Theorem prmrngring

Description: A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010) (Revised by AV, 18-Jun-2026) (Proof shortened by AV, 26-Jun-2026)

	Ref	Expression
Assertion	prmrngring	⊢ ( 𝑅 ∈ PrmRing → 𝑅 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		prmringnzring	⊢ ( 𝑅 ∈ PrmRing → 𝑅 ∈ NzRing )
2		nzrring	⊢ ( 𝑅 ∈ NzRing → 𝑅 ∈ Ring )
3	1 2	syl	⊢ ( 𝑅 ∈ PrmRing → 𝑅 ∈ Ring )