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Theorem oddvds 16166
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 993 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  ZZ )
3 dvdsval3 13652 . . . 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 992 . . . . . 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 15746 . . . . . 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 6202 . . . . . 6  |-  ( ( N  mod  ( O `
 A ) )  =  0  ->  (
( N  mod  ( O `  A )
)  .x.  A )  =  ( 0  .x. 
A ) )
1211eqeq1d 2454 . . . . 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 10853 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  RR )
151nnrpd 11132 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR+ )
16 modlt 11830 . . . . . . . 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 11840 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  e.  NN0 )
1918nn0red 10743 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  e.  RR )
201nnred 10443 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR )
2119, 20ltnled 9627 . . . . . . 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 16158 . . . . . . . . . . 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 11567 . . . . . . . . . . 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 1194 . . . . . . . . 9  |-  ( ( A  e.  X  /\  ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  /\  ( N  mod  ( O `  A ) )  e.  NN )  ->  ( O `  A )  <_  ( N  mod  ( O `  A )
) )
28273expia 1190 . . . . . . . 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 10687 . . . . . . 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 16165 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  .x.  A )  =  ( N  .x.  A ) )
3837eqeq1d 2454 . . 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 4405 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( ( O `
 A )  ||  N 
<->  0  ||  N ) )
42 simpl3 993 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  N  e.  ZZ )
43 0dvds 13666 . . . 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 992 . . . . . 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 6202 . . . . . 6  |-  ( N  =  0  ->  ( N  .x.  A )  =  ( 0  .x.  A
) )
4847eqeq1d 2454 . . . . 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 16164 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  NN )
5150nnne0d 10472 . . . . . . . 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 2676 . . . . 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 16155 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
59583ad2ant2 1010 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  A
)  e.  NN0 )
60 elnn0 10687 . . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   RRcr 9387   0cc0 9388   1c1 9389    < clt 9524    <_ cle 9525   NNcn 10428   NN0cn0 10685   ZZcz 10752   RR+crp 11097   ...cfz 11549    mod cmo 11820    || cdivides 13648   Basecbs 14287   0gc0g 14492   Grpcgrp 15524  .gcmg 15528   odcod 16144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-fl 11754  df-mod 11821  df-seq 11919  df-exp 11978  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-dvds 13649  df-0g 14494  df-mnd 15529  df-grp 15659  df-minusg 15660  df-sbg 15661  df-mulg 15662  df-od 16148
This theorem is referenced by:  oddvdsi  16167  odcong  16168  odeq  16169  odmulgid  16171  odbezout  16175  gexdvds2  16200  gexod  16201  gexcl3  16202  odadd1  16446  odadd2  16447  oddvdssubg  16453  pgpfac1lem3a  16694  chrdvds  18079  dchrfi  22722  dchrabs  22727  dchrptlem2  22732  idomodle  29704
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