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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubcn 23401 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 23402 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 23403 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 23404 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 23405* 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 23406* Version of fsumcn 23405 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 23407* 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 23408* 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 23409* 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 23410 Extend class notation with the unit interval.
class II
 
Syntaxccncf 23411 Extend class notation to include the operation which returns a class of continuous complex functions.
class cn
 
Definitiondf-ii 23412 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 23413* 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 23414 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
II ∈ (TopOn‘(0[,]1))
 
Theoremiitop 23415 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
II ∈ Top
 
Theoremiiuni 23416 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 23417 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
II = ((topGen‘ran (,)) ↾t (0[,]1))
 
Theoremdfii3 23418 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 23419 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐼 = (ℂflds (0[,]1))       II = (TopOpen‘𝐼)
 
Theoremdfii5 23420 The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
 
Theoremiicmp 23421 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
II ∈ Comp
 
Theoremiiconn 23422 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
II ∈ Conn
 
Theoremcncfval 23423* 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 23424* 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 23425* Version of elcncf 23424 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 23426 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)
 
Theoremcncfrss2 23427 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)
 
Theoremcncff 23428 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 23429* 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 23430* 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 23431* 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 23432 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 23433 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 23434 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐵𝐶𝐶 ⊆ ℂ) → (𝐴cn𝐵) ⊆ (𝐴cn𝐶))
 
Theoremclimcncf 23435 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 23436 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
abs ∈ (ℂ–cn→ℝ)
 
Theoremrecncf 23437 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℜ ∈ (ℂ–cn→ℝ)
 
Theoremimcncf 23438 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℑ ∈ (ℂ–cn→ℝ)
 
Theoremcjcncf 23439 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
∗ ∈ (ℂ–cn→ℂ)
 
Theoremmulc1cncf 23440* 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 23441* 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 23442 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 23443 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 23444 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 23445 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 23446* 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 23447* 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 23448* Composition of continuous functions. cn analogue of cnmpt11f 22200. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝐹 ∈ (ℂ–cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝑋cn→ℂ))
 
Theoremcncfmpt2f 23449* Composition of continuous functions. cn analogue of cnmpt12f 22202. (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 23450* 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 23451* 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 23452* 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 23453* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
𝐹 = (𝑥𝐴 ↦ -𝑥)       (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴cn→ℂ))
 
Theoremnegfcncf 23454* 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 23455 Absolute value is continuous. Alternate proof of abscncf 23436. (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 23456 Rewrite cmphaushmeo 22336 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 23457* The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 23407. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥𝑁)) ∈ (ℂ–cn→ℂ))
 
Theoremcnmptre 23458* Lemma for iirevcn 23461 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))
 
Theoremcnmpopc 23459* 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 23460 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1))
 
Theoremiirevcn 23461 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 23462 Map the first half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1))
 
Theoremiihalf1cn 23463 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 23464 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 23465 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 23466 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 23467 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 23468 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 23469* 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 23470 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 23471 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 23472* 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 23473* 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 23474* 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 23475* 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 23476 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 23477* 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)       (𝐺 Isom < , < ((-1[,]1), ℝ*) ∧ 𝐺 ∈ ((𝐽t (-1[,]1))Homeo(ordTop‘ ≤ )))
 
Theoremxrhmph 23478 The extended reals are homeomorphic to the interval [0, 1]. (Contributed by Mario Carneiro, 9-Sep-2015.)
II ≃ (ordTop‘ ≤ )
 
Theoremxrcmp 23479 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 23341), this means that * is a compactification of . (Contributed by Mario Carneiro, 9-Sep-2015.)
(ordTop‘ ≤ ) ∈ Comp
 
Theoremxrconn 23480 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
(ordTop‘ ≤ ) ∈ Conn
 
Theoremicccvx 23481 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 23482* 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 23483* 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 23484* The canonical bijection from (ℝ × ℝ) to described in cnref1o 12372 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 23485* Lemma for cnheibor 23486. (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)
 
Theoremcnheibor 23486* Heine-Borel theorem for complex numbers. A subset of is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝑇 = (𝐽t 𝑋)       (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)))
 
Theoremcnllycmp 23487 The topology on the complex numbers is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ 𝑛-Locally Comp
 
Theoremrellycmp 23488 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
(topGen‘ran (,)) ∈ 𝑛-Locally Comp
 
Theorembndth 23489* The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to -𝐹.) (Contributed by Mario Carneiro, 12-Aug-2014.)
𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥)
 
Theoremevth 23490* The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
 
Theoremevth2 23491* The Extreme Value Theorem, minimum version. A continuous function from a nonempty compact topological space to the reals attains its minimum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑥) ≤ (𝐹𝑦))
 
Theoremlebnumlem1 23492* Lemma for lebnum 23495. The function 𝐹 measures the sum of all of the distances to escape the sets of the cover. Since by assumption it is a cover, there is at least one set which covers a given point, and since it is open, the point is a positive distance from the edge of the set. Thus, the sum is a strictly positive number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝑈)    &   𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))       (𝜑𝐹:𝑋⟶ℝ+)
 
Theoremlebnumlem2 23493* Lemma for lebnum 23495. As a finite sum of point-to-set distance functions, which are continuous by metdscn 23391, the function 𝐹 is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝑈)    &   𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    &   𝐾 = (topGen‘ran (,))       (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremlebnumlem3 23494* Lemma for lebnum 23495. By the previous lemmas, 𝐹 is continuous and positive on a compact set, so it has a positive minimum 𝑟. Then setting 𝑑 = 𝑟 / ♯(𝑈), since for each 𝑢𝑈 we have ball(𝑥, 𝑑) ⊆ 𝑢 iff 𝑑𝑑(𝑥, 𝑋𝑢), if ¬ ball(𝑥, 𝑑) ⊆ 𝑢 for all 𝑢 then summing over 𝑢 yields Σ𝑢𝑈𝑑(𝑥, 𝑋𝑢) = 𝐹(𝑥) < Σ𝑢𝑈𝑑 = 𝑟, in contradiction to the assumption that 𝑟 is the minimum of 𝐹. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) (Revised by AV, 30-Sep-2020.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝑈)    &   𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    &   𝐾 = (topGen‘ran (,))       (𝜑 → ∃𝑑 ∈ ℝ+𝑥𝑋𝑢𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
 
Theoremlebnum 23495* The Lebesgue number lemma, or Lebesgue covering lemma. If 𝑋 is a compact metric space and 𝑈 is an open cover of 𝑋, then there exists a positive real number 𝑑 such that every ball of size 𝑑 (and every subset of a ball of size 𝑑, including every subset of diameter less than 𝑑) is a subset of some member of the cover. (Contributed by Mario Carneiro, 14-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)       (𝜑 → ∃𝑑 ∈ ℝ+𝑥𝑋𝑢𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
 
Theoremxlebnum 23496* Generalize lebnum 23495 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)       (𝜑 → ∃𝑑 ∈ ℝ+𝑥𝑋𝑢𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
 
Theoremlebnumii 23497* Specialize the Lebesgue number lemma lebnum 23495 to the closed unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)
((𝑈 ⊆ II ∧ (0[,]1) = 𝑈) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑢𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢)
 
12.4.12  Path homotopy
 
Syntaxchtpy 23498 Extend class notation with the class of homotopies between two continuous functions.
class Htpy
 
Syntaxcphtpy 23499 Extend class notation with the class of path homotopies between two continuous functions.
class PHtpy
 
Syntaxcphtpc 23500 Extend class notation with the path homotopy relation.
class ph
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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 44801-44834
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