Theorem rlmds

Description: Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015)

	Ref	Expression
Assertion	rlmds	$$⊢\mathrm{dist}\left(R\right)=\mathrm{dist}\left(\mathrm{ringLMod}\left(R\right)\right)$$

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	srads	$$⊢\top \to \mathrm{dist}\left(R\right)=\mathrm{dist}\left(\mathrm{ringLMod}\left(R\right)\right)$$
5	4	mptru	$$⊢\mathrm{dist}\left(R\right)=\mathrm{dist}\left(\mathrm{ringLMod}\left(R\right)\right)$$