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Theorem odmod 16549
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 1002 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  ZZ )
21zred 10976 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  RR )
3 simpr 461 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  NN )
43nnrpd 11266 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR+ )
5 modval 11980 . . . 4  |-  ( ( N  e.  RR  /\  ( O `  A )  e.  RR+ )  ->  ( N  mod  ( O `  A ) )  =  ( N  -  (
( O `  A
)  x.  ( |_
`  ( N  / 
( O `  A
) ) ) ) ) )
62, 4, 5syl2anc 661 . . 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 6296 . 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 1000 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  G  e.  Grp )
93nnzd 10975 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  ZZ )
102, 3nndivred 10591 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  / 
( O `  A
) )  e.  RR )
1110flcld 11917 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( |_ `  ( N  /  ( O `  A )
) )  e.  ZZ )
129, 11zmulcld 10982 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  x.  ( |_ `  ( N  /  ( O `  A ) ) ) )  e.  ZZ )
13 simpl2 1001 . . 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 2443 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
1714, 15, 16mulgsubdir 16152 . . 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 1231 . 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 10551 . . . . . . . 8  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  e.  CC )
20 zcn 10876 . . . . . . . 8  |-  ( ( |_ `  ( N  /  ( O `  A ) ) )  e.  ZZ  ->  ( |_ `  ( N  / 
( O `  A
) ) )  e.  CC )
21 mulcom 9581 . . . . . . . 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 477 . . . . . . 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 661 . . . . . 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 6296 . . . . 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 16151 . . . . . 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 1231 . . . . 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 16541 . . . . . . . 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 6297 . . . . . 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 16142 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( |_ `  ( N  /  ( O `  A ) ) )  e.  ZZ )  -> 
( ( |_ `  ( N  /  ( O `  A )
) )  .x.  .0.  )  =  .0.  )
338, 11, 32syl2anc 661 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( |_
`  ( N  / 
( O `  A
) ) )  .x.  .0.  )  =  .0.  )
3431, 33eqtrd 2484 . . . . 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 2488 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( O `  A )  x.  ( |_ `  ( N  /  ( O `  A )
) ) )  .x.  A )  =  .0.  )
3635oveq2d 6297 . . 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 16138 . . . . 5  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( N  .x.  A )  e.  X )
388, 1, 13, 37syl3anc 1229 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  .x.  A )  e.  X
)
3914, 28, 16grpsubid1 16102 . . . 4  |-  ( ( G  e.  Grp  /\  ( N  .x.  A )  e.  X )  -> 
( ( N  .x.  A ) ( -g `  G )  .0.  )  =  ( N  .x.  A ) )
408, 38, 39syl2anc 661 . . 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 2484 . 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 2488 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494    x. cmul 9500    - cmin 9810    / cdiv 10213   NNcn 10543   ZZcz 10871   RR+crp 11231   |_cfl 11909    mod cmo 11978   Basecbs 14614   0gc0g 14819   Grpcgrp 16032   -gcsg 16034  .gcmg 16035   odcod 16528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-fz 11684  df-fl 11911  df-mod 11979  df-seq 12090  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-sbg 16038  df-mulg 16039  df-od 16532
This theorem is referenced by:  oddvds  16550  odf1o2  16572
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