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Metamath Proof Explorer

Theorem deg1mul

Description: Degree of multiplication of two nonzero polynomials in a domain. (Contributed by metakunt, 6-May-2025)

Ref Expression
Hypotheses deg1mul.1 ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
deg1mul.2 ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
deg1mul.3 ⊢ 𝐵 = ( Base ‘ 𝑃 )
deg1mul.4 ⊢ · = ( .r ‘ 𝑃 )
deg1mul.5 ⊢ 0 = ( 0g ‘ 𝑃 )
deg1mul.6 ⊢ ( 𝜑 → 𝑅 ∈ Domn )
deg1mul.7 ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
deg1mul.8 ⊢ ( 𝜑 → 𝐹 ≠ 0 )
deg1mul.9 ⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
deg1mul.10 ⊢ ( 𝜑 → 𝐺 ≠ 0 )
Assertion deg1mul ( 𝜑 → ( 𝐷 ‘ ( 𝐹 · 𝐺 ) ) = ( ( 𝐷 ‘ 𝐹 ) + ( 𝐷 ‘ 𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 deg1mul.1 ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
2 deg1mul.2 ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
3 deg1mul.3 ⊢ 𝐵 = ( Base ‘ 𝑃 )
4 deg1mul.4 ⊢ · = ( .r ‘ 𝑃 )
5 deg1mul.5 ⊢ 0 = ( 0g ‘ 𝑃 )
6 deg1mul.6 ⊢ ( 𝜑 → 𝑅 ∈ Domn )
7 deg1mul.7 ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
8 deg1mul.8 ⊢ ( 𝜑 → 𝐹 ≠ 0 )
9 deg1mul.9 ⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
10 deg1mul.10 ⊢ ( 𝜑 → 𝐺 ≠ 0 )
11 eqid ⊢ ( RLReg ‘ 𝑅 ) = ( RLReg ‘ 𝑅 )
12 domnring ⊢ ( 𝑅 ∈ Domn → 𝑅 ∈ Ring )
13 6 12 syl ⊢ ( 𝜑 → 𝑅 ∈ Ring )
14 1 2 5 3 deg1nn0cl ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 )
15 13 7 8 14 syl3anc ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 )
16 eqid ⊢ ( coe1 ‘ 𝐹 ) = ( coe1 ‘ 𝐹 )
17 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
18 16 3 2 17 coe1fvalcl ⊢ ( ( 𝐹 ∈ 𝐵 ∧ ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) → ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝑅 ) )
19 7 15 18 syl2anc ⊢ ( 𝜑 → ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝑅 ) )
20 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
21 1 2 5 3 20 16 deg1ldg ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ≠ ( 0g ‘ 𝑅 ) )
22 13 7 8 21 syl3anc ⊢ ( 𝜑 → ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ≠ ( 0g ‘ 𝑅 ) )
23 17 11 20 domnrrg ⊢ ( ( 𝑅 ∈ Domn ∧ ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝑅 ) ∧ ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ≠ ( 0g ‘ 𝑅 ) ) → ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ∈ ( RLReg ‘ 𝑅 ) )
24 6 19 22 23 syl3anc ⊢ ( 𝜑 → ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ∈ ( RLReg ‘ 𝑅 ) )
25 1 2 11 3 4 5 13 7 8 24 9 10 deg1mul2 ⊢ ( 𝜑 → ( 𝐷 ‘ ( 𝐹 · 𝐺 ) ) = ( ( 𝐷 ‘ 𝐹 ) + ( 𝐷 ‘ 𝐺 ) ) )