Theorem oddp1div2z 47036

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 47032	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simprbi 495	1 ⊢ (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2098  (class class class)co 7416  1c1 11139   + caddc 11141   / cdiv 11901  2c2 12297  ℤcz 12588   Odd codd 47028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-iota 6495  df-fv 6551  df-ov 7419  df-odd 47030

This theorem is referenced by:  oddm1div2z  47037  oddp1eveni  47044