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Theorem zdvdsdc 11977
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 9450 . . . . 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 9448 . . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> M e. RR )
54lt0neg1d 8542 . . . . . 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 9336 . . . . 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 11963 . . . 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 11972 . . . . 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 839 . . 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 9337 . . . . 5 |- 0 e. ZZ
17 zdceq 9401 . . . . 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 4036 . . . . . 6 |- ( M = 0 -> ( M || N <-> 0 || N ) )
2120adantl 277 . . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M || N <-> 0 || N ) )
22 0dvds 11976 . . . . . 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 839 . . 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 9336 . . . 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 11963 . . 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 9368 . . 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 1318 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 835   \/ w3o 979   = wceq 1364   e. wcel 2167   class class class wbr 4033   0cc0 7879   < clt 8061   -ucneg 8198   NNcn 8990   ZZcz 9326   || cdvds 11952
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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-n0 9250  df-z 9327  df-q 9694  df-rp 9729  df-fl 10360  df-mod 10415  df-dvds 11953
This theorem is referenced by:  lcmval  12231  lcmcllem  12235  lcmledvds  12238  phiprmpw  12390  pclemdc  12457  pc2dvds  12499  unennn  12614
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