Theorem ridl1

Description: Every ring contains a unit right ideal. (Contributed by AV, 13-Feb-2025)

	Ref	Expression
Hypotheses	ridl0.u	$$⊢U=\mathrm{LIdeal}\left({opp}_r\left(R\right)\right)$$
	ridl1.b	$$⊢B={\mathrm{Base}}_R$$
Assertion	ridl1	$$⊢R\in \mathrm{Ring}\to B\in U$$

Proof

Step	Hyp	Ref	Expression
1		ridl0.u	$$⊢U=\mathrm{LIdeal}\left({opp}_r\left(R\right)\right)$$
2		ridl1.b	$$⊢B={\mathrm{Base}}_R$$
3		eqid	$$⊢{opp}_r\left(R\right)={opp}_r\left(R\right)$$
4	3	opprring	$$⊢R\in \mathrm{Ring}\to {opp}_r\left(R\right)\in \mathrm{Ring}$$
5	3 2	opprbas	$$⊢B={\mathrm{Base}}_{{opp}_r\left(R\right)}$$
6	1 5	lidl1	$$⊢{opp}_r\left(R\right)\in \mathrm{Ring}\to B\in U$$
7	4 6	syl	$$⊢R\in \mathrm{Ring}\to B\in U$$