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Theorem odmod 14960
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 10209 . . . 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 10481 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR+ )
5 modval 11067 . . . 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 5960 . 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 10208 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  ZZ )
102, 3nndivred 9884 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  / 
( O `  A
) )  e.  RR )
1110flcld 11022 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( |_ `  ( N  /  ( O `  A )
) )  e.  ZZ )
129, 11zmulcld 10215 . . 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 2358 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
1714, 15, 16mulgsubdir 14697 . . 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 9844 . . . . . . . 8  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  e.  CC )
20 zcn 10121 . . . . . . . 8  |-  ( ( |_ `  ( N  /  ( O `  A ) ) )  e.  ZZ  ->  ( |_ `  ( N  / 
( O `  A
) ) )  e.  CC )
21 mulcom 8913 . . . . . . . 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 5960 . . . . 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 14696 . . . . . 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 14952 . . . . . . . 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 5961 . . . . . 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 14687 . . . . . . 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 2390 . . . . 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 2394 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( O `  A )  x.  ( |_ `  ( N  /  ( O `  A )
) ) )  .x.  A )  =  .0.  )
3635oveq2d 5961 . . 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 14683 . . . . 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 14650 . . . 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 2390 . 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 2394 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 1642    e. wcel 1710   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826    x. cmul 8832    - cmin 9127    / cdiv 9513   NNcn 9836   ZZcz 10116   RR+crp 10446   |_cfl 11016    mod cmo 11065   Basecbs 13245   0gc0g 13499   Grpcgrp 14461   -gcsg 14464  .gcmg 14465   odcod 14939
This theorem is referenced by:  oddvds  14961  odf1o2  14983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-fz 10875  df-fl 11017  df-mod 11066  df-seq 11139  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-od 14943
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