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Metamath Proof Explorer

Theorem mdegxrf

Description: Functionality of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015) (Proof shortened by AV, 27-Jul-2019)

Ref Expression
Hypotheses mdegxrcl.d ⊢ 𝐷 = ( 𝐼 mDeg 𝑅 )
mdegxrcl.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
mdegxrcl.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
Assertion mdegxrf 𝐷 : 𝐵 ⟶ ℝ*

Proof

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*