P. 258 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-ply 25701*	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 25702	Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ Xp = ( I ↾ ℂ)
Definition	df-coe 25703*	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 25704	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 25705*	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 25706*	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 25707	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 25708*	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 25709*	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 25710	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 25711	The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
Theorem	plyss 25712	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 25713	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 25714*	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 25715*	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 25716*	Lemma for ply1term 25717. (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 25717*	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 25718*	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 25719	A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆))
Theorem	ne0p 25720	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 25721	The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆))
Theorem	plyid 25722	The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆))
Theorem	plyeq0lem 25723*	Lemma for plyeq0 25724. 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 25724*	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 25703 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 25725	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 25726*	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 25727*	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 25728*	Lemma for plyadd 25730. (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 25729*	Lemma for plymul 25731. (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 25730*	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 25731*	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 25732*	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 25733	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 25734	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 25735	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 25736*	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 25737*	Lemma for coeeu 25738. (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 25738*	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 25739*	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 25740*	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 25741	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 25742*	Lemma for dgrcl 25746 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 25743	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 25744	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 25745	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 25746	The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
Theorem	dgrub 25747	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 25748	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 25749	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 25750*	Lemma for coeid 25751. (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 25751*	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 25752*	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 25753*	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 25754*	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 25755*	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 25756*	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 25757*	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 25758	A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)
Theorem	0dgrb 25759	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 25760	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 25761	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 25762	Lemma for coeadd 25764 and dgradd 25780. (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 25763*	Lemma for coemul 25765 and dgrmul 25783. (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 25764	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 25765*	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 25766	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 25767	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 25768	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 25769	The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (coeff‘0𝑝) = (ℕ0 × {0})
Theorem	coesub 25770	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 25771*	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 25772*	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 25773*	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 25774	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	dgr0 25775	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 25746, dgreq0 25778 and coeid 25751 without having to special-case zero, although plydivalg 25811 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (deg‘0𝑝) = 0
Theorem	coeidp 25776	The coefficients of the identity function. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝐴 ∈ ℕ0 → ((coeff‘Xp)‘𝐴) = if(𝐴 = 1, 1, 0))
Theorem	dgrid 25777	The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ (deg‘Xp) = 1
Theorem	dgreq0 25778	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 ↔ (𝐴‘𝑁) = 0))
Theorem	dgrlt 25779	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 = 0𝑝 ∨ 𝑁 < 𝑀) ↔ (𝑁 ≤ 𝑀 ∧ (𝐴‘𝑀) = 0)))
Theorem	dgradd 25780	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‘(𝐹 ∘f + 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
Theorem	dgradd2 25781	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‘(𝐹 ∘f + 𝐺)) = 𝑁)
Theorem	dgrmul2 25782	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‘(𝐹 ∘f · 𝐺)) ≤ (𝑀 + 𝑁))
Theorem	dgrmul 25783	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‘(𝐹 ∘f · 𝐺)) = (𝑀 + 𝑁))
Theorem	dgrmulc 25784	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))
Theorem	dgrsub 25785	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‘(𝐹 ∘f − 𝐺)) ≤ if(𝑀 ≤ 𝑁, 𝑁, 𝑀))
Theorem	dgrcolem1 25786*	The degree of a composition of a monomial with a polynomial. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ 𝑁 = (deg‘𝐺)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    ⇒   ⊢ (𝜑 → (deg‘(𝑥 ∈ ℂ ↦ ((𝐺‘𝑥)↑𝑀))) = (𝑀 · 𝑁))
Theorem	dgrcolem2 25787*	Lemma for dgrco 25788. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ 𝑀 = (deg‘𝐹)    &   ⊢ 𝑁 = (deg‘𝐺)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ 𝐴 = (coeff‘𝐹)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 = (𝐷 + 1))    &   ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((deg‘𝑓) ≤ 𝐷 → (deg‘(𝑓 ∘ 𝐺)) = ((deg‘𝑓) · 𝑁)))    ⇒   ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))
Theorem	dgrco 25788	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)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    ⇒   ⊢ (𝜑 → (deg‘(𝐹 ∘ 𝐺)) = (𝑀 · 𝑁))
Theorem	plycjlem 25789*	Lemma for plycj 25790 and coecj 25791. (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) · (𝑧↑𝑘))))
Theorem	plycj 25790*	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) ∘ ∗)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (∗‘𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
Theorem	coecj 25791	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))
Theorem	plyrecj 25792	A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴)))
Theorem	plymul0or 25793	Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝐹 ∘f · 𝐺) = 0𝑝 ↔ (𝐹 = 0𝑝 ∨ 𝐺 = 0𝑝)))
Theorem	ofmulrt 25794	The set of roots of a product is the union of the roots of the terms. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ) → (◡(𝐹 ∘f · 𝐺) “ {0}) = ((◡𝐹 “ {0}) ∪ (◡𝐺 “ {0})))
Theorem	plyreres 25795	Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ↾ ℝ):ℝ⟶ℝ)
Theorem	dvply1 25796*	Derivative of a polynomial, explicit sum version. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘))))    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)    &   ⊢ 𝐵 = (𝑘 ∈ ℕ0 ↦ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (ℂ D 𝐹) = 𝐺)
Theorem	dvply2g 25797	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))
Theorem	dvply2 25798	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))
Theorem	dvnply2 25799	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))
Theorem	dvnply 25800	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))