# Metamath Proof Explorer

## Theorem evl1muld

Description: Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015)

Ref Expression
Hypotheses evl1addd.q 𝑂 = ( eval1𝑅 )
evl1addd.p 𝑃 = ( Poly1𝑅 )
evl1addd.b 𝐵 = ( Base ‘ 𝑅 )
evl1addd.u 𝑈 = ( Base ‘ 𝑃 )
evl1addd.1 ( 𝜑𝑅 ∈ CRing )
evl1addd.3 ( 𝜑 → ( 𝑀𝑈 ∧ ( ( 𝑂𝑀 ) ‘ 𝑌 ) = 𝑉 ) )
evl1addd.4 ( 𝜑 → ( 𝑁𝑈 ∧ ( ( 𝑂𝑁 ) ‘ 𝑌 ) = 𝑊 ) )
evl1muld.t = ( .r𝑃 )
evl1muld.s · = ( .r𝑅 )
Assertion evl1muld ( 𝜑 → ( ( 𝑀 𝑁 ) ∈ 𝑈 ∧ ( ( 𝑂 ‘ ( 𝑀 𝑁 ) ) ‘ 𝑌 ) = ( 𝑉 · 𝑊 ) ) )

### Proof

Step Hyp Ref Expression
1 evl1addd.q 𝑂 = ( eval1𝑅 )
2 evl1addd.p 𝑃 = ( Poly1𝑅 )
3 evl1addd.b 𝐵 = ( Base ‘ 𝑅 )
4 evl1addd.u 𝑈 = ( Base ‘ 𝑃 )
5 evl1addd.1 ( 𝜑𝑅 ∈ CRing )
7 evl1addd.3 ( 𝜑 → ( 𝑀𝑈 ∧ ( ( 𝑂𝑀 ) ‘ 𝑌 ) = 𝑉 ) )
8 evl1addd.4 ( 𝜑 → ( 𝑁𝑈 ∧ ( ( 𝑂𝑁 ) ‘ 𝑌 ) = 𝑊 ) )
9 evl1muld.t = ( .r𝑃 )
10 evl1muld.s · = ( .r𝑅 )
11 eqid ( 𝑅s 𝐵 ) = ( 𝑅s 𝐵 )
12 1 2 11 3 evl1rhm ( 𝑅 ∈ CRing → 𝑂 ∈ ( 𝑃 RingHom ( 𝑅s 𝐵 ) ) )
13 5 12 syl ( 𝜑𝑂 ∈ ( 𝑃 RingHom ( 𝑅s 𝐵 ) ) )
14 rhmrcl1 ( 𝑂 ∈ ( 𝑃 RingHom ( 𝑅s 𝐵 ) ) → 𝑃 ∈ Ring )
15 13 14 syl ( 𝜑𝑃 ∈ Ring )
16 7 simpld ( 𝜑𝑀𝑈 )
17 8 simpld ( 𝜑𝑁𝑈 )
18 4 9 ringcl ( ( 𝑃 ∈ Ring ∧ 𝑀𝑈𝑁𝑈 ) → ( 𝑀 𝑁 ) ∈ 𝑈 )
19 15 16 17 18 syl3anc ( 𝜑 → ( 𝑀 𝑁 ) ∈ 𝑈 )
20 eqid ( .r ‘ ( 𝑅s 𝐵 ) ) = ( .r ‘ ( 𝑅s 𝐵 ) )
21 4 9 20 rhmmul ( ( 𝑂 ∈ ( 𝑃 RingHom ( 𝑅s 𝐵 ) ) ∧ 𝑀𝑈𝑁𝑈 ) → ( 𝑂 ‘ ( 𝑀 𝑁 ) ) = ( ( 𝑂𝑀 ) ( .r ‘ ( 𝑅s 𝐵 ) ) ( 𝑂𝑁 ) ) )
22 13 16 17 21 syl3anc ( 𝜑 → ( 𝑂 ‘ ( 𝑀 𝑁 ) ) = ( ( 𝑂𝑀 ) ( .r ‘ ( 𝑅s 𝐵 ) ) ( 𝑂𝑁 ) ) )
23 eqid ( Base ‘ ( 𝑅s 𝐵 ) ) = ( Base ‘ ( 𝑅s 𝐵 ) )
24 3 fvexi 𝐵 ∈ V
25 24 a1i ( 𝜑𝐵 ∈ V )
26 4 23 rhmf ( 𝑂 ∈ ( 𝑃 RingHom ( 𝑅s 𝐵 ) ) → 𝑂 : 𝑈 ⟶ ( Base ‘ ( 𝑅s 𝐵 ) ) )
27 13 26 syl ( 𝜑𝑂 : 𝑈 ⟶ ( Base ‘ ( 𝑅s 𝐵 ) ) )
28 27 16 ffvelrnd ( 𝜑 → ( 𝑂𝑀 ) ∈ ( Base ‘ ( 𝑅s 𝐵 ) ) )
29 27 17 ffvelrnd ( 𝜑 → ( 𝑂𝑁 ) ∈ ( Base ‘ ( 𝑅s 𝐵 ) ) )
30 11 23 5 25 28 29 10 20 pwsmulrval ( 𝜑 → ( ( 𝑂𝑀 ) ( .r ‘ ( 𝑅s 𝐵 ) ) ( 𝑂𝑁 ) ) = ( ( 𝑂𝑀 ) ∘f · ( 𝑂𝑁 ) ) )
31 22 30 eqtrd ( 𝜑 → ( 𝑂 ‘ ( 𝑀 𝑁 ) ) = ( ( 𝑂𝑀 ) ∘f · ( 𝑂𝑁 ) ) )
32 31 fveq1d ( 𝜑 → ( ( 𝑂 ‘ ( 𝑀 𝑁 ) ) ‘ 𝑌 ) = ( ( ( 𝑂𝑀 ) ∘f · ( 𝑂𝑁 ) ) ‘ 𝑌 ) )
33 11 3 23 5 25 28 pwselbas ( 𝜑 → ( 𝑂𝑀 ) : 𝐵𝐵 )
34 33 ffnd ( 𝜑 → ( 𝑂𝑀 ) Fn 𝐵 )
35 11 3 23 5 25 29 pwselbas ( 𝜑 → ( 𝑂𝑁 ) : 𝐵𝐵 )
36 35 ffnd ( 𝜑 → ( 𝑂𝑁 ) Fn 𝐵 )
37 fnfvof ( ( ( ( 𝑂𝑀 ) Fn 𝐵 ∧ ( 𝑂𝑁 ) Fn 𝐵 ) ∧ ( 𝐵 ∈ V ∧ 𝑌𝐵 ) ) → ( ( ( 𝑂𝑀 ) ∘f · ( 𝑂𝑁 ) ) ‘ 𝑌 ) = ( ( ( 𝑂𝑀 ) ‘ 𝑌 ) · ( ( 𝑂𝑁 ) ‘ 𝑌 ) ) )
38 34 36 25 6 37 syl22anc ( 𝜑 → ( ( ( 𝑂𝑀 ) ∘f · ( 𝑂𝑁 ) ) ‘ 𝑌 ) = ( ( ( 𝑂𝑀 ) ‘ 𝑌 ) · ( ( 𝑂𝑁 ) ‘ 𝑌 ) ) )
39 7 simprd ( 𝜑 → ( ( 𝑂𝑀 ) ‘ 𝑌 ) = 𝑉 )
40 8 simprd ( 𝜑 → ( ( 𝑂𝑁 ) ‘ 𝑌 ) = 𝑊 )
41 39 40 oveq12d ( 𝜑 → ( ( ( 𝑂𝑀 ) ‘ 𝑌 ) · ( ( 𝑂𝑁 ) ‘ 𝑌 ) ) = ( 𝑉 · 𝑊 ) )
42 32 38 41 3eqtrd ( 𝜑 → ( ( 𝑂 ‘ ( 𝑀 𝑁 ) ) ‘ 𝑌 ) = ( 𝑉 · 𝑊 ) )
43 19 42 jca ( 𝜑 → ( ( 𝑀 𝑁 ) ∈ 𝑈 ∧ ( ( 𝑂 ‘ ( 𝑀 𝑁 ) ) ‘ 𝑌 ) = ( 𝑉 · 𝑊 ) ) )