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Theorem isodd 46069
Description: The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd integer increased by 1 and then divided by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
Assertion
Ref Expression
isodd (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))

Proof of Theorem isodd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 oveq1 7400 . . . 4 (𝑧 = 𝑍 → (𝑧 + 1) = (𝑍 + 1))
21oveq1d 7408 . . 3 (𝑧 = 𝑍 → ((𝑧 + 1) / 2) = ((𝑍 + 1) / 2))
32eleq1d 2817 . 2 (𝑧 = 𝑍 → (((𝑧 + 1) / 2) ∈ ℤ ↔ ((𝑍 + 1) / 2) ∈ ℤ))
4 df-odd 46067 . 2 Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
53, 4elrab2 3682 1 (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  (class class class)co 7393  1c1 11093   + caddc 11095   / cdiv 11853  2c2 12249  cz 12540   Odd codd 46065
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-iota 6484  df-fv 6540  df-ov 7396  df-odd 46067
This theorem is referenced by:  oddz  46071  oddp1div2z  46073  isodd2  46075  evenm1odd  46079  evennodd  46083  oddneven  46084  onego  46086  zeoALTV  46110  oddp1evenALTV  46116  1oddALTV  46130
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