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Theorem oddvds 14878
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 447 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  NN )
2 simpl3 960 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  ZZ )
3 dvdsval3 12551 . . . 4  |-  ( ( ( O `  A
)  e.  NN  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  mod  ( O `
 A ) )  =  0 ) )
41, 2, 3syl2anc 642 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  ||  N 
<->  ( N  mod  ( O `  A )
)  =  0 ) )
5 simpl2 959 . . . . . 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 14588 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  .0.  )
105, 9syl 15 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( 0  .x. 
A )  =  .0.  )
11 oveq1 5881 . . . . . 6  |-  ( ( N  mod  ( O `
 A ) )  =  0  ->  (
( N  mod  ( O `  A )
)  .x.  A )  =  ( 0  .x. 
A ) )
1211eqeq1d 2304 . . . . 5  |-  ( ( N  mod  ( O `
 A ) )  =  0  ->  (
( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  <->  ( 0 
.x.  A )  =  .0.  ) )
1310, 12syl5ibrcom 213 . . . 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 10133 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  RR )
151nnrpd 10405 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR+ )
16 modlt 10997 . . . . . . . 8  |-  ( ( N  e.  RR  /\  ( O `  A )  e.  RR+ )  ->  ( N  mod  ( O `  A ) )  < 
( O `  A
) )
1714, 15, 16syl2anc 642 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  <  ( O `
 A ) )
182, 1zmodcld 11006 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  e.  NN0 )
1918nn0red 10035 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  e.  RR )
201nnred 9777 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR )
2119, 20ltnled 8982 . . . . . . 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 201 . . . . . 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 14870 . . . . . . . . . . 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 10816 . . . . . . . . . . 11  |-  ( ( O `  A )  e.  ( 1 ... ( N  mod  ( O `  A )
) )  ->  ( O `  A )  <_  ( N  mod  ( O `  A )
) )
2624, 25syl 15 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  ( N  mod  ( O `
 A ) )  e.  NN  /\  (
( N  mod  ( O `  A )
)  .x.  A )  =  .0.  )  ->  ( O `  A )  <_  ( N  mod  ( O `  A )
) )
27263com23 1157 . . . . . . . . 9  |-  ( ( A  e.  X  /\  ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  /\  ( N  mod  ( O `  A ) )  e.  NN )  ->  ( O `  A )  <_  ( N  mod  ( O `  A )
) )
28273expia 1153 . . . . . . . 8  |-  ( ( A  e.  X  /\  ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  )  -> 
( ( N  mod  ( O `  A ) )  e.  NN  ->  ( O `  A )  <_  ( N  mod  ( O `  A ) ) ) )
2928con3d 125 . . . . . . 7  |-  ( ( A  e.  X  /\  ( ( N  mod  ( O `  A ) )  .x.  A )  =  .0.  )  -> 
( -.  ( O `
 A )  <_ 
( N  mod  ( O `  A )
)  ->  -.  ( N  mod  ( O `  A ) )  e.  NN ) )
3029impancom 427 . . . . . 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 642 . . . . 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 9983 . . . . . . 7  |-  ( ( N  mod  ( O `
 A ) )  e.  NN0  <->  ( ( N  mod  ( O `  A ) )  e.  NN  \/  ( N  mod  ( O `  A ) )  =  0 ) )
3318, 32sylib 188 . . . . . 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 366 . . . . 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 40 . . . 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 183 . . 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 14877 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  .x.  A )  =  ( N  .x.  A ) )
3837eqeq1d 2304 . . 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 270 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( O `
 A )  ||  N 
<->  ( N  .x.  A
)  =  .0.  )
)
40 simpr 447 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( O `  A )  =  0 )
4140breq1d 4049 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( ( O `
 A )  ||  N 
<->  0  ||  N ) )
42 simpl3 960 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  N  e.  ZZ )
43 0dvds 12565 . . . 4  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
4442, 43syl 15 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( 0  ||  N 
<->  N  =  0 ) )
45 simpl2 959 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  A  e.  X
)
4645, 9syl 15 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( 0  .x. 
A )  =  .0.  )
47 oveq1 5881 . . . . . 6  |-  ( N  =  0  ->  ( N  .x.  A )  =  ( 0  .x.  A
) )
4847eqeq1d 2304 . . . . 5  |-  ( N  =  0  ->  (
( N  .x.  A
)  =  .0.  <->  ( 0 
.x.  A )  =  .0.  ) )
4946, 48syl5ibrcom 213 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( N  =  0  ->  ( N  .x.  A )  =  .0.  ) )
506, 23, 8, 7odnncl 14876 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  NN )
5150nnne0d 9806 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  =/=  0
)
5251expr 598 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  N  =/=  0
)  ->  ( ( N  .x.  A )  =  .0.  ->  ( O `  A )  =/=  0
) )
5352impancom 427 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  .x.  A
)  =  .0.  )  ->  ( N  =/=  0  ->  ( O `  A
)  =/=  0 ) )
5453necon4d 2522 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  .x.  A
)  =  .0.  )  ->  ( ( O `  A )  =  0  ->  N  =  0 ) )
5554impancom 427 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( ( N 
.x.  A )  =  .0.  ->  N  = 
0 ) )
5649, 55impbid 183 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( N  =  0  <->  ( N  .x.  A )  =  .0.  ) )
5741, 44, 563bitrd 270 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  =  0 )  ->  ( ( O `
 A )  ||  N 
<->  ( N  .x.  A
)  =  .0.  )
)
586, 23odcl 14867 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
59583ad2ant2 977 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  A
)  e.  NN0 )
60 elnn0 9983 . . 3  |-  ( ( O `  A )  e.  NN0  <->  ( ( O `
 A )  e.  NN  \/  ( O `
 A )  =  0 ) )
6159, 60sylib 188 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  e.  NN  \/  ( O `  A
)  =  0 ) )
6239, 57, 61mpjaodan 761 1  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883    <_ cle 8884   NNcn 9762   NN0cn0 9981   ZZcz 10040   RR+crp 10370   ...cfz 10798    mod cmo 10989    || cdivides 12547   Basecbs 13164   0gc0g 13416   Grpcgrp 14378  .gcmg 14382   odcod 14856
This theorem is referenced by:  oddvdsi  14879  odcong  14880  odeq  14881  odmulgid  14883  odbezout  14887  gexdvds2  14912  gexod  14913  gexcl3  14914  odadd1  15156  odadd2  15157  oddvdssubg  15163  pgpfac1lem3a  15327  chrdvds  16498  dchrfi  20510  dchrabs  20515  dchrptlem2  20520  idomodle  27615
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-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-od 14860
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