Theorem df2idl2rng

Description: Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025)

	Ref	Expression
Hypotheses	df2idl2rng.u	|- U = ( 2Ideal ` R )
	df2idl2rng.b	|- B = ( Base ` R )
	df2idl2rng.t	|- .x. = ( .r ` R )
Assertion	df2idl2rng	|- ( ( R e. Rng /\ I e. ( SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) ) )

Proof

Step	Hyp	Ref	Expression
1		df2idl2rng.u	|- U = ( 2Ideal ` R )
2		df2idl2rng.b	|- B = ( Base ` R )
3		df2idl2rng.t	|- .x. = ( .r ` R )
4		eqid	|- ( LIdeal ` R ) = ( LIdeal ` R )
5	4 2 3	dflidl2rng	|- ( ( R e. Rng /\ I e. ( SubGrp ` R ) ) -> ( I e. ( LIdeal ` R ) <-> A. x e. B A. y e. I ( x .x. y ) e. I ) )
6		eqid	|- ( LIdeal ` ( oppR ` R ) ) = ( LIdeal ` ( oppR ` R ) )
7	6 2 3	isridlrng	|- ( ( R e. Rng /\ I e. ( SubGrp ` R ) ) -> ( I e. ( LIdeal ` ( oppR ` R ) ) <-> A. x e. B A. y e. I ( y .x. x ) e. I ) )
8	5 7	anbi12d	|- ( ( R e. Rng /\ I e. ( SubGrp ` R ) ) -> ( ( I e. ( LIdeal ` R ) /\ I e. ( LIdeal ` ( oppR ` R ) ) ) <-> ( A. x e. B A. y e. I ( x .x. y ) e. I /\ A. x e. B A. y e. I ( y .x. x ) e. I ) ) )
9		eqid	|- ( oppR ` R ) = ( oppR ` R )
10	4 9 6 1	2idlelb	|- ( I e. U <-> ( I e. ( LIdeal ` R ) /\ I e. ( LIdeal ` ( oppR ` R ) ) ) )
11		r19.26-2	|- ( A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) <-> ( A. x e. B A. y e. I ( x .x. y ) e. I /\ A. x e. B A. y e. I ( y .x. x ) e. I ) )
12	8 10 11	3bitr4g	|- ( ( R e. Rng /\ I e. ( SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) ) )