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Theorem odmulgeq 14886
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 2298 . 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 14867 . . . . . 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 10036 . . . 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 14600 . . . . . . 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 14867 . . . . . 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 10036 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  e.  CC )
16 nnne0 9794 . . . . . 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 14884 . . . . . . . . 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 4042 . . . . . . . 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 10131 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  ZZ )
23 0dvds 12565 . . . . . . . 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 2497 . . . . 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 9470 . . . 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 12713 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  gcd  ( O `  A ) )  e.  NN0 )
3130nn0cnd 10036 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  gcd  ( O `  A ) )  e.  CC )
3215, 31mulcomd 8872 . . . . . 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 14885 . . . . . . 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 2331 . . . . 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 9574 . . . . 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 2304 . . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040    || cdivides 12547    gcd cgcd 12701   Basecbs 13164   Grpcgrp 14378  .gcmg 14382   odcod 14856
This theorem is referenced by:  odngen  14904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-od 14860
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