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Theorem oddvds 16698
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 14002 . . . 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 16274 . . . . . 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 6303 . . . . . 6  |-  ( ( N  mod  ( O `
 A ) )  =  0  ->  (
( N  mod  ( O `  A )
)  .x.  A )  =  ( 0  .x. 
A ) )
1211eqeq1d 2459 . . . . 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 10990 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  N  e.  RR )
151nnrpd 11280 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR+ )
16 modlt 12009 . . . . . . . 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 12019 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  e.  NN0 )
1918nn0red 10874 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( N  mod  ( O `  A ) )  e.  RR )
201nnred 10571 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( O `  A )  e.  RR )
2119, 20ltnled 9749 . . . . . . 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 16690 . . . . . . . . . . 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 11715 . . . . . . . . . . 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 10818 . . . . . . 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 16697 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A
)  e.  NN )  ->  ( ( N  mod  ( O `  A ) )  .x.  A )  =  ( N  .x.  A ) )
3837eqeq1d 2459 . . 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 4466 . . 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 14016 . . . 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 6303 . . . . . 6  |-  ( N  =  0  ->  ( N  .x.  A )  =  ( 0  .x.  A
) )
4847eqeq1d 2459 . . . . 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 16696 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  NN )
5150nnne0d 10601 . . . . . . . 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 2684 . . . . 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 16687 . . . 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 10818 . . 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 786 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 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    < clt 9645    <_ cle 9646   NNcn 10556   NN0cn0 10816   ZZcz 10885   RR+crp 11245   ...cfz 11697    mod cmo 11999    || cdvds 13998   Basecbs 14644   0gc0g 14857   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-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:  oddvdsi  16699  odcong  16700  odeq  16701  odmulgid  16703  odbezout  16707  gexdvds2  16732  gexod  16733  gexcl3  16734  odadd1  16981  odadd2  16982  oddvdssubg  16988  pgpfac1lem3a  17254  chrdvds  18692  dchrfi  23656  dchrabs  23661  dchrptlem2  23666  idomodle  31357
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