Metamath Proof Explorer

Theorem fldidomOLD

Description: Obsolete version of fldidom as of 11-Nov-2024. (Contributed by Mario Carneiro, 29-Mar-2015) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	fldidomOLD	$⊢ R \in Field \to R \in IDomn$

Proof

Step	Hyp	Ref	Expression
1		isfld	$⊢ R \in Field \leftrightarrow (R \in DivRing \land R \in CRing)$
2	1	simprbi	$⊢ R \in Field \to R \in CRing$
3	1	simplbi	$⊢ R \in Field \to R \in DivRing$
4		drngdomn	$⊢ R \in DivRing \to R \in Domn$
5	3 4	syl	$⊢ R \in Field \to R \in Domn$
6		isidom	$⊢ R \in IDomn \leftrightarrow (R \in CRing \land R \in Domn)$
7	2 5 6	sylanbrc	$⊢ R \in Field \to R \in IDomn$