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Theorem mulgneg2 14872
Description: Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)
Hypotheses
Ref Expression
mulgneg2.b  |-  B  =  ( Base `  G
)
mulgneg2.m  |-  .x.  =  (.g
`  G )
mulgneg2.i  |-  I  =  ( inv g `  G )
Assertion
Ref Expression
mulgneg2  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( N  .x.  ( I `  X
) ) )

Proof of Theorem mulgneg2
Dummy variables  x  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 negeq 9254 . . . . . . 7  |-  ( x  =  0  ->  -u x  =  -u 0 )
2 neg0 9303 . . . . . . 7  |-  -u 0  =  0
31, 2syl6eq 2452 . . . . . 6  |-  ( x  =  0  ->  -u x  =  0 )
43oveq1d 6055 . . . . 5  |-  ( x  =  0  ->  ( -u x  .x.  X )  =  ( 0  .x. 
X ) )
5 oveq1 6047 . . . . 5  |-  ( x  =  0  ->  (
x  .x.  ( I `  X ) )  =  ( 0  .x.  (
I `  X )
) )
64, 5eqeq12d 2418 . . . 4  |-  ( x  =  0  ->  (
( -u x  .x.  X
)  =  ( x 
.x.  ( I `  X ) )  <->  ( 0 
.x.  X )  =  ( 0  .x.  (
I `  X )
) ) )
7 negeq 9254 . . . . . 6  |-  ( x  =  n  ->  -u x  =  -u n )
87oveq1d 6055 . . . . 5  |-  ( x  =  n  ->  ( -u x  .x.  X )  =  ( -u n  .x.  X ) )
9 oveq1 6047 . . . . 5  |-  ( x  =  n  ->  (
x  .x.  ( I `  X ) )  =  ( n  .x.  (
I `  X )
) )
108, 9eqeq12d 2418 . . . 4  |-  ( x  =  n  ->  (
( -u x  .x.  X
)  =  ( x 
.x.  ( I `  X ) )  <->  ( -u n  .x.  X )  =  ( n  .x.  ( I `
 X ) ) ) )
11 negeq 9254 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  -u x  =  -u ( n  + 
1 ) )
1211oveq1d 6055 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  ( -u x  .x.  X )  =  ( -u (
n  +  1 ) 
.x.  X ) )
13 oveq1 6047 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
x  .x.  ( I `  X ) )  =  ( ( n  + 
1 )  .x.  (
I `  X )
) )
1412, 13eqeq12d 2418 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( -u x  .x.  X
)  =  ( x 
.x.  ( I `  X ) )  <->  ( -u (
n  +  1 ) 
.x.  X )  =  ( ( n  + 
1 )  .x.  (
I `  X )
) ) )
15 negeq 9254 . . . . . 6  |-  ( x  =  -u n  ->  -u x  =  -u -u n )
1615oveq1d 6055 . . . . 5  |-  ( x  =  -u n  ->  ( -u x  .x.  X )  =  ( -u -u n  .x.  X ) )
17 oveq1 6047 . . . . 5  |-  ( x  =  -u n  ->  (
x  .x.  ( I `  X ) )  =  ( -u n  .x.  ( I `  X
) ) )
1816, 17eqeq12d 2418 . . . 4  |-  ( x  =  -u n  ->  (
( -u x  .x.  X
)  =  ( x 
.x.  ( I `  X ) )  <->  ( -u -u n  .x.  X )  =  (
-u n  .x.  (
I `  X )
) ) )
19 negeq 9254 . . . . . 6  |-  ( x  =  N  ->  -u x  =  -u N )
2019oveq1d 6055 . . . . 5  |-  ( x  =  N  ->  ( -u x  .x.  X )  =  ( -u N  .x.  X ) )
21 oveq1 6047 . . . . 5  |-  ( x  =  N  ->  (
x  .x.  ( I `  X ) )  =  ( N  .x.  (
I `  X )
) )
2220, 21eqeq12d 2418 . . . 4  |-  ( x  =  N  ->  (
( -u x  .x.  X
)  =  ( x 
.x.  ( I `  X ) )  <->  ( -u N  .x.  X )  =  ( N  .x.  ( I `
 X ) ) ) )
23 mulgneg2.b . . . . . . 7  |-  B  =  ( Base `  G
)
24 eqid 2404 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
25 mulgneg2.m . . . . . . 7  |-  .x.  =  (.g
`  G )
2623, 24, 25mulg0 14850 . . . . . 6  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2726adantl 453 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( 0  .x.  X
)  =  ( 0g
`  G ) )
28 mulgneg2.i . . . . . . 7  |-  I  =  ( inv g `  G )
2923, 28grpinvcl 14805 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( I `  X
)  e.  B )
3023, 24, 25mulg0 14850 . . . . . 6  |-  ( ( I `  X )  e.  B  ->  (
0  .x.  ( I `  X ) )  =  ( 0g `  G
) )
3129, 30syl 16 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( 0  .x.  (
I `  X )
)  =  ( 0g
`  G ) )
3227, 31eqtr4d 2439 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( 0  .x.  X
)  =  ( 0 
.x.  ( I `  X ) ) )
33 oveq1 6047 . . . . . 6  |-  ( (
-u n  .x.  X
)  =  ( n 
.x.  ( I `  X ) )  -> 
( ( -u n  .x.  X ) ( +g  `  G ) ( I `
 X ) )  =  ( ( n 
.x.  ( I `  X ) ) ( +g  `  G ) ( I `  X
) ) )
34 nn0cn 10187 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  n  e.  CC )
3534adantl 453 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  n  e.  CC )
36 ax-1cn 9004 . . . . . . . . . 10  |-  1  e.  CC
37 negdi 9314 . . . . . . . . . 10  |-  ( ( n  e.  CC  /\  1  e.  CC )  -> 
-u ( n  + 
1 )  =  (
-u n  +  -u
1 ) )
3835, 36, 37sylancl 644 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  -u ( n  + 
1 )  =  (
-u n  +  -u
1 ) )
3938oveq1d 6055 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( -u (
n  +  1 ) 
.x.  X )  =  ( ( -u n  +  -u 1 )  .x.  X ) )
40 simpll 731 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  G  e.  Grp )
41 nn0negz 10271 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  -u n  e.  ZZ )
4241adantl 453 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  -u n  e.  ZZ )
43 1z 10267 . . . . . . . . . 10  |-  1  e.  ZZ
44 znegcl 10269 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
4543, 44mp1i 12 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  -u 1  e.  ZZ )
46 simplr 732 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  X  e.  B
)
47 eqid 2404 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
4823, 25, 47mulgdir 14870 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( -u n  e.  ZZ  /\  -u 1  e.  ZZ  /\  X  e.  B ) )  ->  ( ( -u n  +  -u 1
)  .x.  X )  =  ( ( -u n  .x.  X ) ( +g  `  G ) ( -u 1  .x. 
X ) ) )
4940, 42, 45, 46, 48syl13anc 1186 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( ( -u n  +  -u 1 ) 
.x.  X )  =  ( ( -u n  .x.  X ) ( +g  `  G ) ( -u
1  .x.  X )
) )
5023, 25, 28mulgm1 14864 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( -u 1  .x. 
X )  =  ( I `  X ) )
5150adantr 452 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( -u 1  .x.  X )  =  ( I `  X ) )
5251oveq2d 6056 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( ( -u n  .x.  X ) ( +g  `  G ) ( -u 1  .x. 
X ) )  =  ( ( -u n  .x.  X ) ( +g  `  G ) ( I `
 X ) ) )
5339, 49, 523eqtrd 2440 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( -u (
n  +  1 ) 
.x.  X )  =  ( ( -u n  .x.  X ) ( +g  `  G ) ( I `
 X ) ) )
54 grpmnd 14772 . . . . . . . . 9  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5554ad2antrr 707 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  G  e.  Mnd )
56 simpr 448 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
5729adantr 452 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( I `  X )  e.  B
)
5823, 25, 47mulgnn0p1 14856 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  n  e.  NN0  /\  (
I `  X )  e.  B )  ->  (
( n  +  1 )  .x.  ( I `
 X ) )  =  ( ( n 
.x.  ( I `  X ) ) ( +g  `  G ) ( I `  X
) ) )
5955, 56, 57, 58syl3anc 1184 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( ( n  +  1 )  .x.  ( I `  X
) )  =  ( ( n  .x.  (
I `  X )
) ( +g  `  G
) ( I `  X ) ) )
6053, 59eqeq12d 2418 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( ( -u ( n  +  1
)  .x.  X )  =  ( ( n  +  1 )  .x.  ( I `  X
) )  <->  ( ( -u n  .x.  X ) ( +g  `  G
) ( I `  X ) )  =  ( ( n  .x.  ( I `  X
) ) ( +g  `  G ) ( I `
 X ) ) ) )
6133, 60syl5ibr 213 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( ( -u n  .x.  X )  =  ( n  .x.  (
I `  X )
)  ->  ( -u (
n  +  1 ) 
.x.  X )  =  ( ( n  + 
1 )  .x.  (
I `  X )
) ) )
6261ex 424 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( n  e.  NN0  ->  ( ( -u n  .x.  X )  =  ( n  .x.  ( I `
 X ) )  ->  ( -u (
n  +  1 ) 
.x.  X )  =  ( ( n  + 
1 )  .x.  (
I `  X )
) ) ) )
63 fveq2 5687 . . . . . 6  |-  ( (
-u n  .x.  X
)  =  ( n 
.x.  ( I `  X ) )  -> 
( I `  ( -u n  .x.  X ) )  =  ( I `
 ( n  .x.  ( I `  X
) ) ) )
64 simpll 731 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  G  e.  Grp )
65 nnnegz 10241 . . . . . . . . 9  |-  ( n  e.  NN  ->  -u n  e.  ZZ )
6665adantl 453 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  -u n  e.  ZZ )
67 simplr 732 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  X  e.  B
)
6823, 25, 28mulgneg 14863 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  -u n  e.  ZZ  /\  X  e.  B )  ->  ( -u -u n  .x.  X )  =  ( I `  ( -u n  .x.  X ) ) )
6964, 66, 67, 68syl3anc 1184 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  ( -u -u n  .x.  X )  =  ( I `  ( -u n  .x.  X ) ) )
70 id 20 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  NN )
7123, 25, 28mulgnegnn 14855 . . . . . . . 8  |-  ( ( n  e.  NN  /\  ( I `  X
)  e.  B )  ->  ( -u n  .x.  ( I `  X
) )  =  ( I `  ( n 
.x.  ( I `  X ) ) ) )
7270, 29, 71syl2anr 465 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  ( -u n  .x.  ( I `  X
) )  =  ( I `  ( n 
.x.  ( I `  X ) ) ) )
7369, 72eqeq12d 2418 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  ( ( -u -u n  .x.  X )  =  ( -u n  .x.  ( I `  X
) )  <->  ( I `  ( -u n  .x.  X ) )  =  ( I `  (
n  .x.  ( I `  X ) ) ) ) )
7463, 73syl5ibr 213 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  ( ( -u n  .x.  X )  =  ( n  .x.  (
I `  X )
)  ->  ( -u -u n  .x.  X )  =  (
-u n  .x.  (
I `  X )
) ) )
7574ex 424 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( n  e.  NN  ->  ( ( -u n  .x.  X )  =  ( n  .x.  ( I `
 X ) )  ->  ( -u -u n  .x.  X )  =  (
-u n  .x.  (
I `  X )
) ) ) )
766, 10, 14, 18, 22, 32, 62, 75zindd 10327 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N  e.  ZZ  ->  ( -u N  .x.  X )  =  ( N  .x.  ( I `
 X ) ) ) )
77763impia 1150 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  N  e.  ZZ )  ->  ( -u N  .x.  X )  =  ( N  .x.  ( I `
 X ) ) )
78773com23 1159 1  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( N  .x.  ( I `  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949   -ucneg 9248   NNcn 9956   NN0cn0 10177   ZZcz 10238   Basecbs 13424   +g cplusg 13484   0gc0g 13678   Mndcmnd 14639   Grpcgrp 14640   inv gcminusg 14641  .gcmg 14644
This theorem is referenced by:  mulgass  14875  mulgsubdi  15407  cyggeninv  15448
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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