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Theorem divides 11422
Description: Define the divides relation. 𝑀𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 12869). As proven in dvdsval3 11424, 𝑀𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 11422 and dvdsval2 11423 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 3900 . . 3 (𝑀𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ∥ )
2 df-dvds 11421 . . . 4 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
32eleq2i 2184 . . 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 5750 . . . . 5 (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀))
65eqeq1d 2126 . . . 4 (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦))
76rexbidv 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))
98rexbidv 2415 . . 3 (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
107, 9opelopab2 4162 . 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 1316  wcel 1465  wrex 2394  cop 3500   class class class wbr 3899  {copab 3958  (class class class)co 5742   · cmul 7593  cz 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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