 Home Metamath Proof ExplorerTheorem List (p. 242 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremquotcl 24101* The quotient of two polynomials in a field 𝑆 is also in the field. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑥 ≠ 0)) → (1 / 𝑥) ∈ 𝑆)    &   (𝜑 → -1 ∈ 𝑆)    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ∈ (Poly‘𝑆))    &   (𝜑𝐺 ≠ 0𝑝)       (𝜑 → (𝐹 quot 𝐺) ∈ (Poly‘𝑆))

Theoremquotcl2 24102 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))

Theoremquotdgr 24103 Remainder property of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝑅 = (𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺)))

Theoremplyremlem 24104 Closure of a linear factor. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐺 = (Xp𝑓 − (ℂ × {𝐴}))       (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (𝐺 “ {0}) = {𝐴}))

Theoremplyrem 24105 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15307). If a polynomial 𝐹 is divided by the linear factor 𝑥𝐴, 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 = (Xp𝑓 − (ℂ × {𝐴}))    &   𝑅 = (𝐹𝑓 − (𝐺𝑓 · (𝐹 quot 𝐺)))       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → 𝑅 = (ℂ × {(𝐹𝐴)}))

Theoremfacth 24106 The factor theorem. If a polynomial 𝐹 has a root at 𝐴, then 𝐺 = 𝑥𝐴 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 = (Xp𝑓 − (ℂ × {𝐴}))       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹𝐴) = 0) → 𝐹 = (𝐺𝑓 · (𝐹 quot 𝐺)))

Theoremfta1lem 24107* Lemma for fta1 24108. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝑅 = (𝐹 “ {0})    &   (𝜑𝐷 ∈ ℕ0)    &   (𝜑𝐹 ∈ ((Poly‘ℂ) ∖ {0𝑝}))    &   (𝜑 → (deg‘𝐹) = (𝐷 + 1))    &   (𝜑𝐴 ∈ (𝐹 “ {0}))    &   (𝜑 → ∀𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝})((deg‘𝑔) = 𝐷 → ((𝑔 “ {0}) ∈ Fin ∧ (#‘(𝑔 “ {0})) ≤ (deg‘𝑔))))       (𝜑 → (𝑅 ∈ Fin ∧ (#‘𝑅) ≤ (deg‘𝐹)))

Theoremfta1 24108 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)))

Theoremquotcan 24109 Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
𝐻 = (𝐹𝑓 · 𝐺)       ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹)

Theoremvieta1lem1 24110* Lemma for vieta1 24112. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   𝑅 = (𝐹 “ {0})    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑 → (#‘𝑅) = 𝑁)    &   (𝜑𝐷 ∈ 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 (Xp𝑓 − (ℂ × {𝑧})))       ((𝜑𝑧𝑅) → (𝑄 ∈ (Poly‘ℂ) ∧ 𝐷 = (deg‘𝑄)))

Theoremvieta1lem2 24111* Lemma for vieta1 24112: inductive step. Let 𝑧 be a root of 𝐹. Then 𝐹 = (Xp𝑧) · 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 Σ𝑥𝑅𝑥 = 𝑧 − (coeff‘𝑄)‘ (𝐷 − 1) / (coeff‘𝑄)‘𝐷. Now the coefficients of 𝐹 are 𝐴‘(𝐷 + 1) = (coeff‘𝑄)‘𝐷 and 𝐴𝐷 = Σ𝑘 ∈ (0...𝐷)(coeff‘Xp𝑧)‘𝑘 · (coeff‘𝑄) ‘(𝐷𝑘), which works out to -𝑧 · (coeff‘𝑄)‘𝐷 + (coeff‘𝑄)‘(𝐷 − 1), so putting it all together we have Σ𝑥𝑅𝑥 = -𝐴𝐷 / 𝐴‘(𝐷 + 1) as we wanted to show. (Contributed by Mario Carneiro, 28-Jul-2014.)
𝐴 = (coeff‘𝐹)    &   𝑁 = (deg‘𝐹)    &   𝑅 = (𝐹 “ {0})    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑 → (#‘𝑅) = 𝑁)    &   (𝜑𝐷 ∈ 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 (Xp𝑓 − (ℂ × {𝑧})))       (𝜑 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))

Theoremvieta1 24112* 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})    &   (𝜑𝐹 ∈ (Poly‘𝑆))    &   (𝜑 → (#‘𝑅) = 𝑁)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))

Theoremplyexmo 24113* 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) ∧ (𝑝𝐷) = 𝐹))

14.1.5  Algebraic numbers

Syntaxcaa 24114 Extend class notation to include the set of algebraic numbers.
class 𝔸

Definitiondf-aa 24115 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})

Theoremelaa 24116* Elementhood in the set of algebraic numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
(𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(𝑓𝐴) = 0))

Theoremaacn 24117 An algebraic number is a complex number. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)

Theoremaasscn 24118 The algebraic numbers are a subset of the complex numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
𝔸 ⊆ ℂ

Theoremelqaalem1 24119* Lemma for elqaa 24122. 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝}))    &   (𝜑 → (𝐹𝐴) = 0)    &   𝐵 = (coeff‘𝐹)    &   𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))    &   𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))       ((𝜑𝐾 ∈ ℕ0) → ((𝑁𝐾) ∈ ℕ ∧ ((𝐵𝐾) · (𝑁𝐾)) ∈ ℤ))

Theoremelqaalem2 24120* Lemma for elqaa 24122. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝}))    &   (𝜑 → (𝐹𝐴) = 0)    &   𝐵 = (coeff‘𝐹)    &   𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))    &   𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))    &   𝑃 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 · 𝑦) mod (𝑁𝐾)))       ((𝜑𝐾 ∈ (0...(deg‘𝐹))) → (𝑅 mod (𝑁𝐾)) = 0)

Theoremelqaalem3 24121* Lemma for elqaa 24122. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐹 ∈ ((Poly‘ℚ) ∖ {0𝑝}))    &   (𝜑 → (𝐹𝐴) = 0)    &   𝐵 = (coeff‘𝐹)    &   𝑁 = (𝑘 ∈ ℕ0 ↦ inf({𝑛 ∈ ℕ ∣ ((𝐵𝑘) · 𝑛) ∈ ℤ}, ℝ, < ))    &   𝑅 = (seq0( · , 𝑁)‘(deg‘𝐹))       (𝜑𝐴 ∈ 𝔸)

Theoremelqaa 24122* 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 24116 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})(𝑓𝐴) = 0))

Theoremqaa 24123 Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
(𝐴 ∈ ℚ → 𝐴 ∈ 𝔸)

Theoremqssaa 24124 The rational numbers are contained in the algebraic numbers. (Contributed by Mario Carneiro, 23-Jul-2014.)
ℚ ⊆ 𝔸

Theoremiaa 24125 The imaginary unit is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
i ∈ 𝔸

Theoremaareccl 24126 The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
((𝐴 ∈ 𝔸 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝔸)

Theoremaacjcl 24127 The conjugate of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝐴 ∈ 𝔸 → (∗‘𝐴) ∈ 𝔸)

Theoremaannenlem1 24128* Lemma for aannen 24131. (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)} (𝑐𝑏) = 0})       (𝐴 ∈ ℕ0 → (𝐻𝐴) ∈ Fin)

Theoremaannenlem2 24129* Lemma for aannen 24131. (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)} (𝑐𝑏) = 0})       𝔸 = ran 𝐻

Theoremaannenlem3 24130* 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)} (𝑐𝑏) = 0})       𝔸 ≈ ℕ

Theoremaannen 24131 The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝔸 ≈ ℕ

14.1.6  Liouville's approximation theorem

Theoremaalioulem1 24132 Lemma for aaliou 24138. An integer polynomial cannot inflate the denominator of a rational by more than its degree. (Contributed by Stefan O'Rear, 12-Nov-2014.)
(𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑋 ∈ ℤ)    &   (𝜑𝑌 ∈ ℕ)       (𝜑 → ((𝐹‘(𝑋 / 𝑌)) · (𝑌↑(deg‘𝐹))) ∈ ℤ)

Theoremaalioulem2 24133* Lemma for aaliou 24138. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Proof shortened by AV, 28-Sep-2020.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) = 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))

Theoremaalioulem3 24134* Lemma for aaliou 24138. (Contributed by Stefan O'Rear, 15-Nov-2014.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (𝐹𝐴) = 0)       (𝜑 → ∃𝑥 ∈ ℝ+𝑟 ∈ ℝ ((abs‘(𝐴𝑟)) ≤ 1 → (𝑥 · (abs‘(𝐹𝑟))) ≤ (abs‘(𝐴𝑟))))

Theoremaalioulem4 24135* Lemma for aaliou 24138. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (𝐹𝐴) = 0)       (𝜑 → ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (((𝐹‘(𝑝 / 𝑞)) ≠ 0 ∧ (abs‘(𝐴 − (𝑝 / 𝑞))) ≤ 1) → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))

Theoremaalioulem5 24136* Lemma for aaliou 24138. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (𝐹𝐴) = 0)       (𝜑 → ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ ((𝐹‘(𝑝 / 𝑞)) ≠ 0 → (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))))

Theoremaalioulem6 24137* Lemma for aaliou 24138. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (𝐹𝐴) = 0)       (𝜑 → ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑁)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞)))))

Theoremaaliou 24138* Liouville's theorem on diophantine approximation: Any algebraic number, being a root of a polynomial 𝐹 in integer coefficients, is not approximable beyond order 𝑁 = deg(𝐹) by rational numbers. In this form, it also applies to rational numbers themselves, which are not well approximable by other rational numbers. This is Metamath 100 proof #18. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝑁 = (deg‘𝐹)    &   (𝜑𝐹 ∈ (Poly‘ℤ))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → (𝐹𝐴) = 0)       (𝜑 → ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑁)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))

Theoremgeolim3 24139* Geometric series convergence with arbitrary shift, radix, and multiplicative constant. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (abs‘𝐵) < 1)    &   (𝜑𝐶 ∈ ℂ)    &   𝐹 = (𝑘 ∈ (ℤ𝐴) ↦ (𝐶 · (𝐵↑(𝑘𝐴))))       (𝜑 → seq𝐴( + , 𝐹) ⇝ (𝐶 / (1 − 𝐵)))

Theoremaaliou2 24140* Liouville's approximation theorem for algebraic numbers per se. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐴 ∈ (𝔸 ∩ ℝ) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))

Theoremaaliou2b 24141* Liouville's approximation theorem extended to complex 𝐴. (Contributed by Stefan O'Rear, 20-Nov-2014.)
(𝐴 ∈ 𝔸 → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ+𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))

Theoremaaliou3lem1 24142* Lemma for aaliou3 24151. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐺 = (𝑐 ∈ (ℤ𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐𝐴))))       ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ𝐴)) → (𝐺𝐵) ∈ ℝ)

Theoremaaliou3lem2 24143* Lemma for aaliou3 24151. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐺 = (𝑐 ∈ (ℤ𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐𝐴))))    &   𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))       ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ𝐴)) → (𝐹𝐵) ∈ (0(,](𝐺𝐵)))

Theoremaaliou3lem3 24144* Lemma for aaliou3 24151. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐺 = (𝑐 ∈ (ℤ𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐𝐴))))    &   𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))       (𝐴 ∈ ℕ → (seq𝐴( + , 𝐹) ∈ dom ⇝ ∧ Σ𝑏 ∈ (ℤ𝐴)(𝐹𝑏) ∈ ℝ+ ∧ Σ𝑏 ∈ (ℤ𝐴)(𝐹𝑏) ≤ (2 · (2↑-(!‘𝐴)))))

Theoremaaliou3lem8 24145* Lemma for aaliou3 24151. (Contributed by Stefan O'Rear, 20-Nov-2014.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℕ (2 · (2↑-(!‘(𝑥 + 1)))) ≤ (𝐵 / ((2↑(!‘𝑥))↑𝐴)))

Theoremaaliou3lem4 24146* Lemma for aaliou3 24151. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    &   𝐿 = Σ𝑏 ∈ ℕ (𝐹𝑏)    &   𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹𝑏))       𝐿 ∈ ℝ

Theoremaaliou3lem5 24147* Lemma for aaliou3 24151. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    &   𝐿 = Σ𝑏 ∈ ℕ (𝐹𝑏)    &   𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹𝑏))       (𝐴 ∈ ℕ → (𝐻𝐴) ∈ ℝ)

Theoremaaliou3lem6 24148* Lemma for aaliou3 24151. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    &   𝐿 = Σ𝑏 ∈ ℕ (𝐹𝑏)    &   𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹𝑏))       (𝐴 ∈ ℕ → ((𝐻𝐴) · (2↑(!‘𝐴))) ∈ ℤ)

Theoremaaliou3lem7 24149* Lemma for aaliou3 24151. (Contributed by Stefan O'Rear, 16-Nov-2014.)
𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    &   𝐿 = Σ𝑏 ∈ ℕ (𝐹𝑏)    &   𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹𝑏))       (𝐴 ∈ ℕ → ((𝐻𝐴) ≠ 𝐿 ∧ (abs‘(𝐿 − (𝐻𝐴))) ≤ (2 · (2↑-(!‘(𝐴 + 1))))))

Theoremaaliou3lem9 24150* Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 20-Nov-2014.)
𝐹 = (𝑎 ∈ ℕ ↦ (2↑-(!‘𝑎)))    &   𝐿 = Σ𝑏 ∈ ℕ (𝐹𝑏)    &   𝐻 = (𝑐 ∈ ℕ ↦ Σ𝑏 ∈ (1...𝑐)(𝐹𝑏))        ¬ 𝐿 ∈ 𝔸

Theoremaaliou3 24151 Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 23-Nov-2014.)
Σ𝑘 ∈ ℕ (2↑-(!‘𝑘)) ∉ 𝔸

14.2  Sequences and series

14.2.1  Taylor polynomials and Taylor's theorem

Syntaxctayl 24152 Taylor polynomial of a function.
class Tayl

Syntaxcana 24153 The class of analytic functions.
class Ana

Definitiondf-tayl 24154* Define the Taylor polynomial or Taylor series of a function. TODO-AV: 𝑛 ∈ (ℕ0 ∪ {+∞}) can/should be replaced by 𝑛 ∈ ℕ0*. (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl = (𝑠 ∈ {ℝ, ℂ}, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ (𝑛 ∈ (ℕ0 ∪ {+∞}), 𝑎 𝑘 ∈ ((0[,]𝑛) ∩ ℤ)dom ((𝑠 D𝑛 𝑓)‘𝑘) ↦ 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑛) ∩ ℤ) ↦ (((((𝑠 D𝑛 𝑓)‘𝑘)‘𝑎) / (!‘𝑘)) · ((𝑥𝑎)↑𝑘)))))))

Definitiondf-ana 24155* Define the set of analytic functions, which are functions such that the Taylor series of the function at each point converges to the function in some neighborhood of the point. (Contributed by Mario Carneiro, 31-Dec-2016.)
Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))})

Theoremtaylfvallem1 24156* Lemma for taylfval 24158. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))       (((𝜑𝑋 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)) ∈ ℂ)

Theoremtaylfvallem 24157* Lemma for taylfval 24158. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))       ((𝜑𝑋 ∈ ℂ) → (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)))) ⊆ ℂ)

Theoremtaylfval 24158* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 𝑆 is the base set with respect to evaluate the derivatives (generally or ), 𝐹 is the function we are approximating, at point 𝐵, to order 𝑁. The result is a polynomial function of 𝑥.

This "extended" version of taylpfval 24164 additionally handles the case 𝑁 = +∞, in which case this is not a polynomial but an infinite series, the Taylor series of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)

(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇 = 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥𝐵)↑𝑘))))))

Theoremeltayl 24159* Value of the Taylor series as a relation (elementhood in the domain here expresses that the series is convergent). (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑 → (𝑋𝑇𝑌 ↔ (𝑋 ∈ ℂ ∧ 𝑌 ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)))))))

Theoremtaylf 24160* The Taylor series defines a function on a subset of the complex numbers. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇:dom 𝑇⟶ℂ)

Theoremtayl0 24161* The Taylor series is always defined at the basepoint, with value equal to the value of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → (𝑁 ∈ ℕ0𝑁 = +∞))    &   ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑 → (𝐵 ∈ dom 𝑇 ∧ (𝑇𝐵) = (𝐹𝐵)))

Theoremtaylplem1 24162* Lemma for taylpfval 24164 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))       ((𝜑𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘))

Theoremtaylplem2 24163* Lemma for taylpfval 24164 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))       (((𝜑𝑋 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)) ∈ ℂ)

Theoremtaylpfval 24164* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 𝑆 is the base set with respect to evaluate the derivatives (generally or ), 𝐹 is the function we are approximating, at point 𝐵, to order 𝑁. The result is a polynomial function of 𝑥. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥𝐵)↑𝑘))))

Theoremtaylpf 24165 The Taylor polynomial is a function on the complex numbers (even if the base set of the original function is the reals). (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑𝑇:ℂ⟶ℂ)

Theoremtaylpval 24166* Value of the Taylor polynomial. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)    &   (𝜑𝑋 ∈ ℂ)       (𝜑 → (𝑇𝑋) = Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑋𝐵)↑𝑘)))

Theoremtaylply2 24167* The Taylor polynomial is a polynomial of degree (at most) 𝑁. This version of taylply 24168 shows that the coefficients of 𝑇 are in a subring of the complex numbers. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)    &   (𝜑𝐷 ∈ (SubRing‘ℂfld))    &   (𝜑𝐵𝐷)    &   ((𝜑𝑘 ∈ (0...𝑁)) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ 𝐷)       (𝜑 → (𝑇 ∈ (Poly‘𝐷) ∧ (deg‘𝑇) ≤ 𝑁))

Theoremtaylply 24168 The Taylor polynomial is a polynomial of degree (at most) 𝑁. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁))

Theoremdvtaylp 24169 The derivative of the Taylor polynomial is the Taylor polynomial of the derivative of the function. (Contributed by Mario Carneiro, 31-Dec-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 1)))       (𝜑 → (ℂ D ((𝑁 + 1)(𝑆 Tayl 𝐹)𝐵)) = (𝑁(𝑆 Tayl (𝑆 D 𝐹))𝐵))

Theoremdvntaylp 24170 The 𝑀-th derivative of the Taylor polynomial is the Taylor polynomial of the 𝑀-th derivative of the function. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)))       (𝜑 → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))

Theoremdvntaylp0 24171 The first 𝑁 derivatives of the Taylor polynomial at 𝐵 match the derivatives of the function from which it is derived. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝑀 ∈ (0...𝑁))    &   (𝜑𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)       (𝜑 → (((ℂ D𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑆 D𝑛 𝐹)‘𝑀)‘𝐵))

Theoremtaylthlem1 24172* Lemma for taylth 24174. This is the main part of Taylor's theorem, except for the induction step, which is supposed to be proven using L'Hôpital's rule. However, since our proof of L'Hôpital assumes that 𝑆 = ℝ, we can only do this part generically, and for taylth 24174 itself we must restrict to . (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) = 𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐵𝐴)    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)    &   𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹𝑥) − (𝑇𝑥)) / ((𝑥𝐵)↑𝑁)))    &   ((𝜑 ∧ (𝑛 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D𝑛 𝐹)‘(𝑁𝑛))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁𝑛))‘𝑦)) / ((𝑦𝐵)↑𝑛))) lim 𝐵))) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥𝐵)↑(𝑛 + 1)))) lim 𝐵))       (𝜑 → 0 ∈ (𝑅 lim 𝐵))

Theoremtaylthlem2 24173* Lemma for taylth 24174. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐵𝐴)    &   𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵)    &   (𝜑𝑀 ∈ (1..^𝑁))    &   (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁𝑀))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁𝑀))‘𝑥)) / ((𝑥𝐵)↑𝑀))) lim 𝐵))       (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) / ((𝑥𝐵)↑(𝑀 + 1)))) lim 𝐵))

Theoremtaylth 24174* Taylor's theorem. The Taylor polynomial of a 𝑁-times differentiable function is such that the error term goes to zero faster than (𝑥𝐵)↑𝑁. This is Metamath 100 proof #35. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐵𝐴)    &   𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵)    &   𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹𝑥) − (𝑇𝑥)) / ((𝑥𝐵)↑𝑁)))       (𝜑 → 0 ∈ (𝑅 lim 𝐵))

14.2.2  Uniform convergence

Syntaxculm 24175 Extend class notation to include the uniform convergence predicate.
class 𝑢

Definitiondf-ulm 24176* Define the uniform convergence of a sequence of functions. Here 𝐹(⇝𝑢𝑆)𝐺 if 𝐹 is a sequence of functions 𝐹(𝑛), 𝑛 ∈ ℕ defined on 𝑆 and 𝐺 is a function on 𝑆, and for every 0 < 𝑥 there is a 𝑗 such that the functions 𝐹(𝑘) for 𝑗𝑘 are all uniformly within 𝑥 of 𝐺 on the domain 𝑆. Compare with df-clim 14263. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝑢 = (𝑠 ∈ V ↦ {⟨𝑓, 𝑦⟩ ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑠 (abs‘(((𝑓𝑘)‘𝑧) − (𝑦𝑧))) < 𝑥)})

Theoremulmrel 24177 The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)
Rel (⇝𝑢𝑆)

Theoremulmscl 24178 Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(⇝𝑢𝑆)𝐺𝑆 ∈ V)

Theoremulmval 24179* Express the predicate: The sequence of functions 𝐹 converges uniformly to 𝐺 on 𝑆. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝑆𝑉 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))

Theoremulmcl 24180 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(⇝𝑢𝑆)𝐺𝐺:𝑆⟶ℂ)

Theoremulmf 24181* Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(⇝𝑢𝑆)𝐺 → ∃𝑛 ∈ ℤ 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))

Theoremulmpm 24182 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝐹(⇝𝑢𝑆)𝐺𝐹 ∈ ((ℂ ↑𝑚 𝑆) ↑pm ℤ))

Theoremulmf2 24183 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.)
((𝐹 Fn 𝑍𝐹(⇝𝑢𝑆)𝐺) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))

Theoremulm2 24184* Simplify ulmval 24179 when 𝐹 and 𝐺 are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → ((𝐹𝑘)‘𝑧) = 𝐵)    &   ((𝜑𝑧𝑆) → (𝐺𝑧) = 𝐴)    &   (𝜑𝐺:𝑆⟶ℂ)    &   (𝜑𝑆𝑉)       (𝜑 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))

Theoremulmi 24185* The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → ((𝐹𝑘)‘𝑧) = 𝐵)    &   ((𝜑𝑧𝑆) → (𝐺𝑧) = 𝐴)    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝐶)

Theoremulmclm 24186* A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   (𝜑𝐴𝑆)    &   (𝜑𝐻𝑊)    &   ((𝜑𝑘𝑍) → ((𝐹𝑘)‘𝐴) = (𝐻𝑘))    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)       (𝜑𝐻 ⇝ (𝐺𝐴))

Theoremulmres 24187 A sequence of functions converges iff the tail of the sequence converges (for any finite cutoff). (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ𝑁)    &   (𝜑𝑁𝑍)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))       (𝜑 → (𝐹(⇝𝑢𝑆)𝐺 ↔ (𝐹𝑊)(⇝𝑢𝑆)𝐺))

Theoremulmshftlem 24188* Lemma for ulmshft 24189. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ‘(𝑀 + 𝐾))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐾 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   (𝜑𝐻 = (𝑛𝑊 ↦ (𝐹‘(𝑛𝐾))))       (𝜑 → (𝐹(⇝𝑢𝑆)𝐺𝐻(⇝𝑢𝑆)𝐺))

Theoremulmshft 24189* A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑍 = (ℤ𝑀)    &   𝑊 = (ℤ‘(𝑀 + 𝐾))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐾 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   (𝜑𝐻 = (𝑛𝑊 ↦ (𝐹‘(𝑛𝐾))))       (𝜑 → (𝐹(⇝𝑢𝑆)𝐺𝐻(⇝𝑢𝑆)𝐺))

Theoremulm0 24190 Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   (𝜑𝐺:𝑆⟶ℂ)       ((𝜑𝑆 = ∅) → 𝐹(⇝𝑢𝑆)𝐺)

Theoremulmuni 24191 An sequence of functions uniformly converges to at most one limit. (Contributed by Mario Carneiro, 5-Jul-2017.)
((𝐹(⇝𝑢𝑆)𝐺𝐹(⇝𝑢𝑆)𝐻) → 𝐺 = 𝐻)

Theoremulmdm 24192 Two ways to express that a function has a limit. (The expression ((⇝𝑢𝑆)‘𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Mario Carneiro, 5-Jul-2017.)
(𝐹 ∈ dom (⇝𝑢𝑆) ↔ 𝐹(⇝𝑢𝑆)((⇝𝑢𝑆)‘𝐹))

Theoremulmcaulem 24193* Lemma for ulmcau 24194 and ulmcau2 24195: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 14139. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))       (𝜑 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − ((𝐹𝑗)‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑘)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − ((𝐹𝑚)‘𝑧))) < 𝑥))

Theoremulmcau 24194* A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every 0 < 𝑥 there is a 𝑗 such that for all 𝑗𝑘 the functions 𝐹(𝑘) and 𝐹(𝑗) are uniformly within 𝑥 of each other on 𝑆. This is the four-quantifier version, see ulmcau2 24195 for the more conventional five-quantifier version. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))       (𝜑 → (𝐹 ∈ dom (⇝𝑢𝑆) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − ((𝐹𝑗)‘𝑧))) < 𝑥))

Theoremulmcau2 24195* A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every 0 < 𝑥 there is a 𝑗 such that for all 𝑗𝑘, 𝑚 the functions 𝐹(𝑘) and 𝐹(𝑚) are uniformly within 𝑥 of each other on 𝑆. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))       (𝜑 → (𝐹 ∈ dom (⇝𝑢𝑆) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑘)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − ((𝐹𝑚)‘𝑧))) < 𝑥))

Theoremulmss 24196* A uniform limit of functions is still a uniform limit if restricted to a subset. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑇𝑆)    &   ((𝜑𝑥𝑍) → 𝐴𝑊)    &   (𝜑 → (𝑥𝑍𝐴)(⇝𝑢𝑆)𝐺)       (𝜑 → (𝑥𝑍 ↦ (𝐴𝑇))(⇝𝑢𝑇)(𝐺𝑇))

Theoremulmbdd 24197* A uniform limit of bounded functions is bounded. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑆))    &   ((𝜑𝑘𝑍) → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘((𝐹𝑘)‘𝑧)) ≤ 𝑥)    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧𝑆 (abs‘(𝐺𝑧)) ≤ 𝑥)

Theoremulmcn 24198 A uniform limit of continuous functions is continuous. (Contributed by Mario Carneiro, 27-Feb-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(𝑆cn→ℂ))    &   (𝜑𝐹(⇝𝑢𝑆)𝐺)       (𝜑𝐺 ∈ (𝑆cn→ℂ))

Theoremulmdvlem1 24199* Lemma for ulmdv 24202. (Contributed by Mario Carneiro, 3-Mar-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑋))    &   (𝜑𝐺:𝑋⟶ℂ)    &   ((𝜑𝑧𝑋) → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧))    &   (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻)    &   ((𝜑𝜓) → 𝐶𝑋)    &   ((𝜑𝜓) → 𝑅 ∈ ℝ+)    &   ((𝜑𝜓) → 𝑈 ∈ ℝ+)    &   ((𝜑𝜓) → 𝑊 ∈ ℝ+)    &   ((𝜑𝜓) → 𝑈 < 𝑊)    &   ((𝜑𝜓) → (𝐶(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑈) ⊆ 𝑋)    &   ((𝜑𝜓) → (abs‘(𝑌𝐶)) < 𝑈)    &   ((𝜑𝜓) → 𝑁𝑍)    &   ((𝜑𝜓) → ∀𝑚 ∈ (ℤ𝑁)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑁))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑅 / 2) / 2))    &   ((𝜑𝜓) → (abs‘(((𝑆 D (𝐹𝑁))‘𝐶) − (𝐻𝐶))) < (𝑅 / 2))    &   ((𝜑𝜓) → 𝑌𝑋)    &   ((𝜑𝜓) → 𝑌𝐶)    &   ((𝜑𝜓) → ((abs‘(𝑌𝐶)) < 𝑊 → (abs‘(((((𝐹𝑁)‘𝑌) − ((𝐹𝑁)‘𝐶)) / (𝑌𝐶)) − ((𝑆 D (𝐹𝑁))‘𝐶))) < ((𝑅 / 2) / 2)))       ((𝜑𝜓) → (abs‘((((𝐺𝑌) − (𝐺𝐶)) / (𝑌𝐶)) − (𝐻𝐶))) < 𝑅)

Theoremulmdvlem2 24200* Lemma for ulmdv 24202. (Contributed by Mario Carneiro, 8-May-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑋))    &   (𝜑𝐺:𝑋⟶ℂ)    &   ((𝜑𝑧𝑋) → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧))    &   (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻)       ((𝜑𝑘𝑍) → dom (𝑆 D (𝐹𝑘)) = 𝑋)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >