P. 263 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	r1pid2 26201	Identity law for polynomial remainder operation: it leaves a polynomial 𝐴 unchanged iff the degree of 𝐴 is less than the degree of the divisor 𝐵. (Contributed by Thierry Arnoux, 2-Apr-2025.) Generalize to domains. (Revised by SN, 21-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (Base‘𝑃)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑁)    ⇒   ⊢ (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ (𝐷‘𝐴) < (𝐷‘𝐵)))
Theorem	dvdsq1p 26202	Divisibility in a polynomial ring is witnessed by the quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ ∥ = (∥r‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    &   ⊢ · = (.r‘𝑃)    &   ⊢ 𝑄 = (quot1p‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐺 ∥ 𝐹 ↔ 𝐹 = ((𝐹𝑄𝐺) · 𝐺)))
Theorem	dvdsr1p 26203	Divisibility in a polynomial ring in terms of the remainder. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ ∥ = (∥r‘𝑃)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐸 = (rem1p‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐺 ∥ 𝐹 ↔ (𝐹𝐸𝐺) = 0 ))
Theorem	ply1remlem 26204	A term of the form 𝑥 − 𝑁 is linear, monic, and has exactly one zero. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘𝑁))    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐾)    &   ⊢ 𝑈 = (Monic1p‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝑈 ∧ (𝐷‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ { 0 }) = {𝑁}))
Theorem	ply1rem 26205	The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16580). If a polynomial 𝐹 is divided by the linear factor 𝑥 − 𝐴, the remainder is equal to 𝐹(𝐴), the evaluation of the polynomial at 𝐴 (interpreted as a constant polynomial). (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘𝑁))    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 𝐸 = (rem1p‘𝑅)    ⇒   ⊢ (𝜑 → (𝐹𝐸𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁)))
Theorem	facth1 26206	The factor theorem and its converse. A polynomial 𝐹 has a root at 𝐴 iff 𝐺 = 𝑥 − 𝐴 is a factor of 𝐹. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘𝑁))    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ∥ = (∥r‘𝑃)    ⇒   ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ ((𝑂‘𝐹)‘𝑁) = 0 ))
Theorem	fta1glem1 26207	Lemma for fta1g 26209. (Contributed by Mario Carneiro, 7-Jun-2016.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑊 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘𝑇))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐷‘𝐹) = (𝑁 + 1))    &   ⊢ (𝜑 → 𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}))    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹(quot1p‘𝑅)𝐺)) = 𝑁)
Theorem	fta1glem2 26208*	Lemma for fta1g 26209. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑊 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑋 − (𝐴‘𝑇))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐷‘𝐹) = (𝑁 + 1))    &   ⊢ (𝜑 → 𝑇 ∈ (◡(𝑂‘𝐹) “ {𝑊}))    &   ⊢ (𝜑 → ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑁 → (♯‘(◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))    ⇒   ⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))
Theorem	fta1g 26209	The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 27123, which is only true in ℂ and other algebraically closed fields, this is true in any integral domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑊 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ≠ 0 )    ⇒   ⊢ (𝜑 → (♯‘(◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))
Theorem	fta1blem 26210	Lemma for fta1b 26211. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑊 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐾)    &   ⊢ (𝜑 → (𝑀 × 𝑁) = 𝑊)    &   ⊢ (𝜑 → 𝑀 ≠ 𝑊)    &   ⊢ (𝜑 → ((𝑀 · 𝑋) ∈ (𝐵 ∖ { 0 }) → (♯‘(◡(𝑂‘(𝑀 · 𝑋)) “ {𝑊})) ≤ (𝐷‘(𝑀 · 𝑋))))    ⇒   ⊢ (𝜑 → 𝑁 = 𝑊)
Theorem	fta1b 26211*	The assumption that 𝑅 be a domain in fta1g 26209 is necessary. Here we show that the statement is strong enough to prove that 𝑅 is a domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑂 = (eval1‘𝑅)    &   ⊢ 𝑊 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ NzRing ∧ ∀𝑓 ∈ (𝐵 ∖ { 0 })(♯‘(◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
Theorem	idomrootle 26212*	No element of an integral domain can have more than 𝑁 𝑁-th roots. (Contributed by Stefan O'Rear, 11-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    ⇒   ⊢ ((𝑅 ∈ IDomn ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ) → (♯‘{𝑦 ∈ 𝐵 ∣ (𝑁 ↑ 𝑦) = 𝑋}) ≤ 𝑁)
Theorem	drnguc1p 26213	Over a division ring, all nonzero polynomials are unitic. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → 𝐹 ∈ 𝐶)
Theorem	ig1peu 26214*	There is a unique monic polynomial of minimal degree in any nonzero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ { 0 }) → ∃!𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))
Theorem	ig1pval 26215*	Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (idlGen1p‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) = if(𝐼 = { 0 }, 0 , (℩𝑔 ∈ (𝐼 ∩ 𝑀)(𝐷‘𝑔) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))))
Theorem	ig1pval2 26216	Generator of the zero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Sep-2020.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (idlGen1p‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐺‘{ 0 }) = 0 )
Theorem	ig1pval3 26217	Characterizing properties of the monic generator of a nonzero ideal of polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (idlGen1p‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ { 0 }) → ((𝐺‘𝐼) ∈ 𝐼 ∧ (𝐺‘𝐼) ∈ 𝑀 ∧ (𝐷‘(𝐺‘𝐼)) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )))
Theorem	ig1pcl 26218	The monic generator of an ideal is always in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Sep-2020.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (idlGen1p‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) ∈ 𝐼)
Theorem	ig1pdvds 26219	The monic generator of an ideal divides all elements of the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Sep-2020.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (idlGen1p‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ ∥ = (∥r‘𝑃)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝐺‘𝐼) ∥ 𝑋)
Theorem	ig1prsp 26220	Any ideal of polynomials over a division ring is generated by the ideal's canonical generator. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐺 = (idlGen1p‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 𝐾 = (RSpan‘𝑃)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → 𝐼 = (𝐾‘{(𝐺‘𝐼)}))
Theorem	ply1lpir 26221	The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ LPIR)
Theorem	ply1pid 26222	The polynomials over a field are a PID. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ Field → 𝑃 ∈ PID)
14.1.3  Elementary properties of complex polynomials
Syntax	cply 26223	Extend class notation to include the set of complex polynomials.
class Poly
Syntax	cidp 26224	Extend class notation to include the identity polynomial.
class Xp
Syntax	ccoe 26225	Extend class notation to include the coefficient function on polynomials.
class coeff
Syntax	cdgr 26226	Extend class notation to include the degree function on polynomials.
class deg
Definition	df-ply 26227*	Define the set of polynomials on the complex numbers with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
Definition	df-idp 26228	Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ Xp = ( I ↾ ℂ)
Definition	df-coe 26229*	Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ coeff = (𝑓 ∈ (Poly‘ℂ) ↦ (℩𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
Definition	df-dgr 26230	Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))
Theorem	plyco0 26231*	Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)))
Theorem	plyval 26232*	Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝑆 ⊆ ℂ → (Poly‘𝑆) = {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
Theorem	plybss 26233	Reverse closure of the parameter 𝑆 of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
Theorem	elply 26234*	Definition of a polynomial with coefficients in 𝑆. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
Theorem	elply2 26235*	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 ∈ ℕ0 ∃𝑎 ∈ ((𝑆 ∪ {0}) ↑m ℕ0)((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
Theorem	plyun0 26236	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)
Theorem	plyf 26237	A polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
Theorem	plyss 26238	The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (Poly‘𝑆) ⊆ (Poly‘𝑇))
Theorem	plyssc 26239	Every polynomial ring is contained in the ring of polynomials over ℂ. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
Theorem	elplyr 26240*	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 ∈ ℕ0 ∧ 𝐴:ℕ0⟶𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘𝑆))
Theorem	elplyd 26241*	Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))) ∈ (Poly‘𝑆))
Theorem	ply1termlem 26242*	Lemma for ply1term 26243. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(if(𝑘 = 𝑁, 𝐴, 0) · (𝑧↑𝑘))))
Theorem	ply1term 26243*	A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))    ⇒   ⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆 ∧ 𝑁 ∈ ℕ0) → 𝐹 ∈ (Poly‘𝑆))
Theorem	plypow 26244*	A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆 ∧ 𝑁 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ (𝑧↑𝑁)) ∈ (Poly‘𝑆))
Theorem	plyconst 26245	A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆))
Theorem	ne0p 26246	A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ ((𝐴 ∈ ℂ ∧ (𝐹‘𝐴) ≠ 0) → 𝐹 ≠ 0𝑝)
Theorem	ply0 26247	The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆))
Theorem	plyid 26248	The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆))
Theorem	plyeq0lem 26249*	Lemma for plyeq0 26250. If 𝐴 is the coefficient function for a nonzero polynomial such that 𝑃(𝑧) = Σ𝑘 ∈ ℕ0𝐴(𝑘) · 𝑧↑𝑘 = 0 for every 𝑧 ∈ ℂ and 𝐴(𝑀) is the nonzero leading coefficient, then the function 𝐹(𝑧) = 𝑃(𝑧) / 𝑧↑𝑀 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.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ 𝑀 = sup((◡𝐴 “ (𝑆 ∖ {0})), ℝ, < )    &   ⊢ (𝜑 → (◡𝐴 “ (𝑆 ∖ {0})) ≠ ∅)    ⇒   ⊢ ¬ 𝜑
Theorem	plyeq0 26250*	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 26229 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 0𝑝 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → 𝐴 = (ℕ0 × {0}))
Theorem	plypf1 26251	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 = (ℂfld ↾s 𝑆)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝐸 = (eval1‘ℂfld)    ⇒   ⊢ (𝑆 ∈ (SubRing‘ℂfld) → (Poly‘𝑆) = (𝐸 “ 𝐴))
Theorem	plyaddlem1 26252*	Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ)    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})    &   ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → (𝐹 ∘f + 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...if(𝑀 ≤ 𝑁, 𝑁, 𝑀))(((𝐴 ∘f + 𝐵)‘𝑘) · (𝑧↑𝑘))))
Theorem	plymullem1 26253*	Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → 𝐵:ℕ0⟶ℂ)    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})    &   ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → (𝐹 ∘f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑛 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) · (𝑧↑𝑛))))
Theorem	plyaddlem 26254*	Lemma for plyadd 26256. (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))    &   ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})    &   ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆))
Theorem	plymullem 26255*	Lemma for plymul 26257. (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))    &   ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})    &   ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ (Poly‘𝑆))
Theorem	plyadd 26256*	The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ (Poly‘𝑆))
Theorem	plymul 26257*	The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ (Poly‘𝑆))
Theorem	plysub 26258*	The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 ∘f − 𝐺) ∈ (Poly‘𝑆))
Theorem	plyaddcl 26259	The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f + 𝐺) ∈ (Poly‘ℂ))
Theorem	plymulcl 26260	The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f · 𝐺) ∈ (Poly‘ℂ))
Theorem	plysubcl 26261	The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f − 𝐺) ∈ (Poly‘ℂ))
Theorem	coeval 26262*	Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹) = (℩𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘))))))
Theorem	coeeulem 26263*	Lemma for coeeu 26264. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ (ℂ ↑m ℕ0))    &   ⊢ (𝜑 → 𝐵 ∈ (ℂ ↑m ℕ0))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑀 + 1))) = {0})    &   ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	coeeu 26264*	Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → ∃!𝑎 ∈ (ℂ ↑m ℕ0)∃𝑛 ∈ ℕ0 ((𝑎 “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))
Theorem	coelem 26265*	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 ↑m ℕ0) ∧ ∃𝑛 ∈ ℕ0 (((coeff‘𝐹) “ (ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)(((coeff‘𝐹)‘𝑘) · (𝑧↑𝑘))))))
Theorem	coeeq 26266*	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.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → (coeff‘𝐹) = 𝐴)
Theorem	dgrval 26267	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, < ))
Theorem	dgrlem 26268*	Lemma for dgrcl 26272 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}))𝑥 ≤ 𝑛))
Theorem	coef 26269	The domain and codomain of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ 𝐴 = (coeff‘𝐹)    ⇒   ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
Theorem	coef2 26270	The domain and codomain of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ 𝐴 = (coeff‘𝐹)    ⇒   ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → 𝐴:ℕ0⟶𝑆)
Theorem	coef3 26271	The domain and codomain of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ 𝐴 = (coeff‘𝐹)    ⇒   ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
Theorem	dgrcl 26272	The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
Theorem	dgrub 26273	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 ∧ (𝐴‘𝑀) ≠ 0) → 𝑀 ≤ 𝑁)
Theorem	dgrub2 26274	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})
Theorem	dgrlb 26275	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}) → 𝑁 ≤ 𝑀)
Theorem	coeidlem 26276*	Lemma for coeid 26277. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ 𝐴 = (coeff‘𝐹)    &   ⊢ 𝑁 = (deg‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐵 ∈ ((𝑆 ∪ {0}) ↑m ℕ0))    &   ⊢ (𝜑 → (𝐵 “ (ℤ≥‘(𝑀 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐵‘𝑘) · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))
Theorem	coeid 26277*	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)((𝐴‘𝑘) · (𝑧↑𝑘))))
Theorem	coeid2 26278*	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) → (𝐹‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑋↑𝑘)))
Theorem	coeid3 26279*	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 ∈ (ℤ≥‘𝑁) ∧ 𝑋 ∈ ℂ) → (𝐹‘𝑋) = Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑋↑𝑘)))
Theorem	plyco 26280*	The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘𝑆))
Theorem	coeeq2 26281*	Compute the coefficient function given a sum expression for the polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → (coeff‘𝐹) = (𝑘 ∈ ℕ0 ↦ if(𝑘 ≤ 𝑁, 𝐴, 0)))
Theorem	dgrle 26282*	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.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(𝐴 · (𝑧↑𝑘))))    ⇒   ⊢ (𝜑 → (deg‘𝐹) ≤ 𝑁)
Theorem	dgreq 26283*	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.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})    &   ⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0)    ⇒   ⊢ (𝜑 → (deg‘𝐹) = 𝑁)
Theorem	0dgr 26284	A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)
Theorem	0dgrb 26285	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)})))
Theorem	dgrnznn 26286	A nonzero polynomial with a root has positive degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (((𝑃 ∈ (Poly‘𝑆) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃‘𝐴) = 0)) → (deg‘𝑃) ∈ ℕ)
Theorem	coefv0 26287	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))
Theorem	coeaddlem 26288	Lemma for coeadd 26290 and dgradd 26307. (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‘(𝐹 ∘f + 𝐺)) = (𝐴 ∘f + 𝐵) ∧ (deg‘(𝐹 ∘f + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀)))
Theorem	coemullem 26289*	Lemma for coemul 26291 and dgrmul 26310. (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‘(𝐹 ∘f · 𝐺)) = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) ∧ (deg‘(𝐹 ∘f · 𝐺)) ≤ (𝑀 + 𝑁)))
Theorem	coeadd 26290	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‘(𝐹 ∘f + 𝐺)) = (𝐴 ∘f + 𝐵))
Theorem	coemul 26291*	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‘(𝐹 ∘f · 𝐺))‘𝑁) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝐵‘(𝑁 − 𝑘))))
Theorem	coe11 26292	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 = 𝐺 ↔ 𝐴 = 𝐵))
Theorem	coemulhi 26293	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‘(𝐹 ∘f · 𝐺))‘(𝑀 + 𝑁)) = ((𝐴‘𝑀) · (𝐵‘𝑁)))
Theorem	coemulc 26294	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)))
Theorem	coe0 26295	The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (coeff‘0𝑝) = (ℕ0 × {0})
Theorem	coesub 26296	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‘(𝐹 ∘f − 𝐺)) = (𝐴 ∘f − 𝐵))
Theorem	coe1termlem 26297*	The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((coeff‘𝐹) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 𝑁, 𝐴, 0)) ∧ (𝐴 ≠ 0 → (deg‘𝐹) = 𝑁)))
Theorem	coe1term 26298*	The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → ((coeff‘𝐹)‘𝑀) = if(𝑀 = 𝑁, 𝐴, 0))
Theorem	dgr1term 26299*	The degree of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝐹 = (𝑧 ∈ ℂ ↦ (𝐴 · (𝑧↑𝑁)))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℕ0) → (deg‘𝐹) = 𝑁)
Theorem	plycn 26300	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))