P. 253 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-uc1p 25201*	Define the set of unitic univariate polynomials, as the polynomials with an invertible leading coefficient. This is not a standard concept but is useful to us as the set of polynomials which can be used as the divisor in the polynomial division theorem ply1divalg 25207. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
Definition	df-q1p 25202*	Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 25207. We actually use the reversed version for better harmony with our divisibility df-dvdsr 19798. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ quot1p = (𝑟 ∈ V ↦ ⦋(Poly1‘𝑟) / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))))
Definition	df-r1p 25203*	Define the remainder after dividing two univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ rem1p = (𝑟 ∈ V ↦ ⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))))
Definition	df-ig1p 25204*	Define a choice function for generators of ideals over a division ring; this is the unique monic polynomial of minimal degree in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Sep-2020.)
⊢ idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ if(𝑖 = {(0g‘(Poly1‘𝑟))}, (0g‘(Poly1‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))(( deg1 ‘𝑟)‘𝑔) = inf((( deg1 ‘𝑟) “ (𝑖 ∖ {(0g‘(Poly1‘𝑟))})), ℝ, < )))))
Theorem	ply1divmo 25205*	Uniqueness of a quotient in a polynomial division. For polynomials 𝐹, 𝐺 such that 𝐺 ≠ 0 and the leading coefficient of 𝐺 is not a zero divisor, there is at most one polynomial 𝑞 which satisfies 𝐹 = (𝐺 · 𝑞) + 𝑟 where the degree of 𝑟 is less than the degree of 𝐺. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by NM, 17-Jun-2017.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ≠ 0 )    &   ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝐸)    &   ⊢ 𝐸 = (RLReg‘𝑅)    ⇒   ⊢ (𝜑 → ∃*𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
Theorem	ply1divex 25206*	Lemma for ply1divalg 25207: existence part. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ≠ 0 )    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐾)    &   ⊢ (𝜑 → (((coe1‘𝐺)‘(𝐷‘𝐺)) · 𝐼) = 1 )    ⇒   ⊢ (𝜑 → ∃𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
Theorem	ply1divalg 25207*	The division algorithm for univariate polynomials over a ring. For polynomials 𝐹, 𝐺 such that 𝐺 ≠ 0 and the leading coefficient of 𝐺 is a unit, there are unique polynomials 𝑞 and 𝑟 = 𝐹 − (𝐺 · 𝑞) such that the degree of 𝑟 is less than the degree of 𝐺. (Contributed by Stefan O'Rear, 27-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ≠ 0 )    &   ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝐺 ∙ 𝑞))) < (𝐷‘𝐺))
Theorem	ply1divalg2 25208*	Reverse the order of multiplication in ply1divalg 25207 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ≠ 0 )    &   ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ 𝑈)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 ∙ 𝐺))) < (𝐷‘𝐺))
Theorem	uc1pval 25209*	Value of the set of unitic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)}
Theorem	isuc1p 25210	Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))
Theorem	mon1pval 25211*	Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}
Theorem	ismon1p 25212	Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝑀 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 ))
Theorem	uc1pcl 25213	Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐶 → 𝐹 ∈ 𝐵)
Theorem	mon1pcl 25214	Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝑀 → 𝐹 ∈ 𝐵)
Theorem	uc1pn0 25215	Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐶 → 𝐹 ≠ 0 )
Theorem	mon1pn0 25216	Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝑀 → 𝐹 ≠ 0 )
Theorem	uc1pdeg 25217	Unitic polynomials have nonnegative degrees. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶) → (𝐷‘𝐹) ∈ ℕ0)
Theorem	uc1pldg 25218	Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐶 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)
Theorem	mon1pldg 25219	Unitic polynomials have one leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝑀 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = 1 )
Theorem	mon1puc1p 25220	Monic polynomials are unitic. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐶 = (Unic1p‘𝑅)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑀) → 𝑋 ∈ 𝐶)
Theorem	uc1pmon1p 25221	Make a unitic polynomial monic by multiplying a factor to normalize the leading coefficient. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝐶 = (Unic1p‘𝑅)    &   ⊢ 𝑀 = (Monic1p‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ · = (.r‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐶) → ((𝐴‘(𝐼‘((coe1‘𝑋)‘(𝐷‘𝑋)))) · 𝑋) ∈ 𝑀)
Theorem	deg1submon1p 25222	The difference of two monic polynomials of the same degree is a polynomial of lesser degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ 𝑂 = (Monic1p‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑂)    &   ⊢ (𝜑 → (𝐷‘𝐹) = 𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑂)    &   ⊢ (𝜑 → (𝐷‘𝐺) = 𝑋)    ⇒   ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝑋)
Theorem	q1pval 25223*	Value of the univariate polynomial quotient function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑄 = (quot1p‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ · = (.r‘𝑃)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝑄𝐺) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
Theorem	q1peqb 25224	Characterizing property of the polynomial quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑄 = (quot1p‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)) ↔ (𝐹𝑄𝐺) = 𝑋))
Theorem	q1pcl 25225	Closure of the quotient by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝑄 = (quot1p‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝑄𝐺) ∈ 𝐵)
Theorem	r1pval 25226	Value of the polynomial remainder function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑄 = (quot1p‘𝑅)    &   ⊢ · = (.r‘𝑃)    &   ⊢ − = (-g‘𝑃)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
Theorem	r1pcl 25227	Closure of remainder following division by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝐸𝐺) ∈ 𝐵)
Theorem	r1pdeglt 25228	The remainder has a degree smaller than the divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    &   ⊢ 𝐷 = ( deg1 ‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐷‘(𝐹𝐸𝐺)) < (𝐷‘𝐺))
Theorem	r1pid 25229	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 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    &   ⊢ 𝑄 = (quot1p‘𝑅)    &   ⊢ 𝐸 = (rem1p‘𝑅)    &   ⊢ · = (.r‘𝑃)    &   ⊢ + = (+g‘𝑃)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 = (((𝐹𝑄𝐺) · 𝐺) + (𝐹𝐸𝐺)))
Theorem	dvdsq1p 25230	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 25231	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 25232	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 25233	The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 16179). 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 25234	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 25235	Lemma for fta1g 25237. (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 25236*	Lemma for fta1g 25237. (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 25237	The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 26134, 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 25238	Lemma for fta1b 25239. (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 25239*	The assumption that 𝑅 be a domain in fta1g 25237 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	drnguc1p 25240	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 25241*	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 25242*	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 25243	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 25244	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 25245	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 25246	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 25247	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 25248	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 25249	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 25250	Extend class notation to include the set of complex polynomials.
class Poly
Syntax	cidp 25251	Extend class notation to include the identity polynomial.
class Xp
Syntax	ccoe 25252	Extend class notation to include the coefficient function on polynomials.
class coeff
Syntax	cdgr 25253	Extend class notation to include the degree function on polynomials.
class deg
Definition	df-ply 25254*	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 25255	Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ Xp = ( I ↾ ℂ)
Definition	df-coe 25256*	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 25257	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 25258*	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 25259*	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 25260	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 25261*	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 25262*	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 25263	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 25264	The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
Theorem	plyss 25265	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 25266	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 25267*	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 25268*	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 25269*	Lemma for ply1term 25270. (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 25270*	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 25271*	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 25272	A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 𝐴 ∈ 𝑆) → (ℂ × {𝐴}) ∈ (Poly‘𝑆))
Theorem	ne0p 25273	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 25274	The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ (𝑆 ⊆ ℂ → 0𝑝 ∈ (Poly‘𝑆))
Theorem	plyid 25275	The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
⊢ ((𝑆 ⊆ ℂ ∧ 1 ∈ 𝑆) → Xp ∈ (Poly‘𝑆))
Theorem	plyeq0lem 25276*	Lemma for plyeq0 25277. 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 25277*	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 25256 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 25278	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 25279*	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 25280*	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 25281*	Lemma for plyadd 25283. (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 25282*	Lemma for plymul 25284. (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 25283*	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 25284*	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 25285*	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 25286	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 25287	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 25288	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 25289*	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 25290*	Lemma for coeeu 25291. (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 25291*	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 25292*	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 25293*	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 25294	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 25295*	Lemma for dgrcl 25299 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 25296	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}))
Theorem	coef2 25297	The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ 𝐴 = (coeff‘𝐹)    ⇒   ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → 𝐴:ℕ0⟶𝑆)
Theorem	coef3 25298	The domain and range of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ 𝐴 = (coeff‘𝐹)    ⇒   ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
Theorem	dgrcl 25299	The degree of any polynomial is a nonnegative integer. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
Theorem	dgrub 25300	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)