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Theorem oddvdsnn0 15069
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 10129 . . . . 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 15067 . . . . . 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 1154 . . . . 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 665 . . . 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 1147 . . 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 10124 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  CC )
11103ad2ant3 979 . . . . . 6  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  ->  N  e.  CC )
1211subid1d 9293 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( N  -  0 )  =  N )
1312breq2d 4137 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  ||  ( N  -  0 )  <-> 
( O `  A
)  ||  N )
)
142, 5, 4mulg0 14782 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  .0.  )
15143ad2ant2 978 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( 0  .x.  A
)  =  .0.  )
1615eqeq2d 2377 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( N  .x.  A )  =  ( 0  .x.  A )  <-> 
( N  .x.  A
)  =  .0.  )
)
1713, 16bibi12d 312 . . 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 205 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  e.  NN  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) ) )
19 simpr 447 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( O `  A
)  =  0 )
2019breq1d 4135 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( O `  A )  ||  N  <->  0 
||  N ) )
21 simpl3 961 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  ->  N  e.  NN0 )
22 nn0z 10197 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  ZZ )
23 0dvds 12757 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
2421, 22, 233syl 18 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( 0  ||  N  <->  N  =  0 ) )
2515adantr 451 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( 0  .x.  A
)  =  .0.  )
26 oveq1 5988 . . . . . . 7  |-  ( N  =  0  ->  ( N  .x.  A )  =  ( 0  .x.  A
) )
2726eqeq1d 2374 . . . . . 6  |-  ( N  =  0  ->  (
( N  .x.  A
)  =  .0.  <->  ( 0 
.x.  A )  =  .0.  ) )
2825, 27syl5ibrcom 213 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( N  =  0  ->  ( N  .x.  A )  =  .0.  ) )
292, 3, 4, 5odlem2 15064 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  N  e.  NN  /\  ( N  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... N
) )
30293com23 1158 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  /\  N  e.  NN )  ->  ( O `  A )  e.  ( 1 ... N
) )
31 elfznn 10972 . . . . . . . . . . 11  |-  ( ( O `  A )  e.  ( 1 ... N )  ->  ( O `  A )  e.  NN )
32 nnne0 9925 . . . . . . . . . . 11  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  =/=  0 )
3330, 31, 323syl 18 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  /\  N  e.  NN )  ->  ( O `  A )  =/=  0 )
34333expia 1154 . . . . . . . . 9  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  ->  ( O `  A
)  =/=  0 ) )
35343ad2antl2 1119 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  ->  ( O `  A
)  =/=  0 ) )
3635necon2bd 2578 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( ( O `  A )  =  0  ->  -.  N  e.  NN ) )
37 simpl3 961 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  ->  N  e.  NN0 )
38 elnn0 10116 . . . . . . . . 9  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3937, 38sylib 188 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  \/  N  =  0
) )
4039ord 366 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( -.  N  e.  NN  ->  N  = 
0 ) )
4136, 40syld 40 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( ( O `  A )  =  0  ->  N  =  0 ) )
4241impancom 427 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( N  .x.  A )  =  .0. 
->  N  =  0
) )
4328, 42impbid 183 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( N  =  0  <-> 
( N  .x.  A
)  =  .0.  )
)
4420, 24, 433bitrd 270 . . 3  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4544ex 423 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  =  0  ->  ( ( O `
 A )  ||  N 
<->  ( N  .x.  A
)  =  .0.  )
) )
462, 3odcl 15061 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
47463ad2ant2 978 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( O `  A
)  e.  NN0 )
48 elnn0 10116 . . 3  |-  ( ( O `  A )  e.  NN0  <->  ( ( O `
 A )  e.  NN  \/  ( O `
 A )  =  0 ) )
4947, 48sylib 188 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  e.  NN  \/  ( O `  A
)  =  0 ) )
5018, 45, 49mpjaod 370 1  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( 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 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   CCcc 8882   0cc0 8884   1c1 8885    - cmin 9184   NNcn 9893   NN0cn0 10114   ZZcz 10175   ...cfz 10935    || cdivides 12739   Basecbs 13356   0gc0g 13610   Mndcmnd 14571  .gcmg 14576   odcod 15050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-fz 10936  df-fl 11089  df-mod 11138  df-seq 11211  df-dvds 12740  df-0g 13614  df-mnd 14577  df-mulg 14702  df-od 15054
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