Metamath Proof Explorer

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	$⊢ Z \in Odd \leftrightarrow (Z \in ℤ \land \frac{Z - 1}{2} \in ℤ)$

Proof

Step	Hyp	Ref	Expression
1		isodd	$⊢ Z \in Odd \leftrightarrow (Z \in ℤ \land \frac{Z + 1}{2} \in ℤ)$
2		zob	$⊢ Z \in ℤ \to (\frac{Z + 1}{2} \in ℤ \leftrightarrow \frac{Z - 1}{2} \in ℤ)$
3	2	pm5.32i	$⊢ (Z \in ℤ \land \frac{Z + 1}{2} \in ℤ) \leftrightarrow (Z \in ℤ \land \frac{Z - 1}{2} \in ℤ)$
4	1 3	bitri	$⊢ Z \in Odd \leftrightarrow (Z \in ℤ \land \frac{Z - 1}{2} \in ℤ)$