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Theorem List for Metamath Proof Explorer - 31001-31100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcirctopn 31001* The topology of the unit circle is generated by open intervals of the polar coordinate. (Contributed by Thierry Arnoux, 4-Jan-2020.)
𝐼 = (0[,](2 · π))    &   𝐽 = (topGen‘ran (,))    &   𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))    &   𝐶 = (abs “ {1})       (𝐽 qTop 𝐹) = (TopOpen‘(𝐹sfld))
 
Theoremcirccn 31002* The function gluing the real line into the unit circle is continuous. (Contributed by Thierry Arnoux, 5-Jan-2020.)
𝐼 = (0[,](2 · π))    &   𝐽 = (topGen‘ran (,))    &   𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥)))    &   𝐶 = (abs “ {1})       𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))
 
20.3.12.3  Refinements
 
Theoremreff 31003* For any cover refinement, there exists a function associating with each set in the refinement a set in the original cover containing it. This is sometimes used as a definition of refinement. Note that this definition uses the axiom of choice through ac6sg 9899. (Contributed by Thierry Arnoux, 12-Jan-2020.)
(𝐴𝑉 → (𝐴Ref𝐵 ↔ ( 𝐵 𝐴 ∧ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))))
 
Theoremlocfinreflem 31004* A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-Jan-2020.)
𝑋 = 𝐽    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑉𝐽)    &   (𝜑𝑉Ref𝑈)    &   (𝜑𝑉 ∈ (LocFin‘𝐽))       (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
 
Theoremlocfinref 31005* A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.)
𝑋 = 𝐽    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑉𝐽)    &   (𝜑𝑉Ref𝑈)    &   (𝜑𝑉 ∈ (LocFin‘𝐽))       (𝜑 → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
 
20.3.12.4  Open cover refinement property
 
Syntaxccref 31006 The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴
 
Definitiondf-cref 31007* Define a statement "every open cover has an 𝐴 refinement" , where 𝐴 is a property for refinements like "finite", "countable", "point finite" or "locally finite". (Contributed by Thierry Arnoux, 7-Jan-2020.)
CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗( 𝑗 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑗𝐴)𝑧Ref𝑦)}
 
Theoremiscref 31008* The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = 𝐽       (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝑦)))
 
Theoremcrefeq 31009 Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)
 
Theoremcreftop 31010 A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ CovHasRef𝐴𝐽 ∈ Top)
 
Theoremcrefi 31011* The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = 𝐽       ((𝐽 ∈ CovHasRef𝐴𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ (𝒫 𝐽𝐴)𝑧Ref𝐶)
 
Theoremcrefdf 31012* A formulation of crefi 31011 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = 𝐽    &   𝐵 = CovHasRef𝐴    &   (𝑧𝐴𝜑)       ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))
 
Theoremcrefss 31013 The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)
 
Theoremcmpcref 31014 Equivalent definition of compact space in terms of open cover refinements. Compact spaces are topologies with finite open cover refinements. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Comp = CovHasRefFin
 
Theoremcmpfiref 31015* Every open cover of a Compact space has a finite refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
𝑋 = 𝐽       ((𝐽 ∈ Comp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ Fin ∧ 𝑣Ref𝑈))
 
20.3.12.5  Lindelöf spaces
 
Syntaxcldlf 31016 Extend class notation with the class of all Lindelöf spaces.
class Ldlf
 
Definitiondf-ldlf 31017 Definition of a Lindelöf space. A Lindelöf space is a topological space in which every open cover has a countable subcover. Definition 1 of [BourbakiTop2] p. 195. (Contributed by Thierry Arnoux, 30-Jan-2020.)
Ldlf = CovHasRef{𝑥𝑥 ≼ ω}
 
Theoremldlfcntref 31018* Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
𝑋 = 𝐽       ((𝐽 ∈ Ldlf ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))
 
20.3.12.6  Paracompact spaces
 
Syntaxcpcmp 31019 Extend class notation with the class of all paracompact topologies.
class Paracomp
 
Definitiondf-pcmp 31020 Definition of a paracompact topology. A topology is said to be paracompact iff every open cover has an open refinement that is locally finite. The definition 6 of [BourbakiTop1] p. I.69. also requires the topology to be Hausdorff, but this is dropped here. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Paracomp = {𝑗𝑗 ∈ CovHasRef(LocFin‘𝑗)}
 
Theoremispcmp 31021 The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
 
Theoremcmppcmp 31022 Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Comp → 𝐽 ∈ Paracomp)
 
Theoremdispcmp 31023 Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)
 
Theorempcmplfin 31024* Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement 𝑣 that is locally finite. (Contributed by Thierry Arnoux, 31-Jan-2020.)
𝑋 = 𝐽       ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
 
Theorempcmplfinf 31025* Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement ran 𝑓 that is locally finite, using the same index as the original cover 𝑈. (Contributed by Thierry Arnoux, 31-Jan-2020.)
𝑋 = 𝐽       ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
 
20.3.12.7  Pseudometrics
 
Syntaxcmetid 31026 Extend class notation with the class of metric identifications.
class ~Met
 
Syntaxcpstm 31027 Extend class notation with the metric induced by a pseudometric.
class pstoMet
 
Definitiondf-metid 31028* Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
~Met = (𝑑 ran PsMet ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ dom dom 𝑑𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)})
 
Definitiondf-pstm 31029* Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
pstoMet = (𝑑 ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met𝑑)) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝑑𝑦)}))
 
Theoremmetidval 31030* Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})
 
Theoremmetidss 31031 As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) ⊆ (𝑋 × 𝑋))
 
Theoremmetidv 31032 𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴(~Met𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0))
 
Theoremmetideq 31033 Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met𝐷)𝐵𝐸(~Met𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹))
 
Theoremmetider 31034 The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)
 
Theorempstmval 31035* Value of the metric induced by a pseudometric 𝐷. (Contributed by Thierry Arnoux, 7-Feb-2018.)
= (~Met𝐷)       (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ), 𝑏 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑥𝑎𝑦𝑏 𝑧 = (𝑥𝐷𝑦)}))
 
Theorempstmfval 31036 Function value of the metric induced by a pseudometric 𝐷 (Contributed by Thierry Arnoux, 11-Feb-2018.)
= (~Met𝐷)       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ([𝐴] (pstoMet‘𝐷)[𝐵] ) = (𝐴𝐷𝐵))
 
Theorempstmxmet 31037 The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.)
= (~Met𝐷)       (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / )))
 
20.3.12.8  Continuity - misc additions
 
Theoremhauseqcn 31038 In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.)
𝑋 = 𝐽    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → (𝐹𝐴) = (𝐺𝐴))    &   (𝜑𝐴𝑋)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)       (𝜑𝐹 = 𝐺)
 
20.3.12.9  Topology of the closed unit interval
 
Theoremunitsscn 31039 The closed unit interval is a subset of the set of the complex numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.)
(0[,]1) ⊆ ℂ
 
Theoremelunitrn 31040 The closed unit interval is a subset of the set of the real numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 21-Dec-2016.)
(𝐴 ∈ (0[,]1) → 𝐴 ∈ ℝ)
 
Theoremelunitcn 31041 The closed unit interval is a subset of the set of the complext numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 21-Dec-2016.)
(𝐴 ∈ (0[,]1) → 𝐴 ∈ ℂ)
 
Theoremelunitge0 31042 An element of the closed unit interval is positive. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 20-Dec-2016.)
(𝐴 ∈ (0[,]1) → 0 ≤ 𝐴)
 
Theoremunitssxrge0 31043 The closed unit interval is a subset of the set of the extended nonnegative reals. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.)
(0[,]1) ⊆ (0[,]+∞)
 
Theoremunitdivcld 31044 Necessary conditions for a quotient to be in the closed unit interval. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.)
((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1) ∧ 𝐵 ≠ 0) → (𝐴𝐵 ↔ (𝐴 / 𝐵) ∈ (0[,]1)))
 
Theoremiistmd 31045 The closed unit interval forms a topological monoid under multiplication. (Contributed by Thierry Arnoux, 25-Mar-2017.)
𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1))       𝐼 ∈ TopMnd
 
20.3.12.10  Topology of ` ( RR X. RR ) `
 
Theoremunicls 31046 The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.)
𝐽 ∈ Top    &   𝑋 = 𝐽        (Clsd‘𝐽) = 𝑋
 
Theoremtpr2tp 31047 The usual topology on (ℝ × ℝ) is the product topology of the usual topology on . (Contributed by Thierry Arnoux, 21-Sep-2017.)
𝐽 = (topGen‘ran (,))       (𝐽 ×t 𝐽) ∈ (TopOn‘(ℝ × ℝ))
 
Theoremtpr2uni 31048 The usual topology on (ℝ × ℝ) is the product topology of the usual topology on . (Contributed by Thierry Arnoux, 21-Sep-2017.)
𝐽 = (topGen‘ran (,))        (𝐽 ×t 𝐽) = (ℝ × ℝ)
 
Theoremxpinpreima 31049 Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
(𝐴 × 𝐵) = (((1st ↾ (V × V)) “ 𝐴) ∩ ((2nd ↾ (V × V)) “ 𝐵))
 
Theoremxpinpreima2 31050 Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
((𝐴𝐸𝐵𝐹) → (𝐴 × 𝐵) = (((1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ ((2nd ↾ (𝐸 × 𝐹)) “ 𝐵)))
 
Theoremsqsscirc1 31051 The complex square of side 𝐷 is a subset of the complex circle of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.)
((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ+) → ((𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2)) → (√‘((𝑋↑2) + (𝑌↑2))) < 𝐷))
 
Theoremsqsscirc2 31052 The complex square of side 𝐷 is a subset of the complex disc of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐷 ∈ ℝ+) → (((abs‘(ℜ‘(𝐵𝐴))) < (𝐷 / 2) ∧ (abs‘(ℑ‘(𝐵𝐴))) < (𝐷 / 2)) → (abs‘(𝐵𝐴)) < 𝐷))
 
Theoremcnre2csqlem 31053* Lemma for cnre2csqima 31054. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(𝐺 ↾ (ℝ × ℝ)) = (𝐻𝐹)    &   𝐹 Fn (ℝ × ℝ)    &   𝐺 Fn V    &   (𝑥 ∈ (ℝ × ℝ) → (𝐺𝑥) ∈ ℝ)    &   ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹) → (𝐻‘(𝑥𝑦)) = ((𝐻𝑥) − (𝐻𝑦)))       ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((𝐺 ↾ (ℝ × ℝ)) “ (((𝐺𝑋) − 𝐷)(,)((𝐺𝑋) + 𝐷))) → (abs‘(𝐻‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷))
 
Theoremcnre2csqima 31054* Image of a centered square by the canonical bijection from (ℝ × ℝ) to . (Contributed by Thierry Arnoux, 27-Sep-2017.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))       ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((((1st𝑋) − 𝐷)(,)((1st𝑋) + 𝐷)) × (((2nd𝑋) − 𝐷)(,)((2nd𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹𝑌) − (𝐹𝑋)))) < 𝐷)))
 
Theoremtpr2rico 31055* For any point of an open set of the usual topology on (ℝ × ℝ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the (𝑙↑+∞) norm 𝑋. (Contributed by Thierry Arnoux, 21-Sep-2017.)
𝐽 = (topGen‘ran (,))    &   𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))    &   𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))       ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋𝐴) → ∃𝑟𝐵 (𝑋𝑟𝑟𝐴))
 
20.3.12.11  Order topology - misc. additions
 
Theoremcnvordtrestixx 31056* The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.)
𝐴 ⊆ ℝ*    &   ((𝑥𝐴𝑦𝐴) → (𝑥[,]𝑦) ⊆ 𝐴)       ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))
 
Theoremprsdm 31057 Domain of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       (𝐾 ∈ Proset → dom = 𝐵)
 
Theoremprsrn 31058 Range of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       (𝐾 ∈ Proset → ran = 𝐵)
 
Theoremprsss 31059 Relation of a subproset. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Proset ∧ 𝐴𝐵) → ( ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
 
Theoremprsssdm 31060 Domain of a subproset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Proset ∧ 𝐴𝐵) → dom ( ∩ (𝐴 × 𝐴)) = 𝐴)
 
Theoremordtprsval 31061* Value of the order topology for a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   𝐸 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})    &   𝐹 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})       (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸𝐹)))))
 
Theoremordtprsuni 31062* Value of the order topology. (Contributed by Thierry Arnoux, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   𝐸 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})    &   𝐹 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})       (𝐾 ∈ Proset → 𝐵 = ({𝐵} ∪ (𝐸𝐹)))
 
TheoremordtcnvNEW 31063 The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.) (Revised by Thierry Arnoux, 13-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       (𝐾 ∈ Proset → (ordTop‘ ) = (ordTop‘ ))
 
TheoremordtrestNEW 31064 The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))       ((𝐾 ∈ Proset ∧ 𝐴𝐵) → (ordTop‘( ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ) ↾t 𝐴))
 
Theoremordtrest2NEWlem 31065* Lemma for ordtrest2NEW 31066. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)       (𝜑 → ∀𝑣 ∈ ran (𝑧𝐵 ↦ {𝑤𝐵 ∣ ¬ 𝑤 𝑧})(𝑣𝐴) ∈ (ordTop‘( ∩ (𝐴 × 𝐴))))
 
Theoremordtrest2NEW 31066* An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in , but in other sets like there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝐵 ∣ (𝑥 𝑧𝑧 𝑦)} ⊆ 𝐴)       (𝜑 → (ordTop‘( ∩ (𝐴 × 𝐴))) = ((ordTop‘ ) ↾t 𝐴))
 
Theoremordtconnlem1 31067* Connectedness in the order topology of a toset. This is the "easy" direction of ordtconn 31068. See also reconnlem1 23363. (Contributed by Thierry Arnoux, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   𝐽 = (ordTop‘ )       ((𝐾 ∈ Toset ∧ 𝐴𝐵) → ((𝐽t 𝐴) ∈ Conn → ∀𝑥𝐴𝑦𝐴𝑟𝐵 ((𝑥 𝑟𝑟 𝑦) → 𝑟𝐴)))
 
Theoremordtconn 31068 Connectedness in the order topology of a complete uniform totally ordered space. (Contributed by Thierry Arnoux, 15-Sep-2018.)
𝐵 = (Base‘𝐾)    &    = ((le‘𝐾) ∩ (𝐵 × 𝐵))    &   𝐽 = (ordTop‘ )       
 
20.3.12.12  Continuity in topological spaces - misc. additions
 
Theoremmndpluscn 31069* A mapping that is both a homeomorphism and a monoid homomorphism preserves the "continuousness" of the operation. (Contributed by Thierry Arnoux, 25-Mar-2017.)
𝐹 ∈ (𝐽Homeo𝐾)    &    + :(𝐵 × 𝐵)⟶𝐵    &    :(𝐶 × 𝐶)⟶𝐶    &   𝐽 ∈ (TopOn‘𝐵)    &   𝐾 ∈ (TopOn‘𝐶)    &   ((𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &    + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)        ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
 
Theoremmhmhmeotmd 31070 Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
𝐹 ∈ (𝑆 MndHom 𝑇)    &   𝐹 ∈ ((TopOpen‘𝑆)Homeo(TopOpen‘𝑇))    &   𝑆 ∈ TopMnd    &   𝑇 ∈ TopSp       𝑇 ∈ TopMnd
 
Theoremrmulccn 31071* Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)
𝐽 = (topGen‘ran (,))    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽))
 
Theoremraddcn 31072* Addition in the real numbers is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.)
𝐽 = (topGen‘ran (,))       (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
Theoremxrmulc1cn 31073* The operation multiplying an extended real number by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
𝐽 = (ordTop‘ ≤ )    &   𝐹 = (𝑥 ∈ ℝ* ↦ (𝑥 ·e 𝐶))    &   (𝜑𝐶 ∈ ℝ+)       (𝜑𝐹 ∈ (𝐽 Cn 𝐽))
 
Theoremfmcncfil 31074 The image of a Cauchy filter by a continuous filter map is a Cauchy filter. (Contributed by Thierry Arnoux, 12-Nov-2017.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = (MetOpen‘𝐸)       (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐸 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝐵 ∈ (CauFil‘𝐷)) → ((𝑌 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐸))
 
20.3.12.13  Topology of the extended nonnegative real numbers ordered monoid
 
Theoremxrge0hmph 31075 The extended nonnegative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
II ≃ ((ordTop‘ ≤ ) ↾t (0[,]+∞))
 
Theoremxrge0iifcnv 31076* Define a bijection from [0, 1] onto [0, +∞]. (Contributed by Thierry Arnoux, 29-Mar-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))       (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ 𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))
 
Theoremxrge0iifcv 31077* The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))       (𝑋 ∈ (0(,]1) → (𝐹𝑋) = -(log‘𝑋))
 
Theoremxrge0iifiso 31078* The defined bijection from the closed unit interval onto the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))       𝐹 Isom < , < ((0[,]1), (0[,]+∞))
 
Theoremxrge0iifhmeo 31079* Expose a homeomorphism from the closed unit interval to the extended nonnegative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))       𝐹 ∈ (IIHomeo𝐽)
 
Theoremxrge0iifhom 31080* The defined function from the closed unit interval to the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 5-Apr-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))       ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) +𝑒 (𝐹𝑌)))
 
Theoremxrge0iif1 31081* Condition for the defined function, -(log‘𝑥) to be a monoid homomorphism. (Contributed by Thierry Arnoux, 20-Jun-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))       (𝐹‘1) = 0
 
Theoremxrge0iifmhm 31082* The defined function from the closed unit interval to the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))       𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠s (0[,]+∞)))
 
Theoremxrge0pluscn 31083* The addition operation of the extended nonnegative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    &    + = ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))        + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
Theoremxrge0mulc1cn 31084* The operation multiplying a nonnegative real numbers by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 6-Jul-2017.)
𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    &   𝐹 = (𝑥 ∈ (0[,]+∞) ↦ (𝑥 ·e 𝐶))    &   (𝜑𝐶 ∈ (0[,)+∞))       (𝜑𝐹 ∈ (𝐽 Cn 𝐽))
 
Theoremxrge0tps 31085 The extended nonnegative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.)
(ℝ*𝑠s (0[,]+∞)) ∈ TopSp
 
Theoremxrge0topn 31086 The topology of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.)
(TopOpen‘(ℝ*𝑠s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞))
 
Theoremxrge0haus 31087 The topology of the extended nonnegative real numbers is Hausdorff. (Contributed by Thierry Arnoux, 26-Jul-2017.)
(TopOpen‘(ℝ*𝑠s (0[,]+∞))) ∈ Haus
 
Theoremxrge0tmd 31088 The extended nonnegative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof Shortened by Thierry Arnoux, 21-Jun-2017.)
(ℝ*𝑠s (0[,]+∞)) ∈ TopMnd
 
Theoremxrge0tmdALT 31089 Alternate proof of xrge0tmd 31088. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(ℝ*𝑠s (0[,]+∞)) ∈ TopMnd
 
20.3.12.14  Limits - misc additions
 
Theoremlmlim 31090 Relate a limit in a given topology to a complex number limit, provided that topology agrees with the common topology on on the required subset. (Contributed by Thierry Arnoux, 11-Jul-2017.)
𝐽 ∈ (TopOn‘𝑌)    &   (𝜑𝐹:ℕ⟶𝑋)    &   (𝜑𝑃𝑋)    &   (𝐽t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋)    &   𝑋 ⊆ ℂ       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹𝑃))
 
Theoremlmlimxrge0 31091 Relate a limit in the nonnegative extended reals to a complex limit, provided the considered function is a real function. (Contributed by Thierry Arnoux, 11-Jul-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))    &   (𝜑𝐹:ℕ⟶𝑋)    &   (𝜑𝑃𝑋)    &   𝑋 ⊆ (0[,)+∞)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹𝑃))
 
Theoremrge0scvg 31092 Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 16247. (Contributed by Thierry Arnoux, 28-Jul-2017.)
((𝐹:ℕ⟶(0[,)+∞) ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → sup(ran seq1( + , 𝐹), ℝ, < ) ∈ ℝ)
 
Theoremfsumcvg4 31093 A serie with finite support is a finite sum, and therefore converges. (Contributed by Thierry Arnoux, 6-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝑆 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑆⟶ℂ)    &   (𝜑 → (𝐹 “ (ℂ ∖ {0})) ∈ Fin)       (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
 
Theorempnfneige0 31094* A neighborhood of +∞ contains an unbounded interval based at a real number. See pnfnei 21758. (Contributed by Thierry Arnoux, 31-Jul-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))       ((𝐴𝐽 ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
 
Theoremlmxrge0 31095* Express "sequence 𝐹 converges to plus infinity" (i.e. diverges), for a sequence of nonnegative extended real numbers. (Contributed by Thierry Arnoux, 2-Aug-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))    &   (𝜑𝐹:ℕ⟶(0[,]+∞))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = 𝐴)       (𝜑 → (𝐹(⇝𝑡𝐽)+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)𝑥 < 𝐴))
 
Theoremlmdvg 31096* If a monotonic sequence of real numbers diverges, it is unbounded. (Contributed by Thierry Arnoux, 4-Aug-2017.)
(𝜑𝐹:ℕ⟶(0[,)+∞))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))    &   (𝜑 → ¬ 𝐹 ∈ dom ⇝ )       (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)𝑥 < (𝐹𝑘))
 
Theoremlmdvglim 31097* If a monotonic real number sequence 𝐹 diverges, it converges in the extended real numbers and its limit is plus infinity. (Contributed by Thierry Arnoux, 3-Aug-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))    &   (𝜑𝐹:ℕ⟶(0[,)+∞))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))    &   (𝜑 → ¬ 𝐹 ∈ dom ⇝ )       (𝜑𝐹(⇝𝑡𝐽)+∞)
 
20.3.12.15  Univariate polynomials
 
Theorempl1cn 31098 A univariate polynomial is continuous. (Contributed by Thierry Arnoux, 17-Sep-2018.)
𝑃 = (Poly1𝑅)    &   𝐸 = (eval1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐽 = (TopOpen‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑅 ∈ TopRing)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐸𝐹) ∈ (𝐽 Cn 𝐽))
 
20.3.13  Uniform Stuctures and Spaces
 
20.3.13.1  Hausdorff uniform completion
 
Syntaxchcmp 31099 Extend class notation with the Hausdorff uniform completion relation.
class HCmp
 
Definitiondf-hcmp 31100* Definition of the Hausdorff completion. In this definition, a structure 𝑤 is a Hausdorff completion of a uniform structure 𝑢 if 𝑤 is a complete uniform space, in which 𝑢 is dense, and which admits the same uniform structure. Theorem 3 of [BourbakiTop1] p. II.21. states the existence and uniqueness of such a completion. (Contributed by Thierry Arnoux, 5-Mar-2018.)
HCmp = {⟨𝑢, 𝑤⟩ ∣ ((𝑢 ran UnifOn ∧ 𝑤 ∈ CUnifSp) ∧ ((UnifSt‘𝑤) ↾t dom 𝑢) = 𝑢 ∧ ((cls‘(TopOpen‘𝑤))‘dom 𝑢) = (Base‘𝑤))}
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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