Theorem rlm0

Description: Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014) (Revised by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Assertion	rlm0	$$⊢0_R=0_{\mathrm{ringLMod}\left(R\right)}$$

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		eqidd	$$⊢\top \to 0_R=0_R$$
4		ssidd	$$⊢\top \to {\mathrm{Base}}_R\subseteq {\mathrm{Base}}_R$$
5	2 3 4	sralmod0	$$⊢\top \to 0_R=0_{\mathrm{ringLMod}\left(R\right)}$$
6	5	mptru	$$⊢0_R=0_{\mathrm{ringLMod}\left(R\right)}$$