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Theorem oddvds 16043
Description: The only multiples of  A that are equal to the identity are the multiples of the order of  A. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-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
oddvds  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )

Proof of Theorem oddvds
StepHypRef Expression
1 simpr 458 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  NN )
2 simpl3 988 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  ZZ )
3 dvdsval3 13535 . . . 4  |-  ( ( ( O `  A
)  e.  NN  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  mod  ( O `
 A ) )  =  0 ) )
41, 2, 3syl2anc 656 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  ||  N 
<->  ( N  mod  ( O `  A )
)  =  0 ) )
5 simpl2 987 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  A  e.  X
)
6 odcl.1 . . . . . . 7  |-  X  =  ( Base `  G
)
7 odid.4 . . . . . . 7  |-  .0.  =  ( 0g `  G )
8 odid.3 . . . . . . 7  |-  .x.  =  (.g
`  G )
96, 7, 8mulg0 15625 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  .0.  )
105, 9syl 16 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( 0  .x. 
A )  =  .0.  )
11 oveq1 6097 . . . . . 6  |-  ( ( N  mod  ( O `
 A ) )  =  0  ->  (
( N  mod  ( O `  A )
)  .x.  A )  =  ( 0  .x. 
A ) )
1211eqeq1d 2449 . . . . 5  |-  ( ( N  mod  ( O `
 A ) )  =  0  ->  (
( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  <->  ( 0 
.x.  A )  =  .0.  ) )
1310, 12syl5ibrcom 222 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  =  0  ->  ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  ) )
142zred 10743 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  RR )
151nnrpd 11022 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR+ )
16 modlt 11714 . . . . . . . 8  |-  ( ( N  e.  RR  /\  ( O `  A )  e.  RR+ )  ->  ( N  mod  ( O `  A ) )  < 
( O `  A
) )
1714, 15, 16syl2anc 656 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  <  ( O `
 A ) )
182, 1zmodcld 11724 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  e.  NN0 )
1918nn0red 10633 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  e.  RR )
201nnred 10333 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR )
2119, 20ltnled 9517 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  < 
( O `  A
)  <->  -.  ( O `  A )  <_  ( N  mod  ( O `  A ) ) ) )
2217, 21mpbid 210 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  -.  ( O `  A )  <_  ( N  mod  ( O `  A ) ) )
23 odcl.2 . . . . . . . . . . . 12  |-  O  =  ( od `  G
)
246, 23, 8, 7odlem2 16035 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  ( N  mod  ( O `
 A ) )  e.  NN  /\  (
( N  mod  ( O `  A )
)  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... ( N  mod  ( O `  A ) ) ) )
25 elfzle2 11451 . . . . . . . . . . 11  |-  ( ( O `  A )  e.  ( 1 ... ( N  mod  ( O `  A )
) )  ->  ( O `  A )  <_  ( N  mod  ( O `  A )
) )
2624, 25syl 16 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  ( N  mod  ( O `
 A ) )  e.  NN  /\  (
( N  mod  ( O `  A )
)  .x.  A )  =  .0.  )  ->  ( O `  A )  <_  ( N  mod  ( O `  A )
) )
27263com23 1188 . . . . . . . . 9  |-  ( ( A  e.  X  /\  ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  /\  ( N  mod  ( O `  A ) )  e.  NN )  ->  ( O `  A )  <_  ( N  mod  ( O `  A )
) )
28273expia 1184 . . . . . . . 8  |-  ( ( A  e.  X  /\  ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  )  -> 
( ( N  mod  ( O `  A ) )  e.  NN  ->  ( O `  A )  <_  ( N  mod  ( O `  A ) ) ) )
2928con3d 133 . . . . . . 7  |-  ( ( A  e.  X  /\  ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  )  -> 
( -.  ( O `
 A )  <_ 
( N  mod  ( O `  A )
)  ->  -.  ( N  mod  ( O `  A ) )  e.  NN ) )
3029impancom 438 . . . . . 6  |-  ( ( A  e.  X  /\  -.  ( O `  A
)  <_  ( N  mod  ( O `  A
) ) )  -> 
( ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0. 
->  -.  ( N  mod  ( O `  A ) )  e.  NN ) )
315, 22, 30syl2anc 656 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( N  mod  ( O `
 A ) ) 
.x.  A )  =  .0.  ->  -.  ( N  mod  ( O `  A ) )  e.  NN ) )
32 elnn0 10577 . . . . . . 7  |-  ( ( N  mod  ( O `
 A ) )  e.  NN0  <->  ( ( N  mod  ( O `  A ) )  e.  NN  \/  ( N  mod  ( O `  A ) )  =  0 ) )
3318, 32sylib 196 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  e.  NN  \/  ( N  mod  ( O `  A ) )  =  0 ) )
3433ord 377 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( -.  ( N  mod  ( O `  A ) )  e.  NN  ->  ( N  mod  ( O `  A
) )  =  0 ) )
3531, 34syld 44 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( N  mod  ( O `
 A ) ) 
.x.  A )  =  .0.  ->  ( N  mod  ( O `  A
) )  =  0 ) )
3613, 35impbid 191 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  =  0  <->  ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  ) )
376, 23, 8, 7odmod 16042 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  .x.  A )  =  ( N  .x.  A ) )
3837eqeq1d 2449 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( ( N  mod  ( O `
 A ) ) 
.x.  A )  =  .0.  <->  ( N  .x.  A )  =  .0.  ) )
394, 36, 383bitrd 279 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  ||  N 
<->  ( N  .x.  A
)  =  .0.  )
)
40 simpr 458 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( O `  A )  =  0 )
4140breq1d 4299 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( ( O `
 A )  ||  N 
<->  0  ||  N ) )
42 simpl3 988 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  N  e.  ZZ )
43 0dvds 13549 . . . 4  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
4442, 43syl 16 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( 0  ||  N 
<->  N  =  0 ) )
45 simpl2 987 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  A  e.  X
)
4645, 9syl 16 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( 0  .x. 
A )  =  .0.  )
47 oveq1 6097 . . . . . 6  |-  ( N  =  0  ->  ( N  .x.  A )  =  ( 0  .x.  A
) )
4847eqeq1d 2449 . . . . 5  |-  ( N  =  0  ->  (
( N  .x.  A
)  =  .0.  <->  ( 0 
.x.  A )  =  .0.  ) )
4946, 48syl5ibrcom 222 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( N  =  0  ->  ( N  .x.  A )  =  .0.  ) )
506, 23, 8, 7odnncl 16041 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  NN )
5150nnne0d 10362 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  =/=  0
)
5251expr 612 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( ( N  .x.  A )  =  .0.  ->  ( O `  A )  =/=  0
) )
5352impancom 438 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  .x.  A
)  =  .0.  )  ->  ( N  =/=  0  ->  ( O `  A
)  =/=  0 ) )
5453necon4d 2672 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  .x.  A
)  =  .0.  )  ->  ( ( O `  A )  =  0  ->  N  =  0 ) )
5554impancom 438 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( ( N 
.x.  A )  =  .0.  ->  N  = 
0 ) )
5649, 55impbid 191 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( N  =  0  <->  ( N  .x.  A )  =  .0.  ) )
5741, 44, 563bitrd 279 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( ( O `
 A )  ||  N 
<->  ( N  .x.  A
)  =  .0.  )
)
586, 23odcl 16032 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
59583ad2ant2 1005 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  A
)  e.  NN0 )
60 elnn0 10577 . . 3  |-  ( ( O `  A )  e.  NN0  <->  ( ( O `
 A )  e.  NN  \/  ( O `
 A )  =  0 ) )
6159, 60sylib 196 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  e.  NN  \/  ( O `  A
)  =  0 ) )
6239, 57, 61mpjaodan 779 1  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   RRcr 9277   0cc0 9278   1c1 9279    < clt 9414    <_ cle 9415   NNcn 10318   NN0cn0 10575   ZZcz 10642   RR+crp 10987   ...cfz 11433    mod cmo 11704    || cdivides 13531   Basecbs 14170   0gc0g 14374   Grpcgrp 15406  .gcmg 15410   odcod 16021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-od 16025
This theorem is referenced by:  oddvdsi  16044  odcong  16045  odeq  16046  odmulgid  16048  odbezout  16052  gexdvds2  16077  gexod  16078  gexcl3  16079  odadd1  16323  odadd2  16324  oddvdssubg  16330  pgpfac1lem3a  16567  chrdvds  17918  dchrfi  22553  dchrabs  22558  dchrptlem2  22563  idomodle  29486
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