Theorem evendiv2z 47626

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 47622	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simprbi 496	1 ⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2109  (class class class)co 7410   / cdiv 11899  2c2 12300  ℤcz 12593   Even ceven 47618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-even 47620

This theorem is referenced by:  zefldiv2ALTV  47655  nn0e  47691  nneven  47692