Theorem evenz 48546

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 48544	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simplbi 502	1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  (class class class)co 7413   / cdiv 11895  2c2 12319  ℤcz 12615   Even ceven 48540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7416  df-even 48542

This theorem is used by:  evenm1odd  48555  evenp1odd  48556  bits0eALTV  48596  opeoALTV  48600  omeoALTV  48602  epoo  48619  emoo  48620  epee  48621  emee  48622  evensumeven  48623  evenltle  48633  even3prm2  48635  mogoldbblem  48636  sbgoldbalt  48697  sgoldbeven3prm  48699  mogoldbb  48701  bgoldbachlt  48729  tgblthelfgott  48731