Theorem evendiv2z 48123

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 48119	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simprbi 498	1 ⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2119  (class class class)co 7356   / cdiv 11798  2c2 12227  ℤcz 12515   Even ceven 48115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-even 48117

This theorem is referenced by:  zefldiv2ALTV  48152  nn0e  48188  nneven  48189