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Theorem List for Metamath Proof Explorer - 40101-40200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrgspnssid 40101 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 40102 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 40103 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 40104* 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 40105* 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)
 
20.29.46  Endomorphism algebra
 
Syntaxcmend 40106 Syntax for module endomorphism algebra.
class MEndo
 
Definitiondf-mend 40107* 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 40108 Lemma to shorten proofs of algbase 40109 through algvsca 40113. (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 40109 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 40110 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 40111 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 40112 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 40113 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 40114* 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 40115 Base set of the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐴 = (MEndo‘𝑀)       (𝑀 LMHom 𝑀) = (Base‘𝐴)
 
Theoremmendplusgfval 40116* 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 40117 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 40118* 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 40119 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 40120 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 40121* 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 40122 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 40123 The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (MEndo‘𝑀)       (𝑀 ∈ LMod → 𝐴 ∈ Ring)
 
Theoremmendlmod 40124 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 40125 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)
 
20.29.47  Cyclic groups and order
 
Theoremidomrootle 40126* 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 40127* 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 40128 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 40129* 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 40130 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 40131 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 40132 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}))
 
20.29.48  Cyclotomic polynomials
 
Syntaxccytp 40133 Syntax for the sequence of cyclotomic polynomials.
class CytP
 
Definitiondf-cytp 40134* 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 40135 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 40136 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 40137 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 40138 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 40139 Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CytP Fn ℕ
 
Theoremcytpval 40140* 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)))))
 
20.29.49  Miscellaneous topology
 
Theoremfgraphopab 40141* 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 40142* 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 40143 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 40144 The class of separable toplogies.
class TopSep
 
Syntaxctoplnd 40145 The class of Lindelöf toplogies.
class TopLnd
 
Definitiondf-topsep 40146* 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 40147* 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))}
 
20.30  Mathbox for Jon Pennant
 
Theoremiocunico 40148 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 40149 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 40150 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 40151* 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 24447 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)
 
Theoremarearect 40152 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 40153* 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))
 
20.31  Mathbox for Richard Penner
 
20.31.1  Short Studies
 
20.31.1.1  Additional work on conditional logical operator
 
Theoremifpan123g 40154 Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑𝜒) ∧ (𝜑𝜏)) ∧ ((¬ 𝜓𝜃) ∧ (𝜓𝜂))))
 
Theoremifpan23 40155 Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
((if-(𝜑, 𝜓, 𝜒) ∧ if-(𝜑, 𝜃, 𝜏)) ↔ if-(𝜑, (𝜓𝜃), (𝜒𝜏)))
 
Theoremifpdfor2 40156 Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜑, 𝜓))
 
Theoremifporcor 40157 Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)
(if-(𝜑, 𝜑, 𝜓) ↔ if-(𝜓, 𝜓, 𝜑))
 
Theoremifpdfan2 40158 Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, 𝜑))
 
Theoremifpancor 40159 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜓, 𝜑) ↔ if-(𝜓, 𝜑, 𝜓))
 
Theoremifpdfor 40160 Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, ⊤, 𝜓))
 
Theoremifpdfan 40161 Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ⊥))
 
Theoremifpbi2 40162 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜒, 𝜑, 𝜃) ↔ if-(𝜒, 𝜓, 𝜃)))
 
Theoremifpbi3 40163 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓)))
 
Theoremifpim1 40164 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓))
 
Theoremifpnot 40165 Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
𝜑 ↔ if-(𝜑, ⊥, ⊤))
 
Theoremifpid2 40166 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
(𝜑 ↔ if-(𝜑, ⊤, ⊥))
 
Theoremifpim2 40167 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑))
 
Theoremifpbi23 40168 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃)))
 
Theoremifpbiidcor 40169 Restatement of biid 264. (Contributed by RP, 25-Apr-2020.)
if-(𝜑, 𝜑, ¬ 𝜑)
 
Theoremifpbicor 40170 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜓, ¬ 𝜓) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
 
Theoremifpxorcor 40171 Corollary of commutation of biimplication. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜓, 𝜓) ↔ if-(𝜓, ¬ 𝜑, 𝜑))
 
Theoremifpbi1 40172 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)))
 
Theoremifpnot23 40173 Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
(¬ if-(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, ¬ 𝜓, ¬ 𝜒))
 
Theoremifpnotnotb 40174 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 40175 Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜑, ¬ 𝜓) ↔ if-(𝜓, ¬ 𝜓, ¬ 𝜑))
 
Theoremifpnancor 40176 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, ¬ 𝜓, ¬ 𝜑) ↔ if-(𝜓, ¬ 𝜑, ¬ 𝜓))
 
Theoremifpnot23b 40177 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, ¬ 𝜓, 𝜒) ↔ if-(𝜑, 𝜓, ¬ 𝜒))
 
Theoremifpbiidcor2 40178 Restatement of biid 264. (Contributed by RP, 25-Apr-2020.)
¬ if-(𝜑, ¬ 𝜑, 𝜑)
 
Theoremifpnot23c 40179 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, 𝜓, ¬ 𝜒) ↔ if-(𝜑, ¬ 𝜓, 𝜒))
 
Theoremifpnot23d 40180 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(¬ if-(𝜑, ¬ 𝜓, ¬ 𝜒) ↔ if-(𝜑, 𝜓, 𝜒))
 
Theoremifpdfnan 40181 Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, ⊤))
 
Theoremifpdfxor 40182 Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜓))
 
Theoremifpbi12 40183 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜏)))
 
Theoremifpbi13 40184 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃)))
 
Theoremifpbi123 40185 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → (if-(𝜑, 𝜒, 𝜏) ↔ if-(𝜓, 𝜃, 𝜂)))
 
Theoremifpidg 40186 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜃 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑𝜓) → 𝜃) ∧ ((𝜑𝜃) → 𝜓)) ∧ ((𝜒 → (𝜑𝜃)) ∧ (𝜃 → (𝜑𝜒)))))
 
Theoremifpid3g 40187 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑𝜓) → 𝜒) ∧ ((𝜑𝜒) → 𝜓)))
 
Theoremifpid2g 40188 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜓 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜓 → (𝜑𝜒)) ∧ (𝜒 → (𝜑𝜓))))
 
Theoremifpid1g 40189 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((𝜑 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((𝜒𝜑) ∧ (𝜑𝜓)))
 
Theoremifpim23g 40190 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(((𝜑𝜓) ↔ if-(𝜒, 𝜓, ¬ 𝜑)) ↔ (((𝜑𝜓) → 𝜒) ∧ (𝜒 → (𝜑𝜓))))
 
Theoremifpim3 40191 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜑))
 
Theoremifpnim1 40192 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(¬ (𝜑𝜓) ↔ if-(𝜑, ¬ 𝜓, 𝜑))
 
Theoremifpim4 40193 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((𝜑𝜓) ↔ if-(𝜓, 𝜓, ¬ 𝜑))
 
Theoremifpnim2 40194 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(¬ (𝜑𝜓) ↔ if-(𝜓, ¬ 𝜓, 𝜑))
 
Theoremifpim123g 40195 Implication of conditional logical operators. The right hand side is basically conjunctive normal form which is useful in proofs. (Contributed by RP, 16-Apr-2020.)
((if-(𝜑, 𝜒, 𝜏) → if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜏𝜃))) ∧ (((𝜑𝜓) ∨ (𝜒𝜂)) ∧ ((¬ 𝜓𝜑) ∨ (𝜏𝜂)))))
 
Theoremifpim1g 40196 Implication of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)) ↔ (((𝜓𝜑) ∨ (𝜃𝜒)) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))
 
Theoremifp1bi 40197 Substitute the first element of conditional logical operator. (Contributed by RP, 20-Apr-2020.)
((if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃)) ↔ ((((𝜑𝜓) ∨ (𝜒𝜃)) ∧ ((𝜑𝜓) ∨ (𝜃𝜒))) ∧ (((𝜓𝜑) ∨ (𝜒𝜃)) ∧ ((𝜓𝜑) ∨ (𝜃𝜒)))))
 
Theoremifpbi1b 40198 When the first variable is irrelevant, it can be replaced. (Contributed by RP, 25-Apr-2020.)
(if-(𝜑, 𝜒, 𝜒) ↔ if-(𝜓, 𝜒, 𝜒))
 
Theoremifpimimb 40199 Factor conditional logic operator over implication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
 
Theoremifpororb 40200 Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(𝜑, (𝜓𝜒), (𝜃𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∨ if-(𝜑, 𝜒, 𝜏)))
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