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Theorem evenz 45577
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 45575 . 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 45571
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 45573
This theorem is referenced by:  evenm1odd  45586  evenp1odd  45587  bits0eALTV  45627  opeoALTV  45631  omeoALTV  45633  epoo  45650  emoo  45651  epee  45652  emee  45653  evensumeven  45654  evenltle  45664  even3prm2  45666  mogoldbblem  45667  sbgoldbalt  45728  sgoldbeven3prm  45730  mogoldbb  45732  bgoldbachlt  45760  tgblthelfgott  45762
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