Theorem rngqiprngho

Description: F is a homomorphism of non-unital rings. (Contributed by AV, 21-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 )
	rngqiprngim.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ⟨ [ 𝑥 ] ∼ , ( 1 · 𝑥 ) ⟩ )
Assertion	rngqiprngho	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 RngHom 𝑃 ) )

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		rngqiprngim.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ⟨ [ 𝑥 ] ∼ , ( 1 · 𝑥 ) ⟩ )
13	1 2 3 4 5 6 7 8 9 10 11	rngqiprng	⊢ ( 𝜑 → 𝑃 ∈ Rng )
14	1 2 3 4 5 6 7 8 9 10 11 12	rngqiprngghm	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 GrpHom 𝑃 ) )
15	1 2 3 4 5 6 7 8 9 10 11 12	rngqiprnglin	⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝐹 ‘ ( 𝑎 · 𝑏 ) ) = ( ( 𝐹 ‘ 𝑎 ) ( .r ‘ 𝑃 ) ( 𝐹 ‘ 𝑏 ) ) )
16	14 15	jca	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑅 GrpHom 𝑃 ) ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝐹 ‘ ( 𝑎 · 𝑏 ) ) = ( ( 𝐹 ‘ 𝑎 ) ( .r ‘ 𝑃 ) ( 𝐹 ‘ 𝑏 ) ) ) )
17		eqid	⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 )
18	5 6 17	isrnghm	⊢ ( 𝐹 ∈ ( 𝑅 RngHom 𝑃 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑃 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑃 ) ∧ ∀ 𝑎 ∈ 𝐵 ∀ 𝑏 ∈ 𝐵 ( 𝐹 ‘ ( 𝑎 · 𝑏 ) ) = ( ( 𝐹 ‘ 𝑎 ) ( .r ‘ 𝑃 ) ( 𝐹 ‘ 𝑏 ) ) ) ) )
19	1 13 16 18	syl21anbrc	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 RngHom 𝑃 ) )