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Theorem mulgnegnn 14827
Description: Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
Hypotheses
Ref Expression
mulg1.b  |-  B  =  ( Base `  G
)
mulg1.m  |-  .x.  =  (.g
`  G )
mulgnegnn.i  |-  I  =  ( inv g `  G )
Assertion
Ref Expression
mulgnegnn  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )

Proof of Theorem mulgnegnn
StepHypRef Expression
1 nncn 9940 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  CC )
21negnegd 9334 . . . . 5  |-  ( N  e.  NN  ->  -u -u N  =  N )
32adantr 452 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  B )  -> 
-u -u N  =  N )
43fveq2d 5672 . . 3  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  (  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u -u N
)  =  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) )
54fveq2d 5672 . 2  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u -u N ) )  =  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) ) )
6 nnnegz 10217 . . . 4  |-  ( N  e.  NN  ->  -u N  e.  ZZ )
7 mulg1.b . . . . 5  |-  B  =  ( Base `  G
)
8 eqid 2387 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
9 eqid 2387 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
10 mulgnegnn.i . . . . 5  |-  I  =  ( inv g `  G )
11 mulg1.m . . . . 5  |-  .x.  =  (.g
`  G )
12 eqid 2387 . . . . 5  |-  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
137, 8, 9, 10, 11, 12mulgval 14819 . . . 4  |-  ( (
-u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  if ( -u N  =  0 ,  ( 0g
`  G ) ,  if ( 0  <  -u N ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ,  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u -u N ) ) ) ) )
146, 13sylan 458 . . 3  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  if ( -u N  =  0 ,  ( 0g
`  G ) ,  if ( 0  <  -u N ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ,  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u -u N ) ) ) ) )
15 nnne0 9964 . . . . . . 7  |-  ( N  e.  NN  ->  N  =/=  0 )
16 negeq0 9287 . . . . . . . . 9  |-  ( N  e.  CC  ->  ( N  =  0  <->  -u N  =  0 ) )
1716necon3abid 2583 . . . . . . . 8  |-  ( N  e.  CC  ->  ( N  =/=  0  <->  -.  -u N  =  0 ) )
181, 17syl 16 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  =/=  0  <->  -.  -u N  =  0 ) )
1915, 18mpbid 202 . . . . . 6  |-  ( N  e.  NN  ->  -.  -u N  =  0 )
20 iffalse 3689 . . . . . 6  |-  ( -.  -u N  =  0  ->  if ( -u N  =  0 ,  ( 0g `  G ) ,  if ( 0  <  -u N ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 -u N ) ,  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u -u N ) ) ) )  =  if ( 0  <  -u N ,  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u N
) ,  ( I `
 (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u -u N
) ) ) )
2119, 20syl 16 . . . . 5  |-  ( N  e.  NN  ->  if ( -u N  =  0 ,  ( 0g `  G ) ,  if ( 0  <  -u N ,  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u N
) ,  ( I `
 (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u -u N
) ) ) )  =  if ( 0  <  -u N ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 -u N ) ,  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u -u N ) ) ) )
22 nnre 9939 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  RR )
2322renegcld 9396 . . . . . . 7  |-  ( N  e.  NN  ->  -u N  e.  RR )
24 nngt0 9961 . . . . . . . 8  |-  ( N  e.  NN  ->  0  <  N )
2522lt0neg2d 9529 . . . . . . . 8  |-  ( N  e.  NN  ->  (
0  <  N  <->  -u N  <  0 ) )
2624, 25mpbid 202 . . . . . . 7  |-  ( N  e.  NN  ->  -u N  <  0 )
27 0re 9024 . . . . . . . 8  |-  0  e.  RR
28 ltnsym 9105 . . . . . . . 8  |-  ( (
-u N  e.  RR  /\  0  e.  RR )  ->  ( -u N  <  0  ->  -.  0  <  -u N ) )
2927, 28mpan2 653 . . . . . . 7  |-  ( -u N  e.  RR  ->  (
-u N  <  0  ->  -.  0  <  -u N
) )
3023, 26, 29sylc 58 . . . . . 6  |-  ( N  e.  NN  ->  -.  0  <  -u N )
31 iffalse 3689 . . . . . 6  |-  ( -.  0  <  -u N  ->  if ( 0  <  -u N ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ,  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u -u N ) ) )  =  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 -u -u N ) ) )
3230, 31syl 16 . . . . 5  |-  ( N  e.  NN  ->  if ( 0  <  -u N ,  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u N
) ,  ( I `
 (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u -u N
) ) )  =  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u -u N ) ) )
3321, 32eqtrd 2419 . . . 4  |-  ( N  e.  NN  ->  if ( -u N  =  0 ,  ( 0g `  G ) ,  if ( 0  <  -u N ,  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u N
) ,  ( I `
 (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u -u N
) ) ) )  =  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 -u -u N ) ) )
3433adantr 452 . . 3  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  if ( -u N  =  0 ,  ( 0g `  G ) ,  if ( 0  <  -u N ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 -u N ) ,  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u -u N ) ) ) )  =  ( I `
 (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u -u N
) ) )
3514, 34eqtrd 2419 . 2  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u -u N ) ) )
367, 8, 11, 12mulgnn 14823 . . 3  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( N  .x.  X
)  =  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) )
3736fveq2d 5672 . 2  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( I `  ( N  .x.  X ) )  =  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ) )
385, 35, 373eqtr4d 2429 1  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   ifcif 3682   {csn 3757   class class class wbr 4153    X. cxp 4816   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   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:  mulgsubcl  14831  mulgneg  14835  mulgneg2  14844  cnfldmulg  16656  tgpmulg  18044  xrsmulgzz  24033
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  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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-nel 2553  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-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-z 10215  df-seq 11251  df-mulg 14742
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