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* )