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Theorem isodd 47616
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 7438 . . . 4 (𝑧 = 𝑍 → (𝑧 + 1) = (𝑍 + 1))
21oveq1d 7446 . . 3 (𝑧 = 𝑍 → ((𝑧 + 1) / 2) = ((𝑍 + 1) / 2))
32eleq1d 2826 . 2 (𝑧 = 𝑍 → (((𝑧 + 1) / 2) ∈ ℤ ↔ ((𝑍 + 1) / 2) ∈ ℤ))
4 df-odd 47614 . 2 Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
53, 4elrab2 3695 1 (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  (class class class)co 7431  1c1 11156   + caddc 11158   / cdiv 11920  2c2 12321  cz 12613   Odd codd 47612
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-odd 47614
This theorem is referenced by:  oddz  47618  oddp1div2z  47620  isodd2  47622  evenm1odd  47626  evennodd  47630  oddneven  47631  onego  47633  zeoALTV  47657  oddp1evenALTV  47663  1oddALTV  47677
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