Theorem oddp1div2z 43792

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 43788	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simprbi 499	1 ⊢ (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2110  (class class class)co 7150  1c1 10532   + caddc 10534   / cdiv 11291  2c2 11686  ℤcz 11975   Odd codd 43784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-ov 7153  df-odd 43786

This theorem is referenced by:  oddm1div2z  43793  oddp1eveni  43800