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Theorem mulgnegnn 15736
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  =  ( invg `  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 10428 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  CC )
21negnegd 9808 . . . . 5  |-  ( N  e.  NN  ->  -u -u N  =  N )
32adantr 465 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  B )  -> 
-u -u N  =  N )
43fveq2d 5790 . . 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 5790 . 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 10747 . . . 4  |-  ( N  e.  NN  ->  -u N  e.  ZZ )
7 mulg1.b . . . . 5  |-  B  =  ( Base `  G
)
8 eqid 2451 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
9 eqid 2451 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
10 mulgnegnn.i . . . . 5  |-  I  =  ( invg `  G )
11 mulg1.m . . . . 5  |-  .x.  =  (.g
`  G )
12 eqid 2451 . . . . 5  |-  seq 1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
137, 8, 9, 10, 11, 12mulgval 15728 . . . 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 471 . . 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 10452 . . . . . . 7  |-  ( N  e.  NN  ->  N  =/=  0 )
16 negeq0 9761 . . . . . . . . 9  |-  ( N  e.  CC  ->  ( N  =  0  <->  -u N  =  0 ) )
1716necon3abid 2692 . . . . . . . 8  |-  ( N  e.  CC  ->  ( N  =/=  0  <->  -.  -u N  =  0 ) )
181, 17syl 16 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  =/=  0  <->  -.  -u N  =  0 ) )
1915, 18mpbid 210 . . . . . 6  |-  ( N  e.  NN  ->  -.  -u N  =  0 )
20 iffalse 3894 . . . . . 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 10427 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  RR )
2322renegcld 9873 . . . . . . 7  |-  ( N  e.  NN  ->  -u N  e.  RR )
24 nngt0 10449 . . . . . . . 8  |-  ( N  e.  NN  ->  0  <  N )
2522lt0neg2d 10008 . . . . . . . 8  |-  ( N  e.  NN  ->  (
0  <  N  <->  -u N  <  0 ) )
2624, 25mpbid 210 . . . . . . 7  |-  ( N  e.  NN  ->  -u N  <  0 )
27 0re 9484 . . . . . . . 8  |-  0  e.  RR
28 ltnsym 9571 . . . . . . . 8  |-  ( (
-u N  e.  RR  /\  0  e.  RR )  ->  ( -u N  <  0  ->  -.  0  <  -u N ) )
2927, 28mpan2 671 . . . . . . 7  |-  ( -u N  e.  RR  ->  (
-u N  <  0  ->  -.  0  <  -u N
) )
3023, 26, 29sylc 60 . . . . . 6  |-  ( N  e.  NN  ->  -.  0  <  -u N )
31 iffalse 3894 . . . . . 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 2491 . . . 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 465 . . 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 2491 . 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 15732 . . 3  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( N  .x.  X
)  =  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) )
3736fveq2d 5790 . 2  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( I `  ( N  .x.  X ) )  =  ( I `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ) )
385, 35, 373eqtr4d 2501 1  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2642   ifcif 3886   {csn 3972   class class class wbr 4387    X. cxp 4933   ` cfv 5513  (class class class)co 6187   CCcc 9378   RRcr 9379   0cc0 9380   1c1 9381    < clt 9516   -ucneg 9694   NNcn 10420   ZZcz 10744    seqcseq 11904   Basecbs 14273   +g cplusg 14337   0gc0g 14477   invgcminusg 15510  .gcmg 15513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-inf2 7945  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-z 10745  df-seq 11905  df-mulg 15647
This theorem is referenced by:  mulgsubcl  15740  mulgneg  15744  mulgneg2  15753  cnfldmulg  17954  tgpmulg  19777  xrsmulgzz  26270  archiabllem1b  26340
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