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Mirrors > Home > MPE Home > Th. List > divides | Structured version Visualization version GIF version |
Description: Define the divides relation. 𝑀 ∥ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 30484). As proven in dvdsval3 16290, 𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 16288 and dvdsval2 16289 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 5148 | . . 3 ⊢ (𝑀 ∥ 𝑁 ↔ 〈𝑀, 𝑁〉 ∈ ∥ ) | |
2 | df-dvds 16287 | . . . 4 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)} | |
3 | 2 | eleq2i 2830 | . . 3 ⊢ (〈𝑀, 𝑁〉 ∈ ∥ ↔ 〈𝑀, 𝑁〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}) |
4 | 1, 3 | bitri 275 | . 2 ⊢ (𝑀 ∥ 𝑁 ↔ 〈𝑀, 𝑁〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}) |
5 | oveq2 7438 | . . . . 5 ⊢ (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀)) | |
6 | 5 | eqeq1d 2736 | . . . 4 ⊢ (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦)) |
7 | 6 | rexbidv 3176 | . . 3 ⊢ (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦)) |
8 | eqeq2 2746 | . . . 4 ⊢ (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁)) | |
9 | 8 | rexbidv 3176 | . . 3 ⊢ (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁)) |
10 | 7, 9 | opelopab2 5550 | . 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 | bitrid 283 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 〈cop 4636 class class class wbr 5147 {copab 5209 (class class class)co 7430 · cmul 11157 ℤcz 12610 ∥ cdvds 16286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-iota 6515 df-fv 6570 df-ov 7433 df-dvds 16287 |
This theorem is referenced by: dvdsval2 16289 dvds0lem 16300 dvds1lem 16301 dvds2lem 16302 0dvds 16310 dvdsle 16343 divconjdvds 16348 dvdsexp2im 16360 odd2np1 16374 even2n 16375 oddm1even 16376 opeo 16398 omeo 16399 m1exp1 16409 divalglem4 16429 divalglem9 16434 divalgb 16437 modremain 16441 zeqzmulgcd 16543 bezoutlem4 16575 gcddiv 16584 dvdssqim 16587 dvdsexpim 16588 coprmdvds2 16687 congr 16697 divgcdcoprm0 16698 cncongr2 16701 dvdsnprmd 16723 prmpwdvds 16937 odmulg 19588 gexdvdsi 19615 lgsquadlem2 27439 primrootspoweq0 42087 aks6d1c2 42111 grpods 42175 unitscyglem4 42179 dvdsrabdioph 42797 jm2.26a 42988 coskpi2 45821 cosknegpi 45824 fourierswlem 46185 dfeven2 47573 |
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