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 e. Odd <-> ( Z e. ZZ /\ ( ( Z - 1 ) / 2 ) e. ZZ ) )

Proof

Step	Hyp	Ref	Expression
1		isodd	|- ( Z e. Odd <-> ( Z e. ZZ /\ ( ( Z + 1 ) / 2 ) e. ZZ ) )
2		zob	|- ( Z e. ZZ -> ( ( ( Z + 1 ) / 2 ) e. ZZ <-> ( ( Z - 1 ) / 2 ) e. ZZ ) )
3	2	pm5.32i	|- ( ( Z e. ZZ /\ ( ( Z + 1 ) / 2 ) e. ZZ ) <-> ( Z e. ZZ /\ ( ( Z - 1 ) / 2 ) e. ZZ ) )
4	1 3	bitri	|- ( Z e. Odd <-> ( Z e. ZZ /\ ( ( Z - 1 ) / 2 ) e. ZZ ) )