Theorem isodd 40080

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.

Step	Hyp	Ref	Expression
1		df-odd 40078	. . 3 ⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
2	1	eleq2i 2680	. 2 ⊢ (𝑍 ∈ Odd ↔ 𝑍 ∈ {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ})
3		oveq1 6556	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 + 1) = (𝑍 + 1))
4	3	oveq1d 6564	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑧 + 1) / 2) = ((𝑍 + 1) / 2))
5	4	eleq1d 2672	. . 3 ⊢ (𝑧 = 𝑍 → (((𝑧 + 1) / 2) ∈ ℤ ↔ ((𝑍 + 1) / 2) ∈ ℤ))
6	5	elrab 3331	. 2 ⊢ (𝑍 ∈ {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ} ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
7	2, 6	bitri 263	1 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  (class class class)co 6549  1c1 9816   + caddc 9818   / cdiv 10563  2c2 10947  ℤcz 11254   Odd codd 40076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-odd 40078

This theorem is referenced by:  oddz  40082  oddp1div2z  40084  isodd2  40086  evenm1odd  40090  evennodd  40094  oddneven  40095  onego  40097  zeoALTV  40119  oddp1evenALTV  40125  1oddALTV  40139