MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  odmod Unicode version

Theorem odmod 15139
Description: Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 6-Sep-2015.)
Hypotheses
Ref Expression
odcl.1  |-  X  =  ( Base `  G
)
odcl.2  |-  O  =  ( od `  G
)
odid.3  |-  .x.  =  (.g
`  G )
odid.4  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
odmod  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  .x.  A )  =  ( N  .x.  A ) )

Proof of Theorem odmod
StepHypRef Expression
1 simpl3 962 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  ZZ )
21zred 10331 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  RR )
3 simpr 448 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  NN )
43nnrpd 10603 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR+ )
5 modval 11207 . . . 4  |-  ( ( N  e.  RR  /\  ( O `  A )  e.  RR+ )  ->  ( N  mod  ( O `  A ) )  =  ( N  -  (
( O `  A
)  x.  ( |_
`  ( N  / 
( O `  A
) ) ) ) ) )
62, 4, 5syl2anc 643 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  =  ( N  -  ( ( O `
 A )  x.  ( |_ `  ( N  /  ( O `  A ) ) ) ) ) )
76oveq1d 6055 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  .x.  A )  =  ( ( N  -  (
( O `  A
)  x.  ( |_
`  ( N  / 
( O `  A
) ) ) ) )  .x.  A ) )
8 simpl1 960 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  G  e.  Grp )
93nnzd 10330 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  ZZ )
102, 3nndivred 10004 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  / 
( O `  A
) )  e.  RR )
1110flcld 11162 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( |_ `  ( N  /  ( O `  A )
) )  e.  ZZ )
129, 11zmulcld 10337 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  x.  ( |_ `  ( N  /  ( O `  A ) ) ) )  e.  ZZ )
13 simpl2 961 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  A  e.  X
)
14 odcl.1 . . . 4  |-  X  =  ( Base `  G
)
15 odid.3 . . . 4  |-  .x.  =  (.g
`  G )
16 eqid 2404 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
1714, 15, 16mulgsubdir 14876 . . 3  |-  ( ( G  e.  Grp  /\  ( N  e.  ZZ  /\  ( ( O `  A )  x.  ( |_ `  ( N  / 
( O `  A
) ) ) )  e.  ZZ  /\  A  e.  X ) )  -> 
( ( N  -  ( ( O `  A )  x.  ( |_ `  ( N  / 
( O `  A
) ) ) ) )  .x.  A )  =  ( ( N 
.x.  A ) (
-g `  G )
( ( ( O `
 A )  x.  ( |_ `  ( N  /  ( O `  A ) ) ) )  .x.  A ) ) )
188, 1, 12, 13, 17syl13anc 1186 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  -  ( ( O `
 A )  x.  ( |_ `  ( N  /  ( O `  A ) ) ) ) )  .x.  A
)  =  ( ( N  .x.  A ) ( -g `  G
) ( ( ( O `  A )  x.  ( |_ `  ( N  /  ( O `  A )
) ) )  .x.  A ) ) )
19 nncn 9964 . . . . . . . 8  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  e.  CC )
20 zcn 10243 . . . . . . . 8  |-  ( ( |_ `  ( N  /  ( O `  A ) ) )  e.  ZZ  ->  ( |_ `  ( N  / 
( O `  A
) ) )  e.  CC )
21 mulcom 9032 . . . . . . . 8  |-  ( ( ( O `  A
)  e.  CC  /\  ( |_ `  ( N  /  ( O `  A ) ) )  e.  CC )  -> 
( ( O `  A )  x.  ( |_ `  ( N  / 
( O `  A
) ) ) )  =  ( ( |_
`  ( N  / 
( O `  A
) ) )  x.  ( O `  A
) ) )
2219, 20, 21syl2an 464 . . . . . . 7  |-  ( ( ( O `  A
)  e.  NN  /\  ( |_ `  ( N  /  ( O `  A ) ) )  e.  ZZ )  -> 
( ( O `  A )  x.  ( |_ `  ( N  / 
( O `  A
) ) ) )  =  ( ( |_
`  ( N  / 
( O `  A
) ) )  x.  ( O `  A
) ) )
233, 11, 22syl2anc 643 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  x.  ( |_ `  ( N  /  ( O `  A ) ) ) )  =  ( ( |_ `  ( N  /  ( O `  A ) ) )  x.  ( O `  A ) ) )
2423oveq1d 6055 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( O `  A )  x.  ( |_ `  ( N  /  ( O `  A )
) ) )  .x.  A )  =  ( ( ( |_ `  ( N  /  ( O `  A )
) )  x.  ( O `  A )
)  .x.  A )
)
2514, 15mulgass 14875 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( ( |_ `  ( N  /  ( O `  A )
) )  e.  ZZ  /\  ( O `  A
)  e.  ZZ  /\  A  e.  X )
)  ->  ( (
( |_ `  ( N  /  ( O `  A ) ) )  x.  ( O `  A ) )  .x.  A )  =  ( ( |_ `  ( N  /  ( O `  A ) ) ) 
.x.  ( ( O `
 A )  .x.  A ) ) )
268, 11, 9, 13, 25syl13anc 1186 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( |_ `  ( N  /  ( O `  A ) ) )  x.  ( O `  A ) )  .x.  A )  =  ( ( |_ `  ( N  /  ( O `  A ) ) ) 
.x.  ( ( O `
 A )  .x.  A ) ) )
27 odcl.2 . . . . . . . . 9  |-  O  =  ( od `  G
)
28 odid.4 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
2914, 27, 15, 28odid 15131 . . . . . . . 8  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  .0.  )
3013, 29syl 16 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  .x.  A )  =  .0.  )
3130oveq2d 6056 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( |_
`  ( N  / 
( O `  A
) ) )  .x.  ( ( O `  A )  .x.  A
) )  =  ( ( |_ `  ( N  /  ( O `  A ) ) ) 
.x.  .0.  ) )
3214, 15, 28mulgz 14866 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( |_ `  ( N  /  ( O `  A ) ) )  e.  ZZ )  -> 
( ( |_ `  ( N  /  ( O `  A )
) )  .x.  .0.  )  =  .0.  )
338, 11, 32syl2anc 643 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( |_
`  ( N  / 
( O `  A
) ) )  .x.  .0.  )  =  .0.  )
3431, 33eqtrd 2436 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( |_
`  ( N  / 
( O `  A
) ) )  .x.  ( ( O `  A )  .x.  A
) )  =  .0.  )
3524, 26, 343eqtrd 2440 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( O `  A )  x.  ( |_ `  ( N  /  ( O `  A )
) ) )  .x.  A )  =  .0.  )
3635oveq2d 6056 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N 
.x.  A ) (
-g `  G )
( ( ( O `
 A )  x.  ( |_ `  ( N  /  ( O `  A ) ) ) )  .x.  A ) )  =  ( ( N  .x.  A ) ( -g `  G
)  .0.  ) )
3714, 15mulgcl 14862 . . . . 5  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( N  .x.  A )  e.  X )
388, 1, 13, 37syl3anc 1184 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  .x.  A )  e.  X
)
3914, 28, 16grpsubid1 14829 . . . 4  |-  ( ( G  e.  Grp  /\  ( N  .x.  A )  e.  X )  -> 
( ( N  .x.  A ) ( -g `  G )  .0.  )  =  ( N  .x.  A ) )
408, 38, 39syl2anc 643 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N 
.x.  A ) (
-g `  G )  .0.  )  =  ( N  .x.  A ) )
4136, 40eqtrd 2436 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N 
.x.  A ) (
-g `  G )
( ( ( O `
 A )  x.  ( |_ `  ( N  /  ( O `  A ) ) ) )  .x.  A ) )  =  ( N 
.x.  A ) )
427, 18, 413eqtrd 2440 1  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  .x.  A )  =  ( N  .x.  A ) )
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   RRcr 8945    x. cmul 8951    - cmin 9247    / cdiv 9633   NNcn 9956   ZZcz 10238   RR+crp 10568   |_cfl 11156    mod cmo 11205   Basecbs 13424   0gc0g 13678   Grpcgrp 14640   -gcsg 14643  .gcmg 14644   odcod 15118
This theorem is referenced by:  oddvds  15140  odf1o2  15162
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  ax-pre-sup 9024
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-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-od 15122
  Copyright terms: Public domain W3C validator