Theorem rng2idlnsg

Description: A two-sided ideal of a non-unital ring which is a non-unital ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025)

	Ref	Expression
Hypotheses	rng2idlsubrng.r	$$⊢\phi \to R\in \mathrm{Rng}$$
	rng2idlsubrng.i	$$⊢\phi \to I\in \mathrm{2Ideal}\left(R\right)$$
	rng2idlsubrng.u	$$⊢\phi \to R{\upharpoonright }_{𝑠}I\in \mathrm{Rng}$$
Assertion	rng2idlnsg	$$⊢\phi \to I\in \mathrm{NrmSGrp}\left(R\right)$$

Proof

Step	Hyp	Ref	Expression
1		rng2idlsubrng.r	$$⊢\phi \to R\in \mathrm{Rng}$$
2		rng2idlsubrng.i	$$⊢\phi \to I\in \mathrm{2Ideal}\left(R\right)$$
3		rng2idlsubrng.u	$$⊢\phi \to R{\upharpoonright }_{𝑠}I\in \mathrm{Rng}$$
4	1 2 3	rng2idlsubrng	$$⊢\phi \to I\in \mathrm{SubRng}\left(R\right)$$
5		subrngringnsg	$$⊢I\in \mathrm{SubRng}\left(R\right)\to I\in \mathrm{NrmSGrp}\left(R\right)$$
6	4 5	syl	$$⊢\phi \to I\in \mathrm{NrmSGrp}\left(R\right)$$