Theorem rlmsca2

Description: Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015)

	Ref	Expression
Assertion	rlmsca2	$$⊢\mathrm{I}\left(R\right)=\mathrm{Scalar}\left(\mathrm{ringLMod}\left(R\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		fvi	$$⊢R\in \mathrm{V}\to \mathrm{I}\left(R\right)=R$$
2		eqid	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_R$$
3	2	ressid	$$⊢R\in \mathrm{V}\to R{\upharpoonright }_{𝑠}{\mathrm{Base}}_R=R$$
4	1 3	eqtr4d	$$⊢R\in \mathrm{V}\to \mathrm{I}\left(R\right)=R{\upharpoonright }_{𝑠}{\mathrm{Base}}_R$$
5		fvprc	$$⊢\neg R\in \mathrm{V}\to \mathrm{I}\left(R\right)=\varnothing$$
6		reldmress	$$⊢\mathrm{Rel}\mathrm{dom}{\upharpoonright }_{𝑠}$$
7	6	ovprc1	$$⊢\neg R\in \mathrm{V}\to R{\upharpoonright }_{𝑠}{\mathrm{Base}}_R=\varnothing$$
8	5 7	eqtr4d	$$⊢\neg R\in \mathrm{V}\to \mathrm{I}\left(R\right)=R{\upharpoonright }_{𝑠}{\mathrm{Base}}_R$$
9	4 8	pm2.61i	$$⊢\mathrm{I}\left(R\right)=R{\upharpoonright }_{𝑠}{\mathrm{Base}}_R$$
10		rlmval	$$⊢\mathrm{ringLMod}\left(R\right)=\mathrm{subringAlg}\left(R\right)\left({\mathrm{Base}}_R\right)$$
11	10	a1i	$$⊢\top \to \mathrm{ringLMod}\left(R\right)=\mathrm{subringAlg}\left(R\right)\left({\mathrm{Base}}_R\right)$$
12		ssidd	$$⊢\top \to {\mathrm{Base}}_R\subseteq {\mathrm{Base}}_R$$
13	11 12	srasca	$$⊢\top \to R{\upharpoonright }_{𝑠}{\mathrm{Base}}_R=\mathrm{Scalar}\left(\mathrm{ringLMod}\left(R\right)\right)$$
14	13	mptru	$$⊢R{\upharpoonright }_{𝑠}{\mathrm{Base}}_R=\mathrm{Scalar}\left(\mathrm{ringLMod}\left(R\right)\right)$$
15	9 14	eqtri	$$⊢\mathrm{I}\left(R\right)=\mathrm{Scalar}\left(\mathrm{ringLMod}\left(R\right)\right)$$