P. 432 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43101-43200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hbt 43101	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 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ LNoeR)
21.33.43  Additional material on polynomials [DEPRECATED]
Syntax	cmnc 43102	Extend class notation with the class of monic polynomials.
class Monic
Syntax	cplylt 43103	Extend class notation with the class of limited-degree polynomials.
class Poly<
Definition	df-mnc 43104*	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})
Definition	df-plylt 43105*	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)})
Theorem	dgrsub2 43106	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 43107	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 43108	A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ∈ (Poly‘𝑆))
Theorem	mnccoe 43109	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 43110	A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑃 ≠ 0𝑝)
21.33.44  Degree and minimal polynomial of algebraic numbers
Syntax	cdgraa 43111	Extend class notation to include the degree function for algebraic numbers.
class degAA
Syntax	cmpaa 43112	Extend class notation to include the minimal polynomial for an algebraic number.
class minPolyAA
Definition	df-dgraa 43113*	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 43114*	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 43115*	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 43116*	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 43117	Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) ∈ ℕ)
Theorem	dgraaf 43118	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 43119	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 43120	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 43121*	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 43122*	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 43123	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 43124	Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ (Poly‘ℚ))
Theorem	mpaadgr 43125	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 43126	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 43127	Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1)
21.33.45  Algebraic integers I
Syntax	citgo 43128	Extend class notation with the integral-over predicate.
class IntgOver
Syntax	cza 43129	Extend class notation with the class of algebraic integers.
class ℤ
Definition	df-itgo 43130*	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 43133. 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 43131	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 43132*	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 43133	The standard algebraic numbers 𝔸 are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝔸 = (IntgOver‘ℚ)
Theorem	itgoss 43134	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 43135	All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ (IntgOver‘𝑆) ⊆ ℂ
Theorem	cnsrexpcl 43136	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 43137*	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 43138	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	rgspnid 43139	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 43140*	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 43141*	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)
21.33.46  Endomorphism algebra
Syntax	cmend 43142	Syntax for module endomorphism algebra.
class MEndo
Definition	df-mend 43143*	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 43144	Lemma to shorten proofs of algbase 43145 through algvsca 43149. (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 43145	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 43146	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 43147	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 43148	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 43149	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 43150*	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 43151	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 43152*	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 43153	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 43154*	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 43155	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 43156	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 43157*	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 43158	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 43159	The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    ⇒   ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)
Theorem	mendlmod 43160	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 43161	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)
21.33.47  Cyclic groups and order
Theorem	idomodle 43162*	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 43163	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 43164*	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 43165	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 43166	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 43167	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}))
21.33.48  Cyclotomic polynomials
Syntax	ccytp 43168	Syntax for the sequence of cyclotomic polynomials.
class CytP
Definition	df-cytp 43169*	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 43170	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 43171	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 43172	Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ CytP Fn ℕ
Theorem	cytpval 43173*	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)))))
21.33.49  Miscellaneous topology
Theorem	fgraphopab 43174*	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 43175*	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 43176	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 43177	The class of separable topologies.
class TopSep
Syntax	ctoplnd 43178	The class of Lindelöf topologies.
class TopLnd
Definition	df-topsep 43179*	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 43180*	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))}
21.34  Mathbox for Noam Pasman
Theorem	r1sssucd 43181	Deductive form of r1sssuc 9795. (Contributed by Noam Pasman, 19-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ On)    ⇒   ⊢ (𝜑 → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴))
21.35  Mathbox for Jon Pennant
Theorem	iocunico 43182	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 43183	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 43184	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 43185*	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 25792 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 43186	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 43187*	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))
21.36  Mathbox for Richard Penner
21.36.1  Set Theory and Ordinal Numbers
Theorem	uniel 43188*	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 43189*	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 43190*	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 43191*	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 43192*	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 43193*	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 43194*	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 43195	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 43196	The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 ∈ On)
Theorem	onuniintrab 43197*	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 7793. (Contributed by RP, 28-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	onintunirab 43198*	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 43199	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 43200	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)