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Metamath Proof Explorer

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 ) )