Theorem mnringvscad

Description: The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024) (Proof shortened by AV, 1-Nov-2024)

	Ref	Expression
Hypotheses	mnringvscad.1	$$⊢F=R\mathrm{MndRing}M$$
	mnringvscad.2	$$⊢B={\mathrm{Base}}_M$$
	mnringvscad.3	$$⊢V=R\mathrm{freeLMod}B$$
	mnringvscad.4	$$⊢\phi \to R\in U$$
	mnringvscad.5	$$⊢\phi \to M\in W$$
Assertion	mnringvscad	$$⊢\phi \to {\cdot }_V={\cdot }_F$$

Proof

Step	Hyp	Ref	Expression
1		mnringvscad.1	$$⊢F=R\mathrm{MndRing}M$$
2		mnringvscad.2	$$⊢B={\mathrm{Base}}_M$$
3		mnringvscad.3	$$⊢V=R\mathrm{freeLMod}B$$
4		mnringvscad.4	$$⊢\phi \to R\in U$$
5		mnringvscad.5	$$⊢\phi \to M\in W$$
6		vscaid	$$⊢{\cdot }_{𝑠}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
7		vscandxnmulrndx	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$
8	1 6 7 2 3 4 5	mnringnmulrd	$$⊢\phi \to {\cdot }_V={\cdot }_F$$