Theorem rnghmmul

Description: A homomorphism of non-unital rings preserves multiplication. (Contributed by AV, 23-Feb-2020)

	Ref	Expression
Hypotheses	rnghmmul.x	⊢ 𝑋 = ( Base ‘ 𝑅 )
	rnghmmul.m	⊢ · = ( .r ‘ 𝑅 )
	rnghmmul.n	⊢ × = ( .r ‘ 𝑆 )
Assertion	rnghmmul	⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐴 · 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rnghmmul.x	⊢ 𝑋 = ( Base ‘ 𝑅 )
2		rnghmmul.m	⊢ · = ( .r ‘ 𝑅 )
3		rnghmmul.n	⊢ × = ( .r ‘ 𝑆 )
4	1 2 3	isrnghm	⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ↔ ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) ) ) )
5		fvoveq1	⊢ ( 𝑥 = 𝐴 → ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( 𝐹 ‘ ( 𝐴 · 𝑦 ) ) )
6		fveq2	⊢ ( 𝑥 = 𝐴 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐴 ) )
7	6	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝑦 ) ) )
8	5 7	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝐴 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝑦 ) ) ) )
9		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 · 𝑦 ) = ( 𝐴 · 𝐵 ) )
10	9	fveq2d	⊢ ( 𝑦 = 𝐵 → ( 𝐹 ‘ ( 𝐴 · 𝑦 ) ) = ( 𝐹 ‘ ( 𝐴 · 𝐵 ) ) )
11		fveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
12	11	oveq2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝐵 ) ) )
13	10 12	eqeq12d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐹 ‘ ( 𝐴 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝐹 ‘ ( 𝐴 · 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝐵 ) ) ) )
14	8 13	rspc2va	⊢ ( ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) ) → ( 𝐹 ‘ ( 𝐴 · 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝐵 ) ) )
15	14	expcom	⊢ ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) → ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐴 · 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝐵 ) ) ) )
16	15	ad2antll	⊢ ( ( ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) ∧ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝐹 ‘ ( 𝑥 · 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) × ( 𝐹 ‘ 𝑦 ) ) ) ) → ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐴 · 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝐵 ) ) ) )
17	4 16	sylbi	⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐴 · 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝐵 ) ) ) )
18	17	3impib	⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐴 · 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝐵 ) ) )