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Theorem List for Metamath Proof Explorer - 22701-22800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmetdstri 22701* A generalization of the triangle inequality to the point-set distance function. Under the usual notation where the same symbol 𝑑 denotes the point-point and point-set distance functions, this theorem would be written 𝑑(𝑎, 𝑆) ≤ 𝑑(𝑎, 𝑏) + 𝑑(𝑏, 𝑆). (Contributed by Mario Carneiro, 4-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹𝐵)))
 
Theoremmetdsle 22702* The distance from a point to a set is bounded by the distance to any member of the set. (Contributed by Mario Carneiro, 5-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝐴𝑆𝐵𝑋)) → (𝐹𝐵) ≤ (𝐴𝐷𝐵))
 
Theoremmetdsre 22703* The distance from a point to a nonempty set in a proper metric space is a real number. (Contributed by Mario Carneiro, 5-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       ((𝐷 ∈ (Met‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝐹:𝑋⟶ℝ)
 
Theoremmetdseq0 22704* The distance from a point to a set is zero iff the point is in the closure set. (Contributed by Mario Carneiro, 14-Feb-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋𝐴𝑋) → ((𝐹𝐴) = 0 ↔ 𝐴 ∈ ((cls‘𝐽)‘𝑆)))
 
Theoremmetdscnlem 22705* Lemma for metdscn 22706. (Contributed by Mario Carneiro, 4-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   𝐶 = (dist‘ℝ*𝑠)    &   𝐾 = (MetOpen‘𝐶)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆𝑋)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → (𝐴𝐷𝐵) < 𝑅)       (𝜑 → ((𝐹𝐴) +𝑒 -𝑒(𝐹𝐵)) < 𝑅)
 
Theoremmetdscn 22706* The function 𝐹 which gives the distance from a point to a set is a continuous function into the metric topology of the extended reals. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   𝐶 = (dist‘ℝ*𝑠)    &   𝐾 = (MetOpen‘𝐶)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremmetdscn2 22707* The function 𝐹 which gives the distance from a point to a nonempty set in a metric space is a continuous function into the topology of the complex numbers. (Contributed by Mario Carneiro, 5-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   𝐾 = (TopOpen‘ℂfld)       ((𝐷 ∈ (Met‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremmetnrmlem1a 22708* Lemma for metnrm 22712. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)       ((𝜑𝐴𝑇) → (0 < (𝐹𝐴) ∧ if(1 ≤ (𝐹𝐴), 1, (𝐹𝐴)) ∈ ℝ+))
 
Theoremmetnrmlem1 22709* Lemma for metnrm 22712. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)       ((𝜑 ∧ (𝐴𝑆𝐵𝑇)) → if(1 ≤ (𝐹𝐵), 1, (𝐹𝐵)) ≤ (𝐴𝐷𝐵))
 
Theoremmetnrmlem2 22710* Lemma for metnrm 22712. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)    &   𝑈 = 𝑡𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))       (𝜑 → (𝑈𝐽𝑇𝑈))
 
Theoremmetnrmlem3 22711* Lemma for metnrm 22712. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)    &   𝑈 = 𝑡𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))    &   𝐺 = (𝑥𝑋 ↦ inf(ran (𝑦𝑇 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝑉 = 𝑠𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2))       (𝜑 → ∃𝑧𝐽𝑤𝐽 (𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅))
 
Theoremmetnrm 22712 A metric space is normal. (Contributed by Jeff Hankins, 31-Aug-2013.) (Revised by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm)
 
Theoremmetreg 22713 A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg)
 
Theoremaddcnlem 22714* Lemma for addcn 22715, subcn 22716, and mulcn 22717. (Contributed by Mario Carneiro, 5-May-2014.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)    &    + :(ℂ × ℂ)⟶ℂ    &   ((𝑎 ∈ ℝ+𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ∃𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢𝑏)) < 𝑦 ∧ (abs‘(𝑣𝑐)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝑏 + 𝑐))) < 𝑎))        + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
Theoremaddcn 22715 Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (TopOpen‘ℂfld)        + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
Theoremsubcn 22716 Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (TopOpen‘ℂfld)        − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
Theoremmulcn 22717 Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (TopOpen‘ℂfld)        · ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
Theoremdivcn 22718 Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t (ℂ ∖ {0}))        / ∈ ((𝐽 ×t 𝐾) Cn 𝐽)
 
Theoremcnfldtgp 22719 The complex numbers form a topological group under addition, with the standard topology induced by the absolute value metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
fld ∈ TopGrp
 
Theoremfsumcn 22720* A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for 𝐵 normally contains free variables 𝑘 and 𝑥 to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ Σ𝑘𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
 
Theoremfsum2cn 22721* Version of fsumcn 22720 for two-argument mappings. (Contributed by Mario Carneiro, 6-May-2014.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐿 ∈ (TopOn‘𝑌))    &   ((𝜑𝑘𝐴) → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ Σ𝑘𝐴 𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾))
 
Theoremexpcn 22722* The power function on complex numbers, for fixed exponent 𝑁, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐽 = (TopOpen‘ℂfld)       (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥𝑁)) ∈ (𝐽 Cn 𝐽))
 
Theoremdivccn 22723* Division by a nonzero constant is a continuous operation. (Contributed by Mario Carneiro, 5-May-2014.)
𝐽 = (TopOpen‘ℂfld)       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴)) ∈ (𝐽 Cn 𝐽))
 
Theoremsqcn 22724* The square function on complex numbers is continuous. (Contributed by NM, 13-Jun-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (TopOpen‘ℂfld)       (𝑥 ∈ ℂ ↦ (𝑥↑2)) ∈ (𝐽 Cn 𝐽)
 
12.4.11  Topological definitions using the reals
 
Syntaxcii 22725 Extend class notation with the unit interval.
class II
 
Syntaxccncf 22726 Extend class notation to include the operation which returns a class of continuous complex functions.
class cn
 
Definitiondf-ii 22727 Define the unit interval with the Euclidean topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
 
Definitiondf-cncf 22728* Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)
cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏𝑚 𝑎) ∣ ∀𝑥𝑎𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑦𝑎 ((abs‘(𝑥𝑦)) < 𝑑 → (abs‘((𝑓𝑥) − (𝑓𝑦))) < 𝑒)})
 
Theoremiitopon 22729 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
II ∈ (TopOn‘(0[,]1))
 
Theoremiitop 22730 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
II ∈ Top
 
Theoremiiuni 22731 The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.)
(0[,]1) = II
 
Theoremdfii2 22732 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
II = ((topGen‘ran (,)) ↾t (0[,]1))
 
Theoremdfii3 22733 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       II = (𝐽t (0[,]1))
 
Theoremdfii4 22734 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐼 = (ℂflds (0[,]1))       II = (TopOpen‘𝐼)
 
Theoremdfii5 22735 The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
 
Theoremiicmp 22736 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
II ∈ Comp
 
Theoremiiconn 22737 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
II ∈ Conn
 
Theoremcncfval 22738* The value of the continuous complex function operation is the set of continuous functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
 
Theoremelcncf 22739* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))
 
Theoremelcncf2 22740* Version of elcncf 22739 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝑥)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝑥))) < 𝑦))))
 
Theoremcncfrss 22741 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)
 
Theoremcncfrss2 22742 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)
 
Theoremcncff 22743 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)
 
Theoremcncfi 22744* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
((𝐹 ∈ (𝐴cn𝐵) ∧ 𝐶𝐴𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝐶)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝐶))) < 𝑅))
 
Theoremelcncf1di 22745* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑 → ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+))    &   (𝜑 → (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))       (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵)))
 
Theoremelcncf1ii 22746* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
𝐹:𝐴𝐵    &   ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)    &   (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵))
 
Theoremrescncf 22747 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐶𝐴 → (𝐹 ∈ (𝐴cn𝐵) → (𝐹𝐶) ∈ (𝐶cn𝐵)))
 
Theoremcncffvrn 22748 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴cn𝐵)) → (𝐹 ∈ (𝐴cn𝐶) ↔ 𝐹:𝐴𝐶))
 
Theoremcncfss 22749 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐵𝐶𝐶 ⊆ ℂ) → (𝐴cn𝐵) ⊆ (𝐴cn𝐶))
 
Theoremclimcncf 22750 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐺:𝑍𝐴)    &   (𝜑𝐺𝐷)    &   (𝜑𝐷𝐴)       (𝜑 → (𝐹𝐺) ⇝ (𝐹𝐷))
 
Theoremabscncf 22751 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
abs ∈ (ℂ–cn→ℝ)
 
Theoremrecncf 22752 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℜ ∈ (ℂ–cn→ℝ)
 
Theoremimcncf 22753 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℑ ∈ (ℂ–cn→ℝ)
 
Theoremcjcncf 22754 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
∗ ∈ (ℂ–cn→ℂ)
 
Theoremmulc1cncf 22755* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
 
Theoremdivccncf 22756* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴))       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐹 ∈ (ℂ–cn→ℂ))
 
Theoremcncfco 22757 The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐺 ∈ (𝐵cn𝐶))       (𝜑 → (𝐺𝐹) ∈ (𝐴cn𝐶))
 
Theoremcncfmet 22758 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
𝐶 = ((abs ∘ − ) ↾ (𝐴 × 𝐴))    &   𝐷 = ((abs ∘ − ) ↾ (𝐵 × 𝐵))    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = (𝐽 Cn 𝐾))
 
Theoremcncfcn 22759 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝐴)    &   𝐿 = (𝐽t 𝐵)       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = (𝐾 Cn 𝐿))
 
Theoremcncfcn1 22760 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
 
Theoremcncfmptc 22761* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
((𝐴𝑇𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥𝑆𝐴) ∈ (𝑆cn𝑇))
 
Theoremcncfmptid 22762* The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇))
 
Theoremcncfmpt1f 22763* Composition of continuous functions. cn analogue of cnmpt11f 21515. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝐹 ∈ (ℂ–cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝑋cn→ℂ))
 
Theoremcncfmpt2f 22764* Composition of continuous functions. cn analogue of cnmpt12f 21517. (Contributed by Mario Carneiro, 3-Sep-2014.)
𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋cn→ℂ))
 
Theoremcncfmpt2ss 22765* Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
𝐽 = (TopOpen‘ℂfld)    &   𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn𝑆))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn𝑆))    &   𝑆 ⊆ ℂ    &   ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋cn𝑆))
 
Theoremaddccncf 22766* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 + 𝐴))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
 
Theoremcdivcncf 22767* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)
𝐹 = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))       (𝐴 ∈ ℂ → 𝐹 ∈ ((ℂ ∖ {0})–cn→ℂ))
 
Theoremnegcncf 22768* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
𝐹 = (𝑥𝐴 ↦ -𝑥)       (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴cn→ℂ))
 
Theoremnegfcncf 22769* The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
𝐺 = (𝑥𝐴 ↦ -(𝐹𝑥))       (𝐹 ∈ (𝐴cn→ℂ) → 𝐺 ∈ (𝐴cn→ℂ))
 
TheoremabscncfALT 22770 Absolute value is continuous. Alternate proof of abscncf 22751. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
abs ∈ (ℂ–cn→ℝ)
 
Theoremcncfcnvcn 22771 Rewrite cmphaushmeo 21651 for functions on the complex numbers. (Contributed by Mario Carneiro, 19-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝑋)       ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋cn𝑌)) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑌cn𝑋)))
 
Theoremexpcncf 22772* The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 22722. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥𝑁)) ∈ (ℂ–cn→ℂ))
 
Theoremcnmptre 22773* Lemma for iirevcn 22776 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
𝑅 = (TopOpen‘ℂfld)    &   𝐽 = ((topGen‘ran (,)) ↾t 𝐴)    &   𝐾 = ((topGen‘ran (,)) ↾t 𝐵)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ⊆ ℝ)    &   ((𝜑𝑥𝐴) → 𝐹𝐵)    &   (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))       (𝜑 → (𝑥𝐴𝐹) ∈ (𝐽 Cn 𝐾))
 
Theoremcnmpt2pc 22774* Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
𝑅 = (topGen‘ran (,))    &   𝑀 = (𝑅t (𝐴[,]𝐵))    &   𝑁 = (𝑅t (𝐵[,]𝐶))    &   𝑂 = (𝑅t (𝐴[,]𝐶))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 ∈ (𝐴[,]𝐶))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   ((𝜑 ∧ (𝑥 = 𝐵𝑦𝑋)) → 𝐷 = 𝐸)    &   (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝑋𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))    &   (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦𝑋𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))       (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦𝑋 ↦ if(𝑥𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))
 
Theoremiirev 22775 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1))
 
Theoremiirevcn 22776 The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.)
(𝑥 ∈ (0[,]1) ↦ (1 − 𝑥)) ∈ (II Cn II)
 
Theoremiihalf1 22777 Map the first half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1))
 
Theoremiihalf1cn 22778 The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2)))       (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II)
 
Theoremiihalf2 22779 Map the second half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ ((1 / 2)[,]1) → ((2 · 𝑋) − 1) ∈ (0[,]1))
 
Theoremiihalf2cn 22780 The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1))       (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II)
 
Theoremelii1 22781 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
(𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2)))
 
Theoremelii2 22782 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ((1 / 2)[,]1))
 
Theoremiimulcl 22783 The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1)) → (𝐴 · 𝐵) ∈ (0[,]1))
 
Theoremiimulcn 22784* Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.)
(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥 · 𝑦)) ∈ ((II ×t II) Cn II)
 
Theoremicoopnst 22785 A half-open interval starting at 𝐴 is open in the closed interval from 𝐴 to 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) → (𝐴[,)𝐶) ∈ 𝐽))
 
Theoremiocopnst 22786 A half-open interval ending at 𝐵 is open in the closed interval from 𝐴 to 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶(,]𝐵) ∈ 𝐽))
 
Theoremicchmeo 22787* The natural bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽t (𝐴[,]𝐵))))
 
Theoremicopnfcnv 22788* Define a bijection from [0, 1) to [0, +∞). (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐹 = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))       (𝐹:(0[,)1)–1-1-onto→(0[,)+∞) ∧ 𝐹 = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦))))
 
Theoremicopnfhmeo 22789* The defined bijection from [0, 1) to [0, +∞) is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐹 = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))    &   𝐽 = (TopOpen‘ℂfld)       (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ∧ 𝐹 ∈ ((𝐽t (0[,)1))Homeo(𝐽t (0[,)+∞))))
 
Theoremiccpnfcnv 22790* Define a bijection from [0, 1] to [0, +∞]. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))       (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ 𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
 
Theoremiccpnfhmeo 22791 The defined bijection from [0, 1] to [0, +∞] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))    &   𝐾 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))       (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾))
 
Theoremxrhmeo 22792* The bijection from [-1, 1] to the extended reals is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))    &   𝐺 = (𝑦 ∈ (-1[,]1) ↦ if(0 ≤ 𝑦, (𝐹𝑦), -𝑒(𝐹‘-𝑦)))    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (ordTop‘ ≤ )       (𝐺 Isom < , < ((-1[,]1), ℝ*) ∧ 𝐺 ∈ ((𝐽t (-1[,]1))Homeo(ordTop‘ ≤ )))
 
Theoremxrhmph 22793 The extended reals are homeomorphic to the interval [0, 1]. (Contributed by Mario Carneiro, 9-Sep-2015.)
II ≃ (ordTop‘ ≤ )
 
Theoremxrcmp 22794 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 22656), this means that * is a compactification of . (Contributed by Mario Carneiro, 9-Sep-2015.)
(ordTop‘ ≤ ) ∈ Comp
 
Theoremxrconn 22795 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
(ordTop‘ ≤ ) ∈ Conn
 
Theoremicccvx 22796 A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝐶) + (𝑇 · 𝐷)) ∈ (𝐴[,]𝐵)))
 
Theoremoprpiece1res1 22797* Restriction to the first part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴𝐵    &   𝑅 ∈ V    &   𝑆 ∈ V    &   𝐾 ∈ (𝐴[,]𝐵)    &   𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝐶 ↦ if(𝑥𝐾, 𝑅, 𝑆))    &   𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦𝐶𝑅)       (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺
 
Theoremoprpiece1res2 22798* Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐴𝐵    &   𝑅 ∈ V    &   𝑆 ∈ V    &   𝐾 ∈ (𝐴[,]𝐵)    &   𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦𝐶 ↦ if(𝑥𝐾, 𝑅, 𝑆))    &   (𝑥 = 𝐾𝑅 = 𝑃)    &   (𝑥 = 𝐾𝑆 = 𝑄)    &   (𝑦𝐶𝑃 = 𝑄)    &   𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦𝐶𝑆)       (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺
 
Theoremcnrehmeo 22799* The canonical bijection from (ℝ × ℝ) to described in cnref1o 11865 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if (ℝ × ℝ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐾 = (TopOpen‘ℂfld)       𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)
 
Theoremcnheiborlem 22800* Lemma for cnheibor 22801. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝑇 = (𝐽t 𝑋)    &   𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   𝑌 = (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅)))       ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp)
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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 42801-42879
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