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Theorem zdvdsdc 11786
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 527 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  M  e.  ZZ )
21znegcld 9348 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  -u M  e.  ZZ )
3 simpr 110 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  M  <  0 )
41zred 9346 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  M  e.  RR )
54lt0neg1d 8446 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  ( M  <  0  <->  0  <  -u M
) )
63, 5mpbid 147 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  0  <  -u M )
7 elnnz 9234 . . . . 5  |-  ( -u M  e.  NN  <->  ( -u M  e.  ZZ  /\  0  <  -u M ) )
82, 6, 7sylanbrc 417 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  -u M  e.  NN )
9 simplr 528 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  N  e.  ZZ )
10 dvdsdc 11772 . . . 4  |-  ( (
-u M  e.  NN  /\  N  e.  ZZ )  -> DECID  -u M  ||  N )
118, 9, 10syl2anc 411 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  -> DECID  -u M  ||  N
)
12 negdvdsb 11781 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  -u M  ||  N ) )
1312adantr 276 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  ( M  ||  N  <->  -u M  ||  N
) )
1413dcbid 838 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  ->  (DECID  M  ||  N  <-> DECID  -u M  ||  N ) )
1511, 14mpbird 167 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  <  0
)  -> DECID  M  ||  N )
16 0z 9235 . . . . 5  |-  0  e.  ZZ
17 zdceq 9299 . . . . 5  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  -> DECID  N  =  0 )
1816, 17mpan2 425 . . . 4  |-  ( N  e.  ZZ  -> DECID  N  =  0
)
1918ad2antlr 489 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  -> DECID  N  =  0
)
20 breq1 4001 . . . . . 6  |-  ( M  =  0  ->  ( M  ||  N  <->  0  ||  N ) )
2120adantl 277 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( M  ||  N  <->  0  ||  N
) )
22 0dvds 11785 . . . . . 6  |-  ( N  e.  ZZ  ->  (
0  ||  N  <->  N  = 
0 ) )
2322ad2antlr 489 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( 0 
||  N  <->  N  = 
0 ) )
2421, 23bitrd 188 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( M  ||  N  <->  N  =  0
) )
2524dcbid 838 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  (DECID  M  ||  N  <-> DECID  N  =  0 ) )
2619, 25mpbird 167 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  -> DECID  M  ||  N )
27 simpll 527 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  0  <  M
)  ->  M  e.  ZZ )
28 simpr 110 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  0  <  M
)  ->  0  <  M )
29 elnnz 9234 . . . 4  |-  ( M  e.  NN  <->  ( M  e.  ZZ  /\  0  < 
M ) )
3027, 28, 29sylanbrc 417 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  0  <  M
)  ->  M  e.  NN )
31 simplr 528 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  0  <  M
)  ->  N  e.  ZZ )
32 dvdsdc 11772 . . 3  |-  ( ( M  e.  NN  /\  N  e.  ZZ )  -> DECID  M 
||  N )
3330, 31, 32syl2anc 411 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  0  <  M
)  -> DECID  M  ||  N )
34 ztri3or0 9266 . . 3  |-  ( M  e.  ZZ  ->  ( M  <  0  \/  M  =  0  \/  0  <  M ) )
3534adantr 276 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  0  \/  M  =  0  \/  0  <  M ) )
3615, 26, 33, 35mpjao3dan 1307 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  M 
||  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105  DECID wdc 834    \/ w3o 977    = wceq 1353    e. wcel 2146   class class class wbr 3998   0cc0 7786    < clt 7966   -ucneg 8103   NNcn 8890   ZZcz 9224    || cdvds 11761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-n0 9148  df-z 9225  df-q 9591  df-rp 9623  df-fl 10238  df-mod 10291  df-dvds 11762
This theorem is referenced by:  lcmval  12029  lcmcllem  12033  lcmledvds  12036  phiprmpw  12188  pclemdc  12254  pc2dvds  12295  unennn  12364
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