Metamath Proof Explorer

Theorem prrngorngo

Description: A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010)

		Ref	Expression
	Assertion	prrngorngo	⊢ ( 𝑅 ∈ PrRing → 𝑅 ∈ RingOps )

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 1^st ‘ 𝑅 ) = ( 1^st ‘ 𝑅 )
2		eqid	⊢ ( GId ‘ ( 1^st ‘ 𝑅 ) ) = ( GId ‘ ( 1^st ‘ 𝑅 ) )
3	1 2	isprrngo	⊢ ( 𝑅 ∈ PrRing ↔ ( 𝑅 ∈ RingOps ∧ { ( GId ‘ ( 1^st ‘ 𝑅 ) ) } ∈ ( PrIdl ‘ 𝑅 ) ) )
4	3	simplbi	⊢ ( 𝑅 ∈ PrRing → 𝑅 ∈ RingOps )