Theorem srabase

Description: Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 4-Oct-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Revised by AV, 29-Oct-2024)

	Ref	Expression
Hypotheses	srapart.a	$$⊢\phi \to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
	srapart.s	$$⊢\phi \to S\subseteq {\mathrm{Base}}_W$$
Assertion	srabase	$$⊢\phi \to {\mathrm{Base}}_W={\mathrm{Base}}_A$$

Proof

Step	Hyp	Ref	Expression
1		srapart.a	$$⊢\phi \to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
2		srapart.s	$$⊢\phi \to S\subseteq {\mathrm{Base}}_W$$
3		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
4		scandxnbasendx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
5		vscandxnbasendx	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
6		ipndxnbasendx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
7	1 2 3 4 5 6	sralem	$$⊢\phi \to {\mathrm{Base}}_W={\mathrm{Base}}_A$$