Theorem drngring

Description: A division ring is a ring. (Contributed by NM, 8-Sep-2011)

	Ref	Expression
Assertion	drngring	⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2		eqid	⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 )
3		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
4	1 2 3	isdrng	⊢ ( 𝑅 ∈ DivRing ↔ ( 𝑅 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( ( Base ‘ 𝑅 ) ∖ { ( 0g ‘ 𝑅 ) } ) ) )
5	4	simplbi	⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring )