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Metamath Proof Explorer

Theorem sdrgbas

Description: Base set of a sub-division-ring structure. (Contributed by SN, 19-Feb-2025)

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

Step	Hyp	Ref	Expression
1		sdrgbas.b	$⊢ S = R ↾_{𝑠} A$
2		eqid	$⊢ {Base}_{R} = {Base}_{R}$
3	2	sdrgss	$⊢ A \in SubDRing (R) \to A \subseteq {Base}_{R}$
4	1 2	ressbas2	$⊢ A \subseteq {Base}_{R} \to A = {Base}_{S}$
5	3 4	syl	$⊢ A \in SubDRing (R) \to A = {Base}_{S}$