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Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremply1lpir 24701 The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ DivRing → 𝑃 ∈ LPIR)
 
Theoremply1pid 24702 The polynomials over a field are a PID. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Field → 𝑃 ∈ PID)
 
14.1.3  Elementary properties of complex polynomials
 
Syntaxcply 24703 Extend class notation to include the set of complex polynomials.
class Poly
 
Syntaxcidp 24704 Extend class notation to include the identity polynomial.
class Xp
 
Syntaxccoe 24705 Extend class notation to include the coefficient function on polynomials.
class coeff
 
Syntaxcdgr 24706 Extend class notation to include the degree function on polynomials.
class deg
 
Definitiondf-ply 24707* Define the set of polynomials on the complex numbers with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
 
Definitiondf-idp 24708 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Xp = ( I ↾ ℂ)
 
Definitiondf-coe 24709* Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff = (𝑓 ∈ (Poly‘ℂ) ↦ (𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
 
Definitiondf-dgr 24710 Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup(((coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))
 
Theoremplyco0 24711* Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)
((𝑁 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑁)))
 
Theoremplyval 24712* Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
 
Theoremplybss 24713 Reverse closure of the parameter 𝑆 of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
 
Theoremelply 24714* Definition of a polynomial with coefficients in 𝑆. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
 
Theoremelply2 24715* The coefficient function can be assumed to have zeroes outside 0...𝑛. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑m0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
 
Theoremplyun0 24716 The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.)
(Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
 
Theoremplyf 24717 The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
 
Theoremplyss 24718 The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆𝑇𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))
 
Theoremplyssc 24719 Every polynomial ring is contained in the ring of polynomials over . (Contributed by Mario Carneiro, 22-Jul-2014.)
(Poly‘𝑆) ⊆ (Poly‘ℂ)
 
Theoremelplyr 24720* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0𝐴:ℕ0𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))) ∈ (Poly‘𝑆))
 
Theoremelplyd 24721* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (0...𝑁)) → 𝐴𝑆)       (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧𝑘))) ∈ (Poly‘𝑆))
 
Theoremply1termlem 24722* Lemma for ply1term 24723. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧𝑘))))
 
Theoremply1term 24723* A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝑆 ⊆ ℂ ∧ 𝐴𝑆𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘𝑆))
 
Theoremplypow 24724* A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆𝑁 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ (𝑧𝑁)) ∈ (Poly‘𝑆))
 
Theoremplyconst 24725 A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 𝐴𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆))
 
Theoremne0p 24726 A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
((𝐴 ∈ ℂ ∧ (𝐹𝐴) ≠ 0) → 𝐹 ≠ 0𝑝)
 
Theoremply0 24727 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆))
 
Theoremplyid 24728 The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆))
 
Theoremplyeq0lem 24729* Lemma for plyeq0 24730. If 𝐴 is the coefficient function for a nonzero polynomial such that 𝑃(𝑧) = Σ𝑘 ∈ ℕ0𝐴(𝑘) · 𝑧𝑘 = 0 for every 𝑧 ∈ ℂ and 𝐴(𝑀) is the nonzero leading coefficient, then the function 𝐹(𝑧) = 𝑃(𝑧) / 𝑧𝑀 is a sum of powers of 1 / 𝑧, and so the limit of this function as 𝑧 ⇝ +∞ is the constant term, 𝐴(𝑀). But 𝐹(𝑧) = 0 everywhere, so this limit is also equal to zero so that 𝐴(𝑀) = 0, a contradiction. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ∈ ((𝑆 ∪ {0}) ↑m0))    &   (𝜑 → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))    &   𝑀 = sup((𝐴 “ (𝑆 ∖ {0})), ℝ, < )    &   (𝜑 → (𝐴 “ (𝑆 ∖ {0})) ≠ ∅)        ¬ 𝜑
 
Theoremplyeq0 24730* If a polynomial is zero at every point (or even just zero at the positive integers), then all the coefficients must be zero. This is the basis for the method of equating coefficients of equal polynomials, and ensures that df-coe 24709 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ∈ ((𝑆 ∪ {0}) ↑m0))    &   (𝜑 → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))       (𝜑𝐴 = (ℕ0 × {0}))
 
Theoremplypf1 24731 Write the set of complex polynomials in a subring in terms of the abstract polynomial construction. (Contributed by Mario Carneiro, 3-Jul-2015.) (Proof shortened by AV, 29-Sep-2019.)
𝑅 = (ℂflds 𝑆)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (Base‘𝑃)    &   𝐸 = (eval1‘ℂfld)       (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸𝐴))
 
Theoremplyaddlem1 24732* Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑𝐵:ℕ0⟶ℂ)    &   (𝜑 → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑 → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))       (𝜑 → (𝐹f + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀))(((𝐴f + 𝐵)‘𝑘) · (𝑧𝑘))))
 
Theoremplymullem1 24733* Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑𝐵:ℕ0⟶ℂ)    &   (𝜑 → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑 → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))       (𝜑 → (𝐹f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) · (𝑧𝑛))))
 
Theoremplyaddlem 24734* Lemma for plyadd 24736. (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ∈ ((𝑆 ∪ {0}) ↑m0))    &   (𝜑𝐵 ∈ ((𝑆 ∪ {0}) ↑m0))    &   (𝜑 → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑 → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))       (𝜑 → (𝐹f + 𝐺) ∈ (Poly‘𝑆))
 
Theoremplymullem 24735* Lemma for plymul 24737. (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ∈ ((𝑆 ∪ {0}) ↑m0))    &   (𝜑𝐵 ∈ ((𝑆 ∪ {0}) ↑m0))    &   (𝜑 → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑 → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)       (𝜑 → (𝐹f · 𝐺) ∈ (Poly‘𝑆))
 
Theoremplyadd 24736* The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (𝐹f + 𝐺) ∈ (Poly‘𝑆))
 
Theoremplymul 24737* The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)       (𝜑 → (𝐹f · 𝐺) ∈ (Poly‘𝑆))
 
Theoremplysub 24738* The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)       (𝜑 → (𝐹f𝐺) ∈ (Poly‘𝑆))
 
Theoremplyaddcl 24739 The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f + 𝐺) ∈ (Poly‘ℂ))
 
Theoremplymulcl 24740 The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) ∈ (Poly‘ℂ))
 
Theoremplysubcl 24741 The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f𝐺) ∈ (Poly‘ℂ))
 
Theoremcoeval 24742* Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹) = (𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))
 
Theoremcoeeulem 24743* Lemma for coeeu 24744. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐴 ∈ (ℂ ↑m0))    &   (𝜑𝐵 ∈ (ℂ ↑m0))    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑 → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))       (𝜑𝐴 = 𝐵)
 
Theoremcoeeu 24744* Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝐹 ∈ (Poly‘𝑆) → ∃!𝑎 ∈ (ℂ ↑m0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))
 
Theoremcoelem 24745* Lemma for properties of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝐹 ∈ (Poly‘𝑆) → ((coeff‘𝐹) ∈ (ℂ ↑m0) ∧ ∃𝑛 ∈ ℕ0 (((coeff‘𝐹) “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)(((coeff‘𝐹)‘𝑘) · (𝑧𝑘))))))
 
Theoremcoeeq 24746* If 𝐴 satisfies the properties of the coefficient function, it must be equal to the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))       (𝜑 → (coeff‘𝐹) = 𝐴)
 
Theoremdgrval 24747 Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((𝐴 “ (ℂ ∖ {0})), ℕ0, < ))
 
Theoremdgrlem 24748* Lemma for dgrcl 24752 and similar theorems. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (𝐴 “ (ℂ ∖ {0}))𝑥𝑛))
 
Theoremcoef 24749 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
 
Theoremcoef2 24750 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)       ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → 𝐴:ℕ0𝑆)
 
Theoremcoef3 24751 The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
 
Theoremdgrcl 24752 The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
 
Theoremdgrub 24753 If the 𝑀-th coefficient of 𝐹 is nonzero, then the degree of 𝐹 is at least 𝑀. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴𝑀) ≠ 0) → 𝑀𝑁)
 
Theoremdgrub2 24754 All the coefficients above the degree of 𝐹 are zero. (Contributed by Mario Carneiro, 23-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})
 
Theoremdgrlb 24755 If all the coefficients above 𝑀 are zero, then the degree of 𝐹 is at most 𝑀. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ‘(𝑀 + 1))) = {0}) → 𝑁𝑀)
 
Theoremcoeidlem 24756* Lemma for coeid 24757. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝐵 ∈ ((𝑆 ∪ {0}) ↑m0))    &   (𝜑 → (𝐵 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐵𝑘) · (𝑧𝑘))))       (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))
 
Theoremcoeid 24757* Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))
 
Theoremcoeid2 24758* Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑋 ∈ ℂ) → (𝐹𝑋) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑋𝑘)))
 
Theoremcoeid3 24759* Reconstruct a polynomial as an explicit sum of the coefficient function up to at least the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ (ℤ𝑁) ∧ 𝑋 ∈ ℂ) → (𝐹𝑋) = Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑋𝑘)))
 
Theoremplyco 24760* The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)       (𝜑 → (𝐹𝐺) ∈ (Poly‘𝑆))
 
Theoremcoeeq2 24761* Compute the coefficient function given a sum expression for the polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧𝑘))))       (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘𝑁, 𝐴, 0)))
 
Theoremdgrle 24762* Given an explicit expression for a polynomial, the degree is at most the highest term in the sum. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧𝑘))))       (𝜑 → (deg‘𝐹) ≤ 𝑁)
 
Theoremdgreq 24763* If the highest term in a polynomial expression is nonzero, then the polynomial's degree is completely determined. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑 → (𝐴𝑁) ≠ 0)       (𝜑 → (deg‘𝐹) = 𝑁)
 
Theorem0dgr 24764 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)
 
Theorem0dgrb 24765 A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → ((deg‘𝐹) = 0 ↔ 𝐹 = (ℂ × {(𝐹‘0)})))
 
Theoremdgrnznn 24766 A nonzero polynomial with a root has positive degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃𝐴) = 0)) → (deg‘𝑃) ∈ ℕ)
 
Theoremcoefv0 24767 The result of evaluating a polynomial at zero is the constant term. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → (𝐹‘0) = (𝐴‘0))
 
Theoremcoeaddlem 24768 Lemma for coeadd 24770 and dgradd 24786. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)    &   𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹f + 𝐺)) = (𝐴f + 𝐵) ∧ (deg‘(𝐹f + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀)))
 
Theoremcoemullem 24769* Lemma for coemul 24771 and dgrmul 24789. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)    &   𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹f · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) ∧ (deg‘(𝐹f · 𝐺)) ≤ (𝑀 + 𝑁)))
 
Theoremcoeadd 24770 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f + 𝐺)) = (𝐴f + 𝐵))
 
Theoremcoemul 24771* A coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → ((coeff‘(𝐹f · 𝐺))‘𝑁) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝐵‘(𝑁𝑘))))
 
Theoremcoe11 24772 The coefficient function is one-to-one, so if the coefficients are equal then the functions are equal and vice-versa. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 𝐺𝐴 = 𝐵))
 
Theoremcoemulhi 24773 The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)    &   𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹f · 𝐺))‘(𝑀 + 𝑁)) = ((𝐴𝑀) · (𝐵𝑁)))
 
Theoremcoemulc 24774 The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)) = ((ℕ0 × {𝐴}) ∘f · (coeff‘𝐹)))
 
Theoremcoe0 24775 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
(coeff‘0𝑝) = (ℕ0 × {0})
 
Theoremcoesub 24776 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f𝐺)) = (𝐴f𝐵))
 
Theoremcoe1termlem 24777* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) ∧ (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁)))
 
Theoremcoe1term 24778* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → ((coeff‘𝐹)‘𝑀) = if(𝑀 = 𝑁, 𝐴, 0))
 
Theoremdgr1term 24779* The degree of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℕ0) → (deg‘𝐹) = 𝑁)
 
Theoremplycn 24780 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
 
Theoremdgr0 24781 The degree of the zero polynomial is zero. Note: this differs from some other definitions of the degree of the zero polynomial, such as -1, -∞ or undefined. But it is convenient for us to define it this way, so that we have dgrcl 24752, dgreq0 24784 and coeid 24757 without having to special-case zero, although plydivalg 24817 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
(deg‘0𝑝) = 0
 
Theoremcoeidp 24782 The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝐴 ∈ ℕ0 → ((coeff‘Xp)‘𝐴) = if(𝐴 = 1, 1, 0))
 
Theoremdgrid 24783 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
(deg‘Xp) = 1
 
Theoremdgreq0 24784 The leading coefficient of a polynomial is nonzero, unless the entire polynomial is zero. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Fan Zheng, 21-Jun-2016.)
𝑁 = (deg‘𝐹)    &   𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴𝑁) = 0))
 
Theoremdgrlt 24785 Two ways to say that the degree of 𝐹 is strictly less than 𝑁. (Contributed by Mario Carneiro, 25-Jul-2014.)
𝑁 = (deg‘𝐹)    &   𝐴 = (coeff‘𝐹)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) → ((𝐹 = 0𝑝𝑁 < 𝑀) ↔ (𝑁𝑀 ∧ (𝐴𝑀) = 0)))
 
Theoremdgradd 24786 The degree of a sum of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹f + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
 
Theoremdgradd2 24787 The degree of a sum of polynomials of unequal degrees is the degree of the larger polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹f + 𝐺)) = 𝑁)
 
Theoremdgrmul2 24788 The degree of a product of polynomials is at most the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹f · 𝐺)) ≤ (𝑀 + 𝑁))
 
Theoremdgrmul 24789 The degree of a product of nonzero polynomials is the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) ∧ (𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝)) → (deg‘(𝐹f · 𝐺)) = (𝑀 + 𝑁))
 
Theoremdgrmulc 24790 Scalar multiplication by a nonzero constant does not change the degree of a function. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐹 ∈ (Poly‘𝑆)) → (deg‘((ℂ × {𝐴}) ∘f · 𝐹)) = (deg‘𝐹))
 
Theoremdgrsub 24791 The degree of a difference of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹f𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))
 
Theoremdgrcolem1 24792* The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.)
𝑁 = (deg‘𝐺)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐺 ∈ (Poly‘𝑆))       (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺𝑥)↑𝑀))) = (𝑀 · 𝑁))
 
Theoremdgrcolem2 24793* Lemma for dgrco 24794. (Contributed by Mario Carneiro, 15-Sep-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   𝐴 = (coeff‘𝐹)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝑀 = (𝐷 + 1))    &   (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝐷 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))       (𝜑 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))
 
Theoremdgrco 24794 The degree of a composition of two polynomials is the product of the degrees. (Contributed by Mario Carneiro, 15-Sep-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))       (𝜑 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))
 
Theoremplycjlem 24795* Lemma for plycj 24796 and coecj 24797. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑁 = (deg‘𝐹)    &   𝐺 = ((∗ ∘ 𝐹) ∘ ∗)    &   𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ 𝐴)‘𝑘) · (𝑧𝑘))))
 
Theoremplycj 24796* The double conjugation of a polynomial is a polynomial. (The single conjugation is not because our definition of polynomial includes only holomorphic functions, i.e. no dependence on (∗‘𝑧) independently of 𝑧.) (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑁 = (deg‘𝐹)    &   𝐺 = ((∗ ∘ 𝐹) ∘ ∗)    &   ((𝜑𝑥𝑆) → (∗‘𝑥) ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))       (𝜑𝐺 ∈ (Poly‘𝑆))
 
Theoremcoecj 24797 Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑁 = (deg‘𝐹)    &   𝐺 = ((∗ ∘ 𝐹) ∘ ∗)    &   𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴))
 
Theoremplyrecj 24798 A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹𝐴)) = (𝐹‘(∗‘𝐴)))
 
Theoremplymul0or 24799 Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹f · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝𝐺 = 0𝑝)))
 
Theoremofmulrt 24800 The set of roots of a product is the union of the roots of the terms. (Contributed by Mario Carneiro, 28-Jul-2014.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹f · 𝐺) “ {0}) = ((𝐹 “ {0}) ∪ (𝐺 “ {0})))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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