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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdgrid 24001 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
(deg‘Xp) = 1

Theoremdgreq0 24002 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 24003 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 24004 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 24005 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 24006 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 24007 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 24008 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 24009 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 24010* 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 24011* Lemma for dgrco 24012. (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 24012 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 24013* Lemma for plycj 24014 and coecj 24015. (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 24014* 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 24015 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 24016 A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹𝐴)) = (𝐹‘(∗‘𝐴)))

Theoremplymul0or 24017 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 24018 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})))

Theoremplyreres 24019 Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐹 ∈ (Poly‘ℝ) → (𝐹 ↾ ℝ):ℝ⟶ℝ)

Theoremdvply1 24020* Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵𝑘) · (𝑧𝑘))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝐵 = (𝑘 ∈ ℕ0 ↦ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (ℂ D 𝐹) = 𝐺)

Theoremdvply2g 24021 The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.)
((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆))

Theoremdvply2 24022 The derivative of a polynomial is a polynomial. (Contributed by Stefan O'Rear, 14-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
(𝐹 ∈ (Poly‘𝑆) → (ℂ D 𝐹) ∈ (Poly‘ℂ))

Theoremdvnply2 24023 Polynomials have polynomials as derivatives of all orders. (Contributed by Mario Carneiro, 1-Jan-2017.)
((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))

Theoremdvnply 24024 Polynomials have polynomials as derivatives of all orders. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘ℂ))

Theoremplycpn 24025 Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ran (Cn‘ℂ))

14.1.4  The division algorithm for polynomials

Syntaxcquot 24026 Extend class notation to include the quotient of a polynomial division.
class quot

Definitiondf-quot 24027* Define the quotient function on polynomials. This is the 𝑞 of the expression 𝑓 = 𝑔 · 𝑞 + 𝑟 in the division algorithm. (Contributed by Mario Carneiro, 23-Jul-2014.)
quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (𝑞 ∈ (Poly‘ℂ)[(𝑓𝑓 − (𝑔𝑓 · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))

Theoremquotval 24028* Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))

Theoremplydivlem1 24029* Lemma for plydivalg 24035. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)       (𝜑 → 0 ∈ 𝑆)

Theoremplydivlem2 24030* Lemma for plydivalg 24035. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))       ((𝜑𝑞 ∈ (Poly‘𝑆)) → 𝑅 ∈ (Poly‘𝑆))

Theoremplydivlem3 24031* Lemma for plydivex 24033. Base case of induction. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))    &   (𝜑 → (𝐹 = 0𝑝 ∨ ((deg‘𝐹) − (deg‘𝐺)) < 0))       (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))

Theoremplydivlem4 24032* Lemma for plydivex 24033. Induction step. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑 → (𝑀𝑁) = 𝐷)    &   (𝜑𝐹 ≠ 0𝑝)    &   𝑈 = (𝑓𝑓 − (𝐺𝑓 · 𝑝))    &   𝐻 = (𝑧 ∈ ℂ ↦ (((𝐴𝑀) / (𝐵𝑁)) · (𝑧𝐷)))    &   (𝜑 → ∀𝑓 ∈ (Poly‘𝑆)((𝑓 = 0𝑝 ∨ ((deg‘𝑓) − 𝑁) < 𝐷) → ∃𝑝 ∈ (Poly‘𝑆)(𝑈 = 0𝑝 ∨ (deg‘𝑈) < 𝑁)))    &   𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)    &   𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < 𝑁))

Theoremplydivex 24033* Lemma for plydivalg 24035. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))       (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))

Theoremplydiveu 24034* Lemma for plydivalg 24035. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))    &   (𝜑𝑞 ∈ (Poly‘𝑆))    &   (𝜑 → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))    &   𝑇 = (𝐹𝑓 − (𝐺𝑓 · 𝑝))    &   (𝜑𝑝 ∈ (Poly‘𝑆))    &   (𝜑 → (𝑇 = 0𝑝 ∨ (deg‘𝑇) < (deg‘𝐺)))       (𝜑𝑝 = 𝑞)

Theoremplydivalg 24035* The division algorithm on polynomials over a subfield 𝑆 of the complex numbers. If 𝐹 and 𝐺 ≠ 0 are polynomials over 𝑆, then there is a unique quotient polynomial 𝑞 such that the remainder 𝐹𝐺 · 𝑞 is either zero or has degree less than 𝐺. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))       (𝜑 → ∃!𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))

Theoremquotlem 24036* Lemma for properties of the polynomial quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))       (𝜑 → ((𝐹 quot 𝐺) ∈ (Poly‘𝑆) ∧ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))

Theoremquotcl 24037* The quotient of two polynomials in a field 𝑆 is also in the field. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)       (𝜑 → (𝐹 quot 𝐺) ∈ (Poly‘𝑆))

Theoremquotcl2 24038 Closure of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))

Theoremquotdgr 24039 Remainder property of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝑅 = (𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))

Theoremplyremlem 24040 Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐺 = (Xp𝑓 − (ℂ × {𝐴}))       (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (𝐺 “ {0}) = {𝐴}))

Theoremplyrem 24041 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15241). If a polynomial 𝐹 is divided by the linear factor 𝑥𝐴, the remainder is equal to 𝐹(𝐴), the evaluation of the polynomial at 𝐴 (interpreted as a constant polynomial). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐺 = (Xp𝑓 − (ℂ × {𝐴}))    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹𝐴)}))

Theoremfacth 24042 The factor theorem. If a polynomial 𝐹 has a root at 𝐴, then 𝐺 = 𝑥𝐴 is a factor of 𝐹 (and the other factor is 𝐹 quot 𝐺). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐺 = (Xp𝑓 − (ℂ × {𝐴}))       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0) → 𝐹 = (𝐺𝑓 · (𝐹 quot 𝐺)))

Theoremfta1lem 24043* Lemma for fta1 24044. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝑅 = (𝐹 “ {0})    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}))    &   (𝜑 → (deg‘𝐹) = (𝐷 + 1))    &   (𝜑𝐴 ∈ (𝐹 “ {0}))    &   (𝜑 → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝐷 → ((𝑔 “ {0}) ∈ Fin ∧ (#‘(𝑔 “ {0})) ≤ (deg‘𝑔))))       (𝜑 → (𝑅 ∈ Fin ∧ (#‘𝑅) ≤ (deg‘𝐹)))

Theoremfta1 24044 The easy direction of the Fundamental Theorem of Algebra: A nonzero polynomial has at most deg(𝐹) roots. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝑅 = (𝐹 “ {0})       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (𝑅 ∈ Fin ∧ (#‘𝑅) ≤ (deg‘𝐹)))

Theoremquotcan 24045 Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐻 = (𝐹𝑓 · 𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹)

Theoremvieta1lem1 24046* Lemma for vieta1 24048. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   𝑅 = (𝐹 “ {0})    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑 → (#‘𝑅) = 𝑁)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → (𝐷 + 1) = 𝑁)    &   (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((𝐷 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))    &   𝑄 = (𝐹 quot (Xp𝑓 − (ℂ × {𝑧})))       ((𝜑𝑧𝑅) → (𝑄 ∈ (Poly‘ℂ) ∧ 𝐷 = (deg‘𝑄)))

Theoremvieta1lem2 24047* Lemma for vieta1 24048: inductive step. Let 𝑧 be a root of 𝐹. Then 𝐹 = (Xp𝑧) · 𝑄 for some 𝑄 by the factor theorem, and 𝑄 is a degree- 𝐷 polynomial, so by the induction hypothesis Σ𝑥 ∈ (𝑄 “ 0)𝑥 = -(coeff‘𝑄)‘(𝐷 − 1) / (coeff‘𝑄)‘𝐷, so Σ𝑥𝑅𝑥 = 𝑧 − (coeff‘𝑄)‘ (𝐷 − 1) / (coeff‘𝑄)‘𝐷. Now the coefficients of 𝐹 are 𝐴‘(𝐷 + 1) = (coeff‘𝑄)‘𝐷 and 𝐴𝐷 = Σ𝑘 ∈ (0...𝐷)(coeff‘Xp𝑧)‘𝑘 · (coeff‘𝑄) ‘(𝐷𝑘), which works out to -𝑧 · (coeff‘𝑄)‘𝐷 + (coeff‘𝑄)‘(𝐷 − 1), so putting it all together we have Σ𝑥𝑅𝑥 = -𝐴𝐷 / 𝐴‘(𝐷 + 1) as we wanted to show. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   𝑅 = (𝐹 “ {0})    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑 → (#‘𝑅) = 𝑁)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → (𝐷 + 1) = 𝑁)    &   (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((𝐷 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))    &   𝑄 = (𝐹 quot (Xp𝑓 − (ℂ × {𝑧})))       (𝜑 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))

Theoremvieta1 24048* The first-order Vieta's formula (see http://en.wikipedia.org/wiki/Vieta%27s_formulas). If a polynomial of degree 𝑁 has 𝑁 distinct roots, then the sum over these roots can be calculated as -𝐴(𝑁 − 1) / 𝐴(𝑁). (If the roots are not distinct, then this formula is still true but must double-count some of the roots according to their multiplicities.) (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   𝑅 = (𝐹 “ {0})    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑 → (#‘𝑅) = 𝑁)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))

Theoremplyexmo 24049* An infinite set of values can be extended to a polynomial in at most one way. (Contributed by Stefan O'Rear, 14-Nov-2014.)
((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘𝑆) ∧ (𝑝𝐷) = 𝐹))

14.1.5  Algebraic numbers

Syntaxcaa 24050 Extend class notation to include the set of algebraic numbers.
class 𝔸

Definitiondf-aa 24051 Define the set of algebraic numbers. An algebraic number is a root of a nonzero polynomial over the integers. Here we construct it as the union of all kernels (preimages of {0}) of all polynomials in (Poly‘ℤ), except the zero polynomial 0𝑝. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝔸 = 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓 “ {0})

Theoremelaa 24052* Elementhood in the set of algebraic numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0))

Theoremaacn 24053 An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)

Theoremaasscn 24054 The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
𝔸 ⊆ ℂ

Theoremelqaalem1 24055* Lemma for elqaa 24058. The function 𝑁 represents the denominators of the rational coefficients 𝐵. By multiplying them all together to make 𝑅, we get a number big enough to clear all the denominators and make 𝑅 · 𝐹 an integer polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝}))    &   (𝜑 → (𝐹𝐴) = 0)    &   𝐵 = (coeff‘𝐹)    &   𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))    &   𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))       ((𝜑𝐾 ∈ ℕ0) → ((𝑁𝐾) ∈ ℕ ∧ ((𝐵𝐾) · (𝑁𝐾)) ∈ ℤ))

Theoremelqaalem2 24056* Lemma for elqaa 24058. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝}))    &   (𝜑 → (𝐹𝐴) = 0)    &   𝐵 = (coeff‘𝐹)    &   𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))    &   𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))    &   𝑃 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 · 𝑦) mod (𝑁𝐾)))       ((𝜑𝐾 ∈ (0...(deg‘𝐹))) → (𝑅 mod (𝑁𝐾)) = 0)

Theoremelqaalem3 24057* Lemma for elqaa 24058. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝}))    &   (𝜑 → (𝐹𝐴) = 0)    &   𝐵 = (coeff‘𝐹)    &   𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))    &   𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))       (𝜑𝐴 ∈ 𝔸)

Theoremelqaa 24058* The set of numbers generated by the roots of polynomials in the rational numbers is the same as the set of algebraic numbers, which by elaa 24052 are defined only in terms of polynomials over the integers. (Contributed by Mario Carneiro, 23-Jul-2014.) (Proof shortened by AV, 3-Oct-2020.)
(𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑓𝐴) = 0))

Theoremqaa 24059 Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ ℚ → 𝐴 ∈ 𝔸)

Theoremqssaa 24060 The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
ℚ ⊆ 𝔸

Theoremiaa 24061 The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
i ∈ 𝔸

Theoremaareccl 24062 The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸)

Theoremaacjcl 24063 The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝐴 ∈ 𝔸 → (∗‘𝐴) ∈ 𝔸)

Theoremaannenlem1 24064* Lemma for aannen 24067. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})       (𝐴 ∈ ℕ0 → (𝐻𝐴) ∈ Fin)

Theoremaannenlem2 24065* Lemma for aannen 24067. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})       𝔸 = ran 𝐻

Theoremaannenlem3 24066* The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})       𝔸 ≈ ℕ

Theoremaannen 24067 The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝔸 ≈ ℕ

14.1.6  Liouville's approximation theorem

Theoremaalioulem1 24068 Lemma for aaliou 24074. An integer polynomial cannot inflate the denominator of a rational by more than its degree. (Contributed by Stefan O'Rear, 12-Nov-2014.)
(𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑋 ∈ ℤ)    &   (𝜑𝑌 ∈ ℕ)       (𝜑 → ((𝐹‘(𝑋 / 𝑌)) · (𝑌↑(deg‘𝐹))) ∈ ℤ)

Theoremaalioulem2 24069* Lemma for aaliou 24074. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Proof shortened by AV, 28-Sep-2020.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))

Theoremaalioulem3 24070* Lemma for aaliou 24074. (Contributed by Stefan O'Rear, 15-Nov-2014.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (𝐹𝐴) = 0)       (𝜑 → ∃𝑥 ∈ ℝ+𝑟 ∈ ℝ ((abs‘(𝐴𝑟)) ≤ 1 → (𝑥 · (abs‘(𝐹𝑟))) ≤ (abs‘(𝐴𝑟))))

Theoremaalioulem4 24071* Lemma for aaliou 24074. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (𝐹𝐴) = 0)       (𝜑 → ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (((𝐹‘(𝑝 / 𝑞)) ≠ 0 ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ≤ 1) → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))

Theoremaalioulem5 24072* Lemma for aaliou 24074. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (𝐹𝐴) = 0)       (𝜑 → ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))

Theoremaalioulem6 24073* Lemma for aaliou 24074. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (𝐹𝐴) = 0)       (𝜑 → ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))

Theoremaaliou 24074* Liouville's theorem on diophantine approximation: Any algebraic number, being a root of a polynomial 𝐹 in integer coefficients, is not approximable beyond order 𝑁 = deg(𝐹) by rational numbers. In this form, it also applies to rational numbers themselves, which are not well approximable by other rational numbers. This is Metamath 100 proof #18. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (𝐹𝐴) = 0)       (𝜑 → ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))

Theoremgeolim3 24075* Geometric series convergence with arbitrary shift, radix, and multiplicative constant. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (abs‘𝐵) < 1)    &   (𝜑𝐶 ∈ ℂ)    &   𝐹 = (𝑘 ∈ (ℤ𝐴) ↦ (𝐶 · (𝐵↑(𝑘𝐴))))       (𝜑 → seq𝐴( + , 𝐹) ⇝ (𝐶 / (1 − 𝐵)))

Theoremaaliou2 24076* Liouville's approximation theorem for algebraic numbers per se. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐴 ∈ (𝔸 ∩ ℝ) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))

Theoremaaliou2b 24077* Liouville's approximation theorem extended to complex 𝐴. (Contributed by Stefan O'Rear, 20-Nov-2014.)
(𝐴 ∈ 𝔸 → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))

Theoremaaliou3lem1 24078* Lemma for aaliou3 24087. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐺 = (𝑐 ∈ (ℤ𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐𝐴))))       ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ𝐴)) → (𝐺𝐵) ∈ ℝ)

Theoremaaliou3lem2 24079* Lemma for aaliou3 24087. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐺 = (𝑐 ∈ (ℤ𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐𝐴))))    &   𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))       ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ𝐴)) → (𝐹𝐵) ∈ (0(,](𝐺𝐵)))

Theoremaaliou3lem3 24080* Lemma for aaliou3 24087. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐺 = (𝑐 ∈ (ℤ𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐𝐴))))    &   𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))       (𝐴 ∈ ℕ → (seq𝐴( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ𝐴)(𝐹𝑏) ∈ ℝ+ ∧ Σ𝑏 ∈ (ℤ𝐴)(𝐹𝑏) ≤ (2 · (2↑-(!‘𝐴)))))

Theoremaaliou3lem8 24081* Lemma for aaliou3 24087. (Contributed by Stefan O'Rear, 20-Nov-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℕ (2 · (2↑-(!‘(𝑥 + 1)))) ≤ (𝐵 / ((2↑(!‘𝑥))↑𝐴)))

Theoremaaliou3lem4 24082* Lemma for aaliou3 24087. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    &   𝐿 = Σ𝑏 ∈ ℕ (𝐹𝑏)    &   𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹𝑏))       𝐿 ∈ ℝ

Theoremaaliou3lem5 24083* Lemma for aaliou3 24087. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    &   𝐿 = Σ𝑏 ∈ ℕ (𝐹𝑏)    &   𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹𝑏))       (𝐴 ∈ ℕ → (𝐻𝐴) ∈ ℝ)

Theoremaaliou3lem6 24084* Lemma for aaliou3 24087. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    &   𝐿 = Σ𝑏 ∈ ℕ (𝐹𝑏)    &   𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹𝑏))       (𝐴 ∈ ℕ → ((𝐻𝐴) · (2↑(!‘𝐴))) ∈ ℤ)

Theoremaaliou3lem7 24085* Lemma for aaliou3 24087. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    &   𝐿 = Σ𝑏 ∈ ℕ (𝐹𝑏)    &   𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹𝑏))       (𝐴 ∈ ℕ → ((𝐻𝐴) ≠ 𝐿 ∧ (abs‘(𝐿 − (𝐻𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))

Theoremaaliou3lem9 24086* Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 20-Nov-2014.)
𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    &   𝐿 = Σ𝑏 ∈ ℕ (𝐹𝑏)    &   𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹𝑏))        ¬ 𝐿 ∈ 𝔸

Theoremaaliou3 24087 Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 23-Nov-2014.)
Σ𝑘 ∈ ℕ (2↑-(!‘𝑘)) ∉ 𝔸

14.2  Sequences and series

14.2.1  Taylor polynomials and Taylor's theorem

Syntaxctayl 24088 Taylor polynomial of a function.
class Tayl

Syntaxcana 24089 The class of analytic functions.
class Ana

Definitiondf-tayl 24090* Define the Taylor polynomial or Taylor series of a function. TODO-AV: 𝑛 ∈ (ℕ0 ∪ {+∞}) can/should be replaced by 𝑛 ∈ ℕ0*. (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl = (𝑠 ∈ {ℝ, ℂ}, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ (𝑛 ∈ (ℕ0 ∪ {+∞}), 𝑎 𝑘 ∈ ((0[,]𝑛) ∩ ℤ)dom ((𝑠 D𝑛 𝑓)‘𝑘) ↦ 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑛) ∩ ℤ) ↦ (((((𝑠 D𝑛 𝑓)‘𝑘)‘𝑎) / (!‘𝑘)) · ((𝑥𝑎)↑𝑘)))))))

Definitiondf-ana 24091* Define the set of analytic functions, which are functions such that the Taylor series of the function at each point converges to the function in some neighborhood of the point. (Contributed by Mario Carneiro, 31-Dec-2016.)
Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))})

Theoremtaylfvallem1 24092* Lemma for taylfval 24094. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))       (((𝜑𝑋 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)) ∈ ℂ)

Theoremtaylfvallem 24093* Lemma for taylfval 24094. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))       ((𝜑𝑋 ∈ ℂ) → (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)))) ⊆ ℂ)

Theoremtaylfval 24094* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 𝑆 is the base set with respect to evaluate the derivatives (generally or ), 𝐹 is the function we are approximating, at point 𝐵, to order 𝑁. The result is a polynomial function of 𝑥.

This "extended" version of taylpfval 24100 additionally handles the case 𝑁 = +∞, in which case this is not a polynomial but an infinite series, the Taylor series of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)

(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇 = 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥𝐵)↑𝑘))))))

Theoremeltayl 24095* Value of the Taylor series as a relation (elementhood in the domain here expresses that the series is convergent). (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑 → (𝑋𝑇𝑌 ↔ (𝑋 ∈ ℂ ∧ 𝑌 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)))))))

Theoremtaylf 24096* The Taylor series defines a function on a subset of the complex numbers. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇:dom 𝑇⟶ℂ)

Theoremtayl0 24097* The Taylor series is always defined at the basepoint, with value equal to the value of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑 → (𝐵 ∈ dom 𝑇 ∧ (𝑇𝐵) = (𝐹𝐵)))

Theoremtaylplem1 24098* Lemma for taylpfval 24100 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))       ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))

Theoremtaylplem2 24099* Lemma for taylpfval 24100 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))       (((𝜑𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)) ∈ ℂ)

Theoremtaylpfval 24100* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 𝑆 is the base set with respect to evaluate the derivatives (generally or ), 𝐹 is the function we are approximating, at point 𝐵, to order 𝑁. The result is a polynomial function of 𝑥. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥𝐵)↑𝑘))))

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