Theorem rngqiprng

Description: The product of the quotient with a two-sided ideal and the two-sided ideal is a non-unital ring. (Contributed by AV, 23-Feb-2025)

	Ref	Expression
Hypotheses	rng2idlring.r	⊢ ( 𝜑 → 𝑅 ∈ Rng )
	rng2idlring.i	⊢ ( 𝜑 → 𝐼 ∈ ( 2Ideal ‘ 𝑅 ) )
	rng2idlring.j	⊢ 𝐽 = ( 𝑅 ↾s 𝐼 )
	rng2idlring.u	⊢ ( 𝜑 → 𝐽 ∈ Ring )
	rng2idlring.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rng2idlring.t	⊢ · = ( .r ‘ 𝑅 )
	rng2idlring.1	⊢ 1 = ( 1r ‘ 𝐽 )
	rngqiprngim.g	⊢ ∼ = ( 𝑅 ~QG 𝐼 )
	rngqiprngim.q	⊢ 𝑄 = ( 𝑅 /s ∼ )
	rngqiprngim.c	⊢ 𝐶 = ( Base ‘ 𝑄 )
	rngqiprngim.p	⊢ 𝑃 = ( 𝑄 ×s 𝐽 )
Assertion	rngqiprng	⊢ ( 𝜑 → 𝑃 ∈ Rng )

Proof

Step	Hyp	Ref	Expression
1		rng2idlring.r	⊢ ( 𝜑 → 𝑅 ∈ Rng )
2		rng2idlring.i	⊢ ( 𝜑 → 𝐼 ∈ ( 2Ideal ‘ 𝑅 ) )
3		rng2idlring.j	⊢ 𝐽 = ( 𝑅 ↾s 𝐼 )
4		rng2idlring.u	⊢ ( 𝜑 → 𝐽 ∈ Ring )
5		rng2idlring.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
6		rng2idlring.t	⊢ · = ( .r ‘ 𝑅 )
7		rng2idlring.1	⊢ 1 = ( 1r ‘ 𝐽 )
8		rngqiprngim.g	⊢ ∼ = ( 𝑅 ~QG 𝐼 )
9		rngqiprngim.q	⊢ 𝑄 = ( 𝑅 /s ∼ )
10		rngqiprngim.c	⊢ 𝐶 = ( Base ‘ 𝑄 )
11		rngqiprngim.p	⊢ 𝑃 = ( 𝑄 ×s 𝐽 )
12		ringrng	⊢ ( 𝐽 ∈ Ring → 𝐽 ∈ Rng )
13	4 12	syl	⊢ ( 𝜑 → 𝐽 ∈ Rng )
14	3 13	eqeltrrid	⊢ ( 𝜑 → ( 𝑅 ↾s 𝐼 ) ∈ Rng )
15	1 2 14	rng2idlsubrng	⊢ ( 𝜑 → 𝐼 ∈ ( SubRng ‘ 𝑅 ) )
16		subrngsubg	⊢ ( 𝐼 ∈ ( SubRng ‘ 𝑅 ) → 𝐼 ∈ ( SubGrp ‘ 𝑅 ) )
17	15 16	syl	⊢ ( 𝜑 → 𝐼 ∈ ( SubGrp ‘ 𝑅 ) )
18	8	oveq2i	⊢ ( 𝑅 /s ∼ ) = ( 𝑅 /s ( 𝑅 ~QG 𝐼 ) )
19	9 18	eqtri	⊢ 𝑄 = ( 𝑅 /s ( 𝑅 ~QG 𝐼 ) )
20		eqid	⊢ ( 2Ideal ‘ 𝑅 ) = ( 2Ideal ‘ 𝑅 )
21	19 20	qus2idrng	⊢ ( ( 𝑅 ∈ Rng ∧ 𝐼 ∈ ( 2Ideal ‘ 𝑅 ) ∧ 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) → 𝑄 ∈ Rng )
22	1 2 17 21	syl3anc	⊢ ( 𝜑 → 𝑄 ∈ Rng )
23	11 22 13	xpsrngd	⊢ ( 𝜑 → 𝑃 ∈ Rng )