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Mirrors > Home > ILE Home > Th. List > zdvdsdc | GIF version |
Description: Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.) |
Ref | Expression |
---|---|
zdvdsdc | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ∥ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 497 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → 𝑀 ∈ ℤ) | |
2 | 1 | znegcld 8969 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → -𝑀 ∈ ℤ) |
3 | simpr 109 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → 𝑀 < 0) | |
4 | 1 | zred 8967 | . . . . . . 7 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → 𝑀 ∈ ℝ) |
5 | 4 | lt0neg1d 8090 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → (𝑀 < 0 ↔ 0 < -𝑀)) |
6 | 3, 5 | mpbid 146 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → 0 < -𝑀) |
7 | elnnz 8858 | . . . . 5 ⊢ (-𝑀 ∈ ℕ ↔ (-𝑀 ∈ ℤ ∧ 0 < -𝑀)) | |
8 | 2, 6, 7 | sylanbrc 409 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → -𝑀 ∈ ℕ) |
9 | simplr 498 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → 𝑁 ∈ ℤ) | |
10 | dvdsdc 11247 | . . . 4 ⊢ ((-𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID -𝑀 ∥ 𝑁) | |
11 | 8, 9, 10 | syl2anc 404 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → DECID -𝑀 ∥ 𝑁) |
12 | negdvdsb 11255 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ -𝑀 ∥ 𝑁)) | |
13 | 12 | adantr 271 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → (𝑀 ∥ 𝑁 ↔ -𝑀 ∥ 𝑁)) |
14 | 13 | dcbid 789 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → (DECID 𝑀 ∥ 𝑁 ↔ DECID -𝑀 ∥ 𝑁)) |
15 | 11, 14 | mpbird 166 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 < 0) → DECID 𝑀 ∥ 𝑁) |
16 | 0z 8859 | . . . . 5 ⊢ 0 ∈ ℤ | |
17 | zdceq 8920 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0) | |
18 | 16, 17 | mpan2 417 | . . . 4 ⊢ (𝑁 ∈ ℤ → DECID 𝑁 = 0) |
19 | 18 | ad2antlr 474 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → DECID 𝑁 = 0) |
20 | breq1 3870 | . . . . . 6 ⊢ (𝑀 = 0 → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁)) | |
21 | 20 | adantl 272 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 ∥ 𝑁 ↔ 0 ∥ 𝑁)) |
22 | 0dvds 11259 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (0 ∥ 𝑁 ↔ 𝑁 = 0)) | |
23 | 22 | ad2antlr 474 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (0 ∥ 𝑁 ↔ 𝑁 = 0)) |
24 | 21, 23 | bitrd 187 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 ∥ 𝑁 ↔ 𝑁 = 0)) |
25 | 24 | dcbid 789 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (DECID 𝑀 ∥ 𝑁 ↔ DECID 𝑁 = 0)) |
26 | 19, 25 | mpbird 166 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → DECID 𝑀 ∥ 𝑁) |
27 | simpll 497 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑀) → 𝑀 ∈ ℤ) | |
28 | simpr 109 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑀) → 0 < 𝑀) | |
29 | elnnz 8858 | . . . 4 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 0 < 𝑀)) | |
30 | 27, 28, 29 | sylanbrc 409 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑀) → 𝑀 ∈ ℕ) |
31 | simplr 498 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑀) → 𝑁 ∈ ℤ) | |
32 | dvdsdc 11247 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ∥ 𝑁) | |
33 | 30, 31, 32 | syl2anc 404 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑀) → DECID 𝑀 ∥ 𝑁) |
34 | ztri3or0 8890 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑀 < 0 ∨ 𝑀 = 0 ∨ 0 < 𝑀)) | |
35 | 34 | adantr 271 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 0 ∨ 𝑀 = 0 ∨ 0 < 𝑀)) |
36 | 15, 26, 33, 35 | mpjao3dan 1250 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ∥ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 783 ∨ w3o 926 = wceq 1296 ∈ wcel 1445 class class class wbr 3867 0cc0 7447 < clt 7619 -cneg 7751 ℕcn 8520 ℤcz 8848 ∥ cdvds 11239 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-po 4147 df-iso 4148 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-n0 8772 df-z 8849 df-q 9204 df-rp 9234 df-fl 9826 df-mod 9879 df-dvds 11240 |
This theorem is referenced by: lcmval 11488 lcmcllem 11492 lcmledvds 11495 phiprmpw 11641 unennn 11653 |
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