Theorem idomcringd

Description: An integral domain is a commutative ring with unity. (Contributed by Jeff Madsen, 6-Jan-2011) (Revised by Thierry Arnoux, 4-May-2025) Formerly subproof of idomringd . (Proof shortened by SN, 14-May-2025)

	Ref	Expression
Hypothesis	idomringd.1	⊢ ( 𝜑 → 𝑅 ∈ IDomn )
Assertion	idomcringd	⊢ ( 𝜑 → 𝑅 ∈ CRing )

Proof

Step	Hyp	Ref	Expression
1		idomringd.1	⊢ ( 𝜑 → 𝑅 ∈ IDomn )
2		df-idom	⊢ IDomn = ( CRing ∩ Domn )
3	1 2	eleqtrdi	⊢ ( 𝜑 → 𝑅 ∈ ( CRing ∩ Domn ) )
4	3	elin1d	⊢ ( 𝜑 → 𝑅 ∈ CRing )