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Theorem oddvdsnn0 15187
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 10241 . . . . 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 15185 . . . . . 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 1156 . . . . 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 667 . . . 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 1149 . . 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 10236 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  CC )
11103ad2ant3 981 . . . . . 6  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  ->  N  e.  CC )
1211subid1d 9405 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( N  -  0 )  =  N )
1312breq2d 4227 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  ||  ( N  -  0 )  <-> 
( O `  A
)  ||  N )
)
142, 5, 4mulg0 14900 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  .0.  )
15143ad2ant2 980 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( 0  .x.  A
)  =  .0.  )
1615eqeq2d 2449 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( N  .x.  A )  =  ( 0  .x.  A )  <-> 
( N  .x.  A
)  =  .0.  )
)
1713, 16bibi12d 314 . . 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 207 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  e.  NN  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) ) )
19 simpr 449 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( O `  A
)  =  0 )
2019breq1d 4225 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( O `  A )  ||  N  <->  0 
||  N ) )
21 simpl3 963 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  ->  N  e.  NN0 )
22 nn0z 10309 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  ZZ )
23 0dvds 12875 . . . . 5  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
2421, 22, 233syl 19 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( 0  ||  N  <->  N  =  0 ) )
2515adantr 453 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( 0  .x.  A
)  =  .0.  )
26 oveq1 6091 . . . . . . 7  |-  ( N  =  0  ->  ( N  .x.  A )  =  ( 0  .x.  A
) )
2726eqeq1d 2446 . . . . . 6  |-  ( N  =  0  ->  (
( N  .x.  A
)  =  .0.  <->  ( 0 
.x.  A )  =  .0.  ) )
2825, 27syl5ibrcom 215 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( N  =  0  ->  ( N  .x.  A )  =  .0.  ) )
292, 3, 4, 5odlem2 15182 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  N  e.  NN  /\  ( N  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... N
) )
30293com23 1160 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  /\  N  e.  NN )  ->  ( O `  A )  e.  ( 1 ... N
) )
31 elfznn 11085 . . . . . . . . . . 11  |-  ( ( O `  A )  e.  ( 1 ... N )  ->  ( O `  A )  e.  NN )
32 nnne0 10037 . . . . . . . . . . 11  |-  ( ( O `  A )  e.  NN  ->  ( O `  A )  =/=  0 )
3330, 31, 323syl 19 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  /\  N  e.  NN )  ->  ( O `  A )  =/=  0 )
34333expia 1156 . . . . . . . . 9  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  ->  ( O `  A
)  =/=  0 ) )
35343ad2antl2 1121 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  ->  ( O `  A
)  =/=  0 ) )
3635necon2bd 2655 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( ( O `  A )  =  0  ->  -.  N  e.  NN ) )
37 simpl3 963 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  ->  N  e.  NN0 )
38 elnn0 10228 . . . . . . . . 9  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3937, 38sylib 190 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  \/  N  =  0
) )
4039ord 368 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( -.  N  e.  NN  ->  N  = 
0 ) )
4136, 40syld 43 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( ( O `  A )  =  0  ->  N  =  0 ) )
4241impancom 429 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( N  .x.  A )  =  .0. 
->  N  =  0
) )
4328, 42impbid 185 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( N  =  0  <-> 
( N  .x.  A
)  =  .0.  )
)
4420, 24, 433bitrd 272 . . 3  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4544ex 425 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  =  0  ->  ( ( O `
 A )  ||  N 
<->  ( N  .x.  A
)  =  .0.  )
) )
462, 3odcl 15179 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
47463ad2ant2 980 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( O `  A
)  e.  NN0 )
48 elnn0 10228 . . 3  |-  ( ( O `  A )  e.  NN0  <->  ( ( O `
 A )  e.  NN  \/  ( O `
 A )  =  0 ) )
4947, 48sylib 190 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  e.  NN  \/  ( O `  A
)  =  0 ) )
5018, 45, 49mpjaod 372 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996    - cmin 9296   NNcn 10005   NN0cn0 10226   ZZcz 10287   ...cfz 11048    || cdivides 12857   Basecbs 13474   0gc0g 13728   Mndcmnd 14689  .gcmg 14694   odcod 15168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fl 11207  df-mod 11256  df-seq 11329  df-dvds 12858  df-0g 13732  df-mnd 14695  df-mulg 14820  df-od 15172
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