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Theorem mulgneg 15666
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  =  ( invg `  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 10602 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simpr 461 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  N  e.  NN )
3 simpl3 993 . . . . 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  =  ( invg `  G )
74, 5, 6mulgnegnn 15658 . . . . 5  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
82, 3, 7syl2anc 661 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
9 simpl1 991 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  G  e.  Grp )
10 eqid 2443 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
1110, 6grpinvid 15610 . . . . . 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 461 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  N  = 
0 )
1413oveq1d 6127 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( N  .x.  X )  =  ( 0  .x.  X ) )
15 simpl3 993 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  X  e.  B )
164, 10, 5mulg0 15653 . . . . . . . 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 2475 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( N  .x.  X )  =  ( 0g `  G ) )
1918fveq2d 5716 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( I `  ( N  .x.  X
) )  =  ( I `  ( 0g
`  G ) ) )
2013negeqd 9625 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  -u N  = 
-u 0 )
21 neg0 9676 . . . . . . . 8  |-  -u 0  =  0
2220, 21syl6eq 2491 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  -u N  =  0 )
2322oveq1d 6127 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( 0  .x.  X ) )
2423, 17eqtrd 2475 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( 0g `  G ) )
2512, 19, 243eqtr4rd 2486 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
268, 25jaodan 783 . . 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 475 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN0 )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
28 simpl1 991 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  G  e.  Grp )
29 simprr 756 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
3029nnzd 10767 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
31 simpl3 993 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  X  e.  B )
324, 5mulgcl 15665 . . . . 5  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  e.  B
)
3328, 30, 31, 32syl3anc 1218 . . . 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 15614 . . . 4  |-  ( ( G  e.  Grp  /\  ( -u N  .x.  X
)  e.  B )  ->  ( I `  ( I `  ( -u N  .x.  X ) ) )  =  (
-u N  .x.  X
) )
3528, 33, 34syl2anc 661 . . 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 15658 . . . . . 6  |-  ( (
-u N  e.  NN  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X ) ) )
3729, 31, 36syl2anc 661 . . . . 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 755 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
3938recnd 9433 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4039negnegd 9731 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u -u N  =  N )
4140oveq1d 6127 . . . . 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 2477 . . . 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 5716 . . 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 2477 . 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 989 . . 3  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  N  e.  ZZ )
46 elznn0nn 10681 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4745, 46sylib 196 . 2  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4827, 44, 47mpjaodan 784 1  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N  .x.  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5439  (class class class)co 6112   RRcr 9302   0cc0 9303   -ucneg 9617   NNcn 10343   NN0cn0 10600   ZZcz 10667   Basecbs 14195   0gc0g 14399   Grpcgrp 15431   invgcminusg 15432  .gcmg 15435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-seq 11828  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-mulg 15569
This theorem is referenced by:  mulgm1  15667  mulgz  15669  mulgdirlem  15672  mulgdir  15673  mulgneg2  15675  mulgass  15678  mulgsubdir  15679  cycsubgcl  15728  ghmmulg  15780  odnncl  16069  gexdvds  16104  mulgdi  16335  mulgsubdi  16338  mulgass2  16714  clmmulg  20687  archirngz  26228  archiabllem2c  26234
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