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*