Theorem divides 12558

Description: Define the divides relation. 𝑀 ∥ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 16756). As proven in dvdsval3 12560, 𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 12558 and dvdsval2 12559 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 4131	. . 3 ⊢ (𝑀 ∥ 𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ∥ )
2		df-dvds 12557	. . . 4 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
3	2	eleq2i 2305	. . 3 ⊢ (⟨𝑀, 𝑁⟩ ∈ ∥ ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
4	1, 3	bitri 184	. 2 ⊢ (𝑀 ∥ 𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
5		oveq2 6093	. . . . 5 ⊢ (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀))
6	5	eqeq1d 2247	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦))
7	6	rexbidv 2551	. . 3 ⊢ (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦))
8		eqeq2 2248	. . . 4 ⊢ (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁))
9	8	rexbidv 2551	. . 3 ⊢ (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
10	7, 9	opelopab2 4413	. 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	bitrid 192	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  ⟨cop 3712   class class class wbr 4130  {copab 4191  (class class class)co 6085   · cmul 8184  ℤcz 9646   ∥ cdvds 12556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-iota 5337  df-fv 5385  df-ov 6088  df-dvds 12557

This theorem is used by:  dvdsval2  12559  dvds0lem  12570  dvds1lem  12571  dvds2lem  12572  0dvds  12580  dvdsle  12613  divconjdvds  12618  odd2np1  12642  even2n  12643  oddm1even  12644  opeo  12666  omeo  12667  m1exp1  12670  divalgb  12694  modremain  12698  zeqzmulgcd  12749  gcddiv  12798  dvdssqim  12803  coprmdvds2  12873  congr  12880  divgcdcoprm0  12881  cncongr2  12884  dvdsnprmd  12905  prmpwdvds  13136  lgsquadlem2  16209