Theorem oddp1div2z 45085

Description: The result of dividing an odd number increased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)

Assertion

Ref	Expression
oddp1div2z	⊢ (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)

Proof of Theorem oddp1div2z

Step	Hyp	Ref	Expression
1		isodd 45081	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simprbi 497	1 ⊢ (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2106  (class class class)co 7275  1c1 10872   + caddc 10874   / cdiv 11632  2c2 12028  ℤcz 12319   Odd codd 45077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-odd 45079

This theorem is referenced by:  oddm1div2z  45086  oddp1eveni  45093