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Type | Label | Description |
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Statement | ||
Theorem | ply1divalg 23701* | The division algorithm for univariate polynomials over a ring. For polynomials 𝐹, 𝐺 such that 𝐺 ≠ 0 and the leading coefficient of 𝐺 is a unit, there are unique polynomials 𝑞 and 𝑟 = 𝐹 − (𝐺 · 𝑞) such that the degree of 𝑟 is less than the degree of 𝐺. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ ∙ = (.r‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) & ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) | ||
Theorem | ply1divalg2 23702* | Reverse the order of multiplication in ply1divalg 23701 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ ∙ = (.r‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) & ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺)) | ||
Theorem | uc1pval 23703* | Value of the set of unitic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝐶 = (Unic1p‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} | ||
Theorem | isuc1p 23704 | Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝐶 = (Unic1p‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)) | ||
Theorem | mon1pval 23705* | Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑀 = (Monic1p‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} | ||
Theorem | ismon1p 23706 | Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑀 = (Monic1p‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )) | ||
Theorem | uc1pcl 23707 | Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 → 𝐹 ∈ 𝐵) | ||
Theorem | mon1pcl 23708 | Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑀 = (Monic1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 → 𝐹 ∈ 𝐵) | ||
Theorem | uc1pn0 23709 | Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 → 𝐹 ≠ 0 ) | ||
Theorem | mon1pn0 23710 | Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝑀 = (Monic1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 → 𝐹 ≠ 0 ) | ||
Theorem | uc1pdeg 23711 | Unitic polynomials have nonnegative degrees. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝐶 = (Unic1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶) → (𝐷‘𝐹) ∈ ℕ0) | ||
Theorem | uc1pldg 23712 | Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐶 = (Unic1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐶 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈) | ||
Theorem | mon1pldg 23713 | Unitic polynomials have one leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑀 = (Monic1p‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝑀 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ) | ||
Theorem | mon1puc1p 23714 | Monic polynomials are unitic. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐶 = (Unic1p‘𝑅) & ⊢ 𝑀 = (Monic1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑀) → 𝑋 ∈ 𝐶) | ||
Theorem | uc1pmon1p 23715 | Make a unitic polynomial monic by multiplying a factor to normalize the leading coefficient. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝐶 = (Unic1p‘𝑅) & ⊢ 𝑀 = (Monic1p‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ · = (.r‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐶) → ((𝐴‘(𝐼‘((coe1‘𝑋)‘(𝐷‘𝑋)))) · 𝑋) ∈ 𝑀) | ||
Theorem | deg1submon1p 23716 | The difference of two monic polynomials of the same degree is a polynomial of lesser degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑂 = (Monic1p‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝑂) & ⊢ (𝜑 → (𝐷‘𝐹) = 𝑋) & ⊢ (𝜑 → 𝐺 ∈ 𝑂) & ⊢ (𝜑 → (𝐷‘𝐺) = 𝑋) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝑋) | ||
Theorem | q1pval 23717* | Value of the univariate polynomial quotient function. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑄 = (quot1p‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ · = (.r‘𝑃) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝑄𝐺) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺))) | ||
Theorem | q1peqb 23718 | Characterizing property of the polynomial quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑄 = (quot1p‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)) ↔ (𝐹𝑄𝐺) = 𝑋)) | ||
Theorem | q1pcl 23719 | Closure of the quotient by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑄 = (quot1p‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝑄𝐺) ∈ 𝐵) | ||
Theorem | r1pval 23720 | Value of the polynomial remainder function. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐸 = (rem1p‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑄 = (quot1p‘𝑅) & ⊢ · = (.r‘𝑃) & ⊢ − = (-g‘𝑃) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺))) | ||
Theorem | r1pcl 23721 | Closure of remainder following division by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐸 = (rem1p‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) ∈ 𝐵) | ||
Theorem | r1pdeglt 23722 | The remainder has a degree smaller than the divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐸 = (rem1p‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) & ⊢ 𝐷 = ( deg1 ‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹𝐸𝐺)) < (𝐷‘𝐺)) | ||
Theorem | r1pid 23723 | Express the original polynomial 𝐹 as 𝐹 = (𝑞 · 𝐺) + 𝑟 using the quotient and remainder functions for 𝑞 and 𝑟. (Contributed by Mario Carneiro, 5-Jun-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) & ⊢ 𝑄 = (quot1p‘𝑅) & ⊢ 𝐸 = (rem1p‘𝑅) & ⊢ · = (.r‘𝑃) & ⊢ + = (+g‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 = (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺))) | ||
Theorem | dvdsq1p 23724 | Divisibility in a polynomial ring is witnessed by the quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ ∥ = (∥r‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) & ⊢ · = (.r‘𝑃) & ⊢ 𝑄 = (quot1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐺 ∥ 𝐹 ↔ 𝐹 = ((𝐹𝑄𝐺) · 𝐺))) | ||
Theorem | dvdsr1p 23725 | Divisibility in a polynomial ring in terms of the remainder. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ ∥ = (∥r‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐸 = (rem1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐺 ∥ 𝐹 ↔ (𝐹𝐸𝐺) = 0 )) | ||
Theorem | ply1remlem 23726 | A term of the form 𝑥 − 𝑁 is linear, monic, and has exactly one zero. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐺 = (𝑋 − (𝐴‘𝑁)) & ⊢ 𝑂 = (eval1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑁 ∈ 𝐾) & ⊢ 𝑈 = (Monic1p‘𝑅) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝜑 → (𝐺 ∈ 𝑈 ∧ (𝐷‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ { 0 }) = {𝑁})) | ||
Theorem | ply1rem 23727 | The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15098). If a polynomial 𝐹 is divided by the linear factor 𝑥 − 𝐴, the remainder is equal to 𝐹(𝐴), the evaluation of the polynomial at 𝐴 (interpreted as a constant polynomial). (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐺 = (𝑋 − (𝐴‘𝑁)) & ⊢ 𝑂 = (eval1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑁 ∈ 𝐾) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 𝐸 = (rem1p‘𝑅) ⇒ ⊢ (𝜑 → (𝐹𝐸𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁))) | ||
Theorem | facth1 23728 | The factor theorem and its converse. A polynomial 𝐹 has a root at 𝐴 iff 𝐺 = 𝑥 − 𝐴 is a factor of 𝐹. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐺 = (𝑋 − (𝐴‘𝑁)) & ⊢ 𝑂 = (eval1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑁 ∈ 𝐾) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 0 = (0g‘𝑅) & ⊢ ∥ = (∥r‘𝑃) ⇒ ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) | ||
Theorem | fta1glem1 23729 | Lemma for fta1g 23731. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑂 = (eval1‘𝑅) & ⊢ 𝑊 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐺 = (𝑋 − (𝐴‘𝑇)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) = (𝑁 + 1)) & ⊢ (𝜑 → 𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊})) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁) | ||
Theorem | fta1glem2 23730* | Lemma for fta1g 23731. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑂 = (eval1‘𝑅) & ⊢ 𝑊 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐺 = (𝑋 − (𝐴‘𝑇)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) = (𝑁 + 1)) & ⊢ (𝜑 → 𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊})) & ⊢ (𝜑 → ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑁 → (#‘(◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔))) ⇒ ⊢ (𝜑 → (#‘(◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹)) | ||
Theorem | fta1g 23731 | The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 24606, which is only true in ℂ and other algebraically closed fields, this is true in any integral domain. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑂 = (eval1‘𝑅) & ⊢ 𝑊 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ≠ 0 ) ⇒ ⊢ (𝜑 → (#‘(◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹)) | ||
Theorem | fta1blem 23732 | Lemma for fta1b 23733. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑂 = (eval1‘𝑅) & ⊢ 𝑊 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑀 ∈ 𝐾) & ⊢ (𝜑 → 𝑁 ∈ 𝐾) & ⊢ (𝜑 → (𝑀 × 𝑁) = 𝑊) & ⊢ (𝜑 → 𝑀 ≠ 𝑊) & ⊢ (𝜑 → ((𝑀 · 𝑋) ∈ (𝐵 ∖ { 0 }) → (#‘(◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ (𝐷‘(𝑀 · 𝑋)))) ⇒ ⊢ (𝜑 → 𝑁 = 𝑊) | ||
Theorem | fta1b 23733* | The assumption that 𝑅 be a domain in fta1g 23731 is necessary. Here we show that the statement is strong enough to prove that 𝑅 is a domain. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑂 = (eval1‘𝑅) & ⊢ 𝑊 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ NzRing ∧ ∀𝑓 ∈ (𝐵 ∖ { 0 })(#‘(◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))) | ||
Theorem | drnguc1p 23734 | Over a division ring, all nonzero polynomials are unitic. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐶 = (Unic1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → 𝐹 ∈ 𝐶) | ||
Theorem | ig1peu 23735* | There is a unique monic polynomial of minimal degree in any nonzero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝑀 = (Monic1p‘𝑅) & ⊢ 𝐷 = ( deg1 ‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ { 0 }) → ∃!𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) | ||
Theorem | ig1pval 23736* | Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐺 = (idlGen1p‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑀 = (Monic1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) = if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))) | ||
Theorem | ig1pval2 23737 | Generator of the zero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Sep-2020.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐺 = (idlGen1p‘𝑅) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → (𝐺‘{ 0 }) = 0 ) | ||
Theorem | ig1pval3 23738 | Characterizing properties of the monic generator of a nonzero ideal of polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐺 = (idlGen1p‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝑀 = (Monic1p‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ { 0 }) → ((𝐺‘𝐼) ∈ 𝐼 ∧ (𝐺‘𝐼) ∈ 𝑀 ∧ (𝐷‘(𝐺‘𝐼)) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) | ||
Theorem | ig1pcl 23739 | The monic generator of an ideal is always in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Sep-2020.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐺 = (idlGen1p‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) ∈ 𝐼) | ||
Theorem | ig1pdvds 23740 | The monic generator of an ideal divides all elements of the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Sep-2020.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐺 = (idlGen1p‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ ∥ = (∥r‘𝑃) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝐺‘𝐼) ∥ 𝑋) | ||
Theorem | ig1prsp 23741 | Any ideal of polynomials over a division ring is generated by the ideal's canonical generator. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐺 = (idlGen1p‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑃) & ⊢ 𝐾 = (RSpan‘𝑃) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → 𝐼 = (𝐾‘{(𝐺‘𝐼)})) | ||
Theorem | ply1lpir 23742 | 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 23743 | 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 23744 | Extend class notation to include the set of complex polynomials. |
class Poly | ||
Syntax | cidp 23745 | Extend class notation to include the identity polynomial. |
class Xp | ||
Syntax | ccoe 23746 | Extend class notation to include the coefficient function on polynomials. |
class coeff | ||
Syntax | cdgr 23747 | Extend class notation to include the degree function on polynomials. |
class deg | ||
Definition | df-ply 23748* | 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 23749 | Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ Xp = ( I ↾ ℂ) | ||
Definition | df-coe 23750* | 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 23751 | 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 23752* | 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 23753* | 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 23754 | 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 23755* | 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 23756* | 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 23757 | 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 23758 | The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | ||
Theorem | plyss 23759 | 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 23760 | 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 23761* | 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 23762* | 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 23763* | Lemma for ply1term 23764. (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 23764* | 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 23765* | 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 23766 | A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆)) | ||
Theorem | ne0p 23767 | 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 23768 | The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ (𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆)) | ||
Theorem | plyid 23769 | The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.) |
⊢ ((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆)) | ||
Theorem | plyeq0lem 23770* | Lemma for plyeq0 23771. 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 23771* | 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 23750 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 23772 | 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 23773* | 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 23774* | 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 23775* | Lemma for plyadd 23777. (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 23776* | Lemma for plymul 23778. (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 23777* | 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 23778* | 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 23779* | 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 23780 | 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 23781 | 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 23782 | 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 23783* | 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 23784* | Lemma for coeeu 23785. (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 23785* | 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 23786* | 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 23787* | 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 23788 | 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 23789* | Lemma for dgrcl 23793 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 23790 | 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 23791 | 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 23792 | 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 23793 | The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.) |
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) | ||
Theorem | dgrub 23794 | 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 23795 | 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 23796 | 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 23797* | Lemma for coeid 23798. (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 23798* | 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 23799* | 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 23800* | 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))) |
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