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	⊢ · = ( .r ‘ 𝑅 )
Assertion	df2idl2rng	⊢ ( ( 𝑅 ∈ Rng ∧ 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) → ( 𝐼 ∈ 𝑈 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( ( 𝑥 · 𝑦 ) ∈ 𝐼 ∧ ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df2idl2rng.u	⊢ 𝑈 = ( 2Ideal ‘ 𝑅 )
2		df2idl2rng.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
3		df2idl2rng.t	⊢ · = ( .r ‘ 𝑅 )
4		eqid	⊢ ( LIdeal ‘ 𝑅 ) = ( LIdeal ‘ 𝑅 )
5	4 2 3	dflidl2rng	⊢ ( ( 𝑅 ∈ Rng ∧ 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) → ( 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 · 𝑦 ) ∈ 𝐼 ) )
6		eqid	⊢ ( LIdeal ‘ ( oppr ‘ 𝑅 ) ) = ( LIdeal ‘ ( oppr ‘ 𝑅 ) )
7	6 2 3	isridlrng	⊢ ( ( 𝑅 ∈ Rng ∧ 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) → ( 𝐼 ∈ ( LIdeal ‘ ( oppr ‘ 𝑅 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) )
8	5 7	anbi12d	⊢ ( ( 𝑅 ∈ Rng ∧ 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) → ( ( 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝐼 ∈ ( LIdeal ‘ ( oppr ‘ 𝑅 ) ) ) ↔ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 · 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) )
9		eqid	⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 )
10	4 9 6 1	2idlelb	⊢ ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝐼 ∈ ( LIdeal ‘ ( oppr ‘ 𝑅 ) ) ) )
11		r19.26-2	⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( ( 𝑥 · 𝑦 ) ∈ 𝐼 ∧ ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ↔ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 · 𝑦 ) ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑦 · 𝑥 ) ∈ 𝐼 ) )
12	8 10 11	3bitr4g	⊢ ( ( 𝑅 ∈ Rng ∧ 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) → ( 𝐼 ∈ 𝑈 ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( ( 𝑥 · 𝑦 ) ∈ 𝐼 ∧ ( 𝑦 · 𝑥 ) ∈ 𝐼 ) ) )