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Theorem isodd 47553
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 7437 . . . 4 (𝑧 = 𝑍 → (𝑧 + 1) = (𝑍 + 1))
21oveq1d 7445 . . 3 (𝑧 = 𝑍 → ((𝑧 + 1) / 2) = ((𝑍 + 1) / 2))
32eleq1d 2823 . 2 (𝑧 = 𝑍 → (((𝑧 + 1) / 2) ∈ ℤ ↔ ((𝑍 + 1) / 2) ∈ ℤ))
4 df-odd 47551 . 2 Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
53, 4elrab2 3697 1 (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wcel 2105  (class class class)co 7430  1c1 11153   + caddc 11155   / cdiv 11917  2c2 12318  cz 12610   Odd codd 47549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-odd 47551
This theorem is referenced by:  oddz  47555  oddp1div2z  47557  isodd2  47559  evenm1odd  47563  evennodd  47567  oddneven  47568  onego  47570  zeoALTV  47594  oddp1evenALTV  47600  1oddALTV  47614
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