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Theorem mulgneg 12877
Description: Group multiple (exponentiation) operation at a negative integer. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised 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 9154 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 simpr 110 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  N  e.  NN )
3 simpl3 1002 . . . . 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 12869 . . . . 5  |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
82, 3, 7syl2anc 411 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
9 simpl1 1000 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  G  e.  Grp )
10 eqid 2177 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
1110, 6grpinvid 12807 . . . . . 6  |-  ( G  e.  Grp  ->  (
I `  ( 0g `  G ) )  =  ( 0g `  G
) )
129, 11syl 14 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( I `  ( 0g `  G
) )  =  ( 0g `  G ) )
13 simpr 110 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  N  = 
0 )
1413oveq1d 5883 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( N  .x.  X )  =  ( 0  .x.  X ) )
15 simpl3 1002 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  X  e.  B )
164, 10, 5mulg0 12864 . . . . . . . 8  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
1715, 16syl 14 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( 0 
.x.  X )  =  ( 0g `  G
) )
1814, 17eqtrd 2210 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( N  .x.  X )  =  ( 0g `  G ) )
1918fveq2d 5514 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( I `  ( N  .x.  X
) )  =  ( I `  ( 0g
`  G ) ) )
2013negeqd 8129 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  -u N  = 
-u 0 )
21 neg0 8180 . . . . . . . 8  |-  -u 0  =  0
2220, 21eqtrdi 2226 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  -u N  =  0 )
2322oveq1d 5883 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( 0  .x.  X ) )
2423, 17eqtrd 2210 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( 0g `  G ) )
2512, 19, 243eqtr4rd 2221 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  =  0
)  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
268, 25jaodan 797 . . 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 287 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN0 )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
.x.  X ) ) )
28 simpl1 1000 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  G  e.  Grp )
29 simprr 531 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN )
3029nnzd 9350 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
31 simpl3 1002 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  X  e.  B )
324, 5mulgcl 12876 . . . . 5  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  e.  B
)
3328, 30, 31, 32syl3anc 1238 . . . 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 12813 . . . 4  |-  ( ( G  e.  Grp  /\  ( -u N  .x.  X
)  e.  B )  ->  ( I `  ( I `  ( -u N  .x.  X ) ) )  =  (
-u N  .x.  X
) )
3528, 33, 34syl2anc 411 . . 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 12869 . . . . . 6  |-  ( (
-u N  e.  NN  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( I `  ( -u N  .x.  X ) ) )
3729, 31, 36syl2anc 411 . . . . 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 529 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
3938recnd 7963 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4039negnegd 8236 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u -u N  =  N )
4140oveq1d 5883 . . . . 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 2212 . . . 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 5514 . . 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 2212 . 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 998 . . 3  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  N  e.  ZZ )
46 elznn0nn 9243 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4745, 46sylib 122 . 2  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
4827, 44, 47mpjaodan 798 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    /\ wa 104    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148   ` cfv 5211  (class class class)co 5868   RRcr 7788   0cc0 7789   -ucneg 8106   NNcn 8895   NN0cn0 9152   ZZcz 9229   Basecbs 12432   0gc0g 12640   Grpcgrp 12754   invgcminusg 12755  .gcmg 12859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-2 8954  df-n0 9153  df-z 9230  df-uz 9505  df-seqfrec 10419  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757  df-minusg 12758  df-mulg 12860
This theorem is referenced by:  mulgnegneg  12878  mulgm1  12879  mulgaddcomlem  12881  mulginvcom  12883  mulgz  12886  mulgdirlem  12889  mulgdir  12890  mulgneg2  12892  mulgass  12895  mulgsubdir  12898  mulgass2  13048
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