Step |
Hyp |
Ref |
Expression |
1 |
|
coe1mul3.s |
⊢ 𝑌 = ( Poly1 ‘ 𝑅 ) |
2 |
|
coe1mul3.t |
⊢ ∙ = ( .r ‘ 𝑌 ) |
3 |
|
coe1mul3.u |
⊢ · = ( .r ‘ 𝑅 ) |
4 |
|
coe1mul3.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
5 |
|
coe1mul3.d |
⊢ 𝐷 = ( deg1 ‘ 𝑅 ) |
6 |
|
coe1mul4.z |
⊢ 0 = ( 0g ‘ 𝑌 ) |
7 |
|
coe1mul4.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
8 |
|
coe1mul4.f1 |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
9 |
|
coe1mul4.f2 |
⊢ ( 𝜑 → 𝐹 ≠ 0 ) |
10 |
|
coe1mul4.g1 |
⊢ ( 𝜑 → 𝐺 ∈ 𝐵 ) |
11 |
|
coe1mul4.g2 |
⊢ ( 𝜑 → 𝐺 ≠ 0 ) |
12 |
5 1 6 4
|
deg1nn0cl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) |
13 |
7 8 9 12
|
syl3anc |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) |
14 |
13
|
nn0red |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) ∈ ℝ ) |
15 |
14
|
leidd |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ 𝐹 ) ) |
16 |
5 1 6 4
|
deg1nn0cl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → ( 𝐷 ‘ 𝐺 ) ∈ ℕ0 ) |
17 |
7 10 11 16
|
syl3anc |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐺 ) ∈ ℕ0 ) |
18 |
17
|
nn0red |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐺 ) ∈ ℝ ) |
19 |
18
|
leidd |
⊢ ( 𝜑 → ( 𝐷 ‘ 𝐺 ) ≤ ( 𝐷 ‘ 𝐺 ) ) |
20 |
1 2 3 4 5 7 8 13 15 10 17 19
|
coe1mul3 |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝐹 ∙ 𝐺 ) ) ‘ ( ( 𝐷 ‘ 𝐹 ) + ( 𝐷 ‘ 𝐺 ) ) ) = ( ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) · ( ( coe1 ‘ 𝐺 ) ‘ ( 𝐷 ‘ 𝐺 ) ) ) ) |