P. 254 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dgrub2 25301	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 25302	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 25303*	Lemma for coeid 25304. (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 25304*	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 25305*	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 25306*	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 25307*	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 25308*	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 25309*	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 25310*	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 25311	A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (deg‘(ℂ × {𝐴})) = 0)
Theorem	0dgrb 25312	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 25313	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 25314	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 25315	Lemma for coeadd 25317 and dgradd 25333. (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 25316*	Lemma for coemul 25318 and dgrmul 25336. (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 25317	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 25318*	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 25319	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 25320	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 25321	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 25322	The coefficients of the zero polynomial are zero. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (coeff‘0𝑝) = (ℕ0 × {0})
Theorem	coesub 25323	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 25324*	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 25325*	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 25326*	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 25327	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	dgr0 25328	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 25299, dgreq0 25331 and coeid 25304 without having to special-case zero, although plydivalg 25364 is a little more complicated as a result. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (deg‘0𝑝) = 0
Theorem	coeidp 25329	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 25330	The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ (deg‘Xp) = 1
Theorem	dgreq0 25331	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 25332	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 25333	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 25334	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 25335	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 25336	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 25337	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 25338	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 25339*	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 25340*	Lemma for dgrco 25341. (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 25341	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 25342*	Lemma for plycj 25343 and coecj 25344. (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 25343*	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 25344	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 25345	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 25346	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 25347	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 25348	Real-coefficient polynomials restrict to real functions. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ↾ ℝ):ℝ⟶ℝ)
Theorem	dvply1 25349*	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 25350	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 25351	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 25352	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 25353	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))
Theorem	plycpn 25354	Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ ∩ ran (𝓑C𝑛‘ℂ))
14.1.4  The division algorithm for polynomials
Syntax	cquot 25355	Extend class notation to include the quotient of a polynomial division.
class quot
Definition	df-quot 25356*	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)[(𝑓 ∘f − (𝑔 ∘f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
Theorem	quotval 25357*	Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 𝑞))    ⇒   ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
Theorem	plydivlem1 25358*	Lemma for plydivalg 25364. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    ⇒   ⊢ (𝜑 → 0 ∈ 𝑆)
Theorem	plydivlem2 25359*	Lemma for plydivalg 25364. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ≠ 0𝑝)    &   ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 𝑞))    ⇒   ⊢ ((𝜑 ∧ 𝑞 ∈ (Poly‘𝑆)) → 𝑅 ∈ (Poly‘𝑆))
Theorem	plydivlem3 25360*	Lemma for plydivex 25362. Base case of induction. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ≠ 0𝑝)    &   ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 𝑞))    &   ⊢ (𝜑 → (𝐹 = 0𝑝 ∨ ((deg‘𝐹) − (deg‘𝐺)) < 0))    ⇒   ⊢ (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
Theorem	plydivlem4 25361*	Lemma for plydivex 25362. Induction step. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ≠ 0𝑝)    &   ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 𝑞))    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → (𝑀 − 𝑁) = 𝐷)    &   ⊢ (𝜑 → 𝐹 ≠ 0𝑝)    &   ⊢ 𝑈 = (𝑓 ∘f − (𝐺 ∘f · 𝑝))    &   ⊢ 𝐻 = (𝑧 ∈ ℂ ↦ (((𝐴‘𝑀) / (𝐵‘𝑁)) · (𝑧↑𝐷)))    &   ⊢ (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))
Theorem	plydivex 25362*	Lemma for plydivalg 25364. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ≠ 0𝑝)    &   ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 𝑞))    ⇒   ⊢ (𝜑 → ∃𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
Theorem	plydiveu 25363*	Lemma for plydivalg 25364. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ≠ 0𝑝)    &   ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 𝑞))    &   ⊢ (𝜑 → 𝑞 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))    &   ⊢ 𝑇 = (𝐹 ∘f − (𝐺 ∘f · 𝑝))    &   ⊢ (𝜑 → 𝑝 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → (𝑇 = 0𝑝 ∨ (deg‘𝑇) < (deg‘𝐺)))    ⇒   ⊢ (𝜑 → 𝑝 = 𝑞)
Theorem	plydivalg 25364*	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 𝐹 − 𝐺 · 𝑞 is either zero or has degree less than 𝐺. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ≠ 0𝑝)    &   ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 𝑞))    ⇒   ⊢ (𝜑 → ∃!𝑞 ∈ (Poly‘𝑆)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
Theorem	quotlem 25365*	Lemma for properties of the polynomial quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ≠ 0𝑝)    &   ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))    ⇒   ⊢ (𝜑 → ((𝐹 quot 𝐺) ∈ (Poly‘𝑆) ∧ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
Theorem	quotcl 25366*	The quotient of two polynomials in a field 𝑆 is also in the field. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → -1 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → 𝐺 ≠ 0𝑝)    ⇒   ⊢ (𝜑 → (𝐹 quot 𝐺) ∈ (Poly‘𝑆))
Theorem	quotcl2 25367	Closure of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ))
Theorem	quotdgr 25368	Remainder property of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))    ⇒   ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))
Theorem	plyremlem 25369	Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝐺 = (Xp ∘f − (ℂ × {𝐴}))    ⇒   ⊢ (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴}))
Theorem	plyrem 25370	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). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝐺 = (Xp ∘f − (ℂ × {𝐴}))    &   ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · (𝐹 quot 𝐺)))    ⇒   ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹‘𝐴)}))
Theorem	facth 25371	The factor theorem. If a polynomial 𝐹 has a root at 𝐴, then 𝐺 = 𝑥 − 𝐴 is a factor of 𝐹 (and the other factor is 𝐹 quot 𝐺). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝐺 = (Xp ∘f − (ℂ × {𝐴}))    ⇒   ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 𝐹 = (𝐺 ∘f · (𝐹 quot 𝐺)))
Theorem	fta1lem 25372*	Lemma for fta1 25373. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝑅 = (◡𝐹 “ {0})    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}))    &   ⊢ (𝜑 → (deg‘𝐹) = (𝐷 + 1))    &   ⊢ (𝜑 → 𝐴 ∈ (◡𝐹 “ {0}))    &   ⊢ (𝜑 → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝐷 → ((◡𝑔 “ {0}) ∈ Fin ∧ (♯‘(◡𝑔 “ {0})) ≤ (deg‘𝑔))))    ⇒   ⊢ (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
Theorem	fta1 25373	The easy direction of the Fundamental Theorem of Algebra: A nonzero polynomial has at most deg(𝐹) roots. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝑅 = (◡𝐹 “ {0})    ⇒   ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
Theorem	quotcan 25374	Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ 𝐻 = (𝐹 ∘f · 𝐺)    ⇒   ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹)
Theorem	vieta1lem1 25375*	Lemma for vieta1 25377. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ 𝐴 = (coeff‘𝐹)    &   ⊢ 𝑁 = (deg‘𝐹)    &   ⊢ 𝑅 = (◡𝐹 “ {0})    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → (♯‘𝑅) = 𝑁)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ)    &   ⊢ (𝜑 → (𝐷 + 1) = 𝑁)    &   ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((𝐷 = (deg‘𝑓) ∧ (♯‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))    &   ⊢ 𝑄 = (𝐹 quot (Xp ∘f − (ℂ × {𝑧})))    ⇒   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑅) → (𝑄 ∈ (Poly‘ℂ) ∧ 𝐷 = (deg‘𝑄)))
Theorem	vieta1lem2 25376*	Lemma for vieta1 25377: inductive step. Let 𝑧 be a root of 𝐹. Then 𝐹 = (Xp − 𝑧) · 𝑄 for some 𝑄 by the factor theorem, and 𝑄 is a degree- 𝐷 polynomial, so by the induction hypothesis Σ𝑥 ∈ (◡𝑄 “ 0)𝑥 = -(coeff‘𝑄)‘(𝐷 − 1) / (coeff‘𝑄)‘𝐷, so Σ𝑥 ∈ 𝑅𝑥 = 𝑧 − (coeff‘𝑄)‘ (𝐷 − 1) / (coeff‘𝑄)‘𝐷. Now the coefficients of 𝐹 are 𝐴‘(𝐷 + 1) = (coeff‘𝑄)‘𝐷 and 𝐴‘𝐷 = Σ𝑘 ∈ (0...𝐷)(coeff‘Xp − 𝑧)‘𝑘 · (coeff‘𝑄) ‘(𝐷 − 𝑘), which works out to -𝑧 · (coeff‘𝑄)‘𝐷 + (coeff‘𝑄)‘(𝐷 − 1), so putting it all together we have Σ𝑥 ∈ 𝑅𝑥 = -𝐴‘𝐷 / 𝐴‘(𝐷 + 1) as we wanted to show. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ 𝐴 = (coeff‘𝐹)    &   ⊢ 𝑁 = (deg‘𝐹)    &   ⊢ 𝑅 = (◡𝐹 “ {0})    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → (♯‘𝑅) = 𝑁)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ)    &   ⊢ (𝜑 → (𝐷 + 1) = 𝑁)    &   ⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((𝐷 = (deg‘𝑓) ∧ (♯‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))    &   ⊢ 𝑄 = (𝐹 quot (Xp ∘f − (ℂ × {𝑧})))    ⇒   ⊢ (𝜑 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁)))
Theorem	vieta1 25377*	The first-order Vieta's formula (see http://en.wikipedia.org/wiki/Vieta%27s_formulas). If a polynomial of degree 𝑁 has 𝑁 distinct roots, then the sum over these roots can be calculated as -𝐴(𝑁 − 1) / 𝐴(𝑁). (If the roots are not distinct, then this formula is still true but must double-count some of the roots according to their multiplicities.) (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ 𝐴 = (coeff‘𝐹)    &   ⊢ 𝑁 = (deg‘𝐹)    &   ⊢ 𝑅 = (◡𝐹 “ {0})    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   ⊢ (𝜑 → (♯‘𝑅) = 𝑁)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁)))
Theorem	plyexmo 25378*	An infinite set of values can be extended to a polynomial in at most one way. (Contributed by Stefan O'Rear, 14-Nov-2014.)
⊢ ((𝐷 ⊆ ℂ ∧ ¬ 𝐷 ∈ Fin) → ∃*𝑝(𝑝 ∈ (Poly‘𝑆) ∧ (𝑝 ↾ 𝐷) = 𝐹))
14.1.5  Algebraic numbers
Syntax	caa 25379	Extend class notation to include the set of algebraic numbers.
class 𝔸
Definition	df-aa 25380	Define the set of algebraic numbers. An algebraic number is a root of a nonzero polynomial over the integers. Here we construct it as the union of all kernels (preimages of {0}) of all polynomials in (Poly‘ℤ), except the zero polynomial 0𝑝. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ 𝔸 = ∪ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(◡𝑓 “ {0})
Theorem	elaa 25381*	Elementhood in the set of algebraic numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
Theorem	aacn 25382	An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
Theorem	aasscn 25383	The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ 𝔸 ⊆ ℂ
Theorem	elqaalem1 25384*	Lemma for elqaa 25387. The function 𝑁 represents the denominators of the rational coefficients 𝐵. By multiplying them all together to make 𝑅, we get a number big enough to clear all the denominators and make 𝑅 · 𝐹 an integer polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝}))    &   ⊢ (𝜑 → (𝐹‘𝐴) = 0)    &   ⊢ 𝐵 = (coeff‘𝐹)    &   ⊢ 𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))    &   ⊢ 𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))    ⇒   ⊢ ((𝜑 ∧ 𝐾 ∈ ℕ0) → ((𝑁‘𝐾) ∈ ℕ ∧ ((𝐵‘𝐾) · (𝑁‘𝐾)) ∈ ℤ))
Theorem	elqaalem2 25385*	Lemma for elqaa 25387. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝}))    &   ⊢ (𝜑 → (𝐹‘𝐴) = 0)    &   ⊢ 𝐵 = (coeff‘𝐹)    &   ⊢ 𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))    &   ⊢ 𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))    &   ⊢ 𝑃 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 · 𝑦) mod (𝑁‘𝐾)))    ⇒   ⊢ ((𝜑 ∧ 𝐾 ∈ (0...(deg‘𝐹))) → (𝑅 mod (𝑁‘𝐾)) = 0)
Theorem	elqaalem3 25386*	Lemma for elqaa 25387. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝}))    &   ⊢ (𝜑 → (𝐹‘𝐴) = 0)    &   ⊢ 𝐵 = (coeff‘𝐹)    &   ⊢ 𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵‘𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))    &   ⊢ 𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝔸)
Theorem	elqaa 25387*	The set of numbers generated by the roots of polynomials in the rational numbers is the same as the set of algebraic numbers, which by elaa 25381 are defined only in terms of polynomials over the integers. (Contributed by Mario Carneiro, 23-Jul-2014.) (Proof shortened by AV, 3-Oct-2020.)
⊢ (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑓‘𝐴) = 0))
Theorem	qaa 25388	Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ 𝔸)
Theorem	qssaa 25389	The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ ℚ ⊆ 𝔸
Theorem	iaa 25390	The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ i ∈ 𝔸
Theorem	aareccl 25391	The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸)
Theorem	aacjcl 25392	The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝐴 ∈ 𝔸 → (∗‘𝐴) ∈ 𝔸)
Theorem	aannenlem1 25393*	Lemma for aannen 25396. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})    ⇒   ⊢ (𝐴 ∈ ℕ0 → (𝐻‘𝐴) ∈ Fin)
Theorem	aannenlem2 25394*	Lemma for aannen 25396. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})    ⇒   ⊢ 𝔸 = ∪ ran 𝐻
Theorem	aannenlem3 25395*	The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})    ⇒   ⊢ 𝔸 ≈ ℕ
Theorem	aannen 25396	The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝔸 ≈ ℕ
14.1.6  Liouville's approximation theorem
Theorem	aalioulem1 25397	Lemma for aaliou 25403. An integer polynomial cannot inflate the denominator of a rational by more than its degree. (Contributed by Stefan O'Rear, 12-Nov-2014.)
⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))    &   ⊢ (𝜑 → 𝑋 ∈ ℤ)    &   ⊢ (𝜑 → 𝑌 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((𝐹‘(𝑋 / 𝑌)) · (𝑌↑(deg‘𝐹))) ∈ ℤ)
Theorem	aalioulem2 25398*	Lemma for aaliou 25403. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Proof shortened by AV, 28-Sep-2020.)
⊢ 𝑁 = (deg‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))
Theorem	aalioulem3 25399*	Lemma for aaliou 25403. (Contributed by Stefan O'Rear, 15-Nov-2014.)
⊢ 𝑁 = (deg‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 0)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑟 ∈ ℝ ((abs‘(𝐴 − 𝑟)) ≤ 1 → (𝑥 · (abs‘(𝐹‘𝑟))) ≤ (abs‘(𝐴 − 𝑟))))
Theorem	aalioulem4 25400*	Lemma for aaliou 25403. (Contributed by Stefan O'Rear, 16-Nov-2014.)
⊢ 𝑁 = (deg‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ (Poly‘ℤ))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 0)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (((𝐹‘(𝑝 / 𝑞)) ≠ 0 ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ≤ 1) → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))