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Theorem evenz 40872
 Description: An even number is an integer. (Contributed by AV, 14-Jun-2020.)
Assertion
Ref Expression
evenz (𝑍 ∈ Even → 𝑍 ∈ ℤ)

Proof of Theorem evenz
StepHypRef Expression
1 iseven 40870 . 2 (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
21simplbi 476 1 (𝑍 ∈ Even → 𝑍 ∈ ℤ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1987  (class class class)co 6615   / cdiv 10644  2c2 11030  ℤcz 11337   Even ceven 40866 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-even 40868 This theorem is referenced by:  evenm1odd  40881  evenp1odd  40882  bits0eALTV  40920  opeoALTV  40924  omeoALTV  40926  epoo  40941  emoo  40942  epee  40943  emee  40944  evensumeven  40945  sgoldbalt  40994  bgoldbachlt  41018  tgblthelfgott  41020  bgoldbachltOLD  41025  tgblthelfgottOLD  41027
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