Theorem idomringd

Description: An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025)

	Ref	Expression
Hypothesis	idomringd.1	$$⊢\phi \to R\in \mathrm{IDomn}$$
Assertion	idomringd	$$⊢\phi \to R\in \mathrm{Ring}$$

Proof

Step	Hyp	Ref	Expression
1		idomringd.1	$$⊢\phi \to R\in \mathrm{IDomn}$$
2		df-idom	$$⊢\mathrm{IDomn}=\mathrm{CRing}\cap \mathrm{Domn}$$
3	1 2	eleqtrdi	$$⊢\phi \to R\in \left(\mathrm{CRing}\cap \mathrm{Domn}\right)$$
4	3	elin1d	$$⊢\phi \to R\in \mathrm{CRing}$$
5	4	crngringd	$$⊢\phi \to R\in \mathrm{Ring}$$