P. 442 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44101-44200   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	cmpaa 44101	Extend class notation to include the minimal polynomial for an algebraic number.
class minPolyAA
Definition	df-dgraa 44102*	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 44103*	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 44104*	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 44105*	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 44106	Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → (degAA‘𝐴) ∈ ℕ)
Theorem	dgraaf 44107	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 44108	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 44109	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 44110*	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 44111*	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 44112	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 44113	Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ (Poly‘ℚ))
Theorem	mpaadgr 44114	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 44115	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 44116	Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)
⊢ (𝐴 ∈ 𝔸 → ((coeff‘(minPolyAA‘𝐴))‘(degAA‘𝐴)) = 1)
21.35.45  Algebraic integers I
Syntax	citgo 44117	Extend class notation with the integral-over predicate.
class IntgOver
Syntax	cza 44118	Extend class notation with the class of algebraic integers.
class ℤ
Definition	df-itgo 44119*	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 44122. 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 44120	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 44121*	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 44122	The standard algebraic numbers 𝔸 are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝔸 = (IntgOver‘ℚ)
Theorem	itgoss 44123	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 44124	All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ (IntgOver‘𝑆) ⊆ ℂ
Theorem	cnsrexpcl 44125	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 44126*	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 44127	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 44128	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 44129*	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 44130*	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)
21.35.46  Endomorphism algebra
Syntax	cmend 44131	Syntax for module endomorphism algebra.
class MEndo
Definition	df-mend 44132*	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 44133	Lemma to shorten proofs of algbase 44134 through algvsca 44138. (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 44134	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 44135	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 44136	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 44137	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 44138	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 44139*	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 44140	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 44141*	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 44142	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 44143*	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 44144	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 44145	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 44146*	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 44147	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 44148	The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐴 = (MEndo‘𝑀)    ⇒   ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)
Theorem	mendlmod 44149	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 44150	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)
21.35.47  Cyclic groups and order
Theorem	idomodle 44151*	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 44152	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 44153*	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 44154	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 44155	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 44156	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}))
21.35.48  Cyclotomic polynomials
Syntax	ccytp 44157	Syntax for the sequence of cyclotomic polynomials.
class CytP
Definition	df-cytp 44158*	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 44159	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 44160	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 44161	Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ CytP Fn ℕ
Theorem	cytpval 44162*	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)))))
21.35.49  Miscellaneous topology
Theorem	fgraphopab 44163*	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 44164*	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 44165	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 44166	The class of separable topologies.
class TopSep
Syntax	ctoplnd 44167	The class of Lindelöf topologies.
class TopLnd
Definition	df-topsep 44168*	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 44169*	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))}
21.36  Mathbox for Noam Pasman
Theorem	r1sssucd 44170	Deductive form of r1sssuc 9773. (Contributed by Noam Pasman, 19-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ On)    ⇒   ⊢ (𝜑 → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴))
21.37  Mathbox for Jon Pennant
Theorem	iocunico 44171	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 44172	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 44173	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 44174*	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 26141 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 44175	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 44176*	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))
21.38  Mathbox for Richard Penner
21.38.1  Set Theory and Ordinal Numbers
Theorem	uniel 44177*	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 44178*	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 44179*	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 44180*	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 44181*	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 44182*	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 44183*	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 44184	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 44185	The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 ∈ On)
Theorem	onuniintrab 44186*	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 7804. (Contributed by RP, 28-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	onintunirab 44187*	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 44188	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 44189	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 44190	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)
Theorem	onsupintrab 44191*	The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Definition 2.9 of [Schloeder] p. 5. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	onsupintrab2 44192*	The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	onsupcl3 44193*	The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} ∈ On)
Theorem	onsupex3 44194*	The supremum of a set of ordinals exists. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} ∈ V)
Theorem	onuniintrab2 44195*	The union of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
Theorem	oninfint 44196	The infimum of a non-empty class of ordinals is the intersection of that class. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = ∩ 𝐴)
Theorem	oninfunirab 44197*	The infimum 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, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})
Theorem	oninfcl2 44198*	The infimum of a non-empty class of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ On)
Theorem	onsupmaxb 44199	The union of a class of ordinals is an element is an element of that class if and only if there is a maximum element of that class under the epsilon relation, which is to say that the domain of the restricted epsilon relation is not the whole class. (Contributed by RP, 25-Jan-2025.)
⊢ (𝐴 ⊆ On → (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝐴))
Theorem	onexgt 44200*	For any ordinal, there is always a larger ordinal. (Contributed by RP, 1-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)