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Type | Label | Description |
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Statement | ||
Theorem | dgrsub2 42701 | 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) = 𝑁 ∧ 𝑁 ∈ ℕ ∧ ((coeff‘𝐹)‘𝑁) = ((coeff‘𝐺)‘𝑁))) → (deg‘(𝐹 ∘f − 𝐺)) < 𝑁) | ||
Theorem | elmnc 42702 | Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
⊢ (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1)) | ||
Theorem | mncply 42703 | A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ∈ (Poly‘𝑆)) | ||
Theorem | mnccoe 42704 | A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
⊢ (𝑃 ∈ ( Monic ‘𝑆) → ((coeff‘𝑃)‘(deg‘𝑃)) = 1) | ||
Theorem | mncn0 42705 | A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ≠ 0𝑝) | ||
Syntax | cdgraa 42706 | Extend class notation to include the degree function for algebraic numbers. |
class degAA | ||
Syntax | cmpaa 42707 | Extend class notation to include the minimal polynomial for an algebraic number. |
class minPolyAA | ||
Definition | df-dgraa 42708* | 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 ∧ (𝑝‘𝑥) = 0)}, ℝ, < )) | ||
Definition | df-mpaa 42709* | 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) = (degAA‘𝑥) ∧ (𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑥)) = 1))) | ||
Theorem | dgraaval 42710* | 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 → (degAA‘𝐴) = inf({𝑑 ∈ ℕ ∣ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝐴) = 0)}, ℝ, < )) | ||
Theorem | dgraalem 42711* | 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 → ((degAA‘𝐴) ∈ ℕ ∧ ∃𝑝 ∈ ((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0))) | ||
Theorem | dgraacl 42712 | Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) ∈ ℕ) | ||
Theorem | dgraaf 42713 | 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 | ||
Theorem | dgraaub 42714 | 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 ∧ (𝑃‘𝐴) = 0)) → (degAA‘𝐴) ≤ (deg‘𝑃)) | ||
Theorem | dgraa0p 42715 | 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) < (degAA‘𝐴)) → ((𝑃‘𝐴) = 0 ↔ 𝑃 = 0𝑝)) | ||
Theorem | mpaaeu 42716* | 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) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)) | ||
Theorem | mpaaval 42717* | 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) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))) | ||
Theorem | mpaalem 42718 | 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)) = (degAA‘𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1))) | ||
Theorem | mpaacl 42719 | Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ (Poly‘ℚ)) | ||
Theorem | mpaadgr 42720 | Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → (deg‘(minPolyAA‘𝐴)) = (degAA‘𝐴)) | ||
Theorem | mpaaroot 42721 | 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) | ||
Theorem | mpaamn 42722 | Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
⊢ (𝐴 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1) | ||
Syntax | citgo 42723 | Extend class notation with the integral-over predicate. |
class IntgOver | ||
Syntax | cza 42724 | Extend class notation with the class of algebraic integers. |
class ℤ | ||
Definition | df-itgo 42725* | 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 42728. 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)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}) | ||
Definition | df-za 42726 | 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.) |
⊢ ℤ = (IntgOver‘ℤ) | ||
Theorem | itgoval 42727* | Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑥 ∈ ℂ ∣ ∃𝑝 ∈ (Poly‘𝑆)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}) | ||
Theorem | aaitgo 42728 | The standard algebraic numbers 𝔸 are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝔸 = (IntgOver‘ℚ) | ||
Theorem | itgoss 42729 | An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (IntgOver‘𝑆) ⊆ (IntgOver‘𝑇)) | ||
Theorem | itgocn 42730 | All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ (IntgOver‘𝑆) ⊆ ℂ | ||
Theorem | cnsrexpcl 42731 | Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑋↑𝑌) ∈ 𝑆) | ||
Theorem | fsumcnsrcl 42732* | Finite sums are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) | ||
Theorem | cnsrplycl 42733 | Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) & ⊢ (𝜑 → 𝑃 ∈ (Poly‘𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ⊆ 𝑆) ⇒ ⊢ (𝜑 → (𝑃‘𝑋) ∈ 𝑆) | ||
Theorem | rgspnval 42734* | Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) & ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) ⇒ ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡}) | ||
Theorem | rgspncl 42735 | The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) & ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) ⇒ ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑅)) | ||
Theorem | rgspnssid 42736 | The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) & ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝑈) | ||
Theorem | rgspnmin 42737 | The ring-span is contained in all subspaces which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) & ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) ⇒ ⊢ (𝜑 → 𝑈 ⊆ 𝑆) | ||
Theorem | rgspnid 42738 | The span of a subring is itself. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑆 = ((RingSpan‘𝑅)‘𝐴)) ⇒ ⊢ (𝜑 → 𝑆 = 𝐴) | ||
Theorem | rngunsnply 42739* | Adjoining one element to a ring results in a set of polynomial evaluations. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ (𝜑 → 𝐵 ∈ (SubRing‘ℂfld)) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑆 = ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))) ⇒ ⊢ (𝜑 → (𝑉 ∈ 𝑆 ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝‘𝑋))) | ||
Theorem | flcidc 42740* | Finite linear combinations with an indicator function. (Contributed by Stefan O'Rear, 5-Dec-2014.) |
⊢ (𝜑 → 𝐹 = (𝑗 ∈ 𝑆 ↦ if(𝑗 = 𝐾, 1, 0))) & ⊢ (𝜑 → 𝑆 ∈ Fin) & ⊢ (𝜑 → 𝐾 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑆) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑖 ∈ 𝑆 ((𝐹‘𝑖) · 𝐵) = ⦋𝐾 / 𝑖⦌𝐵) | ||
Syntax | cmend 42741 | Syntax for module endomorphism algebra. |
class MEndo | ||
Definition | df-mend 42742* | 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), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 ∘f (+g‘𝑚)𝑦))〉, 〈(.r‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 ∘ 𝑦))〉} ∪ {〈(Scalar‘ndx), (Scalar‘𝑚)〉, 〈( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦 ∈ 𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘f ( ·𝑠 ‘𝑚)𝑦))〉})) | ||
Theorem | algstr 42743 | Lemma to shorten proofs of algbase 42744 through algvsca 42748. (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〉 | ||
Theorem | algbase 42744 | 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), · 〉}) ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐴)) | ||
Theorem | algaddg 42745 | 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 → + = (+g‘𝐴)) | ||
Theorem | algmulr 42746 | 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 → × = (.r‘𝐴)) | ||
Theorem | algsca 42747 | 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), · 〉}) ⇒ ⊢ (𝑆 ∈ 𝑉 → 𝑆 = (Scalar‘𝐴)) | ||
Theorem | algvsca 42748 | 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 → · = ( ·𝑠 ‘𝐴)) | ||
Theorem | mendval 42749* | Value of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
⊢ 𝐵 = (𝑀 LMHom 𝑀) & ⊢ + = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f (+g‘𝑀)𝑦)) & ⊢ × = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦)) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ · = (𝑥 ∈ (Base‘𝑆), 𝑦 ∈ 𝐵 ↦ (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦)) ⇒ ⊢ (𝑀 ∈ 𝑋 → (MEndo‘𝑀) = ({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), × 〉} ∪ {〈(Scalar‘ndx), 𝑆〉, 〈( ·𝑠 ‘ndx), · 〉})) | ||
Theorem | mendbas 42750 | Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
⊢ 𝐴 = (MEndo‘𝑀) ⇒ ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴) | ||
Theorem | mendplusgfval 42751* | Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.) |
⊢ 𝐴 = (MEndo‘𝑀) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ + = (+g‘𝑀) ⇒ ⊢ (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦)) | ||
Theorem | mendplusg 42752 | A specific addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.) |
⊢ 𝐴 = (MEndo‘𝑀) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ + = (+g‘𝑀) & ⊢ ✚ = (+g‘𝐴) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) = (𝑋 ∘f + 𝑌)) | ||
Theorem | mendmulrfval 42753* | Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.) |
⊢ 𝐴 = (MEndo‘𝑀) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (.r‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦)) | ||
Theorem | mendmulr 42754 | A specific multiplication in the module endormoprhism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.) |
⊢ 𝐴 = (MEndo‘𝑀) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = (.r‘𝐴) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (𝑋 ∘ 𝑌)) | ||
Theorem | mendsca 42755 | The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.) |
⊢ 𝐴 = (MEndo‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) ⇒ ⊢ 𝑆 = (Scalar‘𝐴) | ||
Theorem | mendvscafval 42756* | Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.) |
⊢ 𝐴 = (MEndo‘𝑀) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐸 = (Base‘𝑀) ⇒ ⊢ ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦)) | ||
Theorem | mendvsca 42757 | A specific scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 3-Sep-2015.) |
⊢ 𝐴 = (MEndo‘𝑀) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐸 = (Base‘𝑀) & ⊢ ∙ = ( ·𝑠 ‘𝐴) ⇒ ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = ((𝐸 × {𝑋}) ∘f · 𝑌)) | ||
Theorem | mendring 42758 | The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐴 = (MEndo‘𝑀) ⇒ ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring) | ||
Theorem | mendlmod 42759 | 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) | ||
Theorem | mendassa 42760 | 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) | ||
Theorem | idomodle 42761* | 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) → (♯‘{𝑥 ∈ 𝐵 ∣ (𝑂‘𝑥) ∥ 𝑁}) ≤ 𝑁) | ||
Theorem | fiuneneq 42762 | 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.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → ((𝐴 ∪ 𝐵) ≈ 𝐴 ↔ 𝐴 = 𝐵)) | ||
Theorem | idomsubgmo 42763* | 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) | ||
Theorem | proot1mul 42764 | 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})) | ||
Theorem | proot1hash 42765 | 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)) | ||
Theorem | proot1ex 42766 | 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})) | ||
Syntax | ccytp 42767 | Syntax for the sequence of cyclotomic polynomials. |
class CytP | ||
Definition | df-cytp 42768* | 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))))) | ||
Theorem | mon1psubm 42769 | Monic polynomials are a multiplicative submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑀 = (Monic1p‘𝑅) & ⊢ 𝑈 = (mulGrp‘𝑃) ⇒ ⊢ (𝑅 ∈ NzRing → 𝑀 ∈ (SubMnd‘𝑈)) | ||
Theorem | deg1mhm 42770 | Homomorphic property of the polynomial degree. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
⊢ 𝐷 = ( deg1 ‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘𝑃) & ⊢ 𝑌 = ((mulGrp‘𝑃) ↾s (𝐵 ∖ { 0 })) & ⊢ 𝑁 = (ℂfld ↾s ℕ0) ⇒ ⊢ (𝑅 ∈ Domn → (𝐷 ↾ (𝐵 ∖ { 0 })) ∈ (𝑌 MndHom 𝑁)) | ||
Theorem | cytpfn 42771 | Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ CytP Fn ℕ | ||
Theorem | cytpval 42772* | 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) & ⊢ − = (-g‘𝑃) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (◡𝑂 “ {𝑁}) ↦ (𝑋 − (𝐴‘𝑟))))) | ||
Theorem | fgraphopab 42773* | Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐹‘𝑎) = 𝑏)}) | ||
Theorem | fgraphxp 42774* | Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 = {𝑥 ∈ (𝐴 × 𝐵) ∣ (𝐹‘(1st ‘𝑥)) = (2nd ‘𝑥)}) | ||
Theorem | hausgraph 42775 | 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))) | ||
Syntax | ctopsep 42776 | The class of separable topologies. |
class TopSep | ||
Syntax | ctoplnd 42777 | The class of Lindelöf topologies. |
class TopLnd | ||
Definition | df-topsep 42778* | 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)} | ||
Definition | df-toplnd 42779* | 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))} | ||
Theorem | r1sssucd 42780 | Deductive form of r1sssuc 9808. (Contributed by Noam Pasman, 19-Jan-2025.) |
⊢ (𝜑 → 𝐴 ∈ On) ⇒ ⊢ (𝜑 → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴)) | ||
Theorem | iocunico 42781 | 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 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)) | ||
Theorem | iocinico 42782 | 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 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝐵}) | ||
Theorem | iocmbl 42783 | An open-below, closed-above real interval is measurable. (Contributed by Jon Pennant, 12-Jun-2019.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ∈ dom vol) | ||
Theorem | cnioobibld 42784* | 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 25814 for closed bounded intervals. (Contributed by Jon Pennant, 31-May-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝐿1) | ||
Theorem | arearect 42785 | The area of a rectangle whose sides are parallel to the coordinate axes in (ℝ × ℝ) is its width multiplied by its height. (Contributed by Jon Pennant, 19-Mar-2019.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ & ⊢ 𝐷 ∈ ℝ & ⊢ 𝐴 ≤ 𝐵 & ⊢ 𝐶 ≤ 𝐷 & ⊢ 𝑆 = ((𝐴[,]𝐵) × (𝐶[,]𝐷)) ⇒ ⊢ (area‘𝑆) = ((𝐵 − 𝐴) · (𝐷 − 𝐶)) | ||
Theorem | areaquad 42786* | The area of a quadrilateral with two sides which are parallel to the y-axis in (ℝ × ℝ) is its width multiplied by the average height of its higher edge minus the average height of its lower edge. Co-author TA. (Contributed by Jon Pennant, 31-May-2019.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ & ⊢ 𝐷 ∈ ℝ & ⊢ 𝐸 ∈ ℝ & ⊢ 𝐹 ∈ ℝ & ⊢ 𝐴 < 𝐵 & ⊢ 𝐶 ≤ 𝐸 & ⊢ 𝐷 ≤ 𝐹 & ⊢ 𝑈 = (𝐶 + (((𝑥 − 𝐴) / (𝐵 − 𝐴)) · (𝐷 − 𝐶))) & ⊢ 𝑉 = (𝐸 + (((𝑥 − 𝐴) / (𝐵 − 𝐴)) · (𝐹 − 𝐸))) & ⊢ 𝑆 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝑈[,]𝑉))} ⇒ ⊢ (area‘𝑆) = ((((𝐹 + 𝐸) / 2) − ((𝐷 + 𝐶) / 2)) · (𝐵 − 𝐴)) | ||
Theorem | uniel 42787* | Two ways to say a union is an element of a class. (Contributed by RP, 27-Jan-2025.) |
⊢ (∪ 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)) | ||
Theorem | unielss 42788* | Two ways to say the union of a class is an element of a subclass. (Contributed by RP, 29-Jan-2025.) |
⊢ (𝐴 ⊆ 𝐵 → (∪ 𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)) | ||
Theorem | unielid 42789* | Two ways to say the union of a class is an element of that class. (Contributed by RP, 27-Jan-2025.) |
⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) | ||
Theorem | ssunib 42790* | Two ways to say a class is a subclass of a union. (Contributed by RP, 27-Jan-2025.) |
⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦) | ||
Theorem | rp-intrabeq 42791* | Equality theorem for supremum of sets of ordinals. (Contributed by RP, 23-Jan-2025.) |
⊢ (𝐴 = 𝐵 → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥}) | ||
Theorem | rp-unirabeq 42792* | Equality theorem for infimum of non-empty classes of ordinals. (Contributed by RP, 23-Jan-2025.) |
⊢ (𝐴 = 𝐵 → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦}) | ||
Theorem | onmaxnelsup 42793* | Two ways to say the maximum element of a class of ordinals is also the supremum of that class. (Contributed by RP, 27-Jan-2025.) |
⊢ (𝐴 ⊆ On → (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)) | ||
Theorem | onsupneqmaxlim0 42794 | If the supremum of a class of ordinals is not in that class, then the supremum is a limit ordinal or empty. (Contributed by RP, 27-Jan-2025.) |
⊢ (𝐴 ⊆ On → (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)) | ||
Theorem | onsupcl2 42795 | The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.) |
⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 ∈ On) | ||
Theorem | onuniintrab 42796* | The union of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Closed form of uniordint 7805. (Contributed by RP, 28-Jan-2025.) |
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}) | ||
Theorem | onintunirab 42797* | The intersection of a non-empty class of ordinals is the union of every ordinal less-than-or-equal to every element of that class. (Contributed by RP, 29-Jan-2025.) |
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦}) | ||
Theorem | onsupnmax 42798 | If the union of a class of ordinals is not the maximum element of that class, then the union is a limit ordinal or empty. But this isn't a biconditional since 𝐴 could be a non-empty set where a limit ordinal or the empty set happens to be the largest element. (Contributed by RP, 27-Jan-2025.) |
⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)) | ||
Theorem | onsupuni 42799 | The supremum of a set of ordinals is the union of that set. Lemma 2.10 of [Schloeder] p. 5. (Contributed by RP, 19-Jan-2025.) |
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴) | ||
Theorem | onsupuni2 42800 | The supremum of a set of ordinals is the union of that set. (Contributed by RP, 22-Jan-2025.) |
⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∪ 𝐴) |
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