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Theorem evenz 45613
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 45611 . 2 (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
21simplbi 499 1 (𝑍 ∈ Even → 𝑍 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  (class class class)co 7350   / cdiv 11746  2c2 12142  cz 12433   Even ceven 45607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-iota 6444  df-fv 6500  df-ov 7353  df-even 45609
This theorem is referenced by:  evenm1odd  45622  evenp1odd  45623  bits0eALTV  45663  opeoALTV  45667  omeoALTV  45669  epoo  45686  emoo  45687  epee  45688  emee  45689  evensumeven  45690  evenltle  45700  even3prm2  45702  mogoldbblem  45703  sbgoldbalt  45764  sgoldbeven3prm  45766  mogoldbb  45768  bgoldbachlt  45796  tgblthelfgott  45798
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