ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  zdvdsdc Unicode version
Theorem zdvdsdc 11752
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 519 . . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> M e. ZZ )
21znegcld 9315 . . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> -u M e. ZZ )
3 simpr 109 . . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> M < 0 )
41zred 9313 . . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> M e. RR )
54lt0neg1d 8413 . . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> ( M < 0 <-> 0 < -u M ) )
63, 5mpbid 146 . . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> 0 < -u M )
7 elnnz 9201 . . . . 5 |- ( -u M e. NN <-> ( -u M e. ZZ /\ 0 < -u M ) )
82, 6, 7sylanbrc 414 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> -u M e. NN )
9 simplr 520 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> N e. ZZ )
10 dvdsdc 11738 . . . 4 |- ( ( -u M e. NN /\ N e. ZZ ) -> DECID -u M || N )
118, 9, 10syl2anc 409 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> DECID -u M || N )
12 negdvdsb 11747 . . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> -u M || N ) )
1312adantr 274 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> ( M || N <-> -u M || N ) )
1413dcbid 828 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> (DECID M || N <-> DECID -u M || N ) )
1511, 14mpbird 166 . 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> DECID M || N )
16 0z 9202 . . . . 5 |- 0 e. ZZ
17 zdceq 9266 . . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
1816, 17mpan2 422 . . . 4 |- ( N e. ZZ -> DECID N = 0 )
1918ad2antlr 481 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> DECID N = 0 )
20 breq1 3985 . . . . . 6 |- ( M = 0 -> ( M || N <-> 0 || N ) )
2120adantl 275 . . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M || N <-> 0 || N ) )
22 0dvds 11751 . . . . . 6 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
2322ad2antlr 481 . . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( 0 || N <-> N = 0 ) )
2421, 23bitrd 187 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M || N <-> N = 0 ) )
2524dcbid 828 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> (DECID M || N <-> DECID N = 0 ) )
2619, 25mpbird 166 . 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> DECID M || N )
27 simpll 519 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ 0 < M ) -> M e. ZZ )
28 simpr 109 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ 0 < M ) -> 0 < M )
29 elnnz 9201 . . . 4 |- ( M e. NN <-> ( M e. ZZ /\ 0 < M ) )
3027, 28, 29sylanbrc 414 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ 0 < M ) -> M e. NN )
31 simplr 520 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ 0 < M ) -> N e. ZZ )
32 dvdsdc 11738 . . 3 |- ( ( M e. NN /\ N e. ZZ ) -> DECID M || N )
3330, 31, 32syl2anc 409 . 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ 0 < M ) -> DECID M || N )
34 ztri3or0 9233 . . 3 |- ( M e. ZZ -> ( M < 0 \/ M = 0 \/ 0 < M ) )
3534adantr 274 . 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < 0 \/ M = 0 \/ 0 < M ) )
3615, 26, 33, 35mpjao3dan 1297 1 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID M || N )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 824   \/ w3o 967   = wceq 1343   e. wcel 2136   class class class wbr 3982   0cc0 7753   < clt 7933   -ucneg 8070   NNcn 8857   ZZcz 9191   || cdvds 11727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-q 9558  df-rp 9590  df-fl 10205  df-mod 10258  df-dvds 11728
This theorem is referenced by:  lcmval  11995  lcmcllem  11999  lcmledvds  12002  phiprmpw  12154  pclemdc  12220  pc2dvds  12261  unennn  12330
  Copyright terms: Public domain W3C validator