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Mirrors > Home > ILE Home > Th. List > zdvdsdc | Unicode version |
Description: Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.) |
Ref | Expression |
---|---|
zdvdsdc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 519 | . . . . . 6 | |
2 | 1 | znegcld 9306 | . . . . 5 |
3 | simpr 109 | . . . . . 6 | |
4 | 1 | zred 9304 | . . . . . . 7 |
5 | 4 | lt0neg1d 8404 | . . . . . 6 |
6 | 3, 5 | mpbid 146 | . . . . 5 |
7 | elnnz 9192 | . . . . 5 | |
8 | 2, 6, 7 | sylanbrc 414 | . . . 4 |
9 | simplr 520 | . . . 4 | |
10 | dvdsdc 11724 | . . . 4 DECID | |
11 | 8, 9, 10 | syl2anc 409 | . . 3 DECID |
12 | negdvdsb 11733 | . . . . 5 | |
13 | 12 | adantr 274 | . . . 4 |
14 | 13 | dcbid 828 | . . 3 DECID DECID |
15 | 11, 14 | mpbird 166 | . 2 DECID |
16 | 0z 9193 | . . . . 5 | |
17 | zdceq 9257 | . . . . 5 DECID | |
18 | 16, 17 | mpan2 422 | . . . 4 DECID |
19 | 18 | ad2antlr 481 | . . 3 DECID |
20 | breq1 3979 | . . . . . 6 | |
21 | 20 | adantl 275 | . . . . 5 |
22 | 0dvds 11737 | . . . . . 6 | |
23 | 22 | ad2antlr 481 | . . . . 5 |
24 | 21, 23 | bitrd 187 | . . . 4 |
25 | 24 | dcbid 828 | . . 3 DECID DECID |
26 | 19, 25 | mpbird 166 | . 2 DECID |
27 | simpll 519 | . . . 4 | |
28 | simpr 109 | . . . 4 | |
29 | elnnz 9192 | . . . 4 | |
30 | 27, 28, 29 | sylanbrc 414 | . . 3 |
31 | simplr 520 | . . 3 | |
32 | dvdsdc 11724 | . . 3 DECID | |
33 | 30, 31, 32 | syl2anc 409 | . 2 DECID |
34 | ztri3or0 9224 | . . 3 | |
35 | 34 | adantr 274 | . 2 |
36 | 15, 26, 33, 35 | mpjao3dan 1296 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 DECID wdc 824 w3o 966 wceq 1342 wcel 2135 class class class wbr 3976 cc0 7744 clt 7924 cneg 8061 cn 8848 cz 9182 cdvds 11713 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-n0 9106 df-z 9183 df-q 9549 df-rp 9581 df-fl 10195 df-mod 10248 df-dvds 11714 |
This theorem is referenced by: lcmval 11974 lcmcllem 11978 lcmledvds 11981 phiprmpw 12133 pclemdc 12199 pc2dvds 12240 unennn 12273 |
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