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Mirrors > Home > ILE Home > Th. List > divides | GIF version |
Description: Define the divides relation. 𝑀 ∥ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 12869). As proven in dvdsval3 11424, 𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 11422 and dvdsval2 11423 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
divides | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3900 | . . 3 ⊢ (𝑀 ∥ 𝑁 ↔ 〈𝑀, 𝑁〉 ∈ ∥ ) | |
2 | df-dvds 11421 | . . . 4 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)} | |
3 | 2 | eleq2i 2184 | . . 3 ⊢ (〈𝑀, 𝑁〉 ∈ ∥ ↔ 〈𝑀, 𝑁〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}) |
4 | 1, 3 | bitri 183 | . 2 ⊢ (𝑀 ∥ 𝑁 ↔ 〈𝑀, 𝑁〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}) |
5 | oveq2 5750 | . . . . 5 ⊢ (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀)) | |
6 | 5 | eqeq1d 2126 | . . . 4 ⊢ (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦)) |
7 | 6 | rexbidv 2415 | . . 3 ⊢ (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦)) |
8 | eqeq2 2127 | . . . 4 ⊢ (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁)) | |
9 | 8 | rexbidv 2415 | . . 3 ⊢ (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁)) |
10 | 7, 9 | opelopab2 4162 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (〈𝑀, 𝑁〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)} ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁)) |
11 | 4, 10 | syl5bb 191 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 ∃wrex 2394 〈cop 3500 class class class wbr 3899 {copab 3958 (class class class)co 5742 · cmul 7593 ℤcz 9022 ∥ cdvds 11420 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-iota 5058 df-fv 5101 df-ov 5745 df-dvds 11421 |
This theorem is referenced by: dvdsval2 11423 dvds0lem 11430 dvds1lem 11431 dvds2lem 11432 0dvds 11440 dvdsle 11469 divconjdvds 11474 odd2np1 11497 even2n 11498 oddm1even 11499 opeo 11521 omeo 11522 m1exp1 11525 divalgb 11549 modremain 11553 zeqzmulgcd 11586 gcddiv 11634 dvdssqim 11639 coprmdvds2 11701 congr 11708 divgcdcoprm0 11709 cncongr2 11712 dvdsnprmd 11733 |
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