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Theorem zdvdsdc 10449
Description: Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.)
Assertion
Ref Expression
zdvdsdc  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  M 
||  N )

Proof of Theorem zdvdsdc
StepHypRef Expression
1 simpll 496 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  M  e.  ZZ )
21znegcld 8629 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  -u M  e.  ZZ )
3 simpr 108 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  M  <  0 )
41zred 8627 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  M  e.  RR )
54lt0neg1d 7760 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  ( M  <  0  <->  0  <  -u M
) )
63, 5mpbid 145 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  0  <  -u M )
7 elnnz 8519 . . . . 5  |-  ( -u M  e.  NN  <->  ( -u M  e.  ZZ  /\  0  <  -u M ) )
82, 6, 7sylanbrc 408 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  -u M  e.  NN )
9 simplr 497 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  N  e.  ZZ )
10 dvdsdc 10436 . . . 4  |-  ( (
-u M  e.  NN  /\  N  e.  ZZ )  -> DECID  -u M  ||  N )
118, 9, 10syl2anc 403 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  -> DECID  -u M  ||  N
)
12 negdvdsb 10444 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  -u M  ||  N ) )
1312adantr 270 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  ( M  ||  N  <->  -u M  ||  N
) )
1413dcbid 782 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  (DECID  M  ||  N  <-> DECID  -u M  ||  N ) )
1511, 14mpbird 165 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  -> DECID  M  ||  N )
16 0z 8520 . . . . 5  |-  0  e.  ZZ
17 zdceq 8581 . . . . 5  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
1816, 17mpan2 416 . . . 4  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
1918ad2antlr 473 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  -> DECID  N  =  0
)
20 breq1 3809 . . . . . 6  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
2120adantl 271 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( M  ||  N  <->  0  ||  N
) )
22 0dvds 10448 . . . . . 6  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
2322ad2antlr 473 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( 0 
||  N  <->  N  = 
0 ) )
2421, 23bitrd 186 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( M  ||  N  <->  N  =  0
) )
2524dcbid 782 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  (DECID  M  ||  N  <-> DECID  N  =  0 ) )
2619, 25mpbird 165 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  -> DECID  M  ||  N )
27 simpll 496 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  0  <  M
)  ->  M  e.  ZZ )
28 simpr 108 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  0  <  M
)  ->  0  <  M )
29 elnnz 8519 . . . 4  |-  ( M  e.  NN  <->  ( M  e.  ZZ  /\  0  < 
M ) )
3027, 28, 29sylanbrc 408 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  0  <  M
)  ->  M  e.  NN )
31 simplr 497 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  0  <  M
)  ->  N  e.  ZZ )
32 dvdsdc 10436 . . 3  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  -> DECID  M 
||  N )
3330, 31, 32syl2anc 403 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  0  <  M
)  -> DECID  M  ||  N )
34 ztri3or0 8551 . . 3  |-  ( M  e.  ZZ  ->  ( M  <  0  \/  M  =  0  \/  0  <  M ) )
3534adantr 270 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  0  \/  M  =  0  \/  0  <  M ) )
3615, 26, 33, 35mpjao3dan 1239 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  M 
||  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103  DECID wdc 776    \/ w3o 919    = wceq 1285    e. wcel 1434   class class class wbr 3806   0cc0 7120    < clt 7292   -ucneg 7424   NNcn 8183   ZZcz 8509    || cdvds 10428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7206  ax-resscn 7207  ax-1cn 7208  ax-1re 7209  ax-icn 7210  ax-addcl 7211  ax-addrcl 7212  ax-mulcl 7213  ax-mulrcl 7214  ax-addcom 7215  ax-mulcom 7216  ax-addass 7217  ax-mulass 7218  ax-distr 7219  ax-i2m1 7220  ax-0lt1 7221  ax-1rid 7222  ax-0id 7223  ax-rnegex 7224  ax-precex 7225  ax-cnre 7226  ax-pre-ltirr 7227  ax-pre-ltwlin 7228  ax-pre-lttrn 7229  ax-pre-apti 7230  ax-pre-ltadd 7231  ax-pre-mulgt0 7232  ax-pre-mulext 7233  ax-arch 7234
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-po 4080  df-iso 4081  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-fv 4961  df-riota 5521  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-1st 5820  df-2nd 5821  df-pnf 7294  df-mnf 7295  df-xr 7296  df-ltxr 7297  df-le 7298  df-sub 7425  df-neg 7426  df-reap 7819  df-ap 7826  df-div 7905  df-inn 8184  df-n0 8433  df-z 8510  df-q 8863  df-rp 8893  df-fl 9429  df-mod 9482  df-dvds 10429
This theorem is referenced by:  lcmval  10677  lcmcllem  10681  lcmledvds  10684  phiprmpw  10830  unennn  10842
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