Theorem zringbas

Description: The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017) (Revised by AV, 9-Jun-2019)

	Ref	Expression
Assertion	zringbas	$$⊢\mathbb{Z}={\mathrm{Base}}_{{\mathbb{Z}}_{\mathrm{ring}}}$$

Proof

Step	Hyp	Ref	Expression
1		zsscn	$$⊢\mathbb{Z}\subseteq ℂ$$
2		df-zring	$$⊢{\mathbb{Z}}_{\mathrm{ring}}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Z}$$
3		cnfldbas	$$⊢ℂ={\mathrm{Base}}_{{ℂ}_{\mathrm{fld}}}$$
4	2 3	ressbas2	$$⊢\mathbb{Z}\subseteq ℂ\to \mathbb{Z}={\mathrm{Base}}_{{\mathbb{Z}}_{\mathrm{ring}}}$$
5	1 4	ax-mp	$$⊢\mathbb{Z}={\mathrm{Base}}_{{\mathbb{Z}}_{\mathrm{ring}}}$$