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Theorem mulgval 14884
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 2443 . . 3  |-  ( ( n  =  N  /\  x  =  X )  ->  ( n  =  0  <-> 
N  =  0 ) )
31breq2d 4216 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  ( 0  <  n  <->  0  <  N ) )
4 simpr 448 . . . . . . . . 9  |-  ( ( n  =  N  /\  x  =  X )  ->  x  =  X )
54sneqd 3819 . . . . . . . 8  |-  ( ( n  =  N  /\  x  =  X )  ->  { x }  =  { X } )
65xpeq2d 4894 . . . . . . 7  |-  ( ( n  =  N  /\  x  =  X )  ->  ( NN  X.  {
x } )  =  ( NN  X.  { X } ) )
76seqeq3d 11323 . . . . . 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 2485 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  seq  1 (  .+  ,  ( NN  X.  { x } ) )  =  S )
109, 1fveq12d 5726 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  n
)  =  ( S `
 N ) )
111negeqd 9292 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  -> 
-u n  =  -u N )
129, 11fveq12d 5726 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  (  seq  1 ( 
.+  ,  ( NN 
X.  { x }
) ) `  -u n
)  =  ( S `
 -u N ) )
1312fveq2d 5724 . . . 4  |-  ( ( n  =  N  /\  x  =  X )  ->  ( I `  (  seq  1 (  .+  , 
( NN  X.  {
x } ) ) `
 -u n ) )  =  ( I `  ( S `  -u N
) ) )
143, 10, 13ifbieq12d 3753 . . 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 3751 . 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 14883 . 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 5734 . . . 4  |-  ( 0g
`  G )  e. 
_V
2318, 22eqeltri 2505 . . 3  |-  .0.  e.  _V
24 fvex 5734 . . . 4  |-  ( S `
 N )  e. 
_V
25 fvex 5734 . . . 4  |-  ( I `
 ( S `  -u N ) )  e. 
_V
2624, 25ifex 3789 . . 3  |-  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) )  e.  _V
2723, 26ifex 3789 . 2  |-  if ( N  =  0 ,  .0.  ,  if ( 0  <  N , 
( S `  N
) ,  ( I `
 ( S `  -u N ) ) ) )  e.  _V
2815, 21, 27ovmpt2a 6196 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 1652    e. wcel 1725   _Vcvv 2948   ifcif 3731   {csn 3806   class class class wbr 4204    X. cxp 4868   ` cfv 5446  (class class class)co 6073   0cc0 8982   1c1 8983    < clt 9112   -ucneg 9284   NNcn 9992   ZZcz 10274    seq cseq 11315   Basecbs 13461   +g cplusg 13521   0gc0g 13715   inv gcminusg 14678  .gcmg 14681
This theorem is referenced by:  mulg0  14887  mulgnn  14888  mulgnegnn  14892  subgmulg  14950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-neg 9286  df-z 10275  df-seq 11316  df-mulg 14807
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