Theorem rspssbasd

Description: The span of a set of ring elements is a set of ring elements. (Contributed by SN, 19-Jun-2025)

	Ref	Expression
Hypotheses	rspssbasd.k	$$⊢K=\mathrm{RSpan}\left(R\right)$$
	rspssbasd.b	$$⊢B={\mathrm{Base}}_R$$
	rspssbasd.r	$$⊢\phi \to R\in \mathrm{Ring}$$
	rspssbasd.g	$$⊢\phi \to G\subseteq B$$
Assertion	rspssbasd	$$⊢\phi \to K\left(G\right)\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		rspssbasd.k	$$⊢K=\mathrm{RSpan}\left(R\right)$$
2		rspssbasd.b	$$⊢B={\mathrm{Base}}_R$$
3		rspssbasd.r	$$⊢\phi \to R\in \mathrm{Ring}$$
4		rspssbasd.g	$$⊢\phi \to G\subseteq B$$
5		eqid	$$⊢\mathrm{LIdeal}\left(R\right)=\mathrm{LIdeal}\left(R\right)$$
6	1 2 5	rspcl	$$⊢\left(R\in \mathrm{Ring}\wedge G\subseteq B\right)\to K\left(G\right)\in \mathrm{LIdeal}\left(R\right)$$
7	3 4 6	syl2anc	$$⊢\phi \to K\left(G\right)\in \mathrm{LIdeal}\left(R\right)$$
8	2 5	lidlss	$$⊢K\left(G\right)\in \mathrm{LIdeal}\left(R\right)\to K\left(G\right)\subseteq B$$
9	7 8	syl	$$⊢\phi \to K\left(G\right)\subseteq B$$