MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  odmulgeq Structured version   Unicode version

Theorem odmulgeq 16706
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 2466 . 2  |-  ( ( O `  ( N 
.x.  A ) )  =  ( O `  A )  <->  ( O `  A )  =  ( O `  ( N 
.x.  A ) ) )
2 simpl2 1000 . . . . . 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 16687 . . . . . 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 10875 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  CC )
8 simpl1 999 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  G  e.  Grp )
9 simpl3 1001 . . . . . . 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 16286 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( N  .x.  A )  e.  X )
128, 9, 2, 11syl3anc 1228 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  .x.  A )  e.  X
)
133, 4odcl 16687 . . . . . 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 10875 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  e.  CC )
16 nnne0 10589 . . . . . 6  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  =/=  0 )
1716adantl 466 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  =/=  0
)
183, 4, 10odmulg2 16704 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  ( N  .x.  A ) ) 
||  ( O `  A ) )
1918adantr 465 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  ( N  .x.  A ) )  ||  ( O `
 A ) )
20 breq1 4459 . . . . . . . 8  |-  ( ( O `  ( N 
.x.  A ) )  =  0  ->  (
( O `  ( N  .x.  A ) ) 
||  ( O `  A )  <->  0  ||  ( O `  A ) ) )
2119, 20syl5ibcom 220 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( N  .x.  A ) )  =  0  ->  0  ||  ( O `  A ) ) )
226nn0zd 10988 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  ZZ )
23 0dvds 14016 . . . . . . . 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 214 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 ( N  .x.  A ) )  =  0  ->  ( O `  A )  =  0 ) )
2625necon3d 2681 . . . . 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 10350 . . 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 14168 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  gcd  ( O `  A ) )  e.  NN0 )
3029nn0cnd 10875 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  gcd  ( O `  A ) )  e.  CC )
3115, 30mulcomd 9634 . . . . . 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 16705 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  A
)  =  ( ( N  gcd  ( O `
 A ) )  x.  ( O `  ( N  .x.  A ) ) ) )
3332adantr 465 . . . . . 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 2501 . . . . 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 10363 . . . . 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 232 . . . 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 2459 . . 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 255 . 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 257 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    x. cmul 9514    / cdiv 10227   NNcn 10556   NN0cn0 10816   ZZcz 10885    || cdvds 13998    gcd cgcd 14156   Basecbs 14644   Grpcgrp 16180  .gcmg 16183   odcod 16676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-od 16680
This theorem is referenced by:  odngen  16724
  Copyright terms: Public domain W3C validator