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Theorem oddvds 16444
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 461 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  NN )
2 simpl3 1001 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  ZZ )
3 dvdsval3 13868 . . . 4  |-  ( ( ( O `  A
)  e.  NN  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  mod  ( O `
 A ) )  =  0 ) )
41, 2, 3syl2anc 661 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  ||  N 
<->  ( N  mod  ( O `  A )
)  =  0 ) )
5 simpl2 1000 . . . . . 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 16019 . . . . . 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 6302 . . . . . 6  |-  ( ( N  mod  ( O `
 A ) )  =  0  ->  (
( N  mod  ( O `  A )
)  .x.  A )  =  ( 0  .x. 
A ) )
1211eqeq1d 2469 . . . . 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 10978 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  RR )
151nnrpd 11267 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR+ )
16 modlt 11986 . . . . . . . 8  |-  ( ( N  e.  RR  /\  ( O `  A )  e.  RR+ )  ->  ( N  mod  ( O `  A ) )  < 
( O `  A
) )
1714, 15, 16syl2anc 661 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  <  ( O `
 A ) )
182, 1zmodcld 11996 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  e.  NN0 )
1918nn0red 10865 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  e.  RR )
201nnred 10563 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR )
2119, 20ltnled 9743 . . . . . . 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 16436 . . . . . . . . . . 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 11702 . . . . . . . . . . 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 1202 . . . . . . . . 9  |-  ( ( A  e.  X  /\  ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  /\  ( N  mod  ( O `  A ) )  e.  NN )  ->  ( O `  A )  <_  ( N  mod  ( O `  A )
) )
28273expia 1198 . . . . . . . 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 440 . . . . . 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 661 . . . . 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 10809 . . . . . . 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 16443 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  .x.  A )  =  ( N  .x.  A ) )
3837eqeq1d 2469 . . 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 461 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( O `  A )  =  0 )
4140breq1d 4463 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( ( O `
 A )  ||  N 
<->  0  ||  N ) )
42 simpl3 1001 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  N  e.  ZZ )
43 0dvds 13882 . . . 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 1000 . . . . . 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 6302 . . . . . 6  |-  ( N  =  0  ->  ( N  .x.  A )  =  ( 0  .x.  A
) )
4847eqeq1d 2469 . . . . 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 16442 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  NN )
5150nnne0d 10592 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  =/=  0
)
5251expr 615 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( ( N  .x.  A )  =  .0.  ->  ( O `  A )  =/=  0
) )
5352impancom 440 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  .x.  A
)  =  .0.  )  ->  ( N  =/=  0  ->  ( O `  A
)  =/=  0 ) )
5453necon4d 2694 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  .x.  A
)  =  .0.  )  ->  ( ( O `  A )  =  0  ->  N  =  0 ) )
5554impancom 440 . . . 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 16433 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
59583ad2ant2 1018 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  A
)  e.  NN0 )
60 elnn0 10809 . . 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 784 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504   1c1 9505    < clt 9640    <_ cle 9641   NNcn 10548   NN0cn0 10807   ZZcz 10876   RR+crp 11232   ...cfz 11684    mod cmo 11976    || cdivides 13864   Basecbs 14507   0gc0g 14712   Grpcgrp 15925  .gcmg 15928   odcod 16422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-od 16426
This theorem is referenced by:  oddvdsi  16445  odcong  16446  odeq  16447  odmulgid  16449  odbezout  16453  gexdvds2  16478  gexod  16479  gexcl3  16480  odadd1  16727  odadd2  16728  oddvdssubg  16734  pgpfac1lem3a  16999  chrdvds  18434  dchrfi  23396  dchrabs  23401  dchrptlem2  23406  idomodle  31073
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