Theorem zringcrng

Description: The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019)

	Ref	Expression
Assertion	zringcrng	$$⊢{\mathbb{Z}}_{\mathrm{ring}}\in \mathrm{CRing}$$

Proof

Step	Hyp	Ref	Expression
1		cncrng	$$⊢{ℂ}_{\mathrm{fld}}\in \mathrm{CRing}$$
2		zsubrg	$$⊢\mathbb{Z}\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)$$
3		df-zring	$$⊢{\mathbb{Z}}_{\mathrm{ring}}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Z}$$
4	3	subrgcrng	$$⊢\left({ℂ}_{\mathrm{fld}}\in \mathrm{CRing}\wedge \mathbb{Z}\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)\to {\mathbb{Z}}_{\mathrm{ring}}\in \mathrm{CRing}$$
5	1 2 4	mp2an	$$⊢{\mathbb{Z}}_{\mathrm{ring}}\in \mathrm{CRing}$$