 Home Metamath Proof ExplorerTheorem List (p. 381 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 - 38001-38100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremislnr2 38001* Property of being a left-Noetherian ring in terms of finite generation of ideals (the usual "pure ring theory" definition). (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝑈 = (LIdeal‘𝑅)    &   𝑁 = (RSpan‘𝑅)       (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ ∀𝑖𝑈𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁𝑔)))

Theoremislnr3 38002 Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐵 = (Base‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ 𝑈 ∈ (NoeACS‘𝐵)))

Theoremlnr2i 38003* Given an ideal in a left-Noetherian ring, there is a finite subset which generates it. (Contributed by Stefan O'Rear, 31-Mar-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝑁 = (RSpan‘𝑅)       ((𝑅 ∈ LNoeR ∧ 𝐼𝑈) → ∃𝑔 ∈ (𝒫 𝐼 ∩ Fin)𝐼 = (𝑁𝑔))

Theoremlpirlnr 38004 Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR → 𝑅 ∈ LNoeR)

Theoremlnrfrlm 38005 Finite-dimensional free modules over a Noetherian ring are Noetherian. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑌 = (𝑅 freeLMod 𝐼)       ((𝑅 ∈ LNoeR ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM)

Theoremlnrfg 38006 Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.)
𝑆 = (Scalar‘𝑀)       ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → 𝑀 ∈ LNoeM)

Theoremlnrfgtr 38007 A submodule of a finitely generated module over a Noetherian ring is finitely generated. Often taken as the definition of Noetherian ring. (Contributed by Stefan O'Rear, 7-Feb-2015.)
𝑆 = (Scalar‘𝑀)    &   𝑈 = (LSubSp‘𝑀)    &   𝑁 = (𝑀s 𝑃)       ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR ∧ 𝑃𝑈) → 𝑁 ∈ LFinGen)

20.25.42  Hilbert's Basis Theorem

Syntaxcldgis 38008 The leading ideal sequence used in the Hilbert Basis Theorem.
class ldgIdlSeq

Definitiondf-ldgis 38009* Define a function which carries polynomial ideals to the sequence of coefficient ideals of leading coefficients of degree- 𝑥 elements in the polynomial ideal. The proof that this map is strictly monotone is the core of the Hilbert Basis Theorem hbt 38017. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))

Theoremhbtlem1 38010* Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   𝐷 = ( deg1𝑅)       ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})

Theoremhbtlem2 38011 Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   𝑇 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ 𝑇)

Theoremhbtlem7 38012 Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   𝑇 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼):ℕ0𝑇)

Theoremhbtlem4 38013 The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑈)    &   (𝜑𝑋 ∈ ℕ0)    &   (𝜑𝑌 ∈ ℕ0)    &   (𝜑𝑋𝑌)       (𝜑 → ((𝑆𝐼)‘𝑋) ⊆ ((𝑆𝐼)‘𝑌))

Theoremhbtlem3 38014 The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑈)    &   (𝜑𝐽𝑈)    &   (𝜑𝐼𝐽)    &   (𝜑𝑋 ∈ ℕ0)       (𝜑 → ((𝑆𝐼)‘𝑋) ⊆ ((𝑆𝐽)‘𝑋))

Theoremhbtlem5 38015* The leading ideal function is strictly monotone. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑈)    &   (𝜑𝐽𝑈)    &   (𝜑𝐼𝐽)    &   (𝜑 → ∀𝑥 ∈ ℕ0 ((𝑆𝐽)‘𝑥) ⊆ ((𝑆𝐼)‘𝑥))       (𝜑𝐼 = 𝐽)

Theoremhbtlem6 38016* There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   𝑁 = (RSpan‘𝑃)    &   (𝜑𝑅 ∈ LNoeR)    &   (𝜑𝐼𝑈)    &   (𝜑𝑋 ∈ ℕ0)       (𝜑 → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁𝑘))‘𝑋))

Theoremhbt 38017 The Hilbert Basis Theorem - the ring of univariate polynomials over a Noetherian ring is a Noetherian ring. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝑃 = (Poly1𝑅)       (𝑅 ∈ LNoeR → 𝑃 ∈ LNoeR)

20.25.43  Additional material on polynomials [DEPRECATED]

Syntaxcmnc 38018 Extend class notation with the class of monic polynomials.
class Monic

Syntaxcplylt 38019 Extend class notatin with the class of limited-degree polynomials.
class Poly<

Definitiondf-mnc 38020* Define the class of monic polynomials. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})

Definitiondf-plylt 38021* Define the class of limited-degree polynomials. (Contributed by Stefan O'Rear, 8-Dec-2014.)
Poly< = (𝑠 ∈ 𝒫 ℂ, 𝑥 ∈ ℕ0 ↦ {𝑝 ∈ (Poly‘𝑠) ∣ (𝑝 = 0𝑝 ∨ (deg‘𝑝) < 𝑥)})

Theoremdgrsub2 38022 Subtracting two polynomials with the same degree and top coefficient gives a polynomial of strictly lower degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
𝑁 = (deg‘𝐹)       (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑇)) ∧ ((deg‘𝐺) = 𝑁𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹𝑓𝐺)) < 𝑁)

Theoremelmnc 38023 Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))

Theoremmncply 38024 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ∈ (Poly‘𝑆))

Theoremmnccoe 38025 A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic ‘𝑆) → ((coeff‘𝑃)‘(deg‘𝑃)) = 1)

Theoremmncn0 38026 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ≠ 0𝑝)

20.25.44  Degree and minimal polynomial of algebraic numbers

Syntaxcdgraa 38027 Extend class notation to include the degree function for algebraic numbers.
class degAA

Syntaxcmpaa 38028 Extend class notation to include the minimal polynomial for an algebraic number.
class minPolyAA

Definitiondf-dgraa 38029* Define the degree of an algebraic number as the smallest degree of any nonzero polynomial which has said number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.)
degAA = (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝑥) = 0)}, ℝ, < ))

Definitiondf-mpaa 38030* Define the minimal polynomial of an algebraic number as the unique monic polynomial which achieves the minimum of degAA. (Contributed by Stefan O'Rear, 25-Nov-2014.)
minPolyAA = (𝑥 ∈ 𝔸 ↦ (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝑥) ∧ (𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(degAA𝑥)) = 1)))

Theoremdgraaval 38031* Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.)
(𝐴 ∈ 𝔸 → (degAA𝐴) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝𝐴) = 0)}, ℝ, < ))

Theoremdgraalem 38032* Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.)
(𝐴 ∈ 𝔸 → ((degAA𝐴) ∈ ℕ ∧ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0)))

Theoremdgraacl 38033 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → (degAA𝐴) ∈ ℕ)

Theoremdgraaf 38034 Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.)
degAA:𝔸⟶ℕ

Theoremdgraaub 38035 Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Proof shortened by AV, 29-Sep-2020.)
(((𝑃 ∈ (Poly‘ℚ) ∧ 𝑃 ≠ 0𝑝) ∧ (𝐴 ∈ ℂ ∧ (𝑃𝐴) = 0)) → (degAA𝐴) ≤ (deg‘𝑃))

Theoremdgraa0p 38036 A rational polynomial of degree less than an algebraic number cannot be zero at that number unless it is the zero polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
((𝐴 ∈ 𝔸 ∧ 𝑃 ∈ (Poly‘ℚ) ∧ (deg‘𝑃) < (degAA𝐴)) → ((𝑃𝐴) = 0 ↔ 𝑃 = 0𝑝))

Theoremmpaaeu 38037* An algebraic number has exactly one monic polynomial of the least degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1))

Theoremmpaaval 38038* Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA𝐴) ∧ (𝑝𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA𝐴)) = 1)))

Theoremmpaalem 38039 Properties of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → ((minPolyAA‘𝐴) ∈ (Poly‘ℚ) ∧ ((deg‘(minPolyAA‘𝐴)) = (degAA𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA𝐴)) = 1)))

Theoremmpaacl 38040 Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ (Poly‘ℚ))

Theoremmpaadgr 38041 Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → (deg‘(minPolyAA‘𝐴)) = (degAA𝐴))

Theoremmpaaroot 38042 The minimal polynomial of an algebraic number has the number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → ((minPolyAA‘𝐴)‘𝐴) = 0)

Theoremmpaamn 38043 Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝐴))‘(degAA𝐴)) = 1)

20.25.45  Algebraic integers I

Syntaxcitgo 38044 Extend class notation with the integral-over predicate.
class IntgOver

Syntaxcza 38045 Extend class notation with the class of algebraic integers.
class

Definitiondf-itgo 38046* A complex number is said to be integral over a subset if it is the root of a monic polynomial with coefficients from the subset. This definition is typically not used for fields but it works there, see aaitgo 38049. This definition could work for subsets of an arbitrary ring with a more general definition of polynomials. TODO: use Monic (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑠)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})

Definitiondf-za 38047 Define an algebraic integer as a complex number which is the root of a monic integer polynomial. (Contributed by Stefan O'Rear, 30-Nov-2014.)
= (IntgOver‘ℤ)

Theoremitgoval 38048* Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑆)((𝑝𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)})

Theoremaaitgo 38049 The standard algebraic numbers 𝔸 are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝔸 = (IntgOver‘ℚ)

Theoremitgoss 38050 An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
((𝑆𝑇𝑇 ⊆ ℂ) → (IntgOver‘𝑆) ⊆ (IntgOver‘𝑇))

Theoremitgocn 38051 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(IntgOver‘𝑆) ⊆ ℂ

Theoremcnsrexpcl 38052 Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
(𝜑𝑆 ∈ (SubRing‘ℂfld))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌 ∈ ℕ0)       (𝜑 → (𝑋𝑌) ∈ 𝑆)

Theoremfsumcnsrcl 38053* Finite sums are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
(𝜑𝑆 ∈ (SubRing‘ℂfld))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)       (𝜑 → Σ𝑘𝐴 𝐵𝑆)

Theoremcnsrplycl 38054 Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
(𝜑𝑆 ∈ (SubRing‘ℂfld))    &   (𝜑𝑃 ∈ (Poly‘𝐶))    &   (𝜑𝑋𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (𝑃𝑋) ∈ 𝑆)

Theoremrgspnval 38055* Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
(𝜑𝑅 ∈ Ring)    &   (𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐴𝐵)    &   (𝜑𝑁 = (RingSpan‘𝑅))    &   (𝜑𝑈 = (𝑁𝐴))       (𝜑𝑈 = {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴𝑡})

Theoremrgspncl 38056 The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
(𝜑𝑅 ∈ Ring)    &   (𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐴𝐵)    &   (𝜑𝑁 = (RingSpan‘𝑅))    &   (𝜑𝑈 = (𝑁𝐴))       (𝜑𝑈 ∈ (SubRing‘𝑅))

Theoremrgspnssid 38057 The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-2014.)
(𝜑𝑅 ∈ Ring)    &   (𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐴𝐵)    &   (𝜑𝑁 = (RingSpan‘𝑅))    &   (𝜑𝑈 = (𝑁𝐴))       (𝜑𝐴𝑈)

Theoremrgspnmin 38058 The ring-span is contained in all subspaces which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014.)
(𝜑𝑅 ∈ Ring)    &   (𝜑𝐵 = (Base‘𝑅))    &   (𝜑𝐴𝐵)    &   (𝜑𝑁 = (RingSpan‘𝑅))    &   (𝜑𝑈 = (𝑁𝐴))    &   (𝜑𝑆 ∈ (SubRing‘𝑅))    &   (𝜑𝐴𝑆)       (𝜑𝑈𝑆)

Theoremrgspnid 38059 The span of a subring is itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
(𝜑𝑅 ∈ Ring)    &   (𝜑𝐴 ∈ (SubRing‘𝑅))    &   (𝜑𝑆 = ((RingSpan‘𝑅)‘𝐴))       (𝜑𝑆 = 𝐴)

Theoremrngunsnply 38060* Adjoining one element to a ring results in a set of polynomial evaluations. (Contributed by Stefan O'Rear, 30-Nov-2014.)
(𝜑𝐵 ∈ (SubRing‘ℂfld))    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑆 = ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))       (𝜑 → (𝑉𝑆 ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))

Theoremflcidc 38061* Finite linear combinations with an indicator function. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝜑𝐹 = (𝑗𝑆 ↦ if(𝑗 = 𝐾, 1, 0)))    &   (𝜑𝑆 ∈ Fin)    &   (𝜑𝐾𝑆)    &   ((𝜑𝑖𝑆) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑖𝑆 ((𝐹𝑖) · 𝐵) = 𝐾 / 𝑖𝐵)

20.25.46  Endomorphism algebra

Syntaxcmend 38062 Syntax for module endomorphism algebra.
class MEndo

Definitiondf-mend 38063* Define the endomorphism algebra of a module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
MEndo = (𝑚 ∈ V ↦ (𝑚 LMHom 𝑚) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑓 (+g𝑚)𝑦))⟩, ⟨(.r‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥𝑦))⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑚)⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘𝑓 ( ·𝑠𝑚)𝑦))⟩}))

Theoremalgstr 38064 Lemma to shorten proofs of algbase 38065 through algvsca 38069. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       𝐴 Struct ⟨1, 6⟩

Theoremalgbase 38065 The base set of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (𝐵𝑉𝐵 = (Base‘𝐴))

Theoremalgaddg 38066 The additive operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       ( +𝑉+ = (+g𝐴))

Theoremalgmulr 38067 The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       ( ×𝑉× = (.r𝐴))

Theoremalgsca 38068 The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (𝑆𝑉𝑆 = (Scalar‘𝐴))

Theoremalgvsca 38069 The scalar product operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.)
𝐴 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       ( ·𝑉· = ( ·𝑠𝐴))

Theoremmendval 38070* Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐵 = (𝑀 LMHom 𝑀)    &    + = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑓 (+g𝑀)𝑦))    &    × = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦))    &   𝑆 = (Scalar‘𝑀)    &    · = (𝑥 ∈ (Base‘𝑆), 𝑦𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘𝑓 ( ·𝑠𝑀)𝑦))       (𝑀𝑋 → (MEndo‘𝑀) = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩}))

Theoremmendbas 38071 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)       (𝑀 LMHom 𝑀) = (Base‘𝐴)

Theoremmendplusgfval 38072* Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    &    + = (+g𝑀)       (+g𝐴) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑓 + 𝑦))

Theoremmendplusg 38073 A specific addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    &    + = (+g𝑀)    &    = (+g𝐴)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋𝑓 + 𝑌))

Theoremmendmulrfval 38074* Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)       (.r𝐴) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦))

Theoremmendmulr 38075 A specific multiplication in the module endormoprhism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    &    · = (.r𝐴)       ((𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) = (𝑋𝑌))

Theoremmendsca 38076 The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &   𝑆 = (Scalar‘𝑀)       𝑆 = (Scalar‘𝐴)

Theoremmendvscafval 38077* Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &    · = ( ·𝑠𝑀)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (Scalar‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝐸 = (Base‘𝑀)       ( ·𝑠𝐴) = (𝑥𝐾, 𝑦𝐵 ↦ ((𝐸 × {𝑥}) ∘𝑓 · 𝑦))

Theoremmendvsca 38078 A specific scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &    · = ( ·𝑠𝑀)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (Scalar‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝐸 = (Base‘𝑀)    &    = ( ·𝑠𝐴)       ((𝑋𝐾𝑌𝐵) → (𝑋 𝑌) = ((𝐸 × {𝑋}) ∘𝑓 · 𝑌))

Theoremmendring 38079 The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (MEndo‘𝑀)       (𝑀 ∈ LMod → 𝐴 ∈ Ring)

Theoremmendlmod 38080 The module endomorphism algebra is a left module. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &   𝑆 = (Scalar‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)

Theoremmendassa 38081 The module endomorphism algebra is an algebra. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &   𝑆 = (Scalar‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ AssAlg)

20.25.47  Subfields

Syntaxcsdrg 38082 Syntax for subfields (sub-division-rings).
class SubDRing

Definitiondf-sdrg 38083* A sub-division-ring is a subset of a division ring's set which is a division ring under the induced operation. If the overring is commutative this is a field; no special consideration is made of the fields in the center of a skew field. (Contributed by Stefan O'Rear, 3-Oct-2015.)
SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})

Theoremissdrg 38084 Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
(𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))

Theoremissdrg2 38085* Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐼 = (invr𝑅)    &    0 = (0g𝑅)       (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼𝑥) ∈ 𝑆))

Theoremacsfn1p 38086* Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.)
((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎𝑌)𝐸𝑎} ∈ (ACS‘𝑋))

Theoremsubrgacs 38087 Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵))

Theoremsdrgacs 38088 Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ DivRing → (SubDRing‘𝑅) ∈ (ACS‘𝐵))

Theoremcntzsdrg 38089 Centralizers in division rings/fields are subfields. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   𝑍 = (Cntz‘𝑀)       ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubDRing‘𝑅))

20.25.48  Cyclic groups and order

Theoremidomrootle 38090* No element of an integral domain can have more than 𝑁 𝑁-th roots. (Contributed by Stefan O'Rear, 11-Sep-2015.)
𝐵 = (Base‘𝑅)    &    = (.g‘(mulGrp‘𝑅))       ((𝑅 ∈ IDomn ∧ 𝑋𝐵𝑁 ∈ ℕ) → (#‘{𝑦𝐵 ∣ (𝑁 𝑦) = 𝑋}) ≤ 𝑁)

Theoremidomodle 38091* Limit on the number of 𝑁-th roots of unity in an integral domain. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))    &   𝐵 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → (#‘{𝑥𝐵 ∣ (𝑂𝑥) ∥ 𝑁}) ≤ 𝑁)

Theoremfiuneneq 38092 Two finite sets of equal size have a union of the same size iff they were equal. (Contributed by Stefan O'Rear, 12-Sep-2015.)
((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))

Theoremidomsubgmo 38093* The units of an integral domain have at most one subgroup of any single finite cardinality. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Revised by NM, 17-Jun-2017.)
𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))       ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) → ∃*𝑦 ∈ (SubGrp‘𝐺)(#‘𝑦) = 𝑁)

Theoremproot1mul 38094 Any primitive 𝑁-th root of unity is a multiple of any other. (Contributed by Stefan O'Rear, 2-Nov-2015.)
𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))    &   𝑂 = (od‘𝐺)    &   𝐾 = (mrCls‘(SubGrp‘𝐺))       (((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ) ∧ (𝑋 ∈ (𝑂 “ {𝑁}) ∧ 𝑌 ∈ (𝑂 “ {𝑁}))) → 𝑋 ∈ (𝐾‘{𝑌}))

Theoremproot1hash 38095 If an integral domain has a primitive 𝑁-th root of unity, it has exactly (ϕ‘𝑁) of them. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))    &   𝑂 = (od‘𝐺)       ((𝑅 ∈ IDomn ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ (𝑂 “ {𝑁})) → (#‘(𝑂 “ {𝑁})) = (ϕ‘𝑁))

Theoremproot1ex 38096 The complex field has primitive 𝑁-th roots of unity for all 𝑁. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐺 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    &   𝑂 = (od‘𝐺)       (𝑁 ∈ ℕ → (-1↑𝑐(2 / 𝑁)) ∈ (𝑂 “ {𝑁}))

20.25.49  Cyclotomic polynomials

Syntaxccytp 38097 Syntax for the sequence of cyclotomic polynomials.
class CytP

Definitiondf-cytp 38098* The Nth cyclotomic polynomial is the polynomial which has as its zeros precisely the primitive Nth roots of unity. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CytP = (𝑛 ∈ ℕ ↦ ((mulGrp‘(Poly1‘ℂfld)) Σg (𝑟 ∈ ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)))))

Theoremisdomn3 38099 Nonzero elements form a multiplicative submonoid of any domain. (Contributed by Stefan O'Rear, 11-Sep-2015.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑈 = (mulGrp‘𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ Ring ∧ (𝐵 ∖ { 0 }) ∈ (SubMnd‘𝑈)))

Theoremmon1pid 38100 Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑃 = (Poly1𝑅)    &    1 = (1r𝑃)    &   𝑀 = (Monic1p𝑅)    &   𝐷 = ( deg1𝑅)       (𝑅 ∈ NzRing → ( 1𝑀 ∧ (𝐷1 ) = 0))

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 >