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Theorem oddp1div2z 44077
 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 44073 . 2 (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
21simprbi 500 1 (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115  (class class class)co 7149  1c1 10536   + caddc 10538   / cdiv 11295  2c2 11689  ℤcz 11978   Odd codd 44069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-odd 44071 This theorem is referenced by:  oddm1div2z  44078  oddp1eveni  44085
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