Theorem zringmpg

Description: The multiplicative group of the ring of integers is the restriction of the multiplicative group of the complex numbers to the integers. (Contributed by AV, 15-Jun-2019)

	Ref	Expression
Assertion	zringmpg	$$⊢{\mathrm{mulGrp}}_{{ℂ}_{\mathrm{fld}}}{\upharpoonright }_{𝑠}\mathbb{Z}={\mathrm{mulGrp}}_{{\mathbb{Z}}_{\mathrm{ring}}}$$

Proof

Step	Hyp	Ref	Expression
1		cndrng	$$⊢{ℂ}_{\mathrm{fld}}\in \mathrm{DivRing}$$
2		zex	$$⊢\mathbb{Z}\in \mathrm{V}$$
3		df-zring	$$⊢{\mathbb{Z}}_{\mathrm{ring}}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Z}$$
4		eqid	$$⊢{\mathrm{mulGrp}}_{{ℂ}_{\mathrm{fld}}}={\mathrm{mulGrp}}_{{ℂ}_{\mathrm{fld}}}$$
5	3 4	mgpress	$$⊢\left({ℂ}_{\mathrm{fld}}\in \mathrm{DivRing}\wedge \mathbb{Z}\in \mathrm{V}\right)\to {\mathrm{mulGrp}}_{{ℂ}_{\mathrm{fld}}}{\upharpoonright }_{𝑠}\mathbb{Z}={\mathrm{mulGrp}}_{{\mathbb{Z}}_{\mathrm{ring}}}$$
6	1 2 5	mp2an	$$⊢{\mathrm{mulGrp}}_{{ℂ}_{\mathrm{fld}}}{\upharpoonright }_{𝑠}\mathbb{Z}={\mathrm{mulGrp}}_{{\mathbb{Z}}_{\mathrm{ring}}}$$