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Theorem mulgneg 16277
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 10714 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simpr 459 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  N  e.  NN )
3 simpl3 999 . . . . 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 16269 . . . . 5  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
82, 3, 7syl2anc 659 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
9 simpl1 997 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  G  e.  Grp )
10 eqid 2382 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
1110, 6grpinvid 16218 . . . . . 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 459 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  N  = 
0 )
1413oveq1d 6211 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( N  .x.  X )  =  ( 0  .x.  X ) )
15 simpl3 999 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  X  e.  B )
164, 10, 5mulg0 16264 . . . . . . . 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 2423 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( N  .x.  X )  =  ( 0g `  G ) )
1918fveq2d 5778 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( I `  ( N  .x.  X
) )  =  ( I `  ( 0g
`  G ) ) )
2013negeqd 9727 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  -u N  = 
-u 0 )
21 neg0 9778 . . . . . . . 8  |-  -u 0  =  0
2220, 21syl6eq 2439 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  -u N  =  0 )
2322oveq1d 6211 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( 0  .x.  X ) )
2423, 17eqtrd 2423 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( 0g `  G ) )
2512, 19, 243eqtr4rd 2434 . . . 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 473 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN0 )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
28 simpl1 997 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  G  e.  Grp )
29 simprr 755 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
3029nnzd 10883 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
31 simpl3 999 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  X  e.  B )
324, 5mulgcl 16276 . . . . 5  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  e.  B
)
3328, 30, 31, 32syl3anc 1226 . . . 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 16222 . . . 4  |-  ( ( G  e.  Grp  /\  ( -u N  .x.  X
)  e.  B )  ->  ( I `  ( I `  ( -u N  .x.  X ) ) )  =  (
-u N  .x.  X
) )
3528, 33, 34syl2anc 659 . . 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 16269 . . . . . 6  |-  ( (
-u N  e.  NN  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X ) ) )
3729, 31, 36syl2anc 659 . . . . 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 754 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
3938recnd 9533 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4039negnegd 9835 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u -u N  =  N )
4140oveq1d 6211 . . . . 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 2425 . . . 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 5778 . . 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 2425 . 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 995 . . 3  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  N  e.  ZZ )
46 elznn0nn 10795 . . 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 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   ` cfv 5496  (class class class)co 6196   RRcr 9402   0cc0 9403   -ucneg 9719   NNcn 10452   NN0cn0 10712   ZZcz 10781   Basecbs 14634   0gc0g 14847   Grpcgrp 16170   invgcminusg 16171  .gcmg 16173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-seq 12011  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-mulg 16177
This theorem is referenced by:  mulgm1  16278  mulgz  16280  mulgdirlem  16283  mulgdir  16284  mulgneg2  16286  mulgass  16289  mulgsubdir  16290  cycsubgcl  16344  ghmmulg  16396  odnncl  16686  gexdvds  16721  mulgdi  16952  mulgsubdi  16955  mulgass2  17360  clmmulg  21678  archirngz  27886  archiabllem2c  27892
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