Theorem isrnghmd

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 ) ) )
	isrnghmd.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
	isrnghmd.p	⊢ + = ( +g ‘ 𝑅 )
	isrnghmd.q	⊢ ⨣ = ( +g ‘ 𝑆 )
	isrnghmd.f	⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 )
	isrnghmd.hp	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
Assertion	isrnghmd	⊢ ( 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		isrnghmd.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
8		isrnghmd.p	⊢ + = ( +g ‘ 𝑅 )
9		isrnghmd.q	⊢ ⨣ = ( +g ‘ 𝑆 )
10		isrnghmd.f	⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 )
11		isrnghmd.hp	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ⨣ ( 𝐹 ‘ 𝑦 ) ) )
12		rngabl	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Abel )
13		ablgrp	⊢ ( 𝑅 ∈ Abel → 𝑅 ∈ Grp )
14	4 12 13	3syl	⊢ ( 𝜑 → 𝑅 ∈ Grp )
15		rngabl	⊢ ( 𝑆 ∈ Rng → 𝑆 ∈ Abel )
16		ablgrp	⊢ ( 𝑆 ∈ Abel → 𝑆 ∈ Grp )
17	5 15 16	3syl	⊢ ( 𝜑 → 𝑆 ∈ Grp )
18	1 7 8 9 14 17 10 11	isghmd	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) )
19	1 2 3 4 5 6 18	isrnghm2d	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) )