Theorem rlmvsca

Description: Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015)

	Ref	Expression
Assertion	rlmvsca	$$⊢{\cdot }_R={\cdot }_{\mathrm{ringLMod}\left(R\right)}$$

Proof

Step	Hyp	Ref	Expression
1		rlmval	$$⊢\mathrm{ringLMod}\left(R\right)=\mathrm{subringAlg}\left(R\right)\left({\mathrm{Base}}_R\right)$$
2	1	a1i	$$⊢\top \to \mathrm{ringLMod}\left(R\right)=\mathrm{subringAlg}\left(R\right)\left({\mathrm{Base}}_R\right)$$
3		ssidd	$$⊢\top \to {\mathrm{Base}}_R\subseteq {\mathrm{Base}}_R$$
4	2 3	sravsca	$$⊢\top \to {\cdot }_R={\cdot }_{\mathrm{ringLMod}\left(R\right)}$$
5	4	mptru	$$⊢{\cdot }_R={\cdot }_{\mathrm{ringLMod}\left(R\right)}$$