Theorem zringmulr

Description: The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017) (Revised by AV, 9-Jun-2019)

	Ref	Expression
Assertion	zringmulr	$$⊢\times ={\cdot }_{{\mathbb{Z}}_{\mathrm{ring}}}$$

Proof

Step	Hyp	Ref	Expression
1		zex	$$⊢\mathbb{Z}\in \mathrm{V}$$
2		df-zring	$$⊢{\mathbb{Z}}_{\mathrm{ring}}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Z}$$
3		cnfldmul	$$⊢\times ={\cdot }_{{ℂ}_{\mathrm{fld}}}$$
4	2 3	ressmulr	$$⊢\mathbb{Z}\in \mathrm{V}\to \times ={\cdot }_{{\mathbb{Z}}_{\mathrm{ring}}}$$
5	1 4	ax-mp	$$⊢\times ={\cdot }_{{\mathbb{Z}}_{\mathrm{ring}}}$$