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Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremply1divalg 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 = (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 23702* Reverse the order of multiplication in ply1divalg 23701 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 23703* 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 23704 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 23705* 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 23706 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 23707 Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Unic1p𝑅)       (𝐹𝐶𝐹𝐵)

Theoremmon1pcl 23708 Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑀 = (Monic1p𝑅)       (𝐹𝑀𝐹𝐵)

Theoremuc1pn0 23709 Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝐶 = (Unic1p𝑅)       (𝐹𝐶𝐹0 )

Theoremmon1pn0 23710 Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝑀 = (Monic1p𝑅)       (𝐹𝑀𝐹0 )

Theoremuc1pdeg 23711 Unitic polynomials have nonnegative degrees. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐷 = ( deg1𝑅)    &   𝐶 = (Unic1p𝑅)       ((𝑅 ∈ Ring ∧ 𝐹𝐶) → (𝐷𝐹) ∈ ℕ0)

Theoremuc1pldg 23712 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 23713 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 23714 Monic polynomials are unitic. (Contributed by Stefan O'Rear, 28-Mar-2015.)
𝐶 = (Unic1p𝑅)    &   𝑀 = (Monic1p𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑀) → 𝑋𝐶)

Theoremuc1pmon1p 23715 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 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 = ( deg1𝑅)    &   𝑂 = (Monic1p𝑅)    &   𝑃 = (Poly1𝑅)    &    = (-g𝑃)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹𝑂)    &   (𝜑 → (𝐷𝐹) = 𝑋)    &   (𝜑𝐺𝑂)    &   (𝜑 → (𝐷𝐺) = 𝑋)       (𝜑 → (𝐷‘(𝐹 𝐺)) < 𝑋)

Theoremq1pval 23717* 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 23718 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 23719 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 23720 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 23721 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 23722 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 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 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Unic1p𝑅)    &   𝑄 = (quot1p𝑅)    &   𝐸 = (rem1p𝑅)    &    · = (.r𝑃)    &    + = (+g𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶) → 𝐹 = (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺)))

Theoremdvdsq1p 23724 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 23725 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 23726 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 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 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 23728 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 23729 Lemma for fta1g 23731. (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 23730* Lemma for fta1g 23731. (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 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 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 23732 Lemma for fta1b 23733. (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 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 = (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 23734 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 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 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &    0 = (0g𝑃)    &   𝑀 = (Monic1p𝑅)    &   𝐷 = ( deg1𝑅)       ((𝑅 ∈ DivRing ∧ 𝐼𝑈𝐼 ≠ { 0 }) → ∃!𝑔 ∈ (𝐼𝑀)(𝐷𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))

Theoremig1pval 23736* 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 23737 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 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 = (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 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 = (Poly1𝑅)    &   𝐺 = (idlGen1p𝑅)    &   𝑈 = (LIdeal‘𝑃)       ((𝑅 ∈ DivRing ∧ 𝐼𝑈) → (𝐺𝐼) ∈ 𝐼)

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

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

Theoremply1lpir 23742 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 23743 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 23744 Extend class notation to include the set of complex polynomials.
class Poly

Syntaxcidp 23745 Extend class notation to include the identity polynomial.
class Xp

Syntaxccoe 23746 Extend class notation to include the coefficient function on polynomials.
class coeff

Syntaxcdgr 23747 Extend class notation to include the degree function on polynomials.
class deg

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

Definitiondf-idp 23749 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Xp = ( I ↾ ℂ)

Definitiondf-coe 23750* 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 23751 Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup(((coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))

Theoremplyco0 23752* 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 23753* 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 23754 Reverse closure of the parameter 𝑆 of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)

Theoremelply 23755* 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 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 ∈ ℕ0𝑎 ∈ ((𝑆 ∪ {0}) ↑𝑚0)((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))

Theoremplyun0 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)

Theoremplyf 23758 The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)

Theoremplyss 23759 The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆𝑇𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))

Theoremplyssc 23760 Every polynomial ring is contained in the ring of polynomials over . (Contributed by Mario Carneiro, 22-Jul-2014.)
(Poly‘𝑆) ⊆ (Poly‘ℂ)

Theoremelplyr 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 ∈ ℕ0𝐴:ℕ0𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))) ∈ (Poly‘𝑆))

Theoremelplyd 23762* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (0...𝑁)) → 𝐴𝑆)       (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧𝑘))) ∈ (Poly‘𝑆))

Theoremply1termlem 23763* Lemma for ply1term 23764. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧𝑘))))

Theoremply1term 23764* A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝑆 ⊆ ℂ ∧ 𝐴𝑆𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘𝑆))

Theoremplypow 23765* A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆𝑁 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ (𝑧𝑁)) ∈ (Poly‘𝑆))

Theoremplyconst 23766 A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 𝐴𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆))

Theoremne0p 23767 A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
((𝐴 ∈ ℂ ∧ (𝐹𝐴) ≠ 0) → 𝐹 ≠ 0𝑝)

Theoremply0 23768 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆))

Theoremplyid 23769 The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆))

Theoremplyeq0lem 23770* Lemma for plyeq0 23771. If 𝐴 is the coefficient function for a nonzero polynomial such that 𝑃(𝑧) = Σ𝑘 ∈ ℕ0𝐴(𝑘) · 𝑧𝑘 = 0 for every 𝑧 ∈ ℂ and 𝐴(𝑀) is the nonzero leading coefficient, then the function 𝐹(𝑧) = 𝑃(𝑧) / 𝑧𝑀 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.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚0))    &   (𝜑 → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))    &   𝑀 = sup((𝐴 “ (𝑆 ∖ {0})), ℝ, < )    &   (𝜑 → (𝐴 “ (𝑆 ∖ {0})) ≠ ∅)        ¬ 𝜑

Theoremplyeq0 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.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚0))    &   (𝜑 → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))       (𝜑𝐴 = (ℕ0 × {0}))

Theoremplypf1 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 = (ℂflds 𝑆)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (Base‘𝑃)    &   𝐸 = (eval1‘ℂfld)       (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸𝐴))

Theoremplyaddlem1 23773* Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑𝐵:ℕ0⟶ℂ)    &   (𝜑 → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑 → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))       (𝜑 → (𝐹𝑓 + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀𝑁, 𝑁, 𝑀))(((𝐴𝑓 + 𝐵)‘𝑘) · (𝑧𝑘))))

Theoremplymullem1 23774* Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑𝐵:ℕ0⟶ℂ)    &   (𝜑 → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑 → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))       (𝜑 → (𝐹𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘))) · (𝑧𝑛))))

(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚0))    &   (𝜑𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚0))    &   (𝜑 → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑 → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))       (𝜑 → (𝐹𝑓 + 𝐺) ∈ (Poly‘𝑆))

Theoremplymullem 23776* Lemma for plymul 23778. (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ∈ ((𝑆 ∪ {0}) ↑𝑚0))    &   (𝜑𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚0))    &   (𝜑 → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑 → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)       (𝜑 → (𝐹𝑓 · 𝐺) ∈ (Poly‘𝑆))

Theoremplyadd 23777* The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (𝐹𝑓 + 𝐺) ∈ (Poly‘𝑆))

Theoremplymul 23778* The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)       (𝜑 → (𝐹𝑓 · 𝐺) ∈ (Poly‘𝑆))

Theoremplysub 23779* The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)       (𝜑 → (𝐹𝑓𝐺) ∈ (Poly‘𝑆))

Theoremplyaddcl 23780 The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 + 𝐺) ∈ (Poly‘ℂ))

Theoremplymulcl 23781 The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 · 𝐺) ∈ (Poly‘ℂ))

Theoremplysubcl 23782 The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓𝐺) ∈ (Poly‘ℂ))

Theoremcoeval 23783* Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹) = (𝑎 ∈ (ℂ ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘))))))

Theoremcoeeulem 23784* Lemma for coeeu 23785. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐴 ∈ (ℂ ↑𝑚0))    &   (𝜑𝐵 ∈ (ℂ ↑𝑚0))    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝐴 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑 → (𝐵 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵𝑘) · (𝑧𝑘))))       (𝜑𝐴 = 𝐵)

Theoremcoeeu 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 ↑𝑚0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))))

Theoremcoelem 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 ↑𝑚0) ∧ ∃𝑛 ∈ ℕ0 (((coeff‘𝐹) “ (ℤ‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)(((coeff‘𝐹)‘𝑘) · (𝑧𝑘))))))

Theoremcoeeq 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.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))       (𝜑 → (coeff‘𝐹) = 𝐴)

Theoremdgrval 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, < ))

Theoremdgrlem 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}))𝑥𝑛))

Theoremcoef 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}))

Theoremcoef2 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:ℕ0𝑆)

Theoremcoef3 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)

Theoremdgrcl 23793 The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)

Theoremdgrub 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 ∧ (𝐴𝑀) ≠ 0) → 𝑀𝑁)

Theoremdgrub2 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})

Theoremdgrlb 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}) → 𝑁𝑀)

Theoremcoeidlem 23797* Lemma for coeid 23798. (Contributed by Mario Carneiro, 22-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝐵 ∈ ((𝑆 ∪ {0}) ↑𝑚0))    &   (𝜑 → (𝐵 “ (ℤ‘(𝑀 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐵𝑘) · (𝑧𝑘))))       (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))

Theoremcoeid 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)((𝐴𝑘) · (𝑧𝑘))))

Theoremcoeid2 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) → (𝐹𝑋) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑋𝑘)))

Theoremcoeid3 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 ∈ (ℤ𝑁) ∧ 𝑋 ∈ ℂ) → (𝐹𝑋) = Σ𝑘 ∈ (0...𝑀)((𝐴𝑘) · (𝑋𝑘)))

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