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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 0dgr 24001 | A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0) | ||
Theorem | 0dgrb 24002 | 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)}))) | ||
Theorem | dgrnznn 24003 | A nonzero polynomial with a root has positive degree. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → (deg‘𝑃) ∈ ℕ) | ||
Theorem | coefv0 24004 | 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)) | ||
Theorem | coeaddlem 24005 | Lemma for coeadd 24007 and dgradd 24023. (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‘(𝐹 ∘𝑓 + 𝐺)) = (𝐴 ∘𝑓 + 𝐵) ∧ (deg‘(𝐹 ∘𝑓 + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))) | ||
Theorem | coemullem 24006* | Lemma for coemul 24008 and dgrmul 24026. (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‘(𝐹 ∘𝑓 · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) ∧ (deg‘(𝐹 ∘𝑓 · 𝐺)) ≤ (𝑀 + 𝑁))) | ||
Theorem | coeadd 24007 | 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‘(𝐹 ∘𝑓 + 𝐺)) = (𝐴 ∘𝑓 + 𝐵)) | ||
Theorem | coemul 24008* | 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‘(𝐹 ∘𝑓 · 𝐺))‘𝑁) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝐵‘(𝑁 − 𝑘)))) | ||
Theorem | coe11 24009 | 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 = 𝐺 ↔ 𝐴 = 𝐵)) | ||
Theorem | coemulhi 24010 | 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‘(𝐹 ∘𝑓 · 𝐺))‘(𝑀 + 𝑁)) = ((𝐴‘𝑀) · (𝐵‘𝑁))) | ||
Theorem | coemulc 24011 | 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))) | ||
Theorem | coe0 24012 | The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (coeff‘0𝑝) = (ℕ0 × {0}) | ||
Theorem | coesub 24013 | 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‘(𝐹 ∘𝑓 − 𝐺)) = (𝐴 ∘𝑓 − 𝐵)) | ||
Theorem | coe1termlem 24014* | The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) ∧ (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁))) | ||
Theorem | coe1term 24015* | The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → ((coeff‘𝐹)‘𝑀) = if(𝑀 = 𝑁, 𝐴, 0)) | ||
Theorem | dgr1term 24016* | The degree of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℕ0) → (deg‘𝐹) = 𝑁) | ||
Theorem | plycn 24017 | A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | dgr0 24018 | 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 23989, dgreq0 24021 and coeid 23994 without having to special-case zero, although plydivalg 24054 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (deg‘0𝑝) = 0 | ||
Theorem | coeidp 24019 | The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ (𝐴 ∈ ℕ0 → ((coeff‘Xp)‘𝐴) = if(𝐴 = 1, 1, 0)) | ||
Theorem | dgrid 24020 | The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ (deg‘Xp) = 1 | ||
Theorem | dgreq0 24021 | 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 ↔ (𝐴‘𝑁) = 0)) | ||
Theorem | dgrlt 24022 | 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 = 0𝑝 ∨ 𝑁 < 𝑀) ↔ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0))) | ||
Theorem | dgradd 24023 | 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‘(𝐹 ∘𝑓 + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) | ||
Theorem | dgradd2 24024 | 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‘(𝐹 ∘𝑓 + 𝐺)) = 𝑁) | ||
Theorem | dgrmul2 24025 | 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‘(𝐹 ∘𝑓 · 𝐺)) ≤ (𝑀 + 𝑁)) | ||
Theorem | dgrmul 24026 | 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‘(𝐹 ∘𝑓 · 𝐺)) = (𝑀 + 𝑁)) | ||
Theorem | dgrmulc 24027 | 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)) | ||
Theorem | dgrsub 24028 | 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‘(𝐹 ∘𝑓 − 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)) | ||
Theorem | dgrcolem1 24029* | The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ 𝑁 = (deg‘𝐺) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) ⇒ ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁)) | ||
Theorem | dgrcolem2 24030* | Lemma for dgrco 24031. (Contributed by Mario Carneiro, 15-Sep-2014.) |
⊢ 𝑀 = (deg‘𝐹) & ⊢ 𝑁 = (deg‘𝐺) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ 𝐴 = (coeff‘𝐹) & ⊢ (𝜑 → 𝐷 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 = (𝐷 + 1)) & ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝐷 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁))) ⇒ ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁)) | ||
Theorem | dgrco 24031 | 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) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) ⇒ ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁)) | ||
Theorem | plycjlem 24032* | Lemma for plycj 24033 and coecj 24034. (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) · (𝑧↑𝑘)))) | ||
Theorem | plycj 24033* | 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) ∘ ∗) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (∗‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) ⇒ ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) | ||
Theorem | coecj 24034 | 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)) | ||
Theorem | plyrecj 24035 | A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴))) | ||
Theorem | plymul0or 24036 | Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘𝑓 · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝))) | ||
Theorem | ofmulrt 24037 | The set of roots of a product is the union of the roots of the terms. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (◡(𝐹 ∘𝑓 · 𝐺) “ {0}) = ((◡𝐹 “ {0}) ∪ (◡𝐺 “ {0}))) | ||
Theorem | plyreres 24038 | Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ↾ ℝ):ℝ⟶ℝ) | ||
Theorem | dvply1 24039* | Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) & ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘)))) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ 𝐵 = (𝑘 ∈ ℕ0 ↦ ((𝑘 + 1) · (𝐴‘(𝑘 + 1)))) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (ℂ D 𝐹) = 𝐺) | ||
Theorem | dvply2g 24040 | 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)) | ||
Theorem | dvply2 24041 | 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)) | ||
Theorem | dvnply2 24042 | 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)) | ||
Theorem | dvnply 24043 | 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)) | ||
Theorem | plycpn 24044 | 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)) | ||
Syntax | cquot 24045 | Extend class notation to include the quotient of a polynomial division. |
class quot | ||
Definition | df-quot 24046* | 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)[(𝑓 ∘𝑓 − (𝑔 ∘𝑓 · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)))) | ||
Theorem | quotval 24047* | Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ 𝑅 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · 𝑞)) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))) | ||
Theorem | plydivlem1 24048* | Lemma for plydivalg 24054. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆) & ⊢ (𝜑 → -1 ∈ 𝑆) ⇒ ⊢ (𝜑 → 0 ∈ 𝑆) | ||
Theorem | plydivlem2 24049* | Lemma for plydivalg 24054. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆) & ⊢ (𝜑 → -1 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ≠ 0𝑝) & ⊢ 𝑅 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · 𝑞)) ⇒ ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → 𝑅 ∈ (Poly‘𝑆)) | ||
Theorem | plydivlem3 24050* | Lemma for plydivex 24052. Base case of induction. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆) & ⊢ (𝜑 → -1 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ≠ 0𝑝) & ⊢ 𝑅 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · 𝑞)) & ⊢ (𝜑 → (𝐹 = 0𝑝 ∨ ((deg‘𝐹) − (deg‘𝐺)) < 0)) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) | ||
Theorem | plydivlem4 24051* | Lemma for plydivex 24052. Induction step. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆) & ⊢ (𝜑 → -1 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ≠ 0𝑝) & ⊢ 𝑅 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · 𝑞)) & ⊢ (𝜑 → 𝐷 ∈ ℕ0) & ⊢ (𝜑 → (𝑀 − 𝑁) = 𝐷) & ⊢ (𝜑 → 𝐹 ≠ 0𝑝) & ⊢ 𝑈 = (𝑓 ∘𝑓 − (𝐺 ∘𝑓 · 𝑝)) & ⊢ 𝐻 = (𝑧 ∈ ℂ ↦ (((𝐴‘𝑀) / (𝐵‘𝑁)) · (𝑧↑𝐷))) & ⊢ (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)) | ||
Theorem | plydivex 24052* | Lemma for plydivalg 24054. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆) & ⊢ (𝜑 → -1 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ≠ 0𝑝) & ⊢ 𝑅 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · 𝑞)) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) | ||
Theorem | plydiveu 24053* | Lemma for plydivalg 24054. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆) & ⊢ (𝜑 → -1 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ≠ 0𝑝) & ⊢ 𝑅 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · 𝑞)) & ⊢ (𝜑 → 𝑞 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) & ⊢ 𝑇 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · 𝑝)) & ⊢ (𝜑 → 𝑝 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → (𝑇 = 0𝑝 ∨ (deg‘𝑇) < (deg‘𝐺))) ⇒ ⊢ (𝜑 → 𝑝 = 𝑞) | ||
Theorem | plydivalg 24054* | 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 𝐹 − 𝐺 · 𝑞 is either zero or has degree less than 𝐺. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆) & ⊢ (𝜑 → -1 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ≠ 0𝑝) & ⊢ 𝑅 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · 𝑞)) ⇒ ⊢ (𝜑 → ∃!𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) | ||
Theorem | quotlem 24055* | Lemma for properties of the polynomial quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆) & ⊢ (𝜑 → -1 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ≠ 0𝑝) & ⊢ 𝑅 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) ⇒ ⊢ (𝜑 → ((𝐹 quot 𝐺) ∈ (Poly‘𝑆) ∧ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))) | ||
Theorem | quotcl 24056* | The quotient of two polynomials in a field 𝑆 is also in the field. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆) & ⊢ (𝜑 → -1 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ≠ 0𝑝) ⇒ ⊢ (𝜑 → (𝐹 quot 𝐺) ∈ (Poly‘𝑆)) | ||
Theorem | quotcl2 24057 | 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)) | ||
Theorem | quotdgr 24058 | Remainder property of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ 𝑅 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) | ||
Theorem | plyremlem 24059 | Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ 𝐺 = (Xp ∘𝑓 − (ℂ × {𝐴})) ⇒ ⊢ (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴})) | ||
Theorem | plyrem 24060 | The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15260). If a polynomial 𝐹 is divided by the linear factor 𝑥 − 𝐴, 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 = (Xp ∘𝑓 − (ℂ × {𝐴})) & ⊢ 𝑅 = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹‘𝐴)})) | ||
Theorem | facth 24061 | The factor theorem. If a polynomial 𝐹 has a root at 𝐴, then 𝐺 = 𝑥 − 𝐴 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 = (Xp ∘𝑓 − (ℂ × {𝐴})) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 𝐹 = (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) | ||
Theorem | fta1lem 24062* | Lemma for fta1 24063. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ 𝑅 = (◡𝐹 “ {0}) & ⊢ (𝜑 → 𝐷 ∈ ℕ0) & ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝})) & ⊢ (𝜑 → (deg‘𝐹) = (𝐷 + 1)) & ⊢ (𝜑 → 𝐴 ∈ (◡𝐹 “ {0})) & ⊢ (𝜑 → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝐷 → ((◡𝑔 “ {0}) ∈ Fin ∧ (#‘(◡𝑔 “ {0})) ≤ (deg‘𝑔)))) ⇒ ⊢ (𝜑 → (𝑅 ∈ Fin ∧ (#‘𝑅) ≤ (deg‘𝐹))) | ||
Theorem | fta1 24063 | 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))) | ||
Theorem | quotcan 24064 | Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ 𝐻 = (𝐹 ∘𝑓 · 𝐺) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹) | ||
Theorem | vieta1lem1 24065* | Lemma for vieta1 24067. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ 𝑅 = (◡𝐹 “ {0}) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → (#‘𝑅) = 𝑁) & ⊢ (𝜑 → 𝐷 ∈ 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 (Xp ∘𝑓 − (ℂ × {𝑧}))) ⇒ ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (𝑄 ∈ (Poly‘ℂ) ∧ 𝐷 = (deg‘𝑄))) | ||
Theorem | vieta1lem2 24066* | Lemma for vieta1 24067: inductive step. Let 𝑧 be a root of 𝐹. Then 𝐹 = (Xp − 𝑧) · 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 Σ𝑥 ∈ 𝑅𝑥 = 𝑧 − (coeff‘𝑄)‘ (𝐷 − 1) / (coeff‘𝑄)‘𝐷. Now the coefficients of 𝐹 are 𝐴‘(𝐷 + 1) = (coeff‘𝑄)‘𝐷 and 𝐴‘𝐷 = Σ𝑘 ∈ (0...𝐷)(coeff‘Xp − 𝑧)‘𝑘 · (coeff‘𝑄) ‘(𝐷 − 𝑘), which works out to -𝑧 · (coeff‘𝑄)‘𝐷 + (coeff‘𝑄)‘(𝐷 − 1), so putting it all together we have Σ𝑥 ∈ 𝑅𝑥 = -𝐴‘𝐷 / 𝐴‘(𝐷 + 1) as we wanted to show. (Contributed by Mario Carneiro, 28-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ 𝑅 = (◡𝐹 “ {0}) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → (#‘𝑅) = 𝑁) & ⊢ (𝜑 → 𝐷 ∈ 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 (Xp ∘𝑓 − (ℂ × {𝑧}))) ⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁))) | ||
Theorem | vieta1 24067* | 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}) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → (#‘𝑅) = 𝑁) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁))) | ||
Theorem | plyexmo 24068* | 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) ∧ (𝑝 ↾ 𝐷) = 𝐹)) | ||
Syntax | caa 24069 | Extend class notation to include the set of algebraic numbers. |
class 𝔸 | ||
Definition | df-aa 24070 | 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}) | ||
Theorem | elaa 24071* | Elementhood in the set of algebraic numbers. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0)) | ||
Theorem | aacn 24072 | An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ) | ||
Theorem | aasscn 24073 | The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ 𝔸 ⊆ ℂ | ||
Theorem | elqaalem1 24074* | Lemma for elqaa 24077. 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.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝})) & ⊢ (𝜑 → (𝐹‘𝐴) = 0) & ⊢ 𝐵 = (coeff‘𝐹) & ⊢ 𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < )) & ⊢ 𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹)) ⇒ ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → ((𝑁‘𝐾) ∈ ℕ ∧ ((𝐵‘𝐾) · (𝑁‘𝐾)) ∈ ℤ)) | ||
Theorem | elqaalem2 24075* | Lemma for elqaa 24077. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝})) & ⊢ (𝜑 → (𝐹‘𝐴) = 0) & ⊢ 𝐵 = (coeff‘𝐹) & ⊢ 𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < )) & ⊢ 𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹)) & ⊢ 𝑃 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 · 𝑦) mod (𝑁‘𝐾))) ⇒ ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (𝑅 mod (𝑁‘𝐾)) = 0) | ||
Theorem | elqaalem3 24076* | Lemma for elqaa 24077. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝})) & ⊢ (𝜑 → (𝐹‘𝐴) = 0) & ⊢ 𝐵 = (coeff‘𝐹) & ⊢ 𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < )) & ⊢ 𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹)) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝔸) | ||
Theorem | elqaa 24077* | 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 24071 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})(𝑓‘𝐴) = 0)) | ||
Theorem | qaa 24078 | Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ 𝔸) | ||
Theorem | qssaa 24079 | The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ ℚ ⊆ 𝔸 | ||
Theorem | iaa 24080 | The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ i ∈ 𝔸 | ||
Theorem | aareccl 24081 | The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸) | ||
Theorem | aacjcl 24082 | The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ (𝐴 ∈ 𝔸 → (∗‘𝐴) ∈ 𝔸) | ||
Theorem | aannenlem1 24083* | Lemma for aannen 24086. (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)} (𝑐‘𝑏) = 0}) ⇒ ⊢ (𝐴 ∈ ℕ0 → (𝐻‘𝐴) ∈ Fin) | ||
Theorem | aannenlem2 24084* | Lemma for aannen 24086. (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)} (𝑐‘𝑏) = 0}) ⇒ ⊢ 𝔸 = ∪ ran 𝐻 | ||
Theorem | aannenlem3 24085* | 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)} (𝑐‘𝑏) = 0}) ⇒ ⊢ 𝔸 ≈ ℕ | ||
Theorem | aannen 24086 | The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝔸 ≈ ℕ | ||
Theorem | aalioulem1 24087 | Lemma for aaliou 24093. An integer polynomial cannot inflate the denominator of a rational by more than its degree. (Contributed by Stefan O'Rear, 12-Nov-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ)) & ⊢ (𝜑 → 𝑋 ∈ ℤ) & ⊢ (𝜑 → 𝑌 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝐹‘(𝑋 / 𝑌)) · (𝑌↑(deg‘𝐹))) ∈ ℤ) | ||
Theorem | aalioulem2 24088* | Lemma for aaliou 24093. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Proof shortened by AV, 28-Sep-2020.) |
⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) | ||
Theorem | aalioulem3 24089* | Lemma for aaliou 24093. (Contributed by Stefan O'Rear, 15-Nov-2014.) |
⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (𝐹‘𝐴) = 0) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑟 ∈ ℝ ((abs‘(𝐴 − 𝑟)) ≤ 1 → (𝑥 · (abs‘(𝐹‘𝑟))) ≤ (abs‘(𝐴 − 𝑟)))) | ||
Theorem | aalioulem4 24090* | Lemma for aaliou 24093. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (𝐹‘𝐴) = 0) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (((𝐹‘(𝑝 / 𝑞)) ≠ 0 ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ≤ 1) → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) | ||
Theorem | aalioulem5 24091* | Lemma for aaliou 24093. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (𝐹‘𝐴) = 0) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))) | ||
Theorem | aalioulem6 24092* | Lemma for aaliou 24093. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (𝐹‘𝐴) = 0) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))) | ||
Theorem | aaliou 24093* | 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) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (𝐹‘𝐴) = 0) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) | ||
Theorem | geolim3 24094* | Geometric series convergence with arbitrary shift, radix, and multiplicative constant. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (abs‘𝐵) < 1) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ 𝐹 = (𝑘 ∈ (ℤ≥‘𝐴) ↦ (𝐶 · (𝐵↑(𝑘 − 𝐴)))) ⇒ ⊢ (𝜑 → seq𝐴( + , 𝐹) ⇝ (𝐶 / (1 − 𝐵))) | ||
Theorem | aaliou2 24095* | 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 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) | ||
Theorem | aaliou2b 24096* | 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 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) | ||
Theorem | aaliou3lem1 24097* | Lemma for aaliou3 24106. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝐺 = (𝑐 ∈ (ℤ≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴)))) ⇒ ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐺‘𝐵) ∈ ℝ) | ||
Theorem | aaliou3lem2 24098* | Lemma for aaliou3 24106. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝐺 = (𝑐 ∈ (ℤ≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴)))) & ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) ⇒ ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝐵) ∈ (0(,](𝐺‘𝐵))) | ||
Theorem | aaliou3lem3 24099* | Lemma for aaliou3 24106. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝐺 = (𝑐 ∈ (ℤ≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴)))) & ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎))) ⇒ ⊢ (𝐴 ∈ ℕ → (seq𝐴( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ≥‘𝐴)(𝐹‘𝑏) ∈ ℝ+ ∧ Σ𝑏 ∈ (ℤ≥‘𝐴)(𝐹‘𝑏) ≤ (2 · (2↑-(!‘𝐴))))) | ||
Theorem | aaliou3lem8 24100* | Lemma for aaliou3 24106. (Contributed by Stefan O'Rear, 20-Nov-2014.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℕ (2 · (2↑-(!‘(𝑥 + 1)))) ≤ (𝐵 / ((2↑(!‘𝑥))↑𝐴))) |
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