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Theorem mulgval 15629
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 2451 . . 3  |-  ( ( n  =  N  /\  x  =  X )  ->  ( n  =  0  <-> 
N  =  0 ) )
31breq2d 4304 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  ( 0  <  n  <->  0  <  N ) )
4 simpr 461 . . . . . . . . 9  |-  ( ( n  =  N  /\  x  =  X )  ->  x  =  X )
54sneqd 3889 . . . . . . . 8  |-  ( ( n  =  N  /\  x  =  X )  ->  { x }  =  { X } )
65xpeq2d 4864 . . . . . . 7  |-  ( ( n  =  N  /\  x  =  X )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { X } ) )
76seqeq3d 11814 . . . . . 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 2493 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  seq 1 (  .+  ,  ( NN  X.  { x } ) )  =  S )
109, 1fveq12d 5697 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq 1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
)  =  ( S `
 N ) )
111negeqd 9604 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  -> 
-u n  =  -u N )
129, 11fveq12d 5697 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq 1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  -u n
)  =  ( S `
 -u N ) )
1312fveq2d 5695 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  ( I `  (  seq 1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) )  =  ( I `  ( S `  -u N
) ) )
143, 10, 13ifbieq12d 3816 . . 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 3814 . 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 15628 . 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 5701 . . . 4  |-  ( 0g
`  G )  e. 
_V
2318, 22eqeltri 2513 . . 3  |-  .0.  e.  _V
24 fvex 5701 . . . 4  |-  ( S `
 N )  e. 
_V
25 fvex 5701 . . . 4  |-  ( I `
 ( S `  -u N ) )  e. 
_V
2624, 25ifex 3858 . . 3  |-  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) )  e.  _V
2723, 26ifex 3858 . 2  |-  if ( N  =  0 ,  .0.  ,  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) )  e.  _V
2815, 21, 27ovmpt2a 6221 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 1369    e. wcel 1756   _Vcvv 2972   ifcif 3791   {csn 3877   class class class wbr 4292    X. cxp 4838   ` cfv 5418  (class class class)co 6091   0cc0 9282   1c1 9283    < clt 9418   -ucneg 9596   NNcn 10322   ZZcz 10646    seqcseq 11806   Basecbs 14174   +g cplusg 14238   0gc0g 14378   invgcminusg 15411  .gcmg 15414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-neg 9598  df-z 10647  df-seq 11807  df-mulg 15548
This theorem is referenced by:  mulg0  15632  mulgnn  15633  mulgnegnn  15637  subgmulg  15695
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