Step |
Hyp |
Ref |
Expression |
1 |
|
mdegxrcl.d |
⊢ 𝐷 = ( 𝐼 mDeg 𝑅 ) |
2 |
|
mdegxrcl.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
3 |
|
mdegxrcl.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
5 |
|
eqid |
⊢ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } = { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } |
6 |
|
eqid |
⊢ ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) = ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) |
7 |
1 2 3 4 5 6
|
mdegval |
⊢ ( 𝐹 ∈ 𝐵 → ( 𝐷 ‘ 𝐹 ) = sup ( ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) , ℝ* , < ) ) |
8 |
|
imassrn |
⊢ ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ⊆ ran ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) |
9 |
5 6
|
tdeglem1 |
⊢ ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) : { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ⟶ ℕ0 |
10 |
|
frn |
⊢ ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) : { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ⟶ ℕ0 → ran ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) ⊆ ℕ0 ) |
11 |
9 10
|
mp1i |
⊢ ( 𝐹 ∈ 𝐵 → ran ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) ⊆ ℕ0 ) |
12 |
|
nn0ssre |
⊢ ℕ0 ⊆ ℝ |
13 |
|
ressxr |
⊢ ℝ ⊆ ℝ* |
14 |
12 13
|
sstri |
⊢ ℕ0 ⊆ ℝ* |
15 |
11 14
|
sstrdi |
⊢ ( 𝐹 ∈ 𝐵 → ran ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) ⊆ ℝ* ) |
16 |
8 15
|
sstrid |
⊢ ( 𝐹 ∈ 𝐵 → ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ⊆ ℝ* ) |
17 |
|
supxrcl |
⊢ ( ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) ⊆ ℝ* → sup ( ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) , ℝ* , < ) ∈ ℝ* ) |
18 |
16 17
|
syl |
⊢ ( 𝐹 ∈ 𝐵 → sup ( ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 𝐹 supp ( 0g ‘ 𝑅 ) ) ) , ℝ* , < ) ∈ ℝ* ) |
19 |
7 18
|
eqeltrd |
⊢ ( 𝐹 ∈ 𝐵 → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* ) |