Theorem isodd2

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 number decreased by 1 and then divided by 2 is still an integer. (Contributed by AV, 15-Jun-2020)

	Ref	Expression
Assertion	isodd2	⊢ ( 𝑍 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ( ( 𝑍 − 1 ) / 2 ) ∈ ℤ ) )

Proof

Step	Hyp	Ref	Expression
1		isodd	⊢ ( 𝑍 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ) )
2		zob	⊢ ( 𝑍 ∈ ℤ → ( ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ↔ ( ( 𝑍 − 1 ) / 2 ) ∈ ℤ ) )
3	2	pm5.32i	⊢ ( ( 𝑍 ∈ ℤ ∧ ( ( 𝑍 + 1 ) / 2 ) ∈ ℤ ) ↔ ( 𝑍 ∈ ℤ ∧ ( ( 𝑍 − 1 ) / 2 ) ∈ ℤ ) )
4	1 3	bitri	⊢ ( 𝑍 ∈ Odd ↔ ( 𝑍 ∈ ℤ ∧ ( ( 𝑍 − 1 ) / 2 ) ∈ ℤ ) )