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Metamath Proof Explorer

Theorem subrgbas

Description: Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014)

		Ref	Expression
	Hypothesis	subrgbas.b	$⊢ S = R ↾_{𝑠} A$
	Assertion	subrgbas	$⊢ A \in SubRing (R) \to A = {Base}_{S}$

Step	Hyp	Ref	Expression
1		subrgbas.b	$⊢ S = R ↾_{𝑠} A$
2		subrgsubg	$⊢ A \in SubRing (R) \to A \in SubGrp (R)$
3	1	subgbas	$⊢ A \in SubGrp (R) \to A = {Base}_{S}$
4	2 3	syl	$⊢ A \in SubRing (R) \to A = {Base}_{S}$