Theorem evendiv2z 48535

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 48531	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simprbi 503	1 ⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  (class class class)co 7416   / cdiv 11898  2c2 12322  ℤcz 12618   Even ceven 48527

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 2734

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 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7419  df-even 48529

This theorem is used by:  zefldiv2ALTV  48564  nn0e  48600  nneven  48601