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Theorem odmod 14861
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 960 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  ZZ )
21zred 10117 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  RR )
3 simpr 447 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  NN )
43nnrpd 10389 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR+ )
5 modval 10975 . . . 4  |-  ( ( N  e.  RR  /\  ( O `  A )  e.  RR+ )  ->  ( N  mod  ( O `  A ) )  =  ( N  -  (
( O `  A
)  x.  ( |_
`  ( N  / 
( O `  A
) ) ) ) ) )
62, 4, 5syl2anc 642 . . 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 5873 . 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 958 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  G  e.  Grp )
93nnzd 10116 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  ZZ )
102, 3nndivred 9794 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  / 
( O `  A
) )  e.  RR )
1110flcld 10930 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( |_ `  ( N  /  ( O `  A )
) )  e.  ZZ )
129, 11zmulcld 10123 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  x.  ( |_ `  ( N  /  ( O `  A ) ) ) )  e.  ZZ )
13 simpl2 959 . . 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 2283 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
1714, 15, 16mulgsubdir 14598 . . 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 1184 . 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 9754 . . . . . . . 8  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  e.  CC )
20 zcn 10029 . . . . . . . 8  |-  ( ( |_ `  ( N  /  ( O `  A ) ) )  e.  ZZ  ->  ( |_ `  ( N  / 
( O `  A
) ) )  e.  CC )
21 mulcom 8823 . . . . . . . 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 463 . . . . . . 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 642 . . . . . 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 5873 . . . . 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 14597 . . . . . 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 1184 . . . . 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 14853 . . . . . . . 8  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  .0.  )
3013, 29syl 15 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  .x.  A )  =  .0.  )
3130oveq2d 5874 . . . . . 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 14588 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( |_ `  ( N  /  ( O `  A ) ) )  e.  ZZ )  -> 
( ( |_ `  ( N  /  ( O `  A )
) )  .x.  .0.  )  =  .0.  )
338, 11, 32syl2anc 642 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( |_
`  ( N  / 
( O `  A
) ) )  .x.  .0.  )  =  .0.  )
3431, 33eqtrd 2315 . . . . 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 2319 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( O `  A )  x.  ( |_ `  ( N  /  ( O `  A )
) ) )  .x.  A )  =  .0.  )
3635oveq2d 5874 . . 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 14584 . . . . 5  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( N  .x.  A )  e.  X )
388, 1, 13, 37syl3anc 1182 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  .x.  A )  e.  X
)
3914, 28, 16grpsubid1 14551 . . . 4  |-  ( ( G  e.  Grp  /\  ( N  .x.  A )  e.  X )  -> 
( ( N  .x.  A ) ( -g `  G )  .0.  )  =  ( N  .x.  A ) )
408, 38, 39syl2anc 642 . . 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 2315 . 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 2319 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736    x. cmul 8742    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   RR+crp 10354   |_cfl 10924    mod cmo 10973   Basecbs 13148   0gc0g 13400   Grpcgrp 14362   -gcsg 14365  .gcmg 14366   odcod 14840
This theorem is referenced by:  oddvds  14862  odf1o2  14884
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  ax-pre-sup 8815
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-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-od 14844
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