Theorem rngqiprngghmlem2

Description: Lemma 2 for rngqiprngghm . (Contributed by AV, 25-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 )
Assertion	rngqiprngghmlem2	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ) → ( ( 1 · 𝐴 ) ( +g ‘ 𝐽 ) ( 1 · 𝐶 ) ) ∈ ( Base ‘ 𝐽 ) )

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		ringrng	⊢ ( 𝐽 ∈ Ring → 𝐽 ∈ Rng )
9	4 8	syl	⊢ ( 𝜑 → 𝐽 ∈ Rng )
10	9	adantr	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ) → 𝐽 ∈ Rng )
11	1 2 3 4 5 6 7	rngqiprngghmlem1	⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝐵 ) → ( 1 · 𝐴 ) ∈ ( Base ‘ 𝐽 ) )
12	11	adantrr	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ) → ( 1 · 𝐴 ) ∈ ( Base ‘ 𝐽 ) )
13	1 2 3 4 5 6 7	rngqiprngghmlem1	⊢ ( ( 𝜑 ∧ 𝐶 ∈ 𝐵 ) → ( 1 · 𝐶 ) ∈ ( Base ‘ 𝐽 ) )
14	13	adantrl	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ) → ( 1 · 𝐶 ) ∈ ( Base ‘ 𝐽 ) )
15		eqid	⊢ ( Base ‘ 𝐽 ) = ( Base ‘ 𝐽 )
16		eqid	⊢ ( +g ‘ 𝐽 ) = ( +g ‘ 𝐽 )
17	15 16	rngacl	⊢ ( ( 𝐽 ∈ Rng ∧ ( 1 · 𝐴 ) ∈ ( Base ‘ 𝐽 ) ∧ ( 1 · 𝐶 ) ∈ ( Base ‘ 𝐽 ) ) → ( ( 1 · 𝐴 ) ( +g ‘ 𝐽 ) ( 1 · 𝐶 ) ) ∈ ( Base ‘ 𝐽 ) )
18	10 12 14 17	syl3anc	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵 ) ) → ( ( 1 · 𝐴 ) ( +g ‘ 𝐽 ) ( 1 · 𝐶 ) ) ∈ ( Base ‘ 𝐽 ) )