Metamath Proof Explorer

Theorem evlmulval

Description: Polynomial evaluation builder for multiplication. (Contributed by SN, 18-Feb-2025)

Ref Expression
Hypotheses evlmulval.q ⊢ 𝑄 = ( 𝐼 eval 𝑆 )
evlmulval.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑆 )
evlmulval.k ⊢ 𝐾 = ( Base ‘ 𝑆 )
evlmulval.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
evlmulval.g ⊢ ∙ = ( .r ‘ 𝑃 )
evlmulval.f ⊢ · = ( .r ‘ 𝑆 )
evlmulval.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑍 )
evlmulval.s ⊢ ( 𝜑 → 𝑆 ∈ CRing )
evlmulval.a ⊢ ( 𝜑 → 𝐴 ∈ ( 𝐾 ↑m 𝐼 ) )
evlmulval.m ⊢ ( 𝜑 → ( 𝑀 ∈ 𝐵 ∧ ( ( 𝑄 ‘ 𝑀 ) ‘ 𝐴 ) = 𝑉 ) )
evlmulval.n ⊢ ( 𝜑 → ( 𝑁 ∈ 𝐵 ∧ ( ( 𝑄 ‘ 𝑁 ) ‘ 𝐴 ) = 𝑊 ) )
Assertion evlmulval ( 𝜑 → ( ( 𝑀 ∙ 𝑁 ) ∈ 𝐵 ∧ ( ( 𝑄 ‘ ( 𝑀 ∙ 𝑁 ) ) ‘ 𝐴 ) = ( 𝑉 · 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 evlmulval.q ⊢ 𝑄 = ( 𝐼 eval 𝑆 )
2 evlmulval.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑆 )
3 evlmulval.k ⊢ 𝐾 = ( Base ‘ 𝑆 )
4 evlmulval.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
5 evlmulval.g ⊢ ∙ = ( .r ‘ 𝑃 )
6 evlmulval.f ⊢ · = ( .r ‘ 𝑆 )
7 evlmulval.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑍 )
8 evlmulval.s ⊢ ( 𝜑 → 𝑆 ∈ CRing )
9 evlmulval.a ⊢ ( 𝜑 → 𝐴 ∈ ( 𝐾 ↑m 𝐼 ) )
10 evlmulval.m ⊢ ( 𝜑 → ( 𝑀 ∈ 𝐵 ∧ ( ( 𝑄 ‘ 𝑀 ) ‘ 𝐴 ) = 𝑉 ) )
11 evlmulval.n ⊢ ( 𝜑 → ( 𝑁 ∈ 𝐵 ∧ ( ( 𝑄 ‘ 𝑁 ) ‘ 𝐴 ) = 𝑊 ) )
12 eqid ⊢ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) = ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) )
13 1 3 2 12 evlrhm ⊢ ( ( 𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ) → 𝑄 ∈ ( 𝑃 RingHom ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
14 7 8 13 syl2anc ⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 RingHom ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
15 rhmrcl1 ⊢ ( 𝑄 ∈ ( 𝑃 RingHom ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) → 𝑃 ∈ Ring )
16 14 15 syl ⊢ ( 𝜑 → 𝑃 ∈ Ring )
17 10 simpld ⊢ ( 𝜑 → 𝑀 ∈ 𝐵 )
18 11 simpld ⊢ ( 𝜑 → 𝑁 ∈ 𝐵 )
19 4 5 16 17 18 ringcld ⊢ ( 𝜑 → ( 𝑀 ∙ 𝑁 ) ∈ 𝐵 )
20 eqid ⊢ ( .r ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) = ( .r ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) )
21 4 5 20 rhmmul ⊢ ( ( 𝑄 ∈ ( 𝑃 RingHom ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) ∧ 𝑀 ∈ 𝐵 ∧ 𝑁 ∈ 𝐵 ) → ( 𝑄 ‘ ( 𝑀 ∙ 𝑁 ) ) = ( ( 𝑄 ‘ 𝑀 ) ( .r ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) ( 𝑄 ‘ 𝑁 ) ) )
22 14 17 18 21 syl3anc ⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝑀 ∙ 𝑁 ) ) = ( ( 𝑄 ‘ 𝑀 ) ( .r ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) ( 𝑄 ‘ 𝑁 ) ) )
23 eqid ⊢ ( Base ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) = ( Base ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) )
24 ovexd ⊢ ( 𝜑 → ( 𝐾 ↑m 𝐼 ) ∈ V )
25 4 23 rhmf ⊢ ( 𝑄 ∈ ( 𝑃 RingHom ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) → 𝑄 : 𝐵 ⟶ ( Base ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
26 14 25 syl ⊢ ( 𝜑 → 𝑄 : 𝐵 ⟶ ( Base ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
27 26 17 ffvelcdmd ⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) ∈ ( Base ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
28 26 18 ffvelcdmd ⊢ ( 𝜑 → ( 𝑄 ‘ 𝑁 ) ∈ ( Base ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
29 12 23 8 24 27 28 6 20 pwsmulrval ⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑀 ) ( .r ‘ ( 𝑆 ↑s ( 𝐾 ↑m 𝐼 ) ) ) ( 𝑄 ‘ 𝑁 ) ) = ( ( 𝑄 ‘ 𝑀 ) ∘f · ( 𝑄 ‘ 𝑁 ) ) )
30 22 29 eqtrd ⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝑀 ∙ 𝑁 ) ) = ( ( 𝑄 ‘ 𝑀 ) ∘f · ( 𝑄 ‘ 𝑁 ) ) )
31 30 fveq1d ⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝑀 ∙ 𝑁 ) ) ‘ 𝐴 ) = ( ( ( 𝑄 ‘ 𝑀 ) ∘f · ( 𝑄 ‘ 𝑁 ) ) ‘ 𝐴 ) )
32 12 3 23 8 24 27 pwselbas ⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) : ( 𝐾 ↑m 𝐼 ) ⟶ 𝐾 )
33 32 ffnd ⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) Fn ( 𝐾 ↑m 𝐼 ) )
34 12 3 23 8 24 28 pwselbas ⊢ ( 𝜑 → ( 𝑄 ‘ 𝑁 ) : ( 𝐾 ↑m 𝐼 ) ⟶ 𝐾 )
35 34 ffnd ⊢ ( 𝜑 → ( 𝑄 ‘ 𝑁 ) Fn ( 𝐾 ↑m 𝐼 ) )
36 fnfvof ⊢ ( ( ( ( 𝑄 ‘ 𝑀 ) Fn ( 𝐾 ↑m 𝐼 ) ∧ ( 𝑄 ‘ 𝑁 ) Fn ( 𝐾 ↑m 𝐼 ) ) ∧ ( ( 𝐾 ↑m 𝐼 ) ∈ V ∧ 𝐴 ∈ ( 𝐾 ↑m 𝐼 ) ) ) → ( ( ( 𝑄 ‘ 𝑀 ) ∘f · ( 𝑄 ‘ 𝑁 ) ) ‘ 𝐴 ) = ( ( ( 𝑄 ‘ 𝑀 ) ‘ 𝐴 ) · ( ( 𝑄 ‘ 𝑁 ) ‘ 𝐴 ) ) )
37 33 35 24 9 36 syl22anc ⊢ ( 𝜑 → ( ( ( 𝑄 ‘ 𝑀 ) ∘f · ( 𝑄 ‘ 𝑁 ) ) ‘ 𝐴 ) = ( ( ( 𝑄 ‘ 𝑀 ) ‘ 𝐴 ) · ( ( 𝑄 ‘ 𝑁 ) ‘ 𝐴 ) ) )
38 10 simprd ⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑀 ) ‘ 𝐴 ) = 𝑉 )
39 11 simprd ⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑁 ) ‘ 𝐴 ) = 𝑊 )
40 38 39 oveq12d ⊢ ( 𝜑 → ( ( ( 𝑄 ‘ 𝑀 ) ‘ 𝐴 ) · ( ( 𝑄 ‘ 𝑁 ) ‘ 𝐴 ) ) = ( 𝑉 · 𝑊 ) )
41 31 37 40 3eqtrd ⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝑀 ∙ 𝑁 ) ) ‘ 𝐴 ) = ( 𝑉 · 𝑊 ) )
42 19 41 jca ⊢ ( 𝜑 → ( ( 𝑀 ∙ 𝑁 ) ∈ 𝐵 ∧ ( ( 𝑄 ‘ ( 𝑀 ∙ 𝑁 ) ) ‘ 𝐴 ) = ( 𝑉 · 𝑊 ) ) )