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Theorem mulgneg2 14594
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 9044 . . . . . . 7  |-  ( x  =  0  ->  -u x  =  -u 0 )
2 neg0 9093 . . . . . . 7  |-  -u 0  =  0
31, 2syl6eq 2331 . . . . . 6  |-  ( x  =  0  ->  -u x  =  0 )
43oveq1d 5873 . . . . 5  |-  ( x  =  0  ->  ( -u x  .x.  X )  =  ( 0  .x. 
X ) )
5 oveq1 5865 . . . . 5  |-  ( x  =  0  ->  (
x  .x.  ( I `  X ) )  =  ( 0  .x.  (
I `  X )
) )
64, 5eqeq12d 2297 . . . 4  |-  ( x  =  0  ->  (
( -u x  .x.  X
)  =  ( x 
.x.  ( I `  X ) )  <->  ( 0 
.x.  X )  =  ( 0  .x.  (
I `  X )
) ) )
7 negeq 9044 . . . . . 6  |-  ( x  =  n  ->  -u x  =  -u n )
87oveq1d 5873 . . . . 5  |-  ( x  =  n  ->  ( -u x  .x.  X )  =  ( -u n  .x.  X ) )
9 oveq1 5865 . . . . 5  |-  ( x  =  n  ->  (
x  .x.  ( I `  X ) )  =  ( n  .x.  (
I `  X )
) )
108, 9eqeq12d 2297 . . . 4  |-  ( x  =  n  ->  (
( -u x  .x.  X
)  =  ( x 
.x.  ( I `  X ) )  <->  ( -u n  .x.  X )  =  ( n  .x.  ( I `
 X ) ) ) )
11 negeq 9044 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  -u x  =  -u ( n  + 
1 ) )
1211oveq1d 5873 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  ( -u x  .x.  X )  =  ( -u (
n  +  1 ) 
.x.  X ) )
13 oveq1 5865 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
x  .x.  ( I `  X ) )  =  ( ( n  + 
1 )  .x.  (
I `  X )
) )
1412, 13eqeq12d 2297 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( -u x  .x.  X
)  =  ( x 
.x.  ( I `  X ) )  <->  ( -u (
n  +  1 ) 
.x.  X )  =  ( ( n  + 
1 )  .x.  (
I `  X )
) ) )
15 negeq 9044 . . . . . 6  |-  ( x  =  -u n  ->  -u x  =  -u -u n )
1615oveq1d 5873 . . . . 5  |-  ( x  =  -u n  ->  ( -u x  .x.  X )  =  ( -u -u n  .x.  X ) )
17 oveq1 5865 . . . . 5  |-  ( x  =  -u n  ->  (
x  .x.  ( I `  X ) )  =  ( -u n  .x.  ( I `  X
) ) )
1816, 17eqeq12d 2297 . . . 4  |-  ( x  =  -u n  ->  (
( -u x  .x.  X
)  =  ( x 
.x.  ( I `  X ) )  <->  ( -u -u n  .x.  X )  =  (
-u n  .x.  (
I `  X )
) ) )
19 negeq 9044 . . . . . 6  |-  ( x  =  N  ->  -u x  =  -u N )
2019oveq1d 5873 . . . . 5  |-  ( x  =  N  ->  ( -u x  .x.  X )  =  ( -u N  .x.  X ) )
21 oveq1 5865 . . . . 5  |-  ( x  =  N  ->  (
x  .x.  ( I `  X ) )  =  ( N  .x.  (
I `  X )
) )
2220, 21eqeq12d 2297 . . . 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 2283 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
25 mulgneg2.m . . . . . . 7  |-  .x.  =  (.g
`  G )
2623, 24, 25mulg0 14572 . . . . . 6  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2726adantl 452 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( 0  .x.  X
)  =  ( 0g
`  G ) )
28 mulgneg2.i . . . . . . 7  |-  I  =  ( inv g `  G )
2923, 28grpinvcl 14527 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( I `  X
)  e.  B )
3023, 24, 25mulg0 14572 . . . . . 6  |-  ( ( I `  X )  e.  B  ->  (
0  .x.  ( I `  X ) )  =  ( 0g `  G
) )
3129, 30syl 15 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( 0  .x.  (
I `  X )
)  =  ( 0g
`  G ) )
3227, 31eqtr4d 2318 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( 0  .x.  X
)  =  ( 0 
.x.  ( I `  X ) ) )
33 oveq1 5865 . . . . . 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 9975 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  n  e.  CC )
3534adantl 452 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  n  e.  CC )
36 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
37 negdi 9104 . . . . . . . . . 10  |-  ( ( n  e.  CC  /\  1  e.  CC )  -> 
-u ( n  + 
1 )  =  (
-u n  +  -u
1 ) )
3835, 36, 37sylancl 643 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  -u ( n  + 
1 )  =  (
-u n  +  -u
1 ) )
3938oveq1d 5873 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( -u (
n  +  1 ) 
.x.  X )  =  ( ( -u n  +  -u 1 )  .x.  X ) )
40 simpll 730 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  G  e.  Grp )
41 nn0negz 10057 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  -u n  e.  ZZ )
4241adantl 452 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  -u n  e.  ZZ )
43 1z 10053 . . . . . . . . . 10  |-  1  e.  ZZ
44 znegcl 10055 . . . . . . . . . 10  |-  ( 1  e.  ZZ  ->  -u 1  e.  ZZ )
4543, 44mp1i 11 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  -u 1  e.  ZZ )
46 simplr 731 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  X  e.  B
)
47 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
4823, 25, 47mulgdir 14592 . . . . . . . . 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 1184 . . . . . . . 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 14586 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( -u 1  .x. 
X )  =  ( I `  X ) )
5150adantr 451 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( -u 1  .x.  X )  =  ( I `  X ) )
5251oveq2d 5874 . . . . . . . 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 2319 . . . . . . 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 14494 . . . . . . . . 9  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5554ad2antrr 706 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  G  e.  Mnd )
56 simpr 447 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
5729adantr 451 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN0 )  ->  ( I `  X )  e.  B
)
5823, 25, 47mulgnn0p1 14578 . . . . . . . 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 1182 . . . . . . 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 2297 . . . . . 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 212 . . . . 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 423 . . . 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 5525 . . . . . 6  |-  ( (
-u n  .x.  X
)  =  ( n 
.x.  ( I `  X ) )  -> 
( I `  ( -u n  .x.  X ) )  =  ( I `
 ( n  .x.  ( I `  X
) ) ) )
64 simpll 730 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  G  e.  Grp )
65 nnnegz 10027 . . . . . . . . 9  |-  ( n  e.  NN  ->  -u n  e.  ZZ )
6665adantl 452 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  -u n  e.  ZZ )
67 simplr 731 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  X  e.  B
)
6823, 25, 28mulgneg 14585 . . . . . . . 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 1182 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  ( -u -u n  .x.  X )  =  ( I `  ( -u n  .x.  X ) ) )
70 id 19 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  NN )
7123, 25, 28mulgnegnn 14577 . . . . . . . 8  |-  ( ( n  e.  NN  /\  ( I `  X
)  e.  B )  ->  ( -u n  .x.  ( I `  X
) )  =  ( I `  ( n 
.x.  ( I `  X ) ) ) )
7270, 29, 71syl2anr 464 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  B )  /\  n  e.  NN )  ->  ( -u n  .x.  ( I `  X
) )  =  ( I `  ( n 
.x.  ( I `  X ) ) ) )
7369, 72eqeq12d 2297 . . . . . 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 212 . . . . 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 423 . . . 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 10113 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N  e.  ZZ  ->  ( -u N  .x.  X )  =  ( N  .x.  ( I `
 X ) ) ) )
77763impia 1148 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  N  e.  ZZ )  ->  ( -u N  .x.  X )  =  ( N  .x.  ( I `
 X ) ) )
78773com23 1157 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740   -ucneg 9038   NNcn 9746   NN0cn0 9965   ZZcz 10024   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361   Grpcgrp 14362   inv gcminusg 14363  .gcmg 14366
This theorem is referenced by:  mulgass  14597  mulgsubdi  15129  cyggeninv  15170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mulg 14492
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