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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | coefv0 26301 | 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 26302 | Lemma for coeadd 26304 and dgradd 26321. (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 26303* | Lemma for coemul 26305 and dgrmul 26324. (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 26304 | 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 26305* | 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 26306 | 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 26307 | 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 26308 | 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 26309 | The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (coeff‘0𝑝) = (ℕ0 × {0}) | ||
Theorem | coesub 26310 | 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 26311* | 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 26312* | 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 26313* | 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 26314 | A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 11232. (Revised by GG, 16-Mar-2025.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | plycnOLD 26315 | Obsolete version of plycn 26314 as of 10-Apr-2025. (Contributed by Mario Carneiro, 23-Jul-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | dgr0 26316 | 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 26286, dgreq0 26319 and coeid 26291 without having to special-case zero, although plydivalg 26355 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (deg‘0𝑝) = 0 | ||
Theorem | coeidp 26317 | 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 26318 | The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ (deg‘Xp) = 1 | ||
Theorem | dgreq0 26319 | 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 26320 | 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 26321 | 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 26322 | 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 26323 | 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 26324 | 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 26325 | 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 26326 | 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 26327* | 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 26328* | Lemma for dgrco 26329. (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 26329 | 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 26330* | Lemma for plycj 26331 and coecj 26332. (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 26331* | 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.) |
⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (∗‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) ⇒ ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) | ||
Theorem | coecj 26332 | Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) & ⊢ 𝐴 = (coeff‘𝐹) ⇒ ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴)) | ||
Theorem | plycjOLD 26333* | Obsolete version of plycj 26331 as of 22-Sep-2025. 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.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑁 = (deg‘𝐹) & ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (∗‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) ⇒ ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) | ||
Theorem | coecjOLD 26334 | Obsolete version of coecj 26332 as of 22-Sep-2025. Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑁 = (deg‘𝐹) & ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) & ⊢ 𝐴 = (coeff‘𝐹) ⇒ ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴)) | ||
Theorem | plyrecj 26335 | 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 26336 | 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 26337 | 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 26338 | Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ↾ ℝ):ℝ⟶ℝ) | ||
Theorem | dvply1 26339* | 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 26340 | 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.) Avoid ax-mulf 11232. (Revised by GG, 30-Apr-2025.) |
⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆)) | ||
Theorem | dvply2gOLD 26341 | Obsolete version of dvply2g 26340 as of 30-Apr-2025. (Contributed by Mario Carneiro, 1-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆)) | ||
Theorem | dvply2 26342 | 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 26343 | 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 26344 | 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 26345 | Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ ∩ ran (𝓑C𝑛‘ℂ)) | ||
Syntax | cquot 26346 | Extend class notation to include the quotient of a polynomial division. |
class quot | ||
Definition | df-quot 26347* | 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 26348* | 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 26349* | Lemma for plydivalg 26355. (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 26350* | Lemma for plydivalg 26355. (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 26351* | Lemma for plydivex 26353. 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 26352* | Lemma for plydivex 26353. 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 26353* | Lemma for plydivalg 26355. (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 26354* | Lemma for plydivalg 26355. (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 26355* | 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 26356* | 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 26357* | 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 26358 | 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 26359 | 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 26360 | 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 26361 | The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16576). 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 26362 | 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 26363* | Lemma for fta1 26364. (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 26364 | 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 26365 | 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 26366* | Lemma for vieta1 26368. (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 26367* | Lemma for vieta1 26368: 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 26368* | 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 26369* | 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 26370 | Extend class notation to include the set of algebraic numbers. |
class 𝔸 | ||
Definition | df-aa 26371 | 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 26372* | 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 26373 | An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ) | ||
Theorem | aasscn 26374 | The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ 𝔸 ⊆ ℂ | ||
Theorem | elqaalem1 26375* | Lemma for elqaa 26378. 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 26376* | Lemma for elqaa 26378. (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 26377* | Lemma for elqaa 26378. (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 26378* | 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 26372 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 26379 | Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ 𝔸) | ||
Theorem | qssaa 26380 | The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ ℚ ⊆ 𝔸 | ||
Theorem | iaa 26381 | The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ i ∈ 𝔸 | ||
Theorem | aareccl 26382 | The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸) | ||
Theorem | aacjcl 26383 | The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ (𝐴 ∈ 𝔸 → (∗‘𝐴) ∈ 𝔸) | ||
Theorem | aannenlem1 26384* | Lemma for aannen 26387. (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 26385* | Lemma for aannen 26387. (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 26386* | 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 26387 | The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ 𝔸 ≈ ℕ | ||
Theorem | aalioulem1 26388 | Lemma for aaliou 26394. 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 26389* | Lemma for aaliou 26394. (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 26390* | Lemma for aaliou 26394. (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 26391* | Lemma for aaliou 26394. (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 26392* | Lemma for aaliou 26394. (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 26393* | Lemma for aaliou 26394. (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 26394* | 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 26395* | 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 26396* | 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 26397* | 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 26398* | Lemma for aaliou3 26407. (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 26399* | Lemma for aaliou3 26407. (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 26400* | Lemma for aaliou3 26407. (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))))) |
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