Theorem sramulr

Description: Multiplicative operation 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	sramulr	$$⊢\phi \to {\cdot }_W={\cdot }_A$$

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		mulridx	$$⊢{\cdot }_{𝑟}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		scandxnmulrndx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}$$
5		vscandxnmulrndx	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$
6		ipndxnmulrndx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}$$
7	1 2 3 4 5 6	sralem	$$⊢\phi \to {\cdot }_W={\cdot }_A$$