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Theorem evendiv2z 45795
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
StepHypRef Expression
1 iseven 45791 . 2 (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
21simprbi 497 1 (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  (class class class)co 7356   / cdiv 11811  2c2 12207  cz 12498   Even ceven 45787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7359  df-even 45789
This theorem is referenced by:  zefldiv2ALTV  45824  nn0e  45860  nneven  45861
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