Theorem evendiv2z 48281

Description: The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.)

Assertion

Ref	Expression
evendiv2z	⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)

Proof of Theorem evendiv2z

Step	Hyp	Ref	Expression
1		iseven 48277	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simprbi 502	1 ⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2149  (class class class)co 7408   / cdiv 11867  2c2 12291  ℤcz 12587   Even ceven 48273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-ov 7411  df-even 48275

This theorem is referenced by:  zefldiv2ALTV  48310  nn0e  48346  nneven  48347