Theorem divides 11751

Description: Define the divides relation. 𝑀 ∥ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 13765). As proven in dvdsval3 11753, 𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 11751 and dvdsval2 11752 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 3990	. . 3 ⊢ (𝑀 ∥ 𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ∥ )
2		df-dvds 11750	. . . 4 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
3	2	eleq2i 2237	. . 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 5861	. . . . 5 ⊢ (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀))
6	5	eqeq1d 2179	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦))
7	6	rexbidv 2471	. . 3 ⊢ (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦))
8		eqeq2 2180	. . . 4 ⊢ (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁))
9	8	rexbidv 2471	. . 3 ⊢ (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
10	7, 9	opelopab2 4255	. 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 1348   ∈ wcel 2141  ∃wrex 2449  ⟨cop 3586   class class class wbr 3989  {copab 4049  (class class class)co 5853   · cmul 7779  ℤcz 9212   ∥ cdvds 11749

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-iota 5160  df-fv 5206  df-ov 5856  df-dvds 11750

This theorem is referenced by:  dvdsval2  11752  dvds0lem  11763  dvds1lem  11764  dvds2lem  11765  0dvds  11773  dvdsle  11804  divconjdvds  11809  odd2np1  11832  even2n  11833  oddm1even  11834  opeo  11856  omeo  11857  m1exp1  11860  divalgb  11884  modremain  11888  zeqzmulgcd  11925  gcddiv  11974  dvdssqim  11979  coprmdvds2  12047  congr  12054  divgcdcoprm0  12055  cncongr2  12058  dvdsnprmd  12079  prmpwdvds  12307