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Theorem divides 15841
Description: Define the divides relation. 𝑀𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 28563). As proven in dvdsval3 15843, 𝑀𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 15841 and dvdsval2 15842 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 5068 . . 3 (𝑀𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ∥ )
2 df-dvds 15840 . . . 4 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
32eleq2i 2830 . . 3 (⟨𝑀, 𝑁⟩ ∈ ∥ ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
41, 3bitri 278 . 2 (𝑀𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
5 oveq2 7239 . . . . 5 (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀))
65eqeq1d 2740 . . . 4 (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦))
76rexbidv 3224 . . 3 (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦))
8 eqeq2 2750 . . . 4 (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁))
98rexbidv 3224 . . 3 (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
107, 9opelopab2 5436 . 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 286 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  wrex 3063  cop 4561   class class class wbr 5067  {copab 5129  (class class class)co 7231   · cmul 10758  cz 12200  cdvds 15839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-iota 6355  df-fv 6405  df-ov 7234  df-dvds 15840
This theorem is referenced by:  dvdsval2  15842  dvds0lem  15852  dvds1lem  15853  dvds2lem  15854  0dvds  15862  dvdsle  15895  divconjdvds  15900  dvdsexp2im  15912  odd2np1  15926  even2n  15927  oddm1even  15928  opeo  15950  omeo  15951  m1exp1  15961  divalglem4  15981  divalglem9  15986  divalgb  15989  modremain  15993  zeqzmulgcd  16093  bezoutlem4  16126  gcddiv  16135  dvdssqim  16140  coprmdvds2  16235  congr  16245  divgcdcoprm0  16246  cncongr2  16249  dvdsnprmd  16271  prmpwdvds  16481  odmulg  18971  gexdvdsi  18996  lgsquadlem2  26286  dvdsexpim  40064  dvdsrabdioph  40363  jm2.26a  40553  coskpi2  43110  cosknegpi  43113  fourierswlem  43474  dfeven2  44802
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