Theorem rng2idlsubgsubrng

Description: A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025)

	Ref	Expression
Hypotheses	rng2idlsubgsubrng.r	|- ( ph -> R e. Rng )
	rng2idlsubgsubrng.i	|- ( ph -> I e. ( 2Ideal ` R ) )
	rng2idlsubgsubrng.u	|- ( ph -> I e. ( SubGrp ` R ) )
Assertion	rng2idlsubgsubrng	|- ( ph -> I e. ( SubRng ` R ) )

Proof

Step	Hyp	Ref	Expression
1		rng2idlsubgsubrng.r	|- ( ph -> R e. Rng )
2		rng2idlsubgsubrng.i	|- ( ph -> I e. ( 2Ideal ` R ) )
3		rng2idlsubgsubrng.u	|- ( ph -> I e. ( SubGrp ` R ) )
4		eqid	|- ( LIdeal ` R ) = ( LIdeal ` R )
5		eqid	|- ( oppR ` R ) = ( oppR ` R )
6		eqid	|- ( LIdeal ` ( oppR ` R ) ) = ( LIdeal ` ( oppR ` R ) )
7		eqid	|- ( 2Ideal ` R ) = ( 2Ideal ` R )
8	4 5 6 7	2idlelb	|- ( I e. ( 2Ideal ` R ) <-> ( I e. ( LIdeal ` R ) /\ I e. ( LIdeal ` ( oppR ` R ) ) ) )
9	8	simplbi	|- ( I e. ( 2Ideal ` R ) -> I e. ( LIdeal ` R ) )
10	2 9	syl	|- ( ph -> I e. ( LIdeal ` R ) )
11		eqid	|- ( R |`s I ) = ( R |`s I )
12	4 11	rnglidlrng	|- ( ( R e. Rng /\ I e. ( LIdeal ` R ) /\ I e. ( SubGrp ` R ) ) -> ( R |`s I ) e. Rng )
13	1 10 3 12	syl3anc	|- ( ph -> ( R |`s I ) e. Rng )
14	1 2 13	rng2idlsubrng	|- ( ph -> I e. ( SubRng ` R ) )