Theorem zdvdsdc 12436

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 9648	. . . . 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 9646	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> M e. RR )
5	4	lt0neg1d 8737	. . . . . 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 9533	. . . . 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 529	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M < 0 ) -> N e. ZZ )
10		dvdsdc 12422	. . . 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 12431	. . . . 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 846	. . 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 9534	. . . . 5 |- 0 e. ZZ
17		zdceq 9599	. . . . 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 4096	. . . . . 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 12435	. . . . . 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 846	. . 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 9533	. . . 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 529	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ 0 < M ) -> N e. ZZ )
32		dvdsdc 12422	. . 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 9565	. . 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 1344	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID M || N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   \/ w3o 1004   = wceq 1398   e. wcel 2202   class class class wbr 4093   0cc0 8075   < clt 8256   -ucneg 8393   NNcn 9185   ZZcz 9523   || cdvds 12411

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-n0 9445  df-z 9524  df-q 9898  df-rp 9933  df-fl 10576  df-mod 10631  df-dvds 12412

This theorem is referenced by:  lcmval  12698  lcmcllem  12702  lcmledvds  12705  phiprmpw  12857  pclemdc  12924  pc2dvds  12966  unennn  13081