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Theorem List for Metamath Proof Explorer - 37501-37600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlnrring 37501 Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR → 𝐴 ∈ Ring)
 
Theoremlnrlnm 37502 Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)
 
Theoremislnr2 37503* 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 37504 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 37505* 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 37506 Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR → 𝑅 ∈ LNoeR)
 
Theoremlnrfrlm 37507 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 37508 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 37509 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)
 
20.24.42  Hilbert's Basis Theorem
 
Syntaxcldgis 37510 The leading ideal sequence used in the Hilbert Basis Theorem.
class ldgIdlSeq
 
Definitiondf-ldgis 37511* 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 37519. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
 
Theoremhbtlem1 37512* 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 37513 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 37514 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 37515 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 37516 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 37517* 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 37518* 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 37519 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)
 
20.24.43  Additional material on polynomials [DEPRECATED]
 
Syntaxcmnc 37520 Extend class notation with the class of monic polynomials.
class Monic
 
Syntaxcplylt 37521 Extend class notatin with the class of limited-degree polynomials.
class Poly<
 
Definitiondf-mnc 37522* 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 37523* 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 37524 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 37525 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 37526 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ∈ (Poly‘𝑆))
 
Theoremmnccoe 37527 A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic ‘𝑆) → ((coeff‘𝑃)‘(deg‘𝑃)) = 1)
 
Theoremmncn0 37528 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ≠ 0𝑝)
 
20.24.44  Degree and minimal polynomial of algebraic numbers
 
Syntaxcdgraa 37529 Extend class notation to include the degree function for algebraic numbers.
class degAA
 
Syntaxcmpaa 37530 Extend class notation to include the minimal polynomial for an algebraic number.
class minPolyAA
 
Definitiondf-dgraa 37531* 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 37532* 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 37533* 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 37534* 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 37535 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → (degAA𝐴) ∈ ℕ)
 
Theoremdgraaf 37536 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 37537 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 37538 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 37539* 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 37540* 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 37541 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 37542 Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ (Poly‘ℚ))
 
Theoremmpaadgr 37543 Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → (deg‘(minPolyAA‘𝐴)) = (degAA𝐴))
 
Theoremmpaaroot 37544 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 37545 Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝐴))‘(degAA𝐴)) = 1)
 
20.24.45  Algebraic integers I
 
Syntaxcitgo 37546 Extend class notation with the integral-over predicate.
class IntgOver
 
Syntaxcza 37547 Extend class notation with the class of algebraic integers.
class
 
Definitiondf-itgo 37548* 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 37551. 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 37549 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 37550* 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 37551 The standard algebraic numbers 𝔸 are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝔸 = (IntgOver‘ℚ)
 
Theoremitgoss 37552 An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
((𝑆𝑇𝑇 ⊆ ℂ) → (IntgOver‘𝑆) ⊆ (IntgOver‘𝑇))
 
Theoremitgocn 37553 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(IntgOver‘𝑆) ⊆ ℂ
 
Theoremcnsrexpcl 37554 Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
(𝜑𝑆 ∈ (SubRing‘ℂfld))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌 ∈ ℕ0)       (𝜑 → (𝑋𝑌) ∈ 𝑆)
 
Theoremfsumcnsrcl 37555* 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 37556 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 37557* 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 37558 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 37559 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 37560 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 37561 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 37562* 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 37563* 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)
 
20.24.46  Endomorphism algebra
 
Syntaxcmend 37564 Syntax for module endomorphism algebra.
class MEndo
 
Definitiondf-mend 37565* 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 37566 Lemma to shorten proofs of algbase 37567 through algvsca 37571. (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 37567 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 37568 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 37569 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 37570 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 37571 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 37572* 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 37573 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)       (𝑀 LMHom 𝑀) = (Base‘𝐴)
 
Theoremmendplusgfval 37574* 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 37575 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 37576* 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 37577 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 37578 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 37579* 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 37580 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 37581 The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (MEndo‘𝑀)       (𝑀 ∈ LMod → 𝐴 ∈ Ring)
 
Theoremmendlmod 37582 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 37583 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)
 
20.24.47  Subfields
 
Syntaxcsdrg 37584 Syntax for subfields (sub-division-rings).
class SubDRing
 
Definitiondf-sdrg 37585* 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 37586 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 37587* 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 37588* 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 37589 Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵))
 
Theoremsdrgacs 37590 Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ DivRing → (SubDRing‘𝑅) ∈ (ACS‘𝐵))
 
Theoremcntzsdrg 37591 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))
 
20.24.48  Cyclic groups and order
 
Theoremidomrootle 37592* 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 37593* 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 37594 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 37595* 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 37596 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 37597 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 37598 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}))
 
20.24.49  Cyclotomic polynomials
 
Syntaxccytp 37599 Syntax for the sequence of cyclotomic polynomials.
class CytP
 
Definitiondf-cytp 37600* 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)))))
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