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Theorem isodd 46597
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 7420 . . . 4 (𝑧 = 𝑍 → (𝑧 + 1) = (𝑍 + 1))
21oveq1d 7428 . . 3 (𝑧 = 𝑍 → ((𝑧 + 1) / 2) = ((𝑍 + 1) / 2))
32eleq1d 2816 . 2 (𝑧 = 𝑍 → (((𝑧 + 1) / 2) ∈ ℤ ↔ ((𝑍 + 1) / 2) ∈ ℤ))
4 df-odd 46595 . 2 Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
53, 4elrab2 3687 1 (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1539  wcel 2104  (class class class)co 7413  1c1 11115   + caddc 11117   / cdiv 11877  2c2 12273  cz 12564   Odd codd 46593
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7416  df-odd 46595
This theorem is referenced by:  oddz  46599  oddp1div2z  46601  isodd2  46603  evenm1odd  46607  evennodd  46611  oddneven  46612  onego  46614  zeoALTV  46638  oddp1evenALTV  46644  1oddALTV  46658
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