Theorem oddp1div2z 47634

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 47630	. 2 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
2	1	simprbi 496	1 ⊢ (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2109  (class class class)co 7387  1c1 11069   + caddc 11071   / cdiv 11835  2c2 12241  ℤcz 12529   Odd codd 47626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-odd 47628

This theorem is referenced by:  oddm1div2z  47635  oddp1eveni  47642