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Theorem odmulgeq 14870
Description: A multiple of a point of finite order only has the same order if the multiplier is relatively prime. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Hypotheses
Ref Expression
odmulgid.1  |-  X  =  ( Base `  G
)
odmulgid.2  |-  O  =  ( od `  G
)
odmulgid.3  |-  .x.  =  (.g
`  G )
Assertion
Ref Expression
odmulgeq  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( N  .x.  A ) )  =  ( O `  A
)  <->  ( N  gcd  ( O `  A ) )  =  1 ) )

Proof of Theorem odmulgeq
StepHypRef Expression
1 eqcom 2285 . 2  |-  ( ( O `  ( N 
.x.  A ) )  =  ( O `  A )  <->  ( O `  A )  =  ( O `  ( N 
.x.  A ) ) )
2 simpl2 959 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  A  e.  X
)
3 odmulgid.1 . . . . . . 7  |-  X  =  ( Base `  G
)
4 odmulgid.2 . . . . . . 7  |-  O  =  ( od `  G
)
53, 4odcl 14851 . . . . . 6  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
62, 5syl 15 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  NN0 )
76nn0cnd 10020 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  CC )
8 simpl1 958 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  G  e.  Grp )
9 simpl3 960 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  ZZ )
10 odmulgid.3 . . . . . . . 8  |-  .x.  =  (.g
`  G )
113, 10mulgcl 14584 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( N  .x.  A )  e.  X )
128, 9, 2, 11syl3anc 1182 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  .x.  A )  e.  X
)
133, 4odcl 14851 . . . . . 6  |-  ( ( N  .x.  A )  e.  X  ->  ( O `  ( N  .x.  A ) )  e. 
NN0 )
1412, 13syl 15 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  e.  NN0 )
1514nn0cnd 10020 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  e.  CC )
16 nnne0 9778 . . . . . 6  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  =/=  0 )
1716adantl 452 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  =/=  0
)
183, 4, 10odmulg2 14868 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  ( N  .x.  A ) ) 
||  ( O `  A ) )
1918adantr 451 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  ||  ( O `
 A ) )
20 breq1 4026 . . . . . . . 8  |-  ( ( O `  ( N 
.x.  A ) )  =  0  ->  (
( O `  ( N  .x.  A ) ) 
||  ( O `  A )  <->  0  ||  ( O `  A ) ) )
2119, 20syl5ibcom 211 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( N  .x.  A ) )  =  0  ->  0  ||  ( O `  A ) ) )
226nn0zd 10115 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  ZZ )
23 0dvds 12549 . . . . . . . 8  |-  ( ( O `  A )  e.  ZZ  ->  (
0  ||  ( O `  A )  <->  ( O `  A )  =  0 ) )
2422, 23syl 15 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( 0  ||  ( O `  A )  <-> 
( O `  A
)  =  0 ) )
2521, 24sylibd 205 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( N  .x.  A ) )  =  0  ->  ( O `  A )  =  0 ) )
2625necon3d 2484 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  =/=  0  ->  ( O `  ( N  .x.  A
) )  =/=  0
) )
2717, 26mpd 14 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  =/=  0 )
28 diveq1 9454 . . . 4  |-  ( ( ( O `  A
)  e.  CC  /\  ( O `  ( N 
.x.  A ) )  e.  CC  /\  ( O `  ( N  .x.  A ) )  =/=  0 )  ->  (
( ( O `  A )  /  ( O `  ( N  .x.  A ) ) )  =  1  <->  ( O `  A )  =  ( O `  ( N 
.x.  A ) ) ) )
297, 15, 27, 28syl3anc 1182 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( O `  A )  /  ( O `  ( N  .x.  A ) ) )  =  1  <-> 
( O `  A
)  =  ( O `
 ( N  .x.  A ) ) ) )
309, 22gcdcld 12697 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  gcd  ( O `  A ) )  e.  NN0 )
3130nn0cnd 10020 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  gcd  ( O `  A ) )  e.  CC )
3215, 31mulcomd 8856 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( N  .x.  A ) )  x.  ( N  gcd  ( O `  A )
) )  =  ( ( N  gcd  ( O `  A )
)  x.  ( O `
 ( N  .x.  A ) ) ) )
333, 4, 10odmulg 14869 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  A
)  =  ( ( N  gcd  ( O `
 A ) )  x.  ( O `  ( N  .x.  A ) ) ) )
3433adantr 451 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  =  ( ( N  gcd  ( O `  A )
)  x.  ( O `
 ( N  .x.  A ) ) ) )
3532, 34eqtr4d 2318 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( N  .x.  A ) )  x.  ( N  gcd  ( O `  A )
) )  =  ( O `  A ) )
367, 15, 31, 27divmuld 9558 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( O `  A )  /  ( O `  ( N  .x.  A ) ) )  =  ( N  gcd  ( O `
 A ) )  <-> 
( ( O `  ( N  .x.  A ) )  x.  ( N  gcd  ( O `  A ) ) )  =  ( O `  A ) ) )
3735, 36mpbird 223 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  / 
( O `  ( N  .x.  A ) ) )  =  ( N  gcd  ( O `  A ) ) )
3837eqeq1d 2291 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( O `  A )  /  ( O `  ( N  .x.  A ) ) )  =  1  <-> 
( N  gcd  ( O `  A )
)  =  1 ) )
3929, 38bitr3d 246 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  =  ( O `  ( N  .x.  A ) )  <-> 
( N  gcd  ( O `  A )
)  =  1 ) )
401, 39syl5bb 248 1  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( N  .x.  A ) )  =  ( O `  A
)  <->  ( N  gcd  ( O `  A ) )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024    || cdivides 12531    gcd cgcd 12685   Basecbs 13148   Grpcgrp 14362  .gcmg 14366   odcod 14840
This theorem is referenced by:  odngen  14888
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-2 9804  df-3 9805  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-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  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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