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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | plypow 25701* | A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆 ∧ 𝑁 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ (𝑧↑𝑁)) ∈ (Poly‘𝑆)) | ||
Theorem | plyconst 25702 | A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆)) | ||
Theorem | ne0p 25703 | A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (𝐹‘𝐴) ≠ 0) → 𝐹 ≠ 0𝑝) | ||
Theorem | ply0 25704 | The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆)) | ||
Theorem | plyid 25705 | The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆)) | ||
Theorem | plyeq0lem 25706* | Lemma for plyeq0 25707. If 𝐴 is the coefficient function for a nonzero polynomial such that 𝑃(𝑧) = Σ𝑘 ∈ ℕ0𝐴(𝑘) · 𝑧↑𝑘 = 0 for every 𝑧 ∈ ℂ and 𝐴(𝑀) is the nonzero leading coefficient, then the function 𝐹(𝑧) = 𝑃(𝑧) / 𝑧↑𝑀 is a sum of powers of 1 / 𝑧, and so the limit of this function as 𝑧 ⇝ +∞ is the constant term, 𝐴(𝑀). But 𝐹(𝑧) = 0 everywhere, so this limit is also equal to zero so that 𝐴(𝑀) = 0, a contradiction. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) & ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) & ⊢ (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) & ⊢ 𝑀 = sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, < ) & ⊢ (𝜑 → (◡𝐴 “ (𝑆 ∖ {0})) ≠ ∅) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | plyeq0 25707* | If a polynomial is zero at every point (or even just zero at the positive integers), then all the coefficients must be zero. This is the basis for the method of equating coefficients of equal polynomials, and ensures that df-coe 25686 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) & ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) & ⊢ (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) ⇒ ⊢ (𝜑 → 𝐴 = (ℕ0 × {0})) | ||
Theorem | plypf1 25708 | Write the set of complex polynomials in a subring in terms of the abstract polynomial construction. (Contributed by Mario Carneiro, 3-Jul-2015.) (Proof shortened by AV, 29-Sep-2019.) |
⊢ 𝑅 = (ℂfld ↾s 𝑆) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (Base‘𝑃) & ⊢ 𝐸 = (eval1‘ℂfld) ⇒ ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸 “ 𝐴)) | ||
Theorem | plyaddlem1 25709* | Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ) & ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) & ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0}) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) & ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) ⇒ ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘f + 𝐵)‘𝑘) · (𝑧↑𝑘)))) | ||
Theorem | plymullem1 25710* | Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ) & ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) & ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0}) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) & ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛)))) | ||
Theorem | plyaddlem 25711* | Lemma for plyadd 25713. (Contributed by Mario Carneiro, 21-Jul-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) & ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) & ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) & ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0}) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) & ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) ⇒ ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆)) | ||
Theorem | plymullem 25712* | Lemma for plymul 25714. (Contributed by Mario Carneiro, 21-Jul-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) & ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) & ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) & ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0}) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) & ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ (Poly‘𝑆)) | ||
Theorem | plyadd 25713* | The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆)) | ||
Theorem | plymul 25714* | The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ (Poly‘𝑆)) | ||
Theorem | plysub 25715* | The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) & ⊢ (𝜑 → -1 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 ∘f − 𝐺) ∈ (Poly‘𝑆)) | ||
Theorem | plyaddcl 25716 | The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ)) | ||
Theorem | plymulcl 25717 | The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f · 𝐺) ∈ (Poly‘ℂ)) | ||
Theorem | plysubcl 25718 | The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f − 𝐺) ∈ (Poly‘ℂ)) | ||
Theorem | coeval 25719* | Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹) = (℩𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))) | ||
Theorem | coeeulem 25720* | Lemma for coeeu 25721. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ (ℂ ↑m ℕ0)) & ⊢ (𝜑 → 𝐵 ∈ (ℂ ↑m ℕ0)) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) & ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0}) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | coeeu 25721* | Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → ∃!𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) | ||
Theorem | coelem 25722* | Lemma for properties of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → ((coeff‘𝐹) ∈ (ℂ ↑m ℕ0) ∧ ∃𝑛 ∈ ℕ0 (((coeff‘𝐹) “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘)))))) | ||
Theorem | coeeq 25723* | If 𝐴 satisfies the properties of the coefficient function, it must be equal to the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) ⇒ ⊢ (𝜑 → (coeff‘𝐹) = 𝐴) | ||
Theorem | dgrval 25724 | Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) ⇒ ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) = sup((◡𝐴 “ (ℂ ∖ {0})), ℕ0, < )) | ||
Theorem | dgrlem 25725* | Lemma for dgrcl 25729 and similar theorems. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) ⇒ ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ ∃𝑛 ∈ ℤ ∀𝑥 ∈ (◡𝐴 “ (ℂ ∖ {0}))𝑥 ≤ 𝑛)) | ||
Theorem | coef 25726 | The domain and codomain of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) ⇒ ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0})) | ||
Theorem | coef2 25727 | The domain and codomain of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → 𝐴:ℕ0⟶𝑆) | ||
Theorem | coef3 25728 | The domain and codomain of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) ⇒ ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) | ||
Theorem | dgrcl 25729 | The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) | ||
Theorem | dgrub 25730 | If the 𝑀-th coefficient of 𝐹 is nonzero, then the degree of 𝐹 is at least 𝑀. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴‘𝑀) ≠ 0) → 𝑀 ≤ 𝑁) | ||
Theorem | dgrub2 25731 | All the coefficients above the degree of 𝐹 are zero. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) ⇒ ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) | ||
Theorem | dgrlb 25732 | If all the coefficients above 𝑀 are zero, then the degree of 𝐹 is at most 𝑀. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0 ∧ (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0}) → 𝑁 ≤ 𝑀) | ||
Theorem | coeidlem 25733* | Lemma for coeid 25734. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)) & ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑀 + 1))) = {0}) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐵‘𝑘) · (𝑧↑𝑘)))) ⇒ ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) | ||
Theorem | coeid 25734* | Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) ⇒ ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) | ||
Theorem | coeid2 25735* | Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑋 ∈ ℂ) → (𝐹‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑋↑𝑘))) | ||
Theorem | coeid3 25736* | Reconstruct a polynomial as an explicit sum of the coefficient function up to at least the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) & ⊢ 𝑁 = (deg‘𝐹) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑋 ∈ ℂ) → (𝐹‘𝑋) = Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑋↑𝑘))) | ||
Theorem | plyco 25737* | The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘𝑆)) | ||
Theorem | coeeq2 25738* | Compute the coefficient function given a sum expression for the polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘)))) ⇒ ⊢ (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0))) | ||
Theorem | dgrle 25739* | 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.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘)))) ⇒ ⊢ (𝜑 → (deg‘𝐹) ≤ 𝑁) | ||
Theorem | dgreq 25740* | 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.) |
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) & ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) & ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) & ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0) ⇒ ⊢ (𝜑 → (deg‘𝐹) = 𝑁) | ||
Theorem | 0dgr 25741 | A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0) | ||
Theorem | 0dgrb 25742 | 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 25743 | 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 25744 | 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 25745 | Lemma for coeadd 25747 and dgradd 25763. (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 25746* | Lemma for coemul 25748 and dgrmul 25766. (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 25747 | 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 25748* | 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 25749 | 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 25750 | 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 25751 | 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 25752 | The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (coeff‘0𝑝) = (ℕ0 × {0}) | ||
Theorem | coesub 25753 | 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 25754* | 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 25755* | 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 25756* | 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 25757 | A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | dgr0 25758 | 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 25729, dgreq0 25761 and coeid 25734 without having to special-case zero, although plydivalg 25794 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (deg‘0𝑝) = 0 | ||
Theorem | coeidp 25759 | 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 25760 | The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ (deg‘Xp) = 1 | ||
Theorem | dgreq0 25761 | 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 25762 | 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 25763 | 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 25764 | 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 25765 | 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 25766 | 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 25767 | 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 25768 | 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 25769* | 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 25770* | Lemma for dgrco 25771. (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 25771 | 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 25772* | Lemma for plycj 25773 and coecj 25774. (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 25773* | 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 25774 | 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 25775 | 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 25776 | 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 25777 | 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 25778 | Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ↾ ℝ):ℝ⟶ℝ) | ||
Theorem | dvply1 25779* | 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 25780 | 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 25781 | 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 25782 | 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 25783 | 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 25784 | 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 25785 | Extend class notation to include the quotient of a polynomial division. |
class quot | ||
Definition | df-quot 25786* | 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 25787* | 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 25788* | Lemma for plydivalg 25794. (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 25789* | Lemma for plydivalg 25794. (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 25790* | Lemma for plydivex 25792. 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 25791* | Lemma for plydivex 25792. 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 25792* | Lemma for plydivalg 25794. (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 25793* | Lemma for plydivalg 25794. (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 25794* | 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 25795* | 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 25796* | 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 25797 | 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 25798 | 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 25799 | 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 25800 | The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16481). 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)})) |
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