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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ply1lpir 24701 | The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ LPIR) | ||
Theorem | ply1pid 24702 | The polynomials over a field are a PID. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ Field → 𝑃 ∈ PID) | ||
Syntax | cply 24703 | Extend class notation to include the set of complex polynomials. |
class Poly | ||
Syntax | cidp 24704 | Extend class notation to include the identity polynomial. |
class Xp | ||
Syntax | ccoe 24705 | Extend class notation to include the coefficient function on polynomials. |
class coeff | ||
Syntax | cdgr 24706 | Extend class notation to include the degree function on polynomials. |
class deg | ||
Definition | df-ply 24707* | Define the set of polynomials on the complex numbers with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))}) | ||
Definition | df-idp 24708 | Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ Xp = ( I ↾ ℂ) | ||
Definition | df-coe 24709* | Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ coeff = (𝑓 ∈ (Poly‘ℂ) ↦ (℩𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))) | ||
Definition | df-dgr 24710 | Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < )) | ||
Theorem | plyco0 24711* | Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) | ||
Theorem | plyval 24712* | Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))}) | ||
Theorem | plybss 24713 | Reverse closure of the parameter 𝑆 of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | ||
Theorem | elply 24714* | Definition of a polynomial with coefficients in 𝑆. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))) | ||
Theorem | elply2 24715* | The coefficient function can be assumed to have zeroes outside 0...𝑛. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))) | ||
Theorem | plyun0 24716 | The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) | ||
Theorem | plyf 24717 | The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | ||
Theorem | plyss 24718 | The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇)) | ||
Theorem | plyssc 24719 | Every polynomial ring is contained in the ring of polynomials over ℂ. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | ||
Theorem | elplyr 24720* | Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘𝑆)) | ||
Theorem | elplyd 24721* | Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))) ∈ (Poly‘𝑆)) | ||
Theorem | ply1termlem 24722* | Lemma for ply1term 24723. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁))) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘)))) | ||
Theorem | ply1term 24723* | A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁))) ⇒ ⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆 ∧ 𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘𝑆)) | ||
Theorem | plypow 24724* | 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 24725 | A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆)) | ||
Theorem | ne0p 24726 | 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 24727 | The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆)) | ||
Theorem | plyid 24728 | The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆)) | ||
Theorem | plyeq0lem 24729* | Lemma for plyeq0 24730. 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 24730* | 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 24709 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 24731 | 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 24732* | 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 24733* | 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 24734* | Lemma for plyadd 24736. (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 24735* | Lemma for plymul 24737. (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 24736* | 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 24737* | 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 24738* | 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 24739 | 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 24740 | 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 24741 | 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 24742* | 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 24743* | Lemma for coeeu 24744. (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 24744* | 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 24745* | 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 24746* | 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 24747 | 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 24748* | Lemma for dgrcl 24752 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 24749 | The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) ⇒ ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0})) | ||
Theorem | coef2 24750 | The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) ⇒ ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → 𝐴:ℕ0⟶𝑆) | ||
Theorem | coef3 24751 | The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ 𝐴 = (coeff‘𝐹) ⇒ ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) | ||
Theorem | dgrcl 24752 | The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) | ||
Theorem | dgrub 24753 | 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 24754 | 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 24755 | 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 24756* | Lemma for coeid 24757. (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 24757* | 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 24758* | 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 24759* | 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 24760* | 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 24761* | 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 24762* | 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 24763* | 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 24764 | A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0) | ||
Theorem | 0dgrb 24765 | 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 24766 | 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 24767 | 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 24768 | Lemma for coeadd 24770 and dgradd 24786. (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 24769* | Lemma for coemul 24771 and dgrmul 24789. (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 24770 | 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 24771* | 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 24772 | 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 24773 | 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 24774 | 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 24775 | The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (coeff‘0𝑝) = (ℕ0 × {0}) | ||
Theorem | coesub 24776 | 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 24777* | 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 24778* | 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 24779* | 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 24780 | A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | dgr0 24781 | 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 24752, dgreq0 24784 and coeid 24757 without having to special-case zero, although plydivalg 24817 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (deg‘0𝑝) = 0 | ||
Theorem | coeidp 24782 | 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 24783 | The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.) |
⊢ (deg‘Xp) = 1 | ||
Theorem | dgreq0 24784 | 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 24785 | 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 24786 | 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 24787 | 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 24788 | 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 24789 | 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 24790 | 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 24791 | 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 24792* | 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 24793* | Lemma for dgrco 24794. (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 24794 | 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 24795* | Lemma for plycj 24796 and coecj 24797. (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 24796* | 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 24797 | 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 24798 | 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 24799 | 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 24800 | 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}))) |
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