
65xpeq2d 5029 . . . . . . 7  |-  ( ( n  =  N  /\  x  =  X )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { X } ) )
76seqeq3d 12095 . . . . . 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 2526 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  seq 1 (  .+  ,  ( NN  X.  { x } ) )  =  S )
109, 1fveq12d 5878 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq 1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
)  =  ( S `
 N ) )
111negeqd 9826 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  -> 
-u n  =  -u N )
129, 11fveq12d 5878 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq 1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  -u n
)  =  ( S `
 -u N ) )
1312fveq2d 5876 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  ( I `  (  seq 1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) )  =  ( I `  ( S `  -u N
) ) )
143, 10, 13ifbieq12d 3972 . . 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 3970 . 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  =  ( invg `  G )
20 mulgval.t . . 3  |-  .x.  =  (.g
`  G )
2116, 17, 18, 19, 20mulgfval 16015 . 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 5882 . . . 4  |-  ( 0g
`  G )  e. 
_V
2318, 22eqeltri 2551 . . 3  |-  .0.  e.  _V
24 fvex 5882 . . . 4  |-  ( S `
 N )  e. 
_V
25 fvex 5882 . . . 4  |-  ( I `
 ( S `  -u N ) )  e. 
_V
2624, 25ifex 4014 . . 3  |-  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) )  e.  _V
2723, 26ifex 4014 . 2  |-  if ( N  =  0 ,  .0.  ,  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) )  e.  _V
2815, 21, 27ovmpt2a 6428 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   ifcif 3945   {csn 4033   class class class wbr 4453    X. cxp 5003   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505    < clt 9640   -ucneg 9818   NNcn 10548   ZZcz 10876    seqcseq 12087   Basecbs 14507   +g cplusg 14572   0gc0g 14712   invgcminusg 15926  .gcmg 15928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-neg 9820  df-z 10877  df-seq 12088  df-mulg 15932
This theorem is referenced by:  mulg0  16019  mulgnn  16020  mulgnegnn  16024  subgmulg  16087
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