Theorem rlmnm

Description: The norm function in the ring module. (Contributed by AV, 9-Oct-2021)

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

Proof

Step	Hyp	Ref	Expression
1		rlmbas	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_{\mathrm{ringLMod}\left(R\right)}$$
2		id	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_{\mathrm{ringLMod}\left(R\right)}\to {\mathrm{Base}}_R={\mathrm{Base}}_{\mathrm{ringLMod}\left(R\right)}$$
3		rlmplusg	$$⊢{+}_R={+}_{\mathrm{ringLMod}\left(R\right)}$$
4	3	a1i	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_{\mathrm{ringLMod}\left(R\right)}\to {+}_R={+}_{\mathrm{ringLMod}\left(R\right)}$$
5		rlmds	$$⊢\mathrm{dist}\left(R\right)=\mathrm{dist}\left(\mathrm{ringLMod}\left(R\right)\right)$$
6	5	a1i	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_{\mathrm{ringLMod}\left(R\right)}\to \mathrm{dist}\left(R\right)=\mathrm{dist}\left(\mathrm{ringLMod}\left(R\right)\right)$$
7	2 4 6	nmpropd	$$⊢{\mathrm{Base}}_R={\mathrm{Base}}_{\mathrm{ringLMod}\left(R\right)}\to \mathrm{norm}\left(R\right)=\mathrm{norm}\left(\mathrm{ringLMod}\left(R\right)\right)$$
8	1 7	ax-mp	$$⊢\mathrm{norm}\left(R\right)=\mathrm{norm}\left(\mathrm{ringLMod}\left(R\right)\right)$$