Metamath Proof Explorer

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

Proof

Step	Hyp	Ref	Expression
1		oveq1	$⊢ z = Z \to z + 1 = Z + 1$
2	1	oveq1d	$⊢ z = Z \to \frac{z + 1}{2} = \frac{Z + 1}{2}$
3	2	eleq1d	$⊢ z = Z \to (\frac{z + 1}{2} \in ℤ \leftrightarrow \frac{Z + 1}{2} \in ℤ)$
4		df-odd	$⊢ Odd = \{z \in ℤ \| \frac{z + 1}{2} \in ℤ\}$
5	3 4	elrab2	$⊢ Z \in Odd \leftrightarrow (Z \in ℤ \land \frac{Z + 1}{2} \in ℤ)$