HomeTheorem List (p. 261 of 498)
GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 261 of 498) | < Previous Next > |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | Metamath Proof Explorer
(1-30854) |
Hilbert Space Explorer
(30855-32377) |
Users' Mathboxes
(32378-49798) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | deg1n0ima 26001 | Degree image of a set of polynomials which does not include zero. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝑅 ∈ Ring → (𝐷 “ (𝐵 ∖ { 0 })) ⊆ ℕ0) | ||
| Theorem | deg1nn0clb 26002 | A polynomial is nonzero iff it has definite degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → (𝐹 ≠ 0 ↔ (𝐷‘𝐹) ∈ ℕ0)) | ||
| Theorem | deg1lt0 26003 | A polynomial is zero iff it has negative degree. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) < 0 ↔ 𝐹 = 0 )) | ||
| Theorem | deg1ldg 26004 | A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑌 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ≠ 𝑌) | ||
| Theorem | deg1ldgn 26005 | An index at which a polynomial is zero, cannot be its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑌 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ ℕ0) & ⊢ (𝜑 → (𝐴‘𝑋) = 𝑌) ⇒ ⊢ (𝜑 → (𝐷‘𝐹) ≠ 𝑋) | ||
| Theorem | deg1ldgdomn 26006 | A nonzero univariate polynomial over a domain always has a nonzero-divisor leading coefficient. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝑅 ∈ Domn ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐴‘(𝐷‘𝐹)) ∈ 𝐸) | ||
| Theorem | deg1leb 26007* | Property of being of limited degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ*) → ((𝐷‘𝐹) ≤ 𝐺 ↔ ∀𝑥 ∈ ℕ0 (𝐺 < 𝑥 → (𝐴‘𝑥) = 0 ))) | ||
| Theorem | deg1val 26008 | Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐴 supp 0 ), ℝ*, < )) | ||
| Theorem | deg1lt 26009 | If the degree of a univariate polynomial is less than some index, then that coefficient must be zero. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐷‘𝐹) < 𝐺) → (𝐴‘𝐺) = 0 ) | ||
| Theorem | deg1ge 26010 | Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐴 = (coe1‘𝐹) ⇒ ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ (𝐴‘𝐺) ≠ 0 ) → 𝐺 ≤ (𝐷‘𝐹)) | ||
| Theorem | coe1mul3 26011 | The coefficient vector of multiplication in the univariate polynomial ring, at indices high enough that at most one component can be active in the sum. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ ∙ = (.r‘𝑌) & ⊢ · = (.r‘𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐼) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐽) ⇒ ⊢ (𝜑 → ((coe1‘(𝐹 ∙ 𝐺))‘(𝐼 + 𝐽)) = (((coe1‘𝐹)‘𝐼) · ((coe1‘𝐺)‘𝐽))) | ||
| Theorem | coe1mul4 26012 | Value of the "leading" coefficient of a product of two nonzero polynomials. This will fail to actually be the leading coefficient only if it is zero (requiring the basic ring to contain zero divisors). (Contributed by Stefan O'Rear, 25-Mar-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ ∙ = (.r‘𝑌) & ⊢ · = (.r‘𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 0 = (0g‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ≠ 0 ) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) ⇒ ⊢ (𝜑 → ((coe1‘(𝐹 ∙ 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹)) · ((coe1‘𝐺)‘(𝐷‘𝐺)))) | ||
| Theorem | deg1addle 26013 | The degree of a sum is at most the maximum of the degrees of the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) | ||
| Theorem | deg1addle2 26014 | If both factors have degree bounded by 𝐿, then the sum of the polynomials also has degree bounded by 𝐿. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐿 ∈ ℝ*) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐿) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐿) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ 𝐿) | ||
| Theorem | deg1add 26015 | Exact degree of a sum of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → (𝐷‘𝐺) < (𝐷‘𝐹)) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 + 𝐺)) = (𝐷‘𝐹)) | ||
| Theorem | deg1vscale 26016 | The degree of a scalar times a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐷‘𝐺)) | ||
| Theorem | deg1vsca 26017 | The degree of a scalar times a polynomial is exactly the degree of the original polynomial when the scalar is not a zero divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐸) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷‘𝐺)) | ||
| Theorem | deg1invg 26018 | The degree of the negated polynomial is the same as the original. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑁 = (invg‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝑁‘𝐹)) = (𝐷‘𝐹)) | ||
| Theorem | deg1suble 26019 | The degree of a difference of polynomials is bounded by the maximum of degrees. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) ≤ if((𝐷‘𝐹) ≤ (𝐷‘𝐺), (𝐷‘𝐺), (𝐷‘𝐹))) | ||
| Theorem | deg1sub 26020 | Exact degree of a difference of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → (𝐷‘𝐺) < (𝐷‘𝐹)) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) = (𝐷‘𝐹)) | ||
| Theorem | deg1mulle2 26021 | Produce a bound on the product of two univariate polynomials given bounds on the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
| ⊢ 𝑌 = (Poly1‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = (.r‘𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐽) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐾) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾)) | ||
| Theorem | deg1sublt 26022 | Subtraction of two polynomials limited to the same degree with the same leading coefficient gives a polynomial with a smaller degree. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝐿) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝐿) & ⊢ 𝐴 = (coe1‘𝐹) & ⊢ 𝐶 = (coe1‘𝐺) & ⊢ (𝜑 → ((coe1‘𝐹)‘𝐿) = ((coe1‘𝐺)‘𝐿)) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝐿) | ||
| Theorem | deg1le0 26023 | A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵) → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘((coe1‘𝐹)‘0)))) | ||
| Theorem | deg1sclle 26024 | A scalar polynomial has nonpositive degree. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾) → (𝐷‘(𝐴‘𝐹)) ≤ 0) | ||
| Theorem | deg1scl 26025 | A nonzero scalar polynomial has zero degree. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐹 ≠ 0 ) → (𝐷‘(𝐴‘𝐹)) = 0) | ||
| Theorem | deg1mul2 26026 | Degree of multiplication of two nonzero polynomials when the first leads with a nonzero-divisor coefficient. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ≠ 0 ) & ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝐸) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) | ||
| Theorem | deg1mul 26027 | Degree of multiplication of two nonzero polynomials in a domain. (Contributed by metakunt, 6-May-2025.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ≠ 0 ) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺))) | ||
| Theorem | deg1mul3 26028 | Degree of multiplication of a polynomial on the left by a nonzero-dividing scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Jul-2019.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = (𝐷‘𝐺)) | ||
| Theorem | deg1mul3le 26029 | Degree of multiplication of a polynomial on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = (.r‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) ≤ (𝐷‘𝐺)) | ||
| Theorem | deg1tmle 26030 | Limiting degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝑁 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑁) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) ≤ 𝐹) | ||
| Theorem | deg1tm 26031 | Exact degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝑁 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑁) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ∈ 𝐾 ∧ 𝐶 ≠ 0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 ↑ 𝑋))) = 𝐹) | ||
| Theorem | deg1pwle 26032 | Limiting degree of a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑁 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑁) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) ≤ 𝐹) | ||
| Theorem | deg1pw 26033 | Exact degree of a variable power over a nontrivial ring. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
| ⊢ 𝐷 = (deg1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑁 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝑁) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) = 𝐹) | ||
| Theorem | ply1nz 26034 | Univariate polynomials over a nonzero ring are a nonzero ring. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
| ⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing) | ||
| Theorem | ply1nzb 26035 | Univariate polynomials are nonzero iff the base is nonzero. Or in contraposition, the univariate polynomials over the zero ring are also zero. (Contributed by Mario Carneiro, 13-Jun-2015.) |
| ⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ 𝑃 ∈ NzRing)) | ||
| Theorem | ply1domn 26036 | Corollary of deg1mul2 26026: the univariate polynomials over a domain are a domain. This is true for multivariate but with a much more complicated proof. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn → 𝑃 ∈ Domn) | ||
| Theorem | ply1idom 26037 | The ring of univariate polynomials over an integral domain is itself an integral domain. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
| ⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (𝑅 ∈ IDomn → 𝑃 ∈ IDomn) | ||
| Syntax | cmn1 26038 | Monic polynomials. |
| class Monic1p | ||
| Syntax | cuc1p 26039 | Unitic polynomials. |
| class Unic1p | ||
| Syntax | cq1p 26040 | Univariate polynomial quotient. |
| class quot1p | ||
| Syntax | cr1p 26041 | Univariate polynomial remainder. |
| class rem1p | ||
| Syntax | cig1p 26042 | Univariate polynomial ideal generator. |
| class idlGen1p | ||
| Definition | df-mon1 26043* | Define the set of monic univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟))}) | ||
| Definition | df-uc1p 26044* | Define the set of unitic univariate polynomials, as the polynomials with an invertible leading coefficient. This is not a standard concept but is useful to us as the set of polynomials which can be used as the divisor in the polynomial division theorem ply1divalg 26050. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}) | ||
| Definition | df-q1p 26045* | Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 26050. We actually use the reversed version for better harmony with our divisibility df-dvdsr 20273. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ quot1p = (𝑟 ∈ V ↦ ⦋(Poly1‘𝑟) / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 ((deg1‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < ((deg1‘𝑟)‘𝑔)))) | ||
| Definition | df-r1p 26046* | Define the remainder after dividing two univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| ⊢ rem1p = (𝑟 ∈ V ↦ ⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)))) | ||
| Definition | df-ig1p 26047* | Define a choice function for generators of ideals over a division ring; this is the unique monic polynomial of minimal degree in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.) |
| ⊢ idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ if(𝑖 = {(0g‘(Poly1‘𝑟))}, (0g‘(Poly1‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))((deg1‘𝑟)‘𝑔) = inf(((deg1‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < ))))) | ||
| Theorem | ply1divmo 26048* | Uniqueness of a quotient in a polynomial division. For polynomials 𝐹, 𝐺 such that 𝐺 ≠ 0 and the leading coefficient of 𝐺 is not a zero divisor, there is at most one polynomial 𝑞 which satisfies 𝐹 = (𝐺 · 𝑞) + 𝑟 where the degree of 𝑟 is less than the degree of 𝐺. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by NM, 17-Jun-2017.) |
| ⊢ 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)) ∈ 𝐸) & ⊢ 𝐸 = (RLReg‘𝑅) ⇒ ⊢ (𝜑 → ∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) | ||
| Theorem | ply1divex 26049* | Lemma for ply1divalg 26050: existence part. (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 ) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝐾) & ⊢ (𝜑 → (((coe1‘𝐺)‘(𝐷‘𝐺)) · 𝐼) = 1 ) ⇒ ⊢ (𝜑 → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺)) | ||
| Theorem | ply1divalg 26050* | 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 26051* | Reverse the order of multiplication in ply1divalg 26050 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 26052* | 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 26053 | 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 26054* | 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 26055 | 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 26056 | 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 26057 | 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 26058 | 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 26059 | 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 26060 | 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 26061 | 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 26062 | 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 26063 | 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 26064 | 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 26065 | 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 | mon1pid 26066 | Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 1 = (1r‘𝑃) & ⊢ 𝑀 = (Monic1p‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) ⇒ ⊢ (𝑅 ∈ NzRing → ( 1 ∈ 𝑀 ∧ (𝐷‘ 1 ) = 0)) | ||
| Theorem | q1pval 26067* | 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 26068 | 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 26069 | 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 26070 | 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 26071 | 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 26072 | The remainder has a degree less 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 26073 | 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 | r1pid2 26074 | Identity law for polynomial remainder operation: it leaves a polynomial 𝐴 unchanged iff the degree of 𝐴 is less than the degree of the divisor 𝐵. (Contributed by Thierry Arnoux, 2-Apr-2025.) Generalize to domains. (Revised by SN, 21-Jun-2025.) |
| ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ 𝑁 = (Unic1p‘𝑅) & ⊢ 𝐸 = (rem1p‘𝑅) & ⊢ 𝐷 = (deg1‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑁) ⇒ ⊢ (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ (𝐷‘𝐴) < (𝐷‘𝐵))) | ||
| Theorem | dvdsq1p 26075 | 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 26076 | 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 26077 | 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 26078 | The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16520). 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 26079 | 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 26080 | Lemma for fta1g 26082. (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 26081* | Lemma for fta1g 26082. (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 26082 | The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 26997, 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 26083 | Lemma for fta1b 26084. (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 26084* | The assumption that 𝑅 be a domain in fta1g 26082 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 | idomrootle 26085* | No element of an integral domain can have more than 𝑁 𝑁-th roots. (Contributed by Stefan O'Rear, 11-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ↑ = (.g‘(mulGrp‘𝑅)) ⇒ ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (♯‘{𝑦 ∈ 𝐵 ∣ (𝑁 ↑ 𝑦) = 𝑋}) ≤ 𝑁) | ||
| Theorem | drnguc1p 26086 | 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 26087* | 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 26088* | 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 26089 | 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 26090 | 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 26091 | 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 26092 | 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 26093 | 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 26094 | 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 26095 | 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 26096 | Extend class notation to include the set of complex polynomials. |
| class Poly | ||
| Syntax | cidp 26097 | Extend class notation to include the identity polynomial. |
| class Xp | ||
| Syntax | ccoe 26098 | Extend class notation to include the coefficient function on polynomials. |
| class coeff | ||
| Syntax | cdgr 26099 | Extend class notation to include the degree function on polynomials. |
| class deg | ||
| Definition | df-ply 26100* | 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)))}) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |