Theorem evenz 44970

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

Step	Hyp	Ref	Expression
1		iseven 44968	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simplbi 497	1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2108  (class class class)co 7255   / cdiv 11562  2c2 11958  ℤcz 12249   Even ceven 44964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-even 44966

This theorem is referenced by:  evenm1odd  44979  evenp1odd  44980  bits0eALTV  45020  opeoALTV  45024  omeoALTV  45026  epoo  45043  emoo  45044  epee  45045  emee  45046  evensumeven  45047  evenltle  45057  even3prm2  45059  mogoldbblem  45060  sbgoldbalt  45121  sgoldbeven3prm  45123  mogoldbb  45125  bgoldbachlt  45153  tgblthelfgott  45155