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Theorem mulgneg 14863
Description: Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
Hypotheses
Ref Expression
mulgnncl.b  |-  B  =  ( Base `  G
)
mulgnncl.t  |-  .x.  =  (.g
`  G )
mulgneg.i  |-  I  =  ( inv g `  G )
Assertion
Ref Expression
mulgneg  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N  .x.  X ) ) )

Proof of Theorem mulgneg
StepHypRef Expression
1 elnn0 10179 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simpr 448 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  N  e.  NN )
3 simpl3 962 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  X  e.  B
)
4 mulgnncl.b . . . . . 6  |-  B  =  ( Base `  G
)
5 mulgnncl.t . . . . . 6  |-  .x.  =  (.g
`  G )
6 mulgneg.i . . . . . 6  |-  I  =  ( inv g `  G )
74, 5, 6mulgnegnn 14855 . . . . 5  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
82, 3, 7syl2anc 643 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
9 simpl1 960 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  G  e.  Grp )
10 eqid 2404 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
1110, 6grpinvid 14811 . . . . . 6  |-  ( G  e.  Grp  ->  (
I `  ( 0g `  G ) )  =  ( 0g `  G
) )
129, 11syl 16 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( I `  ( 0g `  G
) )  =  ( 0g `  G ) )
13 simpr 448 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  N  = 
0 )
1413oveq1d 6055 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( N  .x.  X )  =  ( 0  .x.  X ) )
15 simpl3 962 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  X  e.  B )
164, 10, 5mulg0 14850 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
1715, 16syl 16 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( 0 
.x.  X )  =  ( 0g `  G
) )
1814, 17eqtrd 2436 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( N  .x.  X )  =  ( 0g `  G ) )
1918fveq2d 5691 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( I `  ( N  .x.  X
) )  =  ( I `  ( 0g
`  G ) ) )
2013negeqd 9256 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  -u N  = 
-u 0 )
21 neg0 9303 . . . . . . . 8  |-  -u 0  =  0
2220, 21syl6eq 2452 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  -u N  =  0 )
2322oveq1d 6055 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( 0  .x.  X ) )
2423, 17eqtrd 2436 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( 0g `  G ) )
2512, 19, 243eqtr4rd 2447 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
268, 25jaodan 761 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( -u N  .x.  X )  =  ( I `  ( N  .x.  X ) ) )
271, 26sylan2b 462 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN0 )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
28 simpl1 960 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  G  e.  Grp )
29 simprr 734 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
3029nnzd 10330 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
31 simpl3 962 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  X  e.  B )
324, 5mulgcl 14862 . . . . 5  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  e.  B
)
3328, 30, 31, 32syl3anc 1184 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( -u N  .x.  X )  e.  B )
344, 6grpinvinv 14813 . . . 4  |-  ( ( G  e.  Grp  /\  ( -u N  .x.  X
)  e.  B )  ->  ( I `  ( I `  ( -u N  .x.  X ) ) )  =  (
-u N  .x.  X
) )
3528, 33, 34syl2anc 643 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
I `  ( I `  ( -u N  .x.  X ) ) )  =  ( -u N  .x.  X ) )
364, 5, 6mulgnegnn 14855 . . . . . 6  |-  ( (
-u N  e.  NN  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X ) ) )
3729, 31, 36syl2anc 643 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X
) ) )
38 simprl 733 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
3938recnd 9070 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4039negnegd 9358 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u -u N  =  N )
4140oveq1d 6055 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( -u -u N  .x.  X )  =  ( N  .x.  X ) )
4237, 41eqtr3d 2438 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
I `  ( -u N  .x.  X ) )  =  ( N  .x.  X
) )
4342fveq2d 5691 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  (
I `  ( I `  ( -u N  .x.  X ) ) )  =  ( I `  ( N  .x.  X ) ) )
4435, 43eqtr3d 2438 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  ( -u N  .x.  X )  =  ( I `  ( N  .x.  X ) ) )
45 simp2 958 . . 3  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  N  e.  ZZ )
46 elznn0nn 10251 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4745, 46sylib 189 . 2  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4827, 44, 47mpjaodan 762 1  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N  .x.  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946   -ucneg 9248   NNcn 9956   NN0cn0 10177   ZZcz 10238   Basecbs 13424   0gc0g 13678   Grpcgrp 14640   inv gcminusg 14641  .gcmg 14644
This theorem is referenced by:  mulgm1  14864  mulgz  14866  mulgdirlem  14869  mulgdir  14870  mulgneg2  14872  mulgass  14875  mulgsubdir  14876  cycsubgcl  14921  ghmmulg  14973  odnncl  15138  gexdvds  15173  mulgdi  15404  mulgsubdi  15407  mulgass2  15665  clmmulg  19071
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-seq 11279  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-mulg 14770
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