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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremplyssc 24001 Every polynomial ring is contained in the ring of polynomials over . (Contributed by Mario Carneiro, 22-Jul-2014.)
(Poly‘𝑆) ⊆ (Poly‘ℂ)

Theoremelplyr 24002* 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 24003* 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 24004* Lemma for ply1term 24005. (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 24005* A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧𝑁)))       ((𝑆 ⊆ ℂ ∧ 𝐴𝑆𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘𝑆))

Theoremplypow 24006* A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆𝑁 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ (𝑧𝑁)) ∈ (Poly‘𝑆))

Theoremplyconst 24007 A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 𝐴𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆))

Theoremne0p 24008 A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
((𝐴 ∈ ℂ ∧ (𝐹𝐴) ≠ 0) → 𝐹 ≠ 0𝑝)

Theoremply0 24009 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
(𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆))

Theoremplyid 24010 The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆))

Theoremplyeq0lem 24011* Lemma for plyeq0 24012. 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 24012* 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 23991 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 24013 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 24014* 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 24015* 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 24017* Lemma for plymul 24019. (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 24018* 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 24019* 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 24020* 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 24021 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 24022 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 24023 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 24024* 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 24025* Lemma for coeeu 24026. (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 24026* 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 24027* 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 24028* 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 24029 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 24030* Lemma for dgrcl 24034 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 24031 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 24032 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 24033 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 24034 The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)

Theoremdgrub 24035 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 24036 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 24037 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 24038* Lemma for coeid 24039. (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 24039* 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 24040* 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 24041* 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 24042* 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 24043* 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 24044* 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 24045* 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 24046 A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)

Theorem0dgrb 24047 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 24048 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 24049 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 24051* Lemma for coemul 24053 and dgrmul 24071. (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 24052 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 24053* 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 24054 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 24055 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 24056 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 24057 The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
(coeff‘0𝑝) = (ℕ0 × {0})

Theoremcoesub 24058 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 24059* 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 24060* 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 24061* 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 24062 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremdgr0 24063 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 24034, dgreq0 24066 and coeid 24039 without having to special-case zero, although plydivalg 24099 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
(deg‘0𝑝) = 0

Theoremcoeidp 24064 The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝐴 ∈ ℕ0 → ((coeff‘Xp)‘𝐴) = if(𝐴 = 1, 1, 0))

Theoremdgrid 24065 The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
(deg‘Xp) = 1

Theoremdgreq0 24066 The leading coefficient of a polynomial is nonzero, unless the entire polynomial is zero. (Contributed by Mario Carneiro, 22-Jul-2014.) (Proof shortened by Fan Zheng, 21-Jun-2016.)
𝑁 = (deg‘𝐹)    &   𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴𝑁) = 0))

Theoremdgrlt 24067 Two ways to say that the degree of 𝐹 is strictly less than 𝑁. (Contributed by Mario Carneiro, 25-Jul-2014.)
𝑁 = (deg‘𝐹)    &   𝐴 = (coeff‘𝐹)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑀 ∈ ℕ0) → ((𝐹 = 0𝑝𝑁 < 𝑀) ↔ (𝑁𝑀 ∧ (𝐴𝑀) = 0)))

Theoremdgradd 24068 The degree of a sum of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹𝑓 + 𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))

Theoremdgradd2 24069 The degree of a sum of polynomials of unequal degrees is the degree of the larger polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝑀 < 𝑁) → (deg‘(𝐹𝑓 + 𝐺)) = 𝑁)

Theoremdgrmul2 24070 The degree of a product of polynomials is at most the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹𝑓 · 𝐺)) ≤ (𝑀 + 𝑁))

Theoremdgrmul 24071 The degree of a product of nonzero polynomials is the sum of degrees. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) ∧ (𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝)) → (deg‘(𝐹𝑓 · 𝐺)) = (𝑀 + 𝑁))

Theoremdgrmulc 24072 Scalar multiplication by a nonzero constant does not change the degree of a function. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐹 ∈ (Poly‘𝑆)) → (deg‘((ℂ × {𝐴}) ∘𝑓 · 𝐹)) = (deg‘𝐹))

Theoremdgrsub 24073 The degree of a difference of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹𝑓𝐺)) ≤ if(𝑀𝑁, 𝑁, 𝑀))

Theoremdgrcolem1 24074* The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.)
𝑁 = (deg‘𝐺)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐺 ∈ (Poly‘𝑆))       (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺𝑥)↑𝑀))) = (𝑀 · 𝑁))

Theoremdgrcolem2 24075* Lemma for dgrco 24076. (Contributed by Mario Carneiro, 15-Sep-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   𝐴 = (coeff‘𝐹)    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝑀 = (𝐷 + 1))    &   (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝐷 → (deg‘(𝑓𝐺)) = ((deg‘𝑓) · 𝑁)))       (𝜑 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))

Theoremdgrco 24076 The degree of a composition of two polynomials is the product of the degrees. (Contributed by Mario Carneiro, 15-Sep-2014.)
𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))       (𝜑 → (deg‘(𝐹𝐺)) = (𝑀 · 𝑁))

Theoremplycjlem 24077* Lemma for plycj 24078 and coecj 24079. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑁 = (deg‘𝐹)    &   𝐺 = ((∗ ∘ 𝐹) ∘ ∗)    &   𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ 𝐴)‘𝑘) · (𝑧𝑘))))

Theoremplycj 24078* The double conjugation of a polynomial is a polynomial. (The single conjugation is not because our definition of polynomial includes only holomorphic functions, i.e. no dependence on (∗‘𝑧) independently of 𝑧.) (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑁 = (deg‘𝐹)    &   𝐺 = ((∗ ∘ 𝐹) ∘ ∗)    &   ((𝜑𝑥𝑆) → (∗‘𝑥) ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))       (𝜑𝐺 ∈ (Poly‘𝑆))

Theoremcoecj 24079 Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.)
𝑁 = (deg‘𝐹)    &   𝐺 = ((∗ ∘ 𝐹) ∘ ∗)    &   𝐴 = (coeff‘𝐹)       (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴))

Theoremplyrecj 24080 A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹𝐴)) = (𝐹‘(∗‘𝐴)))

Theoremplymul0or 24081 Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹𝑓 · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝𝐺 = 0𝑝)))

Theoremofmulrt 24082 The set of roots of a product is the union of the roots of the terms. (Contributed by Mario Carneiro, 28-Jul-2014.)
((𝐴𝑉𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → ((𝐹𝑓 · 𝐺) “ {0}) = ((𝐹 “ {0}) ∪ (𝐺 “ {0})))

Theoremplyreres 24083 Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐹 ∈ (Poly‘ℝ) → (𝐹 ↾ ℝ):ℝ⟶ℝ)

Theoremdvply1 24084* Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑧𝑘))))    &   (𝜑𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵𝑘) · (𝑧𝑘))))    &   (𝜑𝐴:ℕ0⟶ℂ)    &   𝐵 = (𝑘 ∈ ℕ0 ↦ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (ℂ D 𝐹) = 𝐺)

Theoremdvply2g 24085 The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.)
((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆))

Theoremdvply2 24086 The derivative of a polynomial is a polynomial. (Contributed by Stefan O'Rear, 14-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
(𝐹 ∈ (Poly‘𝑆) → (ℂ D 𝐹) ∈ (Poly‘ℂ))

Theoremdvnply2 24087 Polynomials have polynomials as derivatives of all orders. (Contributed by Mario Carneiro, 1-Jan-2017.)
((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘𝑆))

Theoremdvnply 24088 Polynomials have polynomials as derivatives of all orders. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
((𝐹 ∈ (Poly‘𝑆) ∧ 𝑁 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑁) ∈ (Poly‘ℂ))

Theoremplycpn 24089 Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ran (Cn‘ℂ))

14.1.4  The division algorithm for polynomials

Syntaxcquot 24090 Extend class notation to include the quotient of a polynomial division.
class quot

Definitiondf-quot 24091* Define the quotient function on polynomials. This is the 𝑞 of the expression 𝑓 = 𝑔 · 𝑞 + 𝑟 in the division algorithm. (Contributed by Mario Carneiro, 23-Jul-2014.)
quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (𝑞 ∈ (Poly‘ℂ)[(𝑓𝑓 − (𝑔𝑓 · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))

Theoremquotval 24092* Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))

Theoremplydivlem1 24093* Lemma for plydivalg 24099. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)       (𝜑 → 0 ∈ 𝑆)

Theoremplydivlem2 24094* Lemma for plydivalg 24099. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))       ((𝜑𝑞 ∈ (Poly‘𝑆)) → 𝑅 ∈ (Poly‘𝑆))

Theoremplydivlem3 24095* Lemma for plydivex 24097. Base case of induction. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))    &   (𝜑 → (𝐹 = 0𝑝 ∨ ((deg‘𝐹) − (deg‘𝐺)) < 0))       (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))

Theoremplydivlem4 24096* Lemma for plydivex 24097. Induction step. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑 → (𝑀𝑁) = 𝐷)    &   (𝜑𝐹 ≠ 0𝑝)    &   𝑈 = (𝑓𝑓 − (𝐺𝑓 · 𝑝))    &   𝐻 = (𝑧 ∈ ℂ ↦ (((𝐴𝑀) / (𝐵𝑁)) · (𝑧𝐷)))    &   (𝜑 → ∀𝑓 ∈ (Poly‘𝑆)((𝑓 = 0𝑝 ∨ ((deg‘𝑓) − 𝑁) < 𝐷) → ∃𝑝 ∈ (Poly‘𝑆)(𝑈 = 0𝑝 ∨ (deg‘𝑈) < 𝑁)))    &   𝐴 = (coeff‘𝐹)    &   𝐵 = (coeff‘𝐺)    &   𝑀 = (deg‘𝐹)    &   𝑁 = (deg‘𝐺)       (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < 𝑁))

Theoremplydivex 24097* Lemma for plydivalg 24099. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))       (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))

Theoremplydiveu 24098* Lemma for plydivalg 24099. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))    &   (𝜑𝑞 ∈ (Poly‘𝑆))    &   (𝜑 → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))    &   𝑇 = (𝐹𝑓 − (𝐺𝑓 · 𝑝))    &   (𝜑𝑝 ∈ (Poly‘𝑆))    &   (𝜑 → (𝑇 = 0𝑝 ∨ (deg‘𝑇) < (deg‘𝐺)))       (𝜑𝑝 = 𝑞)

Theoremplydivalg 24099* The division algorithm on polynomials over a subfield 𝑆 of the complex numbers. If 𝐹 and 𝐺 ≠ 0 are polynomials over 𝑆, then there is a unique quotient polynomial 𝑞 such that the remainder 𝐹𝐺 · 𝑞 is either zero or has degree less than 𝐺. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · 𝑞))       (𝜑 → ∃!𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))

Theoremquotlem 24100* Lemma for properties of the polynomial quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))       (𝜑 → ((𝐹 quot 𝐺) ∈ (Poly‘𝑆) ∧ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))

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