Theorem oddp1div2z 46301

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 46297	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simprbi 498	1 ⊢ (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2107  (class class class)co 7409  1c1 11111   + caddc 11113   / cdiv 11871  2c2 12267  ℤcz 12558   Odd codd 46293

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-odd 46295

This theorem is referenced by:  oddm1div2z  46302  oddp1eveni  46309