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Theorem odmulgeq 15195
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 2440 . 2  |-  ( ( O `  ( N 
.x.  A ) )  =  ( O `  A )  <->  ( O `  A )  =  ( O `  ( N 
.x.  A ) ) )
2 simpl2 962 . . . . . 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 15176 . . . . . 6  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
62, 5syl 16 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  NN0 )
76nn0cnd 10278 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  CC )
8 simpl1 961 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  G  e.  Grp )
9 simpl3 963 . . . . . . 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 14909 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( N  .x.  A )  e.  X )
128, 9, 2, 11syl3anc 1185 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  .x.  A )  e.  X
)
133, 4odcl 15176 . . . . . 6  |-  ( ( N  .x.  A )  e.  X  ->  ( O `  ( N  .x.  A ) )  e. 
NN0 )
1412, 13syl 16 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  e.  NN0 )
1514nn0cnd 10278 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  e.  CC )
16 nnne0 10034 . . . . . 6  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  =/=  0 )
1716adantl 454 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  =/=  0
)
183, 4, 10odmulg2 15193 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  ( N  .x.  A ) ) 
||  ( O `  A ) )
1918adantr 453 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  ||  ( O `
 A ) )
20 breq1 4217 . . . . . . . 8  |-  ( ( O `  ( N 
.x.  A ) )  =  0  ->  (
( O `  ( N  .x.  A ) ) 
||  ( O `  A )  <->  0  ||  ( O `  A ) ) )
2119, 20syl5ibcom 213 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( N  .x.  A ) )  =  0  ->  0  ||  ( O `  A ) ) )
226nn0zd 10375 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  ZZ )
23 0dvds 12872 . . . . . . . 8  |-  ( ( O `  A )  e.  ZZ  ->  (
0  ||  ( O `  A )  <->  ( O `  A )  =  0 ) )
2422, 23syl 16 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( 0  ||  ( O `  A )  <-> 
( O `  A
)  =  0 ) )
2521, 24sylibd 207 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( N  .x.  A ) )  =  0  ->  ( O `  A )  =  0 ) )
2625necon3d 2641 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  =/=  0  ->  ( O `  ( N  .x.  A
) )  =/=  0
) )
2717, 26mpd 15 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  =/=  0 )
287, 15, 27diveq1ad 9801 . . 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 ) ) ) )
299, 22gcdcld 13020 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  gcd  ( O `  A ) )  e.  NN0 )
3029nn0cnd 10278 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  gcd  ( O `  A ) )  e.  CC )
3115, 30mulcomd 9111 . . . . . 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 ) ) ) )
323, 4, 10odmulg 15194 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  A
)  =  ( ( N  gcd  ( O `
 A ) )  x.  ( O `  ( N  .x.  A ) ) ) )
3332adantr 453 . . . . . 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 ) ) ) )
3431, 33eqtr4d 2473 . . . . 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 ) )
357, 15, 30, 27divmuld 9814 . . . . 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 ) ) )
3634, 35mpbird 225 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  / 
( O `  ( N  .x.  A ) ) )  =  ( N  gcd  ( O `  A ) ) )
3736eqeq1d 2446 . . 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 ) )
3828, 37bitr3d 248 . 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 ) )
391, 38syl5bb 250 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993    x. cmul 8997    / cdiv 9679   NNcn 10002   NN0cn0 10223   ZZcz 10284    || cdivides 12854    gcd cgcd 13008   Basecbs 13471   Grpcgrp 14687  .gcmg 14691   odcod 15165
This theorem is referenced by:  odngen  15213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
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 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-gcd 13009  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-od 15169
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