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Theorem odmod 15189
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 963 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  ZZ )
21zred 10380 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  RR )
3 simpr 449 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  NN )
43nnrpd 10652 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR+ )
5 modval 11257 . . . 4  |-  ( ( N  e.  RR  /\  ( O `  A )  e.  RR+ )  ->  ( N  mod  ( O `  A ) )  =  ( N  -  (
( O `  A
)  x.  ( |_
`  ( N  / 
( O `  A
) ) ) ) ) )
62, 4, 5syl2anc 644 . . 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 6099 . 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 961 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  G  e.  Grp )
93nnzd 10379 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  ZZ )
102, 3nndivred 10053 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  / 
( O `  A
) )  e.  RR )
1110flcld 11212 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( |_ `  ( N  /  ( O `  A )
) )  e.  ZZ )
129, 11zmulcld 10386 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  x.  ( |_ `  ( N  /  ( O `  A ) ) ) )  e.  ZZ )
13 simpl2 962 . . 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 2438 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
1714, 15, 16mulgsubdir 14926 . . 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 1187 . 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 10013 . . . . . . . 8  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  e.  CC )
20 zcn 10292 . . . . . . . 8  |-  ( ( |_ `  ( N  /  ( O `  A ) ) )  e.  ZZ  ->  ( |_ `  ( N  / 
( O `  A
) ) )  e.  CC )
21 mulcom 9081 . . . . . . . 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 465 . . . . . . 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 644 . . . . . 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 6099 . . . . 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 14925 . . . . . 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 1187 . . . . 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 15181 . . . . . . . 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 6100 . . . . . 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 14916 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( |_ `  ( N  /  ( O `  A ) ) )  e.  ZZ )  -> 
( ( |_ `  ( N  /  ( O `  A )
) )  .x.  .0.  )  =  .0.  )
338, 11, 32syl2anc 644 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( |_
`  ( N  / 
( O `  A
) ) )  .x.  .0.  )  =  .0.  )
3431, 33eqtrd 2470 . . . . 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 2474 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( O `  A )  x.  ( |_ `  ( N  /  ( O `  A )
) ) )  .x.  A )  =  .0.  )
3635oveq2d 6100 . . 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 14912 . . . . 5  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( N  .x.  A )  e.  X )
388, 1, 13, 37syl3anc 1185 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  .x.  A )  e.  X
)
3914, 28, 16grpsubid1 14879 . . . 4  |-  ( ( G  e.  Grp  /\  ( N  .x.  A )  e.  X )  -> 
( ( N  .x.  A ) ( -g `  G )  .0.  )  =  ( N  .x.  A ) )
408, 38, 39syl2anc 644 . . 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 2470 . 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 2474 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994    x. cmul 9000    - cmin 9296    / cdiv 9682   NNcn 10005   ZZcz 10287   RR+crp 10617   |_cfl 11206    mod cmo 11255   Basecbs 13474   0gc0g 13728   Grpcgrp 14690   -gcsg 14693  .gcmg 14694   odcod 15168
This theorem is referenced by:  oddvds  15190  odf1o2  15212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fl 11207  df-mod 11256  df-seq 11329  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-od 15172
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