Theorem isevengcd2

Description: The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020) (Revised by AV, 8-Aug-2021)

	Ref	Expression
Assertion	isevengcd2	$$⊢Z\in \mathbb{Z}\to \left(2\parallel Z\leftrightarrow 2\gcd Z=2\right)$$

Proof

Step	Hyp	Ref	Expression
1		2nn	$$⊢2\in \mathbb{N}$$
2		gcdzeq	$$⊢\left(2\in \mathbb{N}\wedge Z\in \mathbb{Z}\right)\to \left(2\gcd Z=2\leftrightarrow 2\parallel Z\right)$$
3	1 2	mpan	$$⊢Z\in \mathbb{Z}\to \left(2\gcd Z=2\leftrightarrow 2\parallel Z\right)$$
4	3	bicomd	$$⊢Z\in \mathbb{Z}\to \left(2\parallel Z\leftrightarrow 2\gcd Z=2\right)$$