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Theorem divides 15191
Description: Define the divides relation. 𝑀𝑁 means 𝑀 divides into 𝑁 with no remainder. For example, 3 ∥ 6 (ex-dvds 27655). As proven in dvdsval3 15193, 𝑀𝑁 ↔ (𝑁 mod 𝑀) = 0. See divides 15191 and dvdsval2 15192 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 4788 . . 3 (𝑀𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ∥ )
2 df-dvds 15190 . . . 4 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)}
32eleq2i 2842 . . 3 (⟨𝑀, 𝑁⟩ ∈ ∥ ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
41, 3bitri 264 . 2 (𝑀𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)})
5 oveq2 6804 . . . . 5 (𝑥 = 𝑀 → (𝑛 · 𝑥) = (𝑛 · 𝑀))
65eqeq1d 2773 . . . 4 (𝑥 = 𝑀 → ((𝑛 · 𝑥) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑦))
76rexbidv 3200 . . 3 (𝑥 = 𝑀 → (∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦))
8 eqeq2 2782 . . . 4 (𝑦 = 𝑁 → ((𝑛 · 𝑀) = 𝑦 ↔ (𝑛 · 𝑀) = 𝑁))
98rexbidv 3200 . . 3 (𝑦 = 𝑁 → (∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑦 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
107, 9opelopab2 5130 . 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 272 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 𝑀) = 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wrex 3062  cop 4323   class class class wbr 4787  {copab 4847  (class class class)co 6796   · cmul 10147  cz 11584  cdvds 15189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-iota 5993  df-fv 6038  df-ov 6799  df-dvds 15190
This theorem is referenced by:  dvdsval2  15192  dvds0lem  15201  dvds1lem  15202  dvds2lem  15203  0dvds  15211  dvdsle  15241  divconjdvds  15246  odd2np1  15273  even2n  15274  oddm1even  15275  opeo  15297  omeo  15298  m1exp1  15301  divalglem4  15327  divalglem9  15332  divalgb  15335  modremain  15340  zeqzmulgcd  15440  bezoutlem4  15467  gcddiv  15476  dvdssqim  15481  coprmdvds2  15575  congr  15585  divgcdcoprm0  15586  cncongr2  15589  dvdsnprmd  15610  prmpwdvds  15815  odmulg  18180  gexdvdsi  18205  lgsquadlem2  25327  dvdspw  31974  dvdsrabdioph  37900  jm2.26a  38093  coskpi2  40592  cosknegpi  40595  fourierswlem  40961  dfeven2  42087
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