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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 26705). As proven in dvdsval3 14825, 𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 14823 and dvdsval2 14824 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 4584 | . . 3 ⊢ (𝑀 ∥ 𝑁 ↔ 〈𝑀, 𝑁〉 ∈ ∥ ) | |
2 | df-dvds 14822 | . . . 4 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)} | |
3 | 2 | eleq2i 2680 | . . 3 ⊢ (〈𝑀, 𝑁〉 ∈ ∥ ↔ 〈𝑀, 𝑁〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}) |
4 | 1, 3 | bitri 263 | . 2 ⊢ (𝑀 ∥ 𝑁 ↔ 〈𝑀, 𝑁〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}) |
5 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀)) | |
6 | 5 | eqeq1d 2612 | . . . 4 ⊢ (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦)) |
7 | 6 | rexbidv 3034 | . . 3 ⊢ (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦)) |
8 | eqeq2 2621 | . . . 4 ⊢ (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁)) | |
9 | 8 | rexbidv 3034 | . . 3 ⊢ (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁)) |
10 | 7, 9 | opelopab2 4921 | . 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 271 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 〈cop 4131 class class class wbr 4583 {copab 4642 (class class class)co 6549 · cmul 9820 ℤcz 11254 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-iota 5768 df-fv 5812 df-ov 6552 df-dvds 14822 |
This theorem is referenced by: dvdsval2 14824 dvds0lem 14830 dvds1lem 14831 dvds2lem 14832 0dvds 14840 dvdsle 14870 divconjdvds 14875 odd2np1 14903 even2n 14904 oddm1even 14905 opeo 14927 omeo 14928 m1exp1 14931 divalglem4 14957 divalglem9 14962 divalgb 14965 modremain 14970 zeqzmulgcd 15070 bezoutlem4 15097 gcddiv 15106 dvdssqim 15111 coprmdvds2 15206 congr 15216 divgcdcoprm0 15217 cncongr2 15220 dvdsnprmd 15241 prmpwdvds 15446 odmulg 17796 gexdvdsi 17821 lgsquadlem2 24906 dvdspw 30889 dvdsrabdioph 36392 jm2.26a 36585 coskpi2 38749 cosknegpi 38752 fourierswlem 39123 dfeven2 40100 |
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