Theorem evendiv2z 48397

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 48393	. 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
2	1	simprbi 502	1 ⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2143  (class class class)co 7410   / cdiv 11866  2c2 12290  ℤcz 12586   Even ceven 48389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-even 48391

This theorem is referenced by:  zefldiv2ALTV  48426  nn0e  48462  nneven  48463