Theorem isodd7

Description: The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020)

	Ref	Expression
Assertion	isodd7	$$⊢Z\in \mathrm{Odd}\leftrightarrow \left(Z\in \mathbb{Z}\wedge 2\gcd Z=1\right)$$

Proof

Step	Hyp	Ref	Expression
1		isodd3	$$⊢Z\in \mathrm{Odd}\leftrightarrow \left(Z\in \mathbb{Z}\wedge \neg 2\parallel Z\right)$$
2		2prm	$$⊢2\in \mathbb{P}$$
3		coprm	$$⊢\left(2\in \mathbb{P}\wedge Z\in \mathbb{Z}\right)\to \left(\neg 2\parallel Z\leftrightarrow 2\gcd Z=1\right)$$
4	2 3	mpan	$$⊢Z\in \mathbb{Z}\to \left(\neg 2\parallel Z\leftrightarrow 2\gcd Z=1\right)$$
5	4	pm5.32i	$$⊢\left(Z\in \mathbb{Z}\wedge \neg 2\parallel Z\right)\leftrightarrow \left(Z\in \mathbb{Z}\wedge 2\gcd Z=1\right)$$
6	1 5	bitri	$$⊢Z\in \mathrm{Odd}\leftrightarrow \left(Z\in \mathbb{Z}\wedge 2\gcd Z=1\right)$$