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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdeg1val 23901 Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       (𝐹𝐵 → (𝐷𝐹) = sup((𝐴 supp 0 ), ℝ*, < ))

Theoremdeg1lt 23902 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 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       ((𝐹𝐵𝐺 ∈ ℕ0 ∧ (𝐷𝐹) < 𝐺) → (𝐴𝐺) = 0 )

Theoremdeg1ge 23903 Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑅)    &   𝐴 = (coe1𝐹)       ((𝐹𝐵𝐺 ∈ ℕ0 ∧ (𝐴𝐺) ≠ 0 ) → 𝐺 ≤ (𝐷𝐹))

Theoremcoe1mul3 23904 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 = (Poly1𝑅)    &    = (.r𝑌)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑌)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) ≤ 𝐼)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑 → (𝐷𝐺) ≤ 𝐽)       (𝜑 → ((coe1‘(𝐹 𝐺))‘(𝐼 + 𝐽)) = (((coe1𝐹)‘𝐼) · ((coe1𝐺)‘𝐽)))

Theoremcoe1mul4 23905 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 = (Poly1𝑅)    &    = (.r𝑌)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑌)    &   𝐷 = ( deg1𝑅)    &    0 = (0g𝑌)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐹0 )    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )       (𝜑 → ((coe1‘(𝐹 𝐺))‘((𝐷𝐹) + (𝐷𝐺))) = (((coe1𝐹)‘(𝐷𝐹)) · ((coe1𝐺)‘(𝐷𝐺))))

Theoremdeg1addle 23906 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ if((𝐷𝐹) ≤ (𝐷𝐺), (𝐷𝐺), (𝐷𝐹)))

Theoremdeg1addle2 23907 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐿 ∈ ℝ*)    &   (𝜑 → (𝐷𝐹) ≤ 𝐿)    &   (𝜑 → (𝐷𝐺) ≤ 𝐿)       (𝜑 → (𝐷‘(𝐹 + 𝐺)) ≤ 𝐿)

Theoremdeg1add 23908 Exact degree of a sum of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑 → (𝐷𝐺) < (𝐷𝐹))       (𝜑 → (𝐷‘(𝐹 + 𝐺)) = (𝐷𝐹))

Theoremdeg1vscale 23909 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝑌)    &   (𝜑𝐹𝐾)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐷𝐺))

Theoremdeg1vsca 23910 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝐸 = (RLReg‘𝑅)    &    · = ( ·𝑠𝑌)    &   (𝜑𝐹𝐸)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷𝐺))

Theoremdeg1invg 23911 The degree of the negated polynomial is the same as the original. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &   𝑁 = (invg𝑌)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐷‘(𝑁𝐹)) = (𝐷𝐹))

Theoremdeg1suble 23912 The degree of a difference of polynomials is bounded by the maximum of degrees. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    = (-g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐷‘(𝐹 𝐺)) ≤ if((𝐷𝐹) ≤ (𝐷𝐺), (𝐷𝐺), (𝐷𝐹)))

Theoremdeg1sub 23913 Exact degree of a difference of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    = (-g𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑 → (𝐷𝐺) < (𝐷𝐹))       (𝜑 → (𝐷‘(𝐹 𝐺)) = (𝐷𝐹))

Theoremdeg1mulle2 23914 Produce a bound on the product of two univariate polynomials given bounds on the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.)
𝑌 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    · = (.r𝑌)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) ≤ 𝐽)    &   (𝜑 → (𝐷𝐺) ≤ 𝐾)       (𝜑 → (𝐷‘(𝐹 · 𝐺)) ≤ (𝐽 + 𝐾))

Theoremdeg1sublt 23915 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 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    = (-g𝑃)    &   (𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑 → (𝐷𝐹) ≤ 𝐿)    &   (𝜑𝐺𝐵)    &   (𝜑 → (𝐷𝐺) ≤ 𝐿)    &   𝐴 = (coe1𝐹)    &   𝐶 = (coe1𝐺)    &   (𝜑 → ((coe1𝐹)‘𝐿) = ((coe1𝐺)‘𝐿))       (𝜑 → (𝐷‘(𝐹 𝐺)) < 𝐿)

Theoremdeg1le0 23916 A polynomial has nonpositive degree iff it is a constant. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵) → ((𝐷𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘((coe1𝐹)‘0))))

Theoremdeg1sclle 23917 A scalar polynomial has nonpositive degree. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐾) → (𝐷‘(𝐴𝐹)) ≤ 0)

Theoremdeg1scl 23918 A nonzero scalar polynomial has zero degree. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐾𝐹0 ) → (𝐷‘(𝐴𝐹)) = 0)

Theoremdeg1mul2 23919 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 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑃)    &    0 = (0g𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐹0 )    &   (𝜑 → ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝐸)    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )       (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷𝐹) + (𝐷𝐺)))

Theoremdeg1mul3 23920 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 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑃)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐸𝐺𝐵) → (𝐷‘((𝐴𝐹) · 𝐺)) = (𝐷𝐺))

Theoremdeg1mul3le 23921 Degree of multiplication of a polynomial on the left by a scalar. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r𝑃)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐾𝐺𝐵) → (𝐷‘((𝐴𝐹) · 𝐺)) ≤ (𝐷𝐺))

Theoremdeg1tmle 23922 Limiting degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 𝑋))) ≤ 𝐹)

Theoremdeg1tm 23923 Exact degree of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ (𝐶𝐾𝐶0 ) ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐶 · (𝐹 𝑋))) = 𝐹)

Theoremdeg1pwle 23924 Limiting degree of a variable power. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       ((𝑅 ∈ Ring ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 𝑋)) ≤ 𝐹)

Theoremdeg1pw 23925 Exact degree of a variable power over a nontrivial ring. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝐷 = ( deg1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑋 = (var1𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)       ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 𝑋)) = 𝐹)

Theoremply1nz 23926 Univariate polynomials over a nonzero ring are a nonzero ring. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ NzRing → 𝑃 ∈ NzRing)

Theoremply1nzb 23927 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 = (Poly1𝑅)       (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ 𝑃 ∈ NzRing))

Theoremply1domn 23928 Corollary of deg1mul2 23919: 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 = (Poly1𝑅)       (𝑅 ∈ Domn → 𝑃 ∈ Domn)

Theoremply1idom 23929 The ring of univariate polynomials over an integral domain is itself an integral domain. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ IDomn → 𝑃 ∈ IDomn)

14.1.2  The division algorithm for univariate polynomials

Syntaxcmn1 23930 Monic polynomials.
class Monic1p

Syntaxcuc1p 23931 Unitic polynomials.
class Unic1p

Syntaxcq1p 23932 Univariate polynomial quotient.
class quot1p

Syntaxcr1p 23933 Univariate polynomial remainder.
class rem1p

Syntaxcig1p 23934 Univariate polynomial ideal generator.
class idlGen1p

Definitiondf-mon1 23935* Define the set of monic univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟))})

Definitiondf-uc1p 23936* 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 23942. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})

Definitiondf-q1p 23937* Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 23942. We actually use the reversed version for better harmony with our divisibility df-dvdsr 18687. (Contributed by Stefan O'Rear, 28-Mar-2015.)
quot1p = (𝑟 ∈ V ↦ (Poly1𝑟) / 𝑝(Base‘𝑝) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑞𝑏 (( deg1𝑟)‘(𝑓(-g𝑝)(𝑞(.r𝑝)𝑔))) < (( deg1𝑟)‘𝑔))))

Definitiondf-r1p 23938* Define the remainder after dividing two univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
rem1p = (𝑟 ∈ V ↦ (Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))))

Definitiondf-ig1p 23939* 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‘(Poly1𝑟)) ↦ if(𝑖 = {(0g‘(Poly1𝑟))}, (0g‘(Poly1𝑟)), (𝑔 ∈ (𝑖 ∩ (Monic1p𝑟))(( deg1𝑟)‘𝑔) = inf((( deg1𝑟) “ (𝑖 ∖ {(0g‘(Poly1𝑟))})), ℝ, < )))))

Theoremply1divmo 23940* 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   𝐵 = (Base‘𝑃)    &    = (-g𝑃)    &    0 = (0g𝑃)    &    = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )    &   (𝜑 → ((coe1𝐺)‘(𝐷𝐺)) ∈ 𝐸)    &   𝐸 = (RLReg‘𝑅)       (𝜑 → ∃*𝑞𝐵 (𝐷‘(𝐹 (𝐺 𝑞))) < (𝐷𝐺))

Theoremply1divex 23941* Lemma for ply1divalg 23942: existence part. (Contributed by Stefan O'Rear, 27-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   𝐵 = (Base‘𝑃)    &    = (-g𝑃)    &    0 = (0g𝑃)    &    = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )    &    1 = (1r𝑅)    &   𝐾 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝐼𝐾)    &   (𝜑 → (((coe1𝐺)‘(𝐷𝐺)) · 𝐼) = 1 )       (𝜑 → ∃𝑞𝐵 (𝐷‘(𝐹 (𝐺 𝑞))) < (𝐷𝐺))

Theoremply1divalg 23942* 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 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   𝐵 = (Base‘𝑃)    &    = (-g𝑃)    &    0 = (0g𝑃)    &    = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )    &   (𝜑 → ((coe1𝐺)‘(𝐷𝐺)) ∈ 𝑈)    &   𝑈 = (Unit‘𝑅)       (𝜑 → ∃!𝑞𝐵 (𝐷‘(𝐹 (𝐺 𝑞))) < (𝐷𝐺))

Theoremply1divalg2 23943* Reverse the order of multiplication in ply1divalg 23942 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐷 = ( deg1𝑅)    &   𝐵 = (Base‘𝑃)    &    = (-g𝑃)    &    0 = (0g𝑃)    &    = (.r𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &   (𝜑𝐺0 )    &   (𝜑 → ((coe1𝐺)‘(𝐷𝐺)) ∈ 𝑈)    &   𝑈 = (Unit‘𝑅)       (𝜑 → ∃!𝑞𝐵 (𝐷‘(𝐹 (𝑞 𝐺))) < (𝐷𝐺))

Theoremuc1pval 23944* Value of the set of unitic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑃)    &   𝐷 = ( deg1𝑅)    &   𝐶 = (Unic1p𝑅)    &   𝑈 = (Unit‘𝑅)       𝐶 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}

Theoremisuc1p 23945 Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑃)    &   𝐷 = ( deg1𝑅)    &   𝐶 = (Unic1p𝑅)    &   𝑈 = (Unit‘𝑅)       (𝐹𝐶 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈))

Theoremmon1pval 23946* Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑀 = (Monic1p𝑅)    &    1 = (1r𝑅)       𝑀 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}

Theoremismon1p 23947 Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑀 = (Monic1p𝑅)    &    1 = (1r𝑅)       (𝐹𝑀 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 ))

Theoremuc1pcl 23948 Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Unic1p𝑅)       (𝐹𝐶𝐹𝐵)

Theoremmon1pcl 23949 Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑀 = (Monic1p𝑅)       (𝐹𝑀𝐹𝐵)

Theoremuc1pn0 23950 Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐶 = (Unic1p𝑅)       (𝐹𝐶𝐹0 )

Theoremmon1pn0 23951 Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝑀 = (Monic1p𝑅)       (𝐹𝑀𝐹0 )

Theoremuc1pdeg 23952 Unitic polynomials have nonnegative degrees. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝐶 = (Unic1p𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐶) → (𝐷𝐹) ∈ ℕ0)

Theoremuc1pldg 23953 Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐶 = (Unic1p𝑅)       (𝐹𝐶 → ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)

Theoremmon1pldg 23954 Unitic polynomials have one leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐷 = ( deg1𝑅)    &    1 = (1r𝑅)    &   𝑀 = (Monic1p𝑅)       (𝐹𝑀 → ((coe1𝐹)‘(𝐷𝐹)) = 1 )

Theoremmon1puc1p 23955 Monic polynomials are unitic. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐶 = (Unic1p𝑅)    &   𝑀 = (Monic1p𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑀) → 𝑋𝐶)

Theoremuc1pmon1p 23956 Make a unitic polynomial monic by multiplying a factor to normalize the leading coefficient. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐶 = (Unic1p𝑅)    &   𝑀 = (Monic1p𝑅)    &   𝑃 = (Poly1𝑅)    &    · = (.r𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐷 = ( deg1𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐶) → ((𝐴‘(𝐼‘((coe1𝑋)‘(𝐷𝑋)))) · 𝑋) ∈ 𝑀)

Theoremdeg1submon1p 23957 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 = ( deg1𝑅)    &   𝑂 = (Monic1p𝑅)    &   𝑃 = (Poly1𝑅)    &    = (-g𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝑂)    &   (𝜑 → (𝐷𝐹) = 𝑋)    &   (𝜑𝐺𝑂)    &   (𝜑 → (𝐷𝐺) = 𝑋)       (𝜑 → (𝐷‘(𝐹 𝐺)) < 𝑋)

Theoremq1pval 23958* Value of the univariate polynomial quotient function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑄 = (quot1p𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐷 = ( deg1𝑅)    &    = (-g𝑃)    &    · = (.r𝑃)       ((𝐹𝐵𝐺𝐵) → (𝐹𝑄𝐺) = (𝑞𝐵 (𝐷‘(𝐹 (𝑞 · 𝐺))) < (𝐷𝐺)))

Theoremq1peqb 23959 Characterizing property of the polynomial quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑄 = (quot1p𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐷 = ( deg1𝑅)    &    = (-g𝑃)    &    · = (.r𝑃)    &   𝐶 = (Unic1p𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶) → ((𝑋𝐵 ∧ (𝐷‘(𝐹 (𝑋 · 𝐺))) < (𝐷𝐺)) ↔ (𝐹𝑄𝐺) = 𝑋))

Theoremq1pcl 23960 Closure of the quotient by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑄 = (quot1p𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Unic1p𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶) → (𝐹𝑄𝐺) ∈ 𝐵)

Theoremr1pval 23961 Value of the polynomial remainder function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐸 = (rem1p𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑄 = (quot1p𝑅)    &    · = (.r𝑃)    &    = (-g𝑃)       ((𝐹𝐵𝐺𝐵) → (𝐹𝐸𝐺) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))

Theoremr1pcl 23962 Closure of remainder following division by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐸 = (rem1p𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Unic1p𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶) → (𝐹𝐸𝐺) ∈ 𝐵)

Theoremr1pdeglt 23963 The remainder has a degree smaller than the divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐸 = (rem1p𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Unic1p𝑅)    &   𝐷 = ( deg1𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶) → (𝐷‘(𝐹𝐸𝐺)) < (𝐷𝐺))

Theoremr1pid 23964 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 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Unic1p𝑅)    &   𝑄 = (quot1p𝑅)    &   𝐸 = (rem1p𝑅)    &    · = (.r𝑃)    &    + = (+g𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶) → 𝐹 = (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺)))

Theoremdvdsq1p 23965 Divisibility in a polynomial ring is witnessed by the quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &    = (∥r𝑃)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Unic1p𝑅)    &    · = (.r𝑃)    &   𝑄 = (quot1p𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶) → (𝐺 𝐹𝐹 = ((𝐹𝑄𝐺) · 𝐺)))

Theoremdvdsr1p 23966 Divisibility in a polynomial ring in terms of the remainder. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &    = (∥r𝑃)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Unic1p𝑅)    &    0 = (0g𝑃)    &   𝐸 = (rem1p𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶) → (𝐺 𝐹 ↔ (𝐹𝐸𝐺) = 0 ))

Theoremply1remlem 23967 A term of the form 𝑥𝑁 is linear, monic, and has exactly one zero. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝑁))    &   𝑂 = (eval1𝑅)    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁𝐾)    &   𝑈 = (Monic1p𝑅)    &   𝐷 = ( deg1𝑅)    &    0 = (0g𝑅)       (𝜑 → (𝐺𝑈 ∧ (𝐷𝐺) = 1 ∧ ((𝑂𝐺) “ { 0 }) = {𝑁}))

Theoremply1rem 23968 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15307). If a polynomial 𝐹 is divided by the linear factor 𝑥𝐴, 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 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝑁))    &   𝑂 = (eval1𝑅)    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁𝐾)    &   (𝜑𝐹𝐵)    &   𝐸 = (rem1p𝑅)       (𝜑 → (𝐹𝐸𝐺) = (𝐴‘((𝑂𝐹)‘𝑁)))

Theoremfacth1 23969 The factor theorem and its converse. A polynomial 𝐹 has a root at 𝐴 iff 𝐺 = 𝑥𝐴 is a factor of 𝐹. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝑁))    &   𝑂 = (eval1𝑅)    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁𝐾)    &   (𝜑𝐹𝐵)    &    0 = (0g𝑅)    &    = (∥r𝑃)       (𝜑 → (𝐺 𝐹 ↔ ((𝑂𝐹)‘𝑁) = 0 ))

Theoremfta1glem1 23970 Lemma for fta1g 23972. (Contributed by Mario Carneiro, 7-Jun-2016.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑂 = (eval1𝑅)    &   𝑊 = (0g𝑅)    &    0 = (0g𝑃)    &   (𝜑𝑅 ∈ IDomn)    &   (𝜑𝐹𝐵)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝑇))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) = (𝑁 + 1))    &   (𝜑𝑇 ∈ ((𝑂𝐹) “ {𝑊}))       (𝜑 → (𝐷‘(𝐹(quot1p𝑅)𝐺)) = 𝑁)

Theoremfta1glem2 23971* Lemma for fta1g 23972. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑂 = (eval1𝑅)    &   𝑊 = (0g𝑅)    &    0 = (0g𝑃)    &   (𝜑𝑅 ∈ IDomn)    &   (𝜑𝐹𝐵)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝑇))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝐷𝐹) = (𝑁 + 1))    &   (𝜑𝑇 ∈ ((𝑂𝐹) “ {𝑊}))    &   (𝜑 → ∀𝑔𝐵 ((𝐷𝑔) = 𝑁 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))       (𝜑 → (#‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))

Theoremfta1g 23972 The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 24851, which is only true in and other algebraically closed fields, this is true in any integral domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑂 = (eval1𝑅)    &   𝑊 = (0g𝑅)    &    0 = (0g𝑃)    &   (𝜑𝑅 ∈ IDomn)    &   (𝜑𝐹𝐵)    &   (𝜑𝐹0 )       (𝜑 → (#‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))

Theoremfta1blem 23973 Lemma for fta1b 23974. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑂 = (eval1𝑅)    &   𝑊 = (0g𝑅)    &    0 = (0g𝑃)    &   𝐾 = (Base‘𝑅)    &    × = (.r𝑅)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑀𝐾)    &   (𝜑𝑁𝐾)    &   (𝜑 → (𝑀 × 𝑁) = 𝑊)    &   (𝜑𝑀𝑊)    &   (𝜑 → ((𝑀 · 𝑋) ∈ (𝐵 ∖ { 0 }) → (#‘((𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ (𝐷‘(𝑀 · 𝑋))))       (𝜑𝑁 = 𝑊)

Theoremfta1b 23974* The assumption that 𝑅 be a domain in fta1g 23972 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 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑂 = (eval1𝑅)    &   𝑊 = (0g𝑅)    &    0 = (0g𝑃)       (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ NzRing ∧ ∀𝑓 ∈ (𝐵 ∖ { 0 })(#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))

Theoremdrnguc1p 23975 Over a division ring, all nonzero polynomials are unitic. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g𝑃)    &   𝐶 = (Unic1p𝑅)       ((𝑅 ∈ DivRing ∧ 𝐹𝐵𝐹0 ) → 𝐹𝐶)

Theoremig1peu 23976* 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 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &    0 = (0g𝑃)    &   𝑀 = (Monic1p𝑅)    &   𝐷 = ( deg1𝑅)       ((𝑅 ∈ DivRing ∧ 𝐼𝑈𝐼 ≠ { 0 }) → ∃!𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))

Theoremig1pval 23977* Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)
𝑃 = (Poly1𝑅)    &   𝐺 = (idlGen1p𝑅)    &    0 = (0g𝑃)    &   𝑈 = (LIdeal‘𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑀 = (Monic1p𝑅)       ((𝑅𝑉𝐼𝑈) → (𝐺𝐼) = if(𝐼 = { 0 }, 0 , (𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))

Theoremig1pval2 23978 Generator of the zero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Sep-2020.)
𝑃 = (Poly1𝑅)    &   𝐺 = (idlGen1p𝑅)    &    0 = (0g𝑃)       (𝑅 ∈ Ring → (𝐺‘{ 0 }) = 0 )

Theoremig1pval3 23979 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 = (Poly1𝑅)    &   𝐺 = (idlGen1p𝑅)    &    0 = (0g𝑃)    &   𝑈 = (LIdeal‘𝑃)    &   𝐷 = ( deg1𝑅)    &   𝑀 = (Monic1p𝑅)       ((𝑅 ∈ DivRing ∧ 𝐼𝑈𝐼 ≠ { 0 }) → ((𝐺𝐼) ∈ 𝐼 ∧ (𝐺𝐼) ∈ 𝑀 ∧ (𝐷‘(𝐺𝐼)) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))

Theoremig1pcl 23980 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 = (Poly1𝑅)    &   𝐺 = (idlGen1p𝑅)    &   𝑈 = (LIdeal‘𝑃)       ((𝑅 ∈ DivRing ∧ 𝐼𝑈) → (𝐺𝐼) ∈ 𝐼)

Theoremig1pdvds 23981 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 = (Poly1𝑅)    &   𝐺 = (idlGen1p𝑅)    &   𝑈 = (LIdeal‘𝑃)    &    = (∥r𝑃)       ((𝑅 ∈ DivRing ∧ 𝐼𝑈𝑋𝐼) → (𝐺𝐼) 𝑋)

Theoremig1prsp 23982 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 = (Poly1𝑅)    &   𝐺 = (idlGen1p𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝐾 = (RSpan‘𝑃)       ((𝑅 ∈ DivRing ∧ 𝐼𝑈) → 𝐼 = (𝐾‘{(𝐺𝐼)}))

Theoremply1lpir 23983 The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ DivRing → 𝑃 ∈ LPIR)

Theoremply1pid 23984 The polynomials over a field are a PID. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ Field → 𝑃 ∈ PID)

14.1.3  Elementary properties of complex polynomials

Syntaxcply 23985 Extend class notation to include the set of complex polynomials.
class Poly

Syntaxcidp 23986 Extend class notation to include the identity polynomial.
class Xp

Syntaxccoe 23987 Extend class notation to include the coefficient function on polynomials.
class coeff

Syntaxcdgr 23988 Extend class notation to include the degree function on polynomials.
class deg

Definitiondf-ply 23989* 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 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})

Definitiondf-idp 23990 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Xp = ( I ↾ ℂ)

Definitiondf-coe 23991* Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff = (𝑓 ∈ (Poly‘ℂ) ↦ (𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))

Definitiondf-dgr 23992 Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup(((coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))

Theoremplyco0 23993* Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)
((𝑁 ∈ ℕ0𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴𝑘) ≠ 0 → 𝑘𝑁)))

Theoremplyval 23994* Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})

Theoremplybss 23995 Reverse closure of the parameter 𝑆 of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)

Theoremelply 23996* Definition of a polynomial with coefficients in 𝑆. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))

Theoremelply2 23997* 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 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))

Theoremplyun0 23998 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)

Theoremplyf 23999 The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)

Theoremplyss 24000 The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆𝑇𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))

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