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Theorem divides 15597
Description: Define the divides relation. 𝑀𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 28162). As proven in dvdsval3 15599, 𝑀𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 15597 and dvdsval2 15598 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
divides ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
Distinct variable groups:   𝑛,𝑀   𝑛,𝑁

Proof of Theorem divides
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5058 . . 3 (𝑀𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ∥ )
2 df-dvds 15596 . . . 4 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
32eleq2i 2901 . . 3 (⟨𝑀, 𝑁⟩ ∈ ∥ ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
41, 3bitri 276 . 2 (𝑀𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
5 oveq2 7153 . . . . 5 (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀))
65eqeq1d 2820 . . . 4 (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦))
76rexbidv 3294 . . 3 (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦))
8 eqeq2 2830 . . . 4 (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁))
98rexbidv 3294 . . 3 (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
107, 9opelopab2 5419 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)} ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
114, 10syl5bb 284 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  cop 4563   class class class wbr 5057  {copab 5119  (class class class)co 7145   · cmul 10530  cz 11969  cdvds 15595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-iota 6307  df-fv 6356  df-ov 7148  df-dvds 15596
This theorem is referenced by:  dvdsval2  15598  dvds0lem  15608  dvds1lem  15609  dvds2lem  15610  0dvds  15618  dvdsle  15648  divconjdvds  15653  odd2np1  15678  even2n  15679  oddm1even  15680  opeo  15702  omeo  15703  m1exp1  15715  divalglem4  15735  divalglem9  15740  divalgb  15743  modremain  15747  zeqzmulgcd  15847  bezoutlem4  15878  gcddiv  15887  dvdssqim  15892  coprmdvds2  15986  congr  15996  divgcdcoprm0  15997  cncongr2  16000  dvdsnprmd  16022  prmpwdvds  16228  odmulg  18612  gexdvdsi  18637  lgsquadlem2  25884  dvdspw  32879  dvdsexpim  39059  dvdsrabdioph  39285  jm2.26a  39475  coskpi2  42023  cosknegpi  42026  fourierswlem  42392  dfeven2  43691
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