Theorem isodd 44147

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		oveq1 7142	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 + 1) = (𝑍 + 1))
2	1	oveq1d 7150	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑧 + 1) / 2) = ((𝑍 + 1) / 2))
3	2	eleq1d 2874	. 2 ⊢ (𝑧 = 𝑍 → (((𝑧 + 1) / 2) ∈ ℤ ↔ ((𝑍 + 1) / 2) ∈ ℤ))
4		df-odd 44145	. 2 ⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
5	3, 4	elrab2 3631	1 ⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  (class class class)co 7135  1c1 10527   + caddc 10529   / cdiv 11286  2c2 11680  ℤcz 11969   Odd codd 44143

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-odd 44145

This theorem is referenced by:  oddz  44149  oddp1div2z  44151  isodd2  44153  evenm1odd  44157  evennodd  44161  oddneven  44162  onego  44164  zeoALTV  44188  oddp1evenALTV  44194  1oddALTV  44208