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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremr1pid 24801 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 24802 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 24803 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 24804 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 24805 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15898). 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 24806 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 24807 Lemma for fta1g 24809. (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 24808* Lemma for fta1g 24809. (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 24809 The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 25706, 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 24810 Lemma for fta1b 24811. (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 24811* The assumption that 𝑅 be a domain in fta1g 24809 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 24812 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 24813* 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 24814* 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 24815 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 24816 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 24817 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 24818 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 24819 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 24820 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 24821 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 24822 Extend class notation to include the set of complex polynomials.
class Poly

Syntaxcidp 24823 Extend class notation to include the identity polynomial.
class Xp

Syntaxccoe 24824 Extend class notation to include the coefficient function on polynomials.
class coeff

Syntaxcdgr 24825 Extend class notation to include the degree function on polynomials.
class deg

Definitiondf-ply 24826* 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 24827 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Xp = ( I ↾ ℂ)

Definitiondf-coe 24828* 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 24829 Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup(((coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))

Theoremplyco0 24830* 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 24831* 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 24832 Reverse closure of the parameter 𝑆 of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)

Theoremelply 24833* 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 24834* 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 24835 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 24836 The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)

Theoremplyss 24837 The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆𝑇𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))

Theoremplyssc 24838 Every polynomial ring is contained in the ring of polynomials over . (Contributed by Mario Carneiro, 22-Jul-2014.)
(Poly‘𝑆) ⊆ (Poly‘ℂ)

Theoremelplyr 24839* 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 24840* 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 24841* Lemma for ply1term 24842. (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 24842* A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝑆 ⊆ ℂ ∧ 𝐴𝑆𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘𝑆))

Theoremplypow 24843* A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆𝑁 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ (𝑧𝑁)) ∈ (Poly‘𝑆))

Theoremplyconst 24844 A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 𝐴𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆))

Theoremne0p 24845 A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
((𝐴 ∈ ℂ ∧ (𝐹𝐴) ≠ 0) → 𝐹 ≠ 0𝑝)

Theoremply0 24846 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆))

Theoremplyid 24847 The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆))

Theoremplyeq0lem 24848* Lemma for plyeq0 24849. 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 24849* 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 24828 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 24850 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 24851* 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 24852* 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 24854* Lemma for plymul 24856. (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 24855* 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 24856* 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 24857* 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 24858 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 24859 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 24860 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 24861* 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 24862* Lemma for coeeu 24863. (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 24863* 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 24864* 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 24865* 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 24866 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 24867* Lemma for dgrcl 24871 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 24868 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 24869 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 24870 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 24871 The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)

Theoremdgrub 24872 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 24873 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 24874 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 24875* Lemma for coeid 24876. (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 24876* 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 24877* 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 24878* 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)))

Theoremplyco 24879* The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)       (𝜑 → (𝐹𝐺) ∈ (Poly‘𝑆))

Theoremcoeeq2 24880* Compute the coefficient function given a sum expression for the polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧𝑘))))       (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘𝑁, 𝐴, 0)))

Theoremdgrle 24881* Given an explicit expression for a polynomial, the degree is at most the highest term in the sum. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧𝑘))))       (𝜑 → (deg‘𝐹) ≤ 𝑁)

Theoremdgreq 24882* If the highest term in a polynomial expression is nonzero, then the polynomial's degree is completely determined. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴:ℕ0⟶ℂ)    &   (𝜑 → (𝐴 “ (ℤ‘(𝑁 + 1))) = {0})    &   (𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑 → (𝐴𝑁) ≠ 0)       (𝜑 → (deg‘𝐹) = 𝑁)

Theorem0dgr 24883 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)

Theorem0dgrb 24884 A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → ((deg‘𝐹) = 0 ↔ 𝐹 = (ℂ × {(𝐹‘0)})))

Theoremdgrnznn 24885 A nonzero polynomial with a root has positive degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃𝐴) = 0)) → (deg‘𝑃) ∈ ℕ)

Theoremcoefv0 24886 The result of evaluating a polynomial at zero is the constant term. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → (𝐹‘0) = (𝐴‘0))

𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)    &   𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹f + 𝐺)) = (𝐴f + 𝐵) ∧ (deg‘(𝐹f + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀)))

Theoremcoemullem 24888* Lemma for coemul 24890 and dgrmul 24908. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)    &   𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹f · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴𝑘) · (𝐵‘(𝑛𝑘)))) ∧ (deg‘(𝐹f · 𝐺)) ≤ (𝑀 + 𝑁)))

Theoremcoeadd 24889 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f + 𝐺)) = (𝐴f + 𝐵))

Theoremcoemul 24890* A coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → ((coeff‘(𝐹f · 𝐺))‘𝑁) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝐵‘(𝑁𝑘))))

Theoremcoe11 24891 The coefficient function is one-to-one, so if the coefficients are equal then the functions are equal and vice-versa. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 = 𝐺𝐴 = 𝐵))

Theoremcoemulhi 24892 The leading coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)    &   𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹f · 𝐺))‘(𝑀 + 𝑁)) = ((𝐴𝑀) · (𝐵𝑁)))

Theoremcoemulc 24893 The coefficient function is linear under scalar multiplication. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)) = ((ℕ0 × {𝐴}) ∘f · (coeff‘𝐹)))

Theoremcoe0 24894 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
(coeff‘0𝑝) = (ℕ0 × {0})

Theoremcoesub 24895 The coefficient function of a sum is the sum of coefficients. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹f𝐺)) = (𝐴f𝐵))

Theoremcoe1termlem 24896* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) ∧ (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁)))

Theoremcoe1term 24897* The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → ((coeff‘𝐹)‘𝑀) = if(𝑀 = 𝑁, 𝐴, 0))

Theoremdgr1term 24898* The degree of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℕ0) → (deg‘𝐹) = 𝑁)

Theoremplycn 24899 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremdgr0 24900 The degree of the zero polynomial is zero. Note: this differs from some other definitions of the degree of the zero polynomial, such as -1, -∞ or undefined. But it is convenient for us to define it this way, so that we have dgrcl 24871, dgreq0 24903 and coeid 24876 without having to special-case zero, although plydivalg 24936 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
(deg‘0𝑝) = 0

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