Theorem rlmsca

Description: Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014)

	Ref	Expression
Assertion	rlmsca	$$⊢R\in X\to R=\mathrm{Scalar}\left(\mathrm{ringLMod}\left(R\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_R$$
2	1	ressid	$$⊢R\in X\to R{\upharpoonright }_{𝑠}{\mathrm{Base}}_R=R$$
3		rlmval	$$⊢\mathrm{ringLMod}\left(R\right)=\mathrm{subringAlg}\left(R\right)\left({\mathrm{Base}}_R\right)$$
4	3	a1i	$$⊢R\in X\to \mathrm{ringLMod}\left(R\right)=\mathrm{subringAlg}\left(R\right)\left({\mathrm{Base}}_R\right)$$
5		ssidd	$$⊢R\in X\to {\mathrm{Base}}_R\subseteq {\mathrm{Base}}_R$$
6	4 5	srasca	$$⊢R\in X\to R{\upharpoonright }_{𝑠}{\mathrm{Base}}_R=\mathrm{Scalar}\left(\mathrm{ringLMod}\left(R\right)\right)$$
7	2 6	eqtr3d	$$⊢R\in X\to R=\mathrm{Scalar}\left(\mathrm{ringLMod}\left(R\right)\right)$$