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Metamath Proof Explorer

Theorem subrngbas

Description: Base set of a subring structure. (Contributed by AV, 14-Feb-2025)

		Ref	Expression
	Hypothesis	subrng0.1	$⊢ S = R ↾_{𝑠} A$
	Assertion	subrngbas	$⊢ A \in SubRng (R) \to A = {Base}_{S}$

Step	Hyp	Ref	Expression
1		subrng0.1	$⊢ S = R ↾_{𝑠} A$
2		subrngsubg	$⊢ A \in SubRng (R) \to A \in SubGrp (R)$
3	1	subgbas	$⊢ A \in SubGrp (R) \to A = {Base}_{S}$
4	2 3	syl	$⊢ A \in SubRng (R) \to A = {Base}_{S}$