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Metamath Proof Explorer

Theorem sdrgdrng

Description: A sub-division-ring is a division ring. (Contributed by SN, 19-Feb-2025)

		Ref	Expression
	Hypothesis	sdrgdrng.1	$⊢ S = R ↾_{𝑠} A$
	Assertion	sdrgdrng	$⊢ A \in SubDRing (R) \to S \in DivRing$

Step	Hyp	Ref	Expression
1		sdrgdrng.1	$⊢ S = R ↾_{𝑠} A$
2		issdrg	$⊢ A \in SubDRing (R) \leftrightarrow (R \in DivRing \land A \in SubRing (R) \land R ↾_{𝑠} A \in DivRing)$
3	2	simp3bi	$⊢ A \in SubDRing (R) \to R ↾_{𝑠} A \in DivRing$
4	1 3	eqeltrid	$⊢ A \in SubDRing (R) \to S \in DivRing$