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Theorem mulgnegnn 14579
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 9756 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  CC )
21negnegd 9150 . . . . 5  |-  ( N  e.  NN  ->  -u -u N  =  N )
32adantr 451 . . . 4  |-  ( ( N  e.  NN  /\  X  e.  B )  -> 
-u -u N  =  N )
43fveq2d 5531 . . 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 5531 . 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 10029 . . . 4  |-  ( N  e.  NN  ->  -u N  e.  ZZ )
7 mulg1.b . . . . 5  |-  B  =  ( Base `  G
)
8 eqid 2285 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
9 eqid 2285 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
10 mulgnegnn.i . . . . 5  |-  I  =  ( inv g `  G )
11 mulg1.m . . . . 5  |-  .x.  =  (.g
`  G )
12 eqid 2285 . . . . 5  |-  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
137, 8, 9, 10, 11, 12mulgval 14571 . . . 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 457 . . 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 9780 . . . . . . 7  |-  ( N  e.  NN  ->  N  =/=  0 )
16 negeq0 9103 . . . . . . . . 9  |-  ( N  e.  CC  ->  ( N  =  0  <->  -u N  =  0 ) )
1716necon3abid 2481 . . . . . . . 8  |-  ( N  e.  CC  ->  ( N  =/=  0  <->  -.  -u N  =  0 ) )
181, 17syl 15 . . . . . . 7  |-  ( N  e.  NN  ->  ( N  =/=  0  <->  -.  -u N  =  0 ) )
1915, 18mpbid 201 . . . . . 6  |-  ( N  e.  NN  ->  -.  -u N  =  0 )
20 iffalse 3574 . . . . . 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 15 . . . . 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 9755 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  RR )
2322renegcld 9212 . . . . . . 7  |-  ( N  e.  NN  ->  -u N  e.  RR )
24 nngt0 9777 . . . . . . . 8  |-  ( N  e.  NN  ->  0  <  N )
2522lt0neg2d 9345 . . . . . . . 8  |-  ( N  e.  NN  ->  (
0  <  N  <->  -u N  <  0 ) )
2624, 25mpbid 201 . . . . . . 7  |-  ( N  e.  NN  ->  -u N  <  0 )
27 0re 8840 . . . . . . . 8  |-  0  e.  RR
28 ltnsym 8921 . . . . . . . 8  |-  ( (
-u N  e.  RR  /\  0  e.  RR )  ->  ( -u N  <  0  ->  -.  0  <  -u N ) )
2927, 28mpan2 652 . . . . . . 7  |-  ( -u N  e.  RR  ->  (
-u N  <  0  ->  -.  0  <  -u N
) )
3023, 26, 29sylc 56 . . . . . 6  |-  ( N  e.  NN  ->  -.  0  <  -u N )
31 iffalse 3574 . . . . . 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 15 . . . . 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 2317 . . . 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 451 . . 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 2317 . 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 14575 . . 3  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( N  .x.  X
)  =  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) )
3736fveq2d 5531 . 2  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( I `  ( N  .x.  X ) )  =  ( I `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ) )
385, 35, 373eqtr4d 2327 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 176    /\ wa 358    = wceq 1625    e. wcel 1686    =/= wne 2448   ifcif 3567   {csn 3642   class class class wbr 4025    X. cxp 4689   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    < clt 8869   -ucneg 9040   NNcn 9748   ZZcz 10026    seq cseq 11048   Basecbs 13150   +g cplusg 13210   0gc0g 13402   inv gcminusg 14365  .gcmg 14368
This theorem is referenced by:  mulgsubcl  14583  mulgneg  14587  mulgneg2  14596  cnfldmulg  16408  tgpmulg  17778  xrsmulgzz  23309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-z 10027  df-seq 11049  df-mulg 14494
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