Theorem evenz 48835

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 48833	. 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 2144  (class class class)co 7422   / cdiv 11978  2c2 12402  ℤcz 12698   Even ceven 48829

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6496  df-fv 6549  df-ov 7425  df-even 48831

This theorem is used by:  evenm1odd  48844  evenp1odd  48845  bits0eALTV  48885  opeoALTV  48889  omeoALTV  48891  epoo  48908  emoo  48909  epee  48910  emee  48911  evensumeven  48912  evenltle  48922  even3prm2  48924  mogoldbblem  48925  sbgoldbalt  48986  sgoldbeven3prm  48988  mogoldbb  48990  bgoldbachlt  49018  tgblthelfgott  49020