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Theorem mulgneg 14684
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 10059 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simpr 447 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  N  e.  NN )
3 simpl3 960 . . . . 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 14676 . . . . 5  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
82, 3, 7syl2anc 642 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
9 simpl1 958 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  G  e.  Grp )
10 eqid 2358 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
1110, 6grpinvid 14632 . . . . . 6  |-  ( G  e.  Grp  ->  (
I `  ( 0g `  G ) )  =  ( 0g `  G
) )
129, 11syl 15 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( I `  ( 0g `  G
) )  =  ( 0g `  G ) )
13 simpr 447 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  N  = 
0 )
1413oveq1d 5960 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( N  .x.  X )  =  ( 0  .x.  X ) )
15 simpl3 960 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  X  e.  B )
164, 10, 5mulg0 14671 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
1715, 16syl 15 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( 0 
.x.  X )  =  ( 0g `  G
) )
1814, 17eqtrd 2390 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( N  .x.  X )  =  ( 0g `  G ) )
1918fveq2d 5612 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( I `  ( N  .x.  X
) )  =  ( I `  ( 0g
`  G ) ) )
2013negeqd 9136 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  -u N  = 
-u 0 )
21 neg0 9183 . . . . . . . 8  |-  -u 0  =  0
2220, 21syl6eq 2406 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  -u N  =  0 )
2322oveq1d 5960 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( 0  .x.  X ) )
2423, 17eqtrd 2390 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( 0g `  G ) )
2512, 19, 243eqtr4rd 2401 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
268, 25jaodan 760 . . 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 461 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN0 )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
28 simpl1 958 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  G  e.  Grp )
29 simprr 733 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
3029nnzd 10208 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
31 simpl3 960 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  X  e.  B )
324, 5mulgcl 14683 . . . . 5  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  e.  B
)
3328, 30, 31, 32syl3anc 1182 . . . 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 14634 . . . 4  |-  ( ( G  e.  Grp  /\  ( -u N  .x.  X
)  e.  B )  ->  ( I `  ( I `  ( -u N  .x.  X ) ) )  =  (
-u N  .x.  X
) )
3528, 33, 34syl2anc 642 . . 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 14676 . . . . . 6  |-  ( (
-u N  e.  NN  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X ) ) )
3729, 31, 36syl2anc 642 . . . . 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 732 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
3938recnd 8951 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4039negnegd 9238 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u -u N  =  N )
4140oveq1d 5960 . . . . 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 2392 . . . 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 5612 . . 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 2392 . 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 956 . . 3  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  N  e.  ZZ )
46 elznn0nn 10129 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4745, 46sylib 188 . 2  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4827, 44, 47mpjaodan 761 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 357    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   ` cfv 5337  (class class class)co 5945   RRcr 8826   0cc0 8827   -ucneg 9128   NNcn 9836   NN0cn0 10057   ZZcz 10116   Basecbs 13245   0gc0g 13499   Grpcgrp 14461   inv gcminusg 14462  .gcmg 14465
This theorem is referenced by:  mulgm1  14685  mulgz  14687  mulgdirlem  14690  mulgdir  14691  mulgneg2  14693  mulgass  14696  mulgsubdir  14697  cycsubgcl  14742  ghmmulg  14794  odnncl  14959  gexdvds  14994  mulgdi  15225  mulgsubdi  15228  mulgass2  15486  clmmulg  18695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-seq 11139  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589  df-mulg 14591
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