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 |
|
xrltso |
⊢ < Or ℝ* |
5 |
4
|
supex |
⊢ sup ( ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 𝑧 supp ( 0g ‘ 𝑅 ) ) ) , ℝ* , < ) ∈ V |
6 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
7 |
|
eqid |
⊢ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } = { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } |
8 |
|
eqid |
⊢ ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) = ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) |
9 |
1 2 3 6 7 8
|
mdegfval |
⊢ 𝐷 = ( 𝑧 ∈ 𝐵 ↦ sup ( ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 𝑧 supp ( 0g ‘ 𝑅 ) ) ) , ℝ* , < ) ) |
10 |
5 9
|
fnmpti |
⊢ 𝐷 Fn 𝐵 |
11 |
1 2 3
|
mdegxrcl |
⊢ ( 𝑓 ∈ 𝐵 → ( 𝐷 ‘ 𝑓 ) ∈ ℝ* ) |
12 |
11
|
rgen |
⊢ ∀ 𝑓 ∈ 𝐵 ( 𝐷 ‘ 𝑓 ) ∈ ℝ* |
13 |
|
ffnfv |
⊢ ( 𝐷 : 𝐵 ⟶ ℝ* ↔ ( 𝐷 Fn 𝐵 ∧ ∀ 𝑓 ∈ 𝐵 ( 𝐷 ‘ 𝑓 ) ∈ ℝ* ) ) |
14 |
10 12 13
|
mpbir2an |
⊢ 𝐷 : 𝐵 ⟶ ℝ* |