Theorem isodd

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)

	Ref	Expression
Assertion	isodd	⊢ ( 𝑍 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ) )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝑧 = 𝑍 → ( 𝑧 + 1 ) = ( 𝑍 + 1 ) )
2	1	oveq1d	⊢ ( 𝑧 = 𝑍 → ( ( 𝑧 + 1 ) / 2 ) = ( ( 𝑍 + 1 ) / 2 ) )
3	2	eleq1d	⊢ ( 𝑧 = 𝑍 → ( ( ( 𝑧 + 1 ) / 2 ) ∈ ℤ ↔ ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ) )
4		df-odd	⊢ Odd = { 𝑧 ∈ ℤ ∣ ( ( 𝑧 + 1 ) / 2 ) ∈ ℤ }
5	3 4	elrab2	⊢ ( 𝑍 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ) )