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Theorem mulgval 14569
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 443 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  n  =  N )
21eqeq1d 2291 . . 3  |-  ( ( n  =  N  /\  x  =  X )  ->  ( n  =  0  <-> 
N  =  0 ) )
31breq2d 4035 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  ( 0  <  n  <->  0  <  N ) )
4 simpr 447 . . . . . . . . 9  |-  ( ( n  =  N  /\  x  =  X )  ->  x  =  X )
54sneqd 3653 . . . . . . . 8  |-  ( ( n  =  N  /\  x  =  X )  ->  { x }  =  { X } )
65xpeq2d 4713 . . . . . . 7  |-  ( ( n  =  N  /\  x  =  X )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { X } ) )
76seqeq3d 11054 . . . . . 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 2333 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  seq  1 (  .+  ,  ( NN  X.  { x } ) )  =  S )
109, 1fveq12d 5531 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
)  =  ( S `
 N ) )
111negeqd 9046 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  -> 
-u n  =  -u N )
129, 11fveq12d 5531 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  -u n
)  =  ( S `
 -u N ) )
1312fveq2d 5529 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  ( I `  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) )  =  ( I `  ( S `  -u N
) ) )
143, 10, 13ifbieq12d 3587 . . 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 3585 . 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 14568 . 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 5539 . . . 4  |-  ( 0g
`  G )  e. 
_V
2318, 22eqeltri 2353 . . 3  |-  .0.  e.  _V
24 fvex 5539 . . . 4  |-  ( S `
 N )  e. 
_V
25 fvex 5539 . . . 4  |-  ( I `
 ( S `  -u N ) )  e. 
_V
2624, 25ifex 3623 . . 3  |-  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) )  e.  _V
2723, 26ifex 3623 . 2  |-  if ( N  =  0 ,  .0.  ,  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) )  e.  _V
2815, 21, 27ovmpt2a 5978 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   ifcif 3565   {csn 3640   class class class wbr 4023    X. cxp 4687   ` cfv 5255  (class class class)co 5858   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:  mulg0  14572  mulgnn  14573  mulgnegnn  14577  subgmulg  14635
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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