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Metamath Proof Explorer

Theorem mdegxrcl

Description: Closure 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 mdegxrcl ( 𝐹 ∈ 𝐵 → ( 𝐷 ‘ 𝐹 ) ∈ 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 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* )