Theorem divides 11729

Description: Define the divides relation. 𝑀 ∥ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 13611). As proven in dvdsval3 11731, 𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 11729 and dvdsval2 11730 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
divides	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃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.

Step	Hyp	Ref	Expression
1		df-br 3983	. . 3 ⊢ (𝑀 ∥ 𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ∥ )
2		df-dvds 11728	. . . 4 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
3	2	eleq2i 2233	. . 3 ⊢ (⟨𝑀, 𝑁⟩ ∈ ∥ ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
4	1, 3	bitri 183	. 2 ⊢ (𝑀 ∥ 𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
5		oveq2 5850	. . . . 5 ⊢ (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀))
6	5	eqeq1d 2174	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦))
7	6	rexbidv 2467	. . 3 ⊢ (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦))
8		eqeq2 2175	. . . 4 ⊢ (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁))
9	8	rexbidv 2467	. . 3 ⊢ (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
10	7, 9	opelopab2 4248	. 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 191	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ∃wrex 2445  ⟨cop 3579   class class class wbr 3982  {copab 4042  (class class class)co 5842   · cmul 7758  ℤcz 9191   ∥ cdvds 11727

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-iota 5153  df-fv 5196  df-ov 5845  df-dvds 11728

This theorem is referenced by:  dvdsval2  11730  dvds0lem  11741  dvds1lem  11742  dvds2lem  11743  0dvds  11751  dvdsle  11782  divconjdvds  11787  odd2np1  11810  even2n  11811  oddm1even  11812  opeo  11834  omeo  11835  m1exp1  11838  divalgb  11862  modremain  11866  zeqzmulgcd  11903  gcddiv  11952  dvdssqim  11957  coprmdvds2  12025  congr  12032  divgcdcoprm0  12033  cncongr2  12036  dvdsnprmd  12057  prmpwdvds  12285