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Metamath Proof Explorer

Theorem idomcringd

Description: An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025) Formerly subproof of idomringd . (Proof shortened by SN, 14-May-2025)

		Ref	Expression
	Hypothesis	idomringd.1	$⊢ φ \to R \in IDomn$
	Assertion	idomcringd	$⊢ φ \to R \in CRing$

Proof

Step	Hyp	Ref	Expression
1		idomringd.1	$⊢ φ \to R \in IDomn$
2		df-idom	$⊢ IDomn = CRing \cap Domn$
3	1 2	eleqtrdi	$⊢ φ \to R \in (CRing \cap Domn)$
4	3	elin1d	$⊢ φ \to R \in CRing$