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Theorem zdvdsdc 12604
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 531 . . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> M e. ZZ )
21znegcld 9781 . . . . 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 9779 . . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> M e. RR )
54lt0neg1d 8850 . . . . . 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 9665 . . . . 5 |- ( -u M e. NN <-> ( -u M e. ZZ /\ 0 < -u M ) )
82, 6, 7sylanbrc 421 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> -u M e. NN )
9 simplr 533 . . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> N e. ZZ )
10 dvdsdc 12590 . . . 4 |- ( ( -u M e. NN /\ N e. ZZ ) -> DECID -u M || N )
118, 9, 10syl2anc 415 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> DECID -u M || N )
12 negdvdsb 12599 . . . . 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 850 . . 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 9666 . . . . 5 |- 0 e. ZZ
17 zdceq 9731 . . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
1816, 17mpan2 429 . . . 4 |- ( N e. ZZ -> DECID N = 0 )
1918ad2antlr 493 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> DECID N = 0 )
20 breq1 4134 . . . . . 6 |- ( M = 0 -> ( M || N <-> 0 || N ) )
2120adantl 277 . . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M || N <-> 0 || N ) )
22 0dvds 12603 . . . . . 6 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
2322ad2antlr 493 . . . . 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 850 . . 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 531 . . . 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 9665 . . . 4 |- ( M e. NN <-> ( M e. ZZ /\ 0 < M ) )
3027, 28, 29sylanbrc 421 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ 0 < M ) -> M e. NN )
31 simplr 533 . . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ 0 < M ) -> N e. ZZ )
32 dvdsdc 12590 . . 3 |- ( ( M e. NN /\ N e. ZZ ) -> DECID M || N )
3330, 31, 32syl2anc 415 . 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ 0 < M ) -> DECID M || N )
34 ztri3or0 9697 . . 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 1348 1 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID M || N )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   \/ w3o 1008   = wceq 1402   e. wcel 2209   class class class wbr 4131   0cc0 8185   < clt 8366   -ucneg 8505   NNcn 9313   ZZcz 9655   || cdvds 12579
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304
This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-n0 9575  df-z 9656  df-q 10036  df-rp 10072  df-fl 10722  df-mod 10781  df-dvds 12580
This theorem is used by:  lcmval  12866  lcmcllem  12870  lcmledvds  12873  phiprmpw  13029  pclemdc  13096  pc2dvds  13138  unennn  13346
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