Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  evendiv2z Structured version   Visualization version   GIF version

Theorem evendiv2z 44165
 Description: The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
Assertion
Ref Expression
evendiv2z (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)

Proof of Theorem evendiv2z
StepHypRef Expression
1 iseven 44161 . 2 (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
21simprbi 500 1 (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  (class class class)co 7135   / cdiv 11288  2c2 11682  ℤcz 11971   Even ceven 44157 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-even 44159 This theorem is referenced by:  zefldiv2ALTV  44194  nn0e  44230  nneven  44231
 Copyright terms: Public domain W3C validator