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Theorem oddvdsnn0 17192
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 10891 . . . . 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 17190 . . . . . 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 1207 . . . . 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 688 . . . 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 1200 . . 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 10886 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  CC )
11103ad2ant3 1028 . . . . . 6  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  ->  N  e.  CC )
1211subid1d 9982 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( N  -  0 )  =  N )
1312breq2d 4435 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  ||  ( N  -  0 )  <-> 
( O `  A
)  ||  N )
)
142, 5, 4mulg0 16762 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  .0.  )
15143ad2ant2 1027 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( 0  .x.  A
)  =  .0.  )
1615eqeq2d 2436 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( N  .x.  A )  =  ( 0  .x.  A )  <-> 
( N  .x.  A
)  =  .0.  )
)
1713, 16bibi12d 322 . . 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 217 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  e.  NN  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) ) )
19 simpr 462 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( O `  A
)  =  0 )
2019breq1d 4433 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( O `  A )  ||  N  <->  0 
||  N ) )
21 simpl3 1010 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  ->  N  e.  NN0 )
22 nn0z 10967 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  ZZ )
23 0dvds 14322 . . . . 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 466 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( 0  .x.  A
)  =  .0.  )
26 oveq1 6312 . . . . . . 7  |-  ( N  =  0  ->  ( N  .x.  A )  =  ( 0  .x.  A
) )
2726eqeq1d 2424 . . . . . 6  |-  ( N  =  0  ->  (
( N  .x.  A
)  =  .0.  <->  ( 0 
.x.  A )  =  .0.  ) )
2825, 27syl5ibrcom 225 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( N  =  0  ->  ( N  .x.  A )  =  .0.  ) )
292, 3, 4, 5odlem2 17187 . . . . . . . . . . . 12  |-  ( ( A  e.  X  /\  N  e.  NN  /\  ( N  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... N
) )
30293com23 1211 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  /\  N  e.  NN )  ->  ( O `  A )  e.  ( 1 ... N
) )
31 elfznn 11835 . . . . . . . . . . 11  |-  ( ( O `  A )  e.  ( 1 ... N )  ->  ( O `  A )  e.  NN )
32 nnne0 10649 . . . . . . . . . . 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 1207 . . . . . . . . 9  |-  ( ( A  e.  X  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  ->  ( O `  A
)  =/=  0 ) )
35343ad2antl2 1168 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  ->  ( O `  A
)  =/=  0 ) )
3635necon2bd 2635 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( ( O `  A )  =  0  ->  -.  N  e.  NN ) )
37 simpl3 1010 . . . . . . . . 9  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  ->  N  e.  NN0 )
38 elnn0 10878 . . . . . . . . 9  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
3937, 38sylib 199 . . . . . . . 8  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( N  e.  NN  \/  N  =  0
) )
4039ord 378 . . . . . . 7  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( -.  N  e.  NN  ->  N  = 
0 ) )
4136, 40syld 45 . . . . . 6  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( N  .x.  A )  =  .0.  )  -> 
( ( O `  A )  =  0  ->  N  =  0 ) )
4241impancom 441 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( N  .x.  A )  =  .0. 
->  N  =  0
) )
4328, 42impbid 193 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( N  =  0  <-> 
( N  .x.  A
)  =  .0.  )
)
4420, 24, 433bitrd 282 . . 3  |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  =  0 )  -> 
( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4544ex 435 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  =  0  ->  ( ( O `
 A )  ||  N 
<->  ( N  .x.  A
)  =  .0.  )
) )
462, 3odcl 17184 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
47463ad2ant2 1027 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( O `  A
)  e.  NN0 )
48 elnn0 10878 . . 3  |-  ( ( O `  A )  e.  NN0  <->  ( ( O `
 A )  e.  NN  \/  ( O `
 A )  =  0 ) )
4947, 48sylib 199 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  e.  NN  \/  ( O `  A
)  =  0 ) )
5018, 45, 49mpjaod 382 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601  (class class class)co 6305   CCcc 9544   0cc0 9546   1c1 9547    - cmin 9867   NNcn 10616   NN0cn0 10876   ZZcz 10944   ...cfz 11791    || cdvds 14304   Basecbs 15120   0gc0g 15337   Mndcmnd 16534  .gcmg 16671   odcod 17164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-inf 7966  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-fl 12034  df-mod 12103  df-seq 12220  df-dvds 14305  df-0g 15339  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-mulg 16675  df-od 17171
This theorem is referenced by: (None)
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