Theorem isrnghm2d

Description: Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020)

	Ref	Expression
Hypotheses	isrnghmd.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	isrnghmd.t	⊢ · = ( .r ‘ 𝑅 )
	isrnghmd.u	⊢ × = ( .r ‘ 𝑆 )
	isrnghmd.r	⊢ ( 𝜑 → 𝑅 ∈ Rng )
	isrnghmd.s	⊢ ( 𝜑 → 𝑆 ∈ Rng )
	isrnghmd.ht	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) )
	isrnghm2d.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )
Assertion	isrnghm2d	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		isrnghmd.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		isrnghmd.t	⊢ · = ( .r ‘ 𝑅 )
3		isrnghmd.u	⊢ × = ( .r ‘ 𝑆 )
4		isrnghmd.r	⊢ ( 𝜑 → 𝑅 ∈ Rng )
5		isrnghmd.s	⊢ ( 𝜑 → 𝑆 ∈ Rng )
6		isrnghmd.ht	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) )
7		isrnghm2d.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )
8	4 5	jca	⊢ ( 𝜑 → ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) )
9	6	ralrimivva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) )
10	7 9	jca	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) ) )
11	1 2 3	isrnghm	⊢ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) ) ) )
12	8 10 11	sylanbrc	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) )