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Theorem List for Metamath Proof Explorer - 39601-39700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
20.28.42  Hilbert's Basis Theorem
 
Syntaxcldgis 39601 The leading ideal sequence used in the Hilbert Basis Theorem.
class ldgIdlSeq
 
Definitiondf-ldgis 39602* 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 39610. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
 
Theoremhbtlem1 39603* 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 39604 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 39605 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 39606 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 39607 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 39608* 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 39609* 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 39610 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)
 
20.28.43  Additional material on polynomials [DEPRECATED]
 
Syntaxcmnc 39611 Extend class notation with the class of monic polynomials.
class Monic
 
Syntaxcplylt 39612 Extend class notatin with the class of limited-degree polynomials.
class Poly<
 
Definitiondf-mnc 39613* 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 39614* 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 39615 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 39616 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 39617 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ∈ (Poly‘𝑆))
 
Theoremmnccoe 39618 A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic ‘𝑆) → ((coeff‘𝑃)‘(deg‘𝑃)) = 1)
 
Theoremmncn0 39619 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ≠ 0𝑝)
 
20.28.44  Degree and minimal polynomial of algebraic numbers
 
Syntaxcdgraa 39620 Extend class notation to include the degree function for algebraic numbers.
class degAA
 
Syntaxcmpaa 39621 Extend class notation to include the minimal polynomial for an algebraic number.
class minPolyAA
 
Definitiondf-dgraa 39622* 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 39623* 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 39624* 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 39625* 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 39626 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → (degAA𝐴) ∈ ℕ)
 
Theoremdgraaf 39627 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 39628 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 39629 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 39630* 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 39631* 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 39632 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 39633 Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ (Poly‘ℚ))
 
Theoremmpaadgr 39634 Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → (deg‘(minPolyAA‘𝐴)) = (degAA𝐴))
 
Theoremmpaaroot 39635 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 39636 Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
(𝐴 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝐴))‘(degAA𝐴)) = 1)
 
20.28.45  Algebraic integers I
 
Syntaxcitgo 39637 Extend class notation with the integral-over predicate.
class IntgOver
 
Syntaxcza 39638 Extend class notation with the class of algebraic integers.
class
 
Definitiondf-itgo 39639* 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 39642. 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 39640 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 39641* 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 39642 The standard algebraic numbers 𝔸 are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝔸 = (IntgOver‘ℚ)
 
Theoremitgoss 39643 An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
((𝑆𝑇𝑇 ⊆ ℂ) → (IntgOver‘𝑆) ⊆ (IntgOver‘𝑇))
 
Theoremitgocn 39644 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(IntgOver‘𝑆) ⊆ ℂ
 
Theoremcnsrexpcl 39645 Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
(𝜑𝑆 ∈ (SubRing‘ℂfld))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌 ∈ ℕ0)       (𝜑 → (𝑋𝑌) ∈ 𝑆)
 
Theoremfsumcnsrcl 39646* 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 39647 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 39648* 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 39649 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 39650 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 39651 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 39652 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 39653* 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 39654* 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)
 
20.28.46  Endomorphism algebra
 
Syntaxcmend 39655 Syntax for module endomorphism algebra.
class MEndo
 
Definitiondf-mend 39656* 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 39657 Lemma to shorten proofs of algbase 39658 through algvsca 39662. (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 39658 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 39659 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 39660 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 39661 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 39662 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 39663* 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 39664 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)       (𝑀 LMHom 𝑀) = (Base‘𝐴)
 
Theoremmendplusgfval 39665* Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 3-Mar-2024.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    &    + = (+g𝑀)       (+g𝐴) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥f + 𝑦))
 
Theoremmendplusg 39666 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 39667* Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 3-Mar-2024.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)       (.r𝐴) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦))
 
Theoremmendmulr 39668 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 39669 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 39670* Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 3-Mar-2024.)
𝐴 = (MEndo‘𝑀)    &    · = ( ·𝑠𝑀)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (Scalar‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝐸 = (Base‘𝑀)       ( ·𝑠𝐴) = (𝑥𝐾, 𝑦𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))
 
Theoremmendvsca 39671 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 39672 The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (MEndo‘𝑀)       (𝑀 ∈ LMod → 𝐴 ∈ Ring)
 
Theoremmendlmod 39673 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 39674 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)
 
20.28.47  Cyclic groups and order
 
Theoremidomrootle 39675* 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 39676* 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 39677 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 39678* 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 39679 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 39680 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 39681 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}))
 
20.28.48  Cyclotomic polynomials
 
Syntaxccytp 39682 Syntax for the sequence of cyclotomic polynomials.
class CytP
 
Definitiondf-cytp 39683* 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 39684 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 39685 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))
 
Theoremmon1psubm 39686 Monic polynomials are a multiplicative submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝑃 = (Poly1𝑅)    &   𝑀 = (Monic1p𝑅)    &   𝑈 = (mulGrp‘𝑃)       (𝑅 ∈ NzRing → 𝑀 ∈ (SubMnd‘𝑈))
 
Theoremdeg1mhm 39687 Homomorphic property of the polynomial degree. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐷 = ( deg1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑃)    &   𝑌 = ((mulGrp‘𝑃) ↾s (𝐵 ∖ { 0 }))    &   𝑁 = (ℂflds0)       (𝑅 ∈ Domn → (𝐷 ↾ (𝐵 ∖ { 0 })) ∈ (𝑌 MndHom 𝑁))
 
Theoremcytpfn 39688 Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CytP Fn ℕ
 
Theoremcytpval 39689* Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝑇 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    &   𝑂 = (od‘𝑇)    &   𝑃 = (Poly1‘ℂfld)    &   𝑋 = (var1‘ℂfld)    &   𝑄 = (mulGrp‘𝑃)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)       (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))))
 
20.28.49  Miscellaneous topology
 
Theoremfgraphopab 39690* Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹:𝐴𝐵𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝐴𝑏𝐵) ∧ (𝐹𝑎) = 𝑏)})
 
Theoremfgraphxp 39691* Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹:𝐴𝐵𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st𝑥)) = (2nd𝑥)})
 
Theoremhausgraph 39692 The graph of a continuous function into a Hausdorff space is closed. (Contributed by Stefan O'Rear, 25-Jan-2015.)
((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (Clsd‘(𝐽 ×t 𝐾)))
 
Syntaxctopsep 39693 The class of separable toplogies.
class TopSep
 
Syntaxctoplnd 39694 The class of Lindelöf toplogies.
class TopLnd
 
Definitiondf-topsep 39695* A topology is separable iff it has a countable dense subset. (Contributed by Stefan O'Rear, 8-Jan-2015.)
TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = 𝑗)}
 
Definitiondf-toplnd 39696* A topology is Lindelöf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.)
TopLnd = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 = 𝑦 → ∃𝑧 ∈ 𝒫 𝑥(𝑧 ≼ ω ∧ 𝑥 = 𝑧))}
 
20.29  Mathbox for Jon Pennant
 
Theoremiocunico 39697 Split an open interval into two pieces at point B, Co-author TA. (Contributed by Jon Pennant, 8-Jun-2019.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶))
 
Theoremiocinico 39698 The intersection of two sets that meet at a point is that point. (Contributed by Jon Pennant, 12-Jun-2019.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝐵})
 
Theoremiocmbl 39699 An open-below, closed-above real interval is measurable. (Contributed by Jon Pennant, 12-Jun-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → (𝐴(,]𝐵) ∈ dom vol)
 
Theoremcnioobibld 39700* A bounded, continuous function on an open bounded interval is integrable. The function must be bounded. For a counterexample, consider 𝐹 = (𝑥 ∈ (0(,)1) ↦ (1 / 𝑥)). See cniccibl 24370 for closed bounded intervals. (Contributed by Jon Pennant, 31-May-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹𝑦)) ≤ 𝑥)       (𝜑𝐹 ∈ 𝐿1)
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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 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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