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Theorem List for Metamath Proof Explorer - 38201-38300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-za 38201 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 38202* 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 38203 The standard algebraic numbers 𝔸 are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝔸 = (IntgOver‘ℚ)

Theoremitgoss 38204 An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
((𝑆𝑇𝑇 ⊆ ℂ) → (IntgOver‘𝑆) ⊆ (IntgOver‘𝑇))

Theoremitgocn 38205 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(IntgOver‘𝑆) ⊆ ℂ

Theoremcnsrexpcl 38206 Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
(𝜑𝑆 ∈ (SubRing‘ℂfld))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌 ∈ ℕ0)       (𝜑 → (𝑋𝑌) ∈ 𝑆)

Theoremfsumcnsrcl 38207* 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 38208 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 38209* 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 38210 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 38211 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 38212 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 38213 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 38214* 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 38215* 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)

20.25.46  Endomorphism algebra

Syntaxcmend 38216 Syntax for module endomorphism algebra.
class MEndo

Definitiondf-mend 38217* 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 38218 Lemma to shorten proofs of algbase 38219 through algvsca 38223. (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 38219 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 38220 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 38221 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 38222 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 38223 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 38224* 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 38225 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)       (𝑀 LMHom 𝑀) = (Base‘𝐴)

Theoremmendplusgfval 38226* Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    &    + = (+g𝑀)       (+g𝐴) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑓 + 𝑦))

Theoremmendplusg 38227 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 38228* Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)       (.r𝐴) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦))

Theoremmendmulr 38229 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 38230 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 38231* Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)    &    · = ( ·𝑠𝑀)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (Scalar‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝐸 = (Base‘𝑀)       ( ·𝑠𝐴) = (𝑥𝐾, 𝑦𝐵 ↦ ((𝐸 × {𝑥}) ∘𝑓 · 𝑦))

Theoremmendvsca 38232 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 38233 The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (MEndo‘𝑀)       (𝑀 ∈ LMod → 𝐴 ∈ Ring)

Theoremmendlmod 38234 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 38235 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)

20.25.47  Subfields

Syntaxcsdrg 38236 Syntax for subfields (sub-division-rings).
class SubDRing

Definitiondf-sdrg 38237* A sub-division-ring is a subset of a division ring's set which is a division ring under the induced operation. If the overring is commutative this is a field; no special consideration is made of the fields in the center of a skew field. (Contributed by Stefan O'Rear, 3-Oct-2015.)
SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤s 𝑠) ∈ DivRing})

Theoremissdrg 38238 Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
(𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅s 𝑆) ∈ DivRing))

Theoremissdrg2 38239* Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐼 = (invr𝑅)    &    0 = (0g𝑅)       (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼𝑥) ∈ 𝑆))

Theoremacsfn1p 38240* Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.)
((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎𝑌)𝐸𝑎} ∈ (ACS‘𝑋))

Theoremsubrgacs 38241 Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵))

Theoremsdrgacs 38242 Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ DivRing → (SubDRing‘𝑅) ∈ (ACS‘𝐵))

Theoremcntzsdrg 38243 Centralizers in division rings/fields are subfields. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   𝑍 = (Cntz‘𝑀)       ((𝑅 ∈ DivRing ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubDRing‘𝑅))

20.25.48  Cyclic groups and order

Theoremidomrootle 38244* 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 38245* 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 38246 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 38247* 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 38248 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 38249 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 38250 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}))

20.25.49  Cyclotomic polynomials

Syntaxccytp 38251 Syntax for the sequence of cyclotomic polynomials.
class CytP

Definitiondf-cytp 38252* 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 38253 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 38254 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 38255 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 38256 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 38257 Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CytP Fn ℕ

Theoremcytpval 38258* 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)))))

20.25.50  Miscellaneous topology

Theoremfgraphopab 38259* 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 38260* 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 38261 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 38262 The class of separable toplogies.
class TopSep

Syntaxctoplnd 38263 The class of Lindelöf toplogies.
class TopLnd

Definitiondf-topsep 38264* 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 38265* 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))}

20.26  Mathbox for Jon Pennant

Theoremioounsn 38266 The closure of the upper end of an open real interval. (Contributed by Jon Pennant, 8-Jun-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))

Theoremiocunico 38267 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 38268 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 38269 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 38270* 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 23777 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)

Theoremitgpowd 38271* The integral of a monomial on a closed bounded interval of the real line. Co-authors TA and MC. (Contributed by Jon Pennant, 31-May-2019.) (Revised by Thierry Arnoux, 14-Jun-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ∫(𝐴[,]𝐵)(𝑥𝑁) d𝑥 = (((𝐵↑(𝑁 + 1)) − (𝐴↑(𝑁 + 1))) / (𝑁 + 1)))

Theoremarearect 38272 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    &   𝐴𝐵    &   𝐶𝐷    &   𝑆 = ((𝐴[,]𝐵) × (𝐶[,]𝐷))       (area‘𝑆) = ((𝐵𝐴) · (𝐷𝐶))

Theoremareaquad 38273* 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    &   𝐶𝐸    &   𝐷𝐹    &   𝑈 = (𝐶 + (((𝑥𝐴) / (𝐵𝐴)) · (𝐷𝐶)))    &   𝑉 = (𝐸 + (((𝑥𝐴) / (𝐵𝐴)) · (𝐹𝐸)))    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝑈[,]𝑉))}       (area‘𝑆) = ((((𝐹 + 𝐸) / 2) − ((𝐷 + 𝐶) / 2)) · (𝐵𝐴))

20.27  Mathbox for Richard Penner

20.27.1  Short Studies

20.27.1.1  Additional work on conditional logical operator

Theoremifpan123g 38274 Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑𝜒) ∧ (𝜑𝜏)) ∧ ((¬ 𝜓𝜃) ∧ (𝜓𝜂))))

Theoremifpan23 38275 Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓𝜃), (𝜒𝜏)))

Theoremifpdfor2 38276 Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜑, 𝜓))

Theoremifporcor 38277 Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)
(if-(𝜑, 𝜑, 𝜓) ↔ if-(𝜓, 𝜓, 𝜑))

Theoremifpdfan2 38278 Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, 𝜑))

Theoremifpancor 38279 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))

Theoremifpdfor 38280 Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, ⊤, 𝜓))

Theoremifpdfan 38281 Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ⊥))

Theoremifpbi2 38282 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))

Theoremifpbi3 38283 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))

Theoremifpim1 38284 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))

Theoremifpnot 38285 Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
𝜑 ↔ if-(𝜑, ⊥, ⊤))

Theoremifpid2 38286 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
(𝜑 ↔ if-(𝜑, ⊤, ⊥))

Theoremifpim2 38287 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))

Theoremifpbi23 38288 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))

Theoremifpdfbi 38289 Define biimplication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))

Theoremifpbiidcor 38290 Restatement of biid 251. (Contributed by RP, 25-Apr-2020.)
if-(𝜑, 𝜑, ¬ 𝜑)

Theoremifpbicor 38291 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑))

Theoremifpxorcor 38292 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑))

Theoremifpbi1 38293 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)))

Theoremifpnot23 38294 Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
(¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))

Theoremifpnotnotb 38295 Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ ¬ if-(𝜑, 𝜓, 𝜒))

Theoremifpnorcor 38296 Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜑, ¬ 𝜓) ↔ if-(𝜓, ¬ 𝜓, ¬ 𝜑))

Theoremifpnancor 38297 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜓, ¬ 𝜑) ↔ if-(𝜓, ¬ 𝜑, ¬ 𝜓))

Theoremifpnot23b 38298 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒))

Theoremifpbiidcor2 38299 Restatement of biid 251. (Contributed by RP, 25-Apr-2020.)
¬ if-(𝜑, ¬ 𝜑, 𝜑)

Theoremifpnot23c 38300 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, 𝜒))

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