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Theorem mulgnegnn 14900
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 10008 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  CC )
21negnegd 9402 . . . . 5  |-  ( N  e.  NN  ->  -u -u N  =  N )
32adantr 452 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  B )  -> 
-u -u N  =  N )
43fveq2d 5732 . . 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 5732 . 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 10285 . . . 4  |-  ( N  e.  NN  ->  -u N  e.  ZZ )
7 mulg1.b . . . . 5  |-  B  =  ( Base `  G
)
8 eqid 2436 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
9 eqid 2436 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
10 mulgnegnn.i . . . . 5  |-  I  =  ( inv g `  G )
11 mulg1.m . . . . 5  |-  .x.  =  (.g
`  G )
12 eqid 2436 . . . . 5  |-  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
137, 8, 9, 10, 11, 12mulgval 14892 . . . 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 10032 . . . . . . 7  |-  ( N  e.  NN  ->  N  =/=  0 )
16 negeq0 9355 . . . . . . . . 9  |-  ( N  e.  CC  ->  ( N  =  0  <->  -u N  =  0 ) )
1716necon3abid 2634 . . . . . . . 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 3746 . . . . . 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 10007 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  RR )
2322renegcld 9464 . . . . . . 7  |-  ( N  e.  NN  ->  -u N  e.  RR )
24 nngt0 10029 . . . . . . . 8  |-  ( N  e.  NN  ->  0  <  N )
2522lt0neg2d 9597 . . . . . . . 8  |-  ( N  e.  NN  ->  (
0  <  N  <->  -u N  <  0 ) )
2624, 25mpbid 202 . . . . . . 7  |-  ( N  e.  NN  ->  -u N  <  0 )
27 0re 9091 . . . . . . . 8  |-  0  e.  RR
28 ltnsym 9172 . . . . . . . 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 3746 . . . . . 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 2468 . . . 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 2468 . 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 14896 . . 3  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( N  .x.  X
)  =  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) )
3736fveq2d 5732 . 2  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( I `  ( N  .x.  X ) )  =  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ) )
385, 35, 373eqtr4d 2478 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 1652    e. wcel 1725    =/= wne 2599   ifcif 3739   {csn 3814   class class class wbr 4212    X. cxp 4876   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    < clt 9120   -ucneg 9292   NNcn 10000   ZZcz 10282    seq cseq 11323   Basecbs 13469   +g cplusg 13529   0gc0g 13723   inv gcminusg 14686  .gcmg 14689
This theorem is referenced by:  mulgsubcl  14904  mulgneg  14908  mulgneg2  14917  cnfldmulg  16733  tgpmulg  18123  xrsmulgzz  24200
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-z 10283  df-seq 11324  df-mulg 14815
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