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Theorem evenz 39883
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 39881 . 2 (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
21simplbi 474 1 (𝑍 ∈ Even → 𝑍 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  (class class class)co 6524   / cdiv 10530  2c2 10914  cz 11207   Even ceven 39877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-iota 5751  df-fv 5795  df-ov 6527  df-even 39879
This theorem is referenced by:  evenm1odd  39892  evenp1odd  39893  bits0eALTV  39931  opeoALTV  39935  omeoALTV  39937  epoo  39952  emoo  39953  epee  39954  emee  39955  evensumeven  39956  sgoldbalt  40005  bgoldbachlt  40029  tgblthelfgott  40031  bgoldbachltOLD  40036  tgblthelfgottOLD  40038
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