Theorem zdvdsdc 12318

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

Step	Hyp	Ref	Expression
1		simpll 527	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> M e. ZZ )
2	1	znegcld 9567	. . . . 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 )
4	1	zred 9565	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> M e. RR )
5	4	lt0neg1d 8658	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> ( M < 0 <-> 0 < -u M ) )
6	3, 5	mpbid 147	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> 0 < -u M )
7		elnnz 9452	. . . . 5 |- ( -u M e. NN <-> ( -u M e. ZZ /\ 0 < -u M ) )
8	2, 6, 7	sylanbrc 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 12304	. . . 4 |- ( ( -u M e. NN /\ N e. ZZ ) -> DECID -u M || N )
11	8, 9, 10	syl2anc 411	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> DECID -u M || N )
12		negdvdsb 12313	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> -u M || N ) )
13	12	adantr 276	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> ( M || N <-> -u M || N ) )
14	13	dcbid 843	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> (DECID M || N <-> DECID -u M || N ) )
15	11, 14	mpbird 167	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> DECID M || N )
16		0z 9453	. . . . 5 |- 0 e. ZZ
17		zdceq 9518	. . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
18	16, 17	mpan2 425	. . . 4 |- ( N e. ZZ -> DECID N = 0 )
19	18	ad2antlr 489	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> DECID N = 0 )
20		breq1 4085	. . . . . 6 |- ( M = 0 -> ( M || N <-> 0 || N ) )
21	20	adantl 277	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M || N <-> 0 || N ) )
22		0dvds 12317	. . . . . 6 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
23	22	ad2antlr 489	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( 0 || N <-> N = 0 ) )
24	21, 23	bitrd 188	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M || N <-> N = 0 ) )
25	24	dcbid 843	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> (DECID M || N <-> DECID N = 0 ) )
26	19, 25	mpbird 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 9452	. . . 4 |- ( M e. NN <-> ( M e. ZZ /\ 0 < M ) )
30	27, 28, 29	sylanbrc 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 12304	. . 3 |- ( ( M e. NN /\ N e. ZZ ) -> DECID M || N )
33	30, 31, 32	syl2anc 411	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ 0 < M ) -> DECID M || N )
34		ztri3or0 9484	. . 3 |- ( M e. ZZ -> ( M < 0 \/ M = 0 \/ 0 < M ) )
35	34	adantr 276	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < 0 \/ M = 0 \/ 0 < M ) )
36	15, 26, 33, 35	mpjao3dan 1341	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID M || N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 839   \/ w3o 1001   = wceq 1395   e. wcel 2200   class class class wbr 4082   0cc0 7995   < clt 8177   -ucneg 8314   NNcn 9106   ZZcz 9442   || cdvds 12293

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-n0 9366  df-z 9443  df-q 9811  df-rp 9846  df-fl 10485  df-mod 10540  df-dvds 12294

This theorem is referenced by:  lcmval  12580  lcmcllem  12584  lcmledvds  12587  phiprmpw  12739  pclemdc  12806  pc2dvds  12848  unennn  12963