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Theorem divides 11678
 Description: Define the divides relation. 𝑀 ∥ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 13277). As proven in dvdsval3 11680, 𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 11678 and dvdsval2 11679 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 3966 . . 3 (𝑀𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ∥ )
2 df-dvds 11677 . . . 4 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
32eleq2i 2224 . . 3 (⟨𝑀, 𝑁⟩ ∈ ∥ ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
41, 3bitri 183 . 2 (𝑀𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
5 oveq2 5829 . . . . 5 (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀))
65eqeq1d 2166 . . . 4 (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦))
76rexbidv 2458 . . 3 (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦))
8 eqeq2 2167 . . . 4 (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁))
98rexbidv 2458 . . 3 (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
107, 9opelopab2 4230 . 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 191 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335   ∈ wcel 2128  ∃wrex 2436  ⟨cop 3563   class class class wbr 3965  {copab 4024  (class class class)co 5821   · cmul 7731  ℤcz 9161   ∥ cdvds 11676 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-iota 5134  df-fv 5177  df-ov 5824  df-dvds 11677 This theorem is referenced by:  dvdsval2  11679  dvds0lem  11689  dvds1lem  11690  dvds2lem  11691  0dvds  11699  dvdsle  11728  divconjdvds  11733  odd2np1  11756  even2n  11757  oddm1even  11758  opeo  11780  omeo  11781  m1exp1  11784  divalgb  11808  modremain  11812  zeqzmulgcd  11845  gcddiv  11894  dvdssqim  11899  coprmdvds2  11961  congr  11968  divgcdcoprm0  11969  cncongr2  11972  dvdsnprmd  11993
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