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Metamath Proof Explorer

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 ) )