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Theorem oddvdsnn0 16441
Description: The only multiples of  A that are equal to the identity are the multiples of the order of  A. (Contributed 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
oddvdsnn0  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )

Proof of Theorem oddvdsnn0
StepHypRef Expression
1 0nn0 10822 . . . . 5  |-  0  e.  NN0
2 odcl.1 . . . . . . 7  |-  X  =  ( Base `  G
)
3 odcl.2 . . . . . . 7  |-  O  =  ( od `  G
)
4 odid.3 . . . . . . 7  |-  .x.  =  (.g
`  G )
5 odid.4 . . . . . . 7  |-  .0.  =  ( 0g `  G )
62, 3, 4, 5mndodcong 16439 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  A  e.  X )  /\  ( N  e. 
NN0  /\  0  e.  NN0 )  /\  ( O `
 A )  e.  NN )  ->  (
( O `  A
)  ||  ( N  -  0 )  <->  ( N  .x.  A )  =  ( 0  .x.  A ) ) )
763expia 1198 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X )  /\  ( N  e. 
NN0  /\  0  e.  NN0 ) )  ->  (
( O `  A
)  e.  NN  ->  ( ( O `  A
)  ||  ( N  -  0 )  <->  ( N  .x.  A )  =  ( 0  .x.  A ) ) ) )
81, 7mpanr2 684 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X )  /\  N  e.  NN0 )  ->  ( ( O `
 A )  e.  NN  ->  ( ( O `  A )  ||  ( N  -  0 )  <->  ( N  .x.  A )  =  ( 0  .x.  A ) ) ) )
983impa 1191 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  e.  NN  ->  ( ( O `  A )  ||  ( N  -  0 )  <-> 
( N  .x.  A
)  =  ( 0 
.x.  A ) ) ) )
10 nn0cn 10817 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  CC )
11103ad2ant3 1019 . . . . . 6  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  ->  N  e.  CC )
1211subid1d 9931 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( N  -  0 )  =  N )
1312breq2d 4465 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  ||  ( N  -  0 )  <-> 
( O `  A
)  ||  N )
)
142, 5, 4mulg0 16019 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  .0.  )
15143ad2ant2 1018 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( 0  .x.  A
)  =  .0.  )
1615eqeq2d 2481 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( N  .x.  A )  =  ( 0  .x.  A )  <-> 
( N  .x.  A
)  =  .0.  )
)
1713, 16bibi12d 321 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( ( O `
 A )  ||  ( N  -  0
)  <->  ( N  .x.  A )  =  ( 0  .x.  A ) )  <->  ( ( O `
 A )  ||  N 
<->  ( N  .x.  A
)  =  .0.  )
) )
189, 17sylibd 214 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  e.  NN  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) ) )
19 simpr 461 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( O `  A
)  =  0 )
2019breq1d 4463 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( O `  A )  ||  N  <->  0 
||  N ) )
21 simpl3 1001 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  ->  N  e.  NN0 )
22 nn0z 10899 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  ZZ )
23 0dvds 13882 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
2421, 22, 233syl 20 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( 0  ||  N  <->  N  =  0 ) )
2515adantr 465 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( 0  .x.  A
)  =  .0.  )
26 oveq1 6302 . . . . . . 7  |-  ( N  =  0  ->  ( N  .x.  A )  =  ( 0  .x.  A
) )
2726eqeq1d 2469 . . . . . 6  |-  ( N  =  0  ->  (
( N  .x.  A
)  =  .0.  <->  ( 0 
.x.  A )  =  .0.  ) )
2825, 27syl5ibrcom 222 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( N  =  0  ->  ( N  .x.  A )  =  .0.  ) )
292, 3, 4, 5odlem2 16436 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  N  e.  NN  /\  ( N  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... N
) )
30293com23 1202 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  /\  N  e.  NN )  ->  ( O `  A )  e.  ( 1 ... N
) )
31 elfznn 11726 . . . . . . . . . . 11  |-  ( ( O `  A )  e.  ( 1 ... N )  ->  ( O `  A )  e.  NN )
32 nnne0 10580 . . . . . . . . . . 11  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  =/=  0 )
3330, 31, 323syl 20 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  /\  N  e.  NN )  ->  ( O `  A )  =/=  0 )
34333expia 1198 . . . . . . . . 9  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  ->  ( O `  A
)  =/=  0 ) )
35343ad2antl2 1159 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  ->  ( O `  A
)  =/=  0 ) )
3635necon2bd 2682 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( ( O `  A )  =  0  ->  -.  N  e.  NN ) )
37 simpl3 1001 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  ->  N  e.  NN0 )
38 elnn0 10809 . . . . . . . . 9  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3937, 38sylib 196 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  \/  N  =  0
) )
4039ord 377 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( -.  N  e.  NN  ->  N  = 
0 ) )
4136, 40syld 44 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( ( O `  A )  =  0  ->  N  =  0 ) )
4241impancom 440 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( N  .x.  A )  =  .0. 
->  N  =  0
) )
4328, 42impbid 191 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( N  =  0  <-> 
( N  .x.  A
)  =  .0.  )
)
4420, 24, 433bitrd 279 . . 3  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4544ex 434 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  =  0  ->  ( ( O `
 A )  ||  N 
<->  ( N  .x.  A
)  =  .0.  )
) )
462, 3odcl 16433 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
47463ad2ant2 1018 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( O `  A
)  e.  NN0 )
48 elnn0 10809 . . 3  |-  ( ( O `  A )  e.  NN0  <->  ( ( O `
 A )  e.  NN  \/  ( O `
 A )  =  0 ) )
4947, 48sylib 196 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  e.  NN  \/  ( O `  A
)  =  0 ) )
5018, 45, 49mpjaod 381 1  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( 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   CCcc 9502   0cc0 9504   1c1 9505    - cmin 9817   NNcn 10548   NN0cn0 10807   ZZcz 10876   ...cfz 11684    || cdivides 13864   Basecbs 14507   0gc0g 14712   Mndcmnd 15793  .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-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fl 11909  df-mod 11977  df-seq 12088  df-dvds 13865  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mulg 15932  df-od 16426
This theorem is referenced by: (None)
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