Theorem zring1

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

	Ref	Expression
Assertion	zring1	$$⊢1=1_{{\mathbb{Z}}_{\mathrm{ring}}}$$

Proof

Step	Hyp	Ref	Expression
1		zsubrg	$$⊢\mathbb{Z}\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)$$
2		df-zring	$$⊢{\mathbb{Z}}_{\mathrm{ring}}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Z}$$
3		cnfld1	$$⊢1=1_{{ℂ}_{\mathrm{fld}}}$$
4	2 3	subrg1	$$⊢\mathbb{Z}\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\to 1=1_{{\mathbb{Z}}_{\mathrm{ring}}}$$
5	1 4	ax-mp	$$⊢1=1_{{\mathbb{Z}}_{\mathrm{ring}}}$$