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Theorem evenz 43990
 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 43988 . 2 (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
21simplbi 501 1 (𝑍 ∈ Even → 𝑍 ∈ ℤ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115  (class class class)co 7138   / cdiv 11282  2c2 11678  ℤcz 11967   Even ceven 43984 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-iota 6295  df-fv 6344  df-ov 7141  df-even 43986 This theorem is referenced by:  evenm1odd  43999  evenp1odd  44000  bits0eALTV  44040  opeoALTV  44044  omeoALTV  44046  epoo  44063  emoo  44064  epee  44065  emee  44066  evensumeven  44067  evenltle  44077  even3prm2  44079  mogoldbblem  44080  sbgoldbalt  44141  sgoldbeven3prm  44143  mogoldbb  44145  bgoldbachlt  44173  tgblthelfgott  44175
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