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Theorem oddvdsnn0 16052
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 10599 . . . . 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 16050 . . . . . 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 1189 . . . . 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 1182 . . 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 10594 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  CC )
11103ad2ant3 1011 . . . . . 6  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  ->  N  e.  CC )
1211subid1d 9713 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( N  -  0 )  =  N )
1312breq2d 4309 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  ||  ( N  -  0 )  <-> 
( O `  A
)  ||  N )
)
142, 5, 4mulg0 15637 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  .0.  )
15143ad2ant2 1010 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( 0  .x.  A
)  =  .0.  )
1615eqeq2d 2454 . . . 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 4307 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( O `  A )  ||  N  <->  0 
||  N ) )
21 simpl3 993 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  ->  N  e.  NN0 )
22 nn0z 10674 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  ZZ )
23 0dvds 13558 . . . . 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 6103 . . . . . . 7  |-  ( N  =  0  ->  ( N  .x.  A )  =  ( 0  .x.  A
) )
2726eqeq1d 2451 . . . . . 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 16047 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  N  e.  NN  /\  ( N  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... N
) )
30293com23 1193 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  /\  N  e.  NN )  ->  ( O `  A )  e.  ( 1 ... N
) )
31 elfznn 11483 . . . . . . . . . . 11  |-  ( ( O `  A )  e.  ( 1 ... N )  ->  ( O `  A )  e.  NN )
32 nnne0 10359 . . . . . . . . . . 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 1189 . . . . . . . . 9  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  ->  ( O `  A
)  =/=  0 ) )
35343ad2antl2 1151 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  ->  ( O `  A
)  =/=  0 ) )
3635necon2bd 2665 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( ( O `  A )  =  0  ->  -.  N  e.  NN ) )
37 simpl3 993 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  ->  N  e.  NN0 )
38 elnn0 10586 . . . . . . . . 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 16044 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
47463ad2ant2 1010 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( O `  A
)  e.  NN0 )
48 elnn0 10586 . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   CCcc 9285   0cc0 9287   1c1 9288    - cmin 9600   NNcn 10327   NN0cn0 10584   ZZcz 10651   ...cfz 11442    || cdivides 13540   Basecbs 14179   0gc0g 14383   Mndcmnd 15414  .gcmg 15419   odcod 16033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fl 11647  df-mod 11714  df-seq 11812  df-dvds 13541  df-0g 14385  df-mnd 15420  df-mulg 15553  df-od 16037
This theorem is referenced by: (None)
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