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Theorem List for Metamath Proof Explorer - 23801-23900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremplyco 23801* 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 23802* 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 23803* 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 23804* 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 23805 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)

Theorem0dgrb 23806 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 23807 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 23808 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))

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 23810* Lemma for coemul 23812 and dgrmul 23830. (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 23811 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 23812* 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 23813 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 23814 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 23815 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 23816 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
(coeff‘0𝑝) = (ℕ0 × {0})

Theoremcoesub 23817 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 23818* 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 23819* 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 23820* 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 23821 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremdgr0 23822 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 23793, dgreq0 23825 and coeid 23798 without having to special-case zero, although plydivalg 23858 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
(deg‘0𝑝) = 0

Theoremcoeidp 23823 The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝐴 ∈ ℕ0 → ((coeff‘Xp)‘𝐴) = if(𝐴 = 1, 1, 0))

Theoremdgrid 23824 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
(deg‘Xp) = 1

Theoremdgreq0 23825 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 23826 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 23827 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 23828 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 23829 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 23830 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 23831 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 23832 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 23833* 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 23834* Lemma for dgrco 23835. (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 23835 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 23836* Lemma for plycj 23837 and coecj 23838. (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 23837* 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 23838 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 23839 A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹𝐴)) = (𝐹‘(∗‘𝐴)))

Theoremplymul0or 23840 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 23841 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 23842 Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐹 ∈ (Poly‘ℝ) → (𝐹 ↾ ℝ):ℝ⟶ℝ)

Theoremdvply1 23843* 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 23844 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 23845 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 23846 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 23847 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 23848 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 23849 Extend class notation to include the quotient of a polynomial division.
class quot

Definitiondf-quot 23850* 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 23851* 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 23852* Lemma for plydivalg 23858. (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 23853* Lemma for plydivalg 23858. (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 23854* Lemma for plydivex 23856. 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 23855* Lemma for plydivex 23856. 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 23856* Lemma for plydivalg 23858. (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 23857* Lemma for plydivalg 23858. (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 23858* 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 23859* 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 23860* 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 23861 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 23862 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 23863 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 23864 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15098). 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 23865 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 23866* Lemma for fta1 23867. (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 23867 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 23868 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 23869* Lemma for vieta1 23871. (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 23870* Lemma for vieta1 23871: 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 23871* 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 23872* 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 23873 Extend class notation to include the set of algebraic numbers.
class 𝔸

Definitiondf-aa 23874 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 23875* 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 23876 An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)

Theoremaasscn 23877 The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
𝔸 ⊆ ℂ

Theoremelqaalem1 23878* Lemma for elqaa 23881. 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 23879* Lemma for elqaa 23881. (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 23880* Lemma for elqaa 23881. (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 23881* 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 23875 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 23882 Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ ℚ → 𝐴 ∈ 𝔸)

Theoremqssaa 23883 The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
ℚ ⊆ 𝔸

Theoremiaa 23884 The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
i ∈ 𝔸

Theoremaareccl 23885 The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸)

Theoremaacjcl 23886 The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝐴 ∈ 𝔸 → (∗‘𝐴) ∈ 𝔸)

Theoremaannenlem1 23887* Lemma for aannen 23890. (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 23888* Lemma for aannen 23890. (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 23889* 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 23890 The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝔸 ≈ ℕ

14.1.6  Liouville's approximation theorem

Theoremaalioulem1 23891 Lemma for aaliou 23897. 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 23892* Lemma for aaliou 23897. (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 23893* Lemma for aaliou 23897. (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 23894* Lemma for aaliou 23897. (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 23895* Lemma for aaliou 23897. (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 23896* Lemma for aaliou 23897. (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 23897* 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 23898* 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 23899* 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 23900* 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)))))

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