Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  oddp1div2z Structured version   Visualization version   GIF version

Theorem oddp1div2z 42564
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
StepHypRef Expression
1 isodd 42560 . 2 (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
21simprbi 492 1 (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  (class class class)co 6922  1c1 10273   + caddc 10275   / cdiv 11032  2c2 11430  cz 11728   Odd codd 42556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-odd 42558
This theorem is referenced by:  oddm1div2z  42565  oddp1eveni  42572
  Copyright terms: Public domain W3C validator