Theorem divides 16157

Description: Define the divides relation. 𝑀 ∥ 𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 30426). As proven in dvdsval3 16159, 𝑀 ∥ 𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 16157 and dvdsval2 16158 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 5090	. . 3 ⊢ (𝑀 ∥ 𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ∥ )
2		df-dvds 16156	. . . 4 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
3	2	eleq2i 2821	. . 3 ⊢ (⟨𝑀, 𝑁⟩ ∈ ∥ ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
4	1, 3	bitri 275	. 2 ⊢ (𝑀 ∥ 𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
5		oveq2 7349	. . . . 5 ⊢ (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀))
6	5	eqeq1d 2732	. . . 4 ⊢ (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦))
7	6	rexbidv 3154	. . 3 ⊢ (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦))
8		eqeq2 2742	. . . 4 ⊢ (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁))
9	8	rexbidv 3154	. . 3 ⊢ (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
10	7, 9	opelopab2 5479	. 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 283	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∃wrex 3054  ⟨cop 4580   class class class wbr 5089  {copab 5151  (class class class)co 7341   · cmul 11003  ℤcz 12460   ∥ cdvds 16155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-iota 6433  df-fv 6485  df-ov 7344  df-dvds 16156

This theorem is referenced by:  dvdsval2  16158  dvds0lem  16169  dvds1lem  16170  dvds2lem  16171  0dvds  16179  dvdsle  16213  divconjdvds  16218  dvdsexp2im  16230  odd2np1  16244  even2n  16245  oddm1even  16246  opeo  16268  omeo  16269  m1exp1  16279  divalglem4  16299  divalglem9  16304  divalgb  16307  modremain  16311  zeqzmulgcd  16413  bezoutlem4  16445  gcddiv  16454  dvdssqim  16457  dvdsexpim  16458  coprmdvds2  16557  congr  16567  divgcdcoprm0  16568  cncongr2  16571  dvdsnprmd  16593  prmpwdvds  16808  odmulg  19461  gexdvdsi  19488  lgsquadlem2  27312  primrootspoweq0  42118  aks6d1c2  42142  grpods  42206  unitscyglem4  42210  dvdsrabdioph  42822  jm2.26a  43012  coskpi2  45883  cosknegpi  45886  fourierswlem  46247  dfeven2  47659