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Metamath Proof Explorer

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 RngHom 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 RngHom 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 RngHom 𝑆 ) → ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐴 · 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝐵 ) ) ) )
18 17 3impib ⊢ ( ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐴 · 𝐵 ) ) = ( ( 𝐹 ‘ 𝐴 ) × ( 𝐹 ‘ 𝐵 ) ) )