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Theorem List for Metamath Proof Explorer - 30501-30600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremqtophaus 30501* If an open map's graph in the product space (𝐽 ×t 𝐽) is closed, then its quotient topology is Hausdorff. (Contributed by Thierry Arnoux, 4-Jan-2020.)
𝑋 = 𝐽    &    = (𝐹𝐹)    &   𝐻 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)    &   (𝜑𝐽 ∈ Haus)    &   (𝜑𝐹:𝑋onto𝑌)    &   ((𝜑𝑥𝐽) → (𝐹𝑥) ∈ (𝐽 qTop 𝐹))    &   (𝜑 ∈ (Clsd‘(𝐽 ×t 𝐽)))       (𝜑 → (𝐽 qTop 𝐹) ∈ Haus)

Theoremcirctopn 30502* 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 30503* 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))

20.3.11.3  Refinements

Theoremreff 30504* 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 defintion of refinement. Note that this definition uses the axiom of choice through ac6sg 9645. (Contributed by Thierry Arnoux, 12-Jan-2020.)
(𝐴𝑉 → (𝐴Ref𝐵 ↔ ( 𝐵 𝐴 ∧ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))))

Theoremlocfinreflem 30505* 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 30506* 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)))

20.3.11.4  Open cover refinement property

Syntaxccref 30507 The "every open cover has an 𝐴 refinement" predicate.
class CovHasRef𝐴

Definitiondf-cref 30508* 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 30509* 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 30510 Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵)

Theoremcreftop 30511 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 30512* 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 30513* A formulation of crefi 30512 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)
𝑋 = 𝐽    &   𝐵 = CovHasRef𝐴    &   (𝑧𝐴𝜑)       ((𝐽𝐵𝐶𝐽𝑋 = 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑𝑧Ref𝐶))

Theoremcrefss 30514 The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐴𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)

Theoremcmpcref 30515 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 30516* 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))

20.3.11.5  Lindelöf spaces

Syntaxcldlf 30517 Extend class notation with the class of all Lindelöf spaces.
class Ldlf

Definitiondf-ldlf 30518 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 30519* 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))

20.3.11.6  Paracompact spaces

Syntaxcpcmp 30520 Extend class notation with the class of all paracompact topologies.
class Paracomp

Definitiondf-pcmp 30521 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 30522 The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))

Theoremcmppcmp 30523 Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝐽 ∈ Comp → 𝐽 ∈ Paracomp)

Theoremdispcmp 30524 Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
(𝑋𝑉 → 𝒫 𝑋 ∈ Paracomp)

Theorempcmplfin 30525* 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 30526* 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)))

20.3.11.7  Pseudometrics

Syntaxcmetid 30527 Extend class notation with the class of metric identifications.
class ~Met

Syntaxcpstm 30528 Extend class notation with the metric induced by a pseudometric.
class pstoMet

Definitiondf-metid 30529* 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 30530* 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 30531* Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑋𝑦𝑋) ∧ (𝑥𝐷𝑦) = 0)})

Theoremmetidss 30532 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 30533 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 30534 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 30535 The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)

Theorempstmval 30536* 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 30537 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 30538 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 / )))

Theoremhauseqcn 30539 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)

20.3.11.9  Topology of the closed unit interval

Theoremunitsscn 30540 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 30541 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 30542 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 30543 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 30544 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 30545 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 30546 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

20.3.11.10  Topology of ` ( RR X. RR ) `

Theoremunicls 30547 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 30548 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 30549 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 30550 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 30551 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 30552 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 30553 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 30554* Lemma for cnre2csqima 30555. (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 30555* 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 30556* 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))

20.3.11.11  Order topology - misc. additions

Theoremcnvordtrestixx 30557* 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 30558 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 30559 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 30560 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 30561 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 30562* 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 30563* 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 30564 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 30565 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 30566* Lemma for ordtrest2NEW 30567. (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 30567* 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 30568* Connectedness in the order topology of a toset. This is the "easy" direction of ordtconn 30569. See also reconnlem1 23037. (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 30569 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‘ )

20.3.11.12  Continuity in topological spaces - misc. additions

Theoremmndpluscn 30570* 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 30571 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 30572* 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 30573* 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 30574* 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 30575 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))

20.3.11.13  Topology of the extended nonnegative real numbers ordered monoid

Theoremxrge0hmph 30576 The extended nonnegative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
II ≃ ((ordTop‘ ≤ ) ↾t (0[,]+∞))

Theoremxrge0iifcnv 30577* 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 30578* 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 30579* 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 30580* 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 30581* 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 30582* 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 30583* 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 30584* 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 30585* 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 30586 The extended nonnegative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.)
(ℝ*𝑠s (0[,]+∞)) ∈ TopSp

Theoremxrge0topn 30587 The topology of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.)
(TopOpen‘(ℝ*𝑠s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞))

Theoremxrge0haus 30588 The topology of the extended nonnegative real numbers is Hausdorff. (Contributed by Thierry Arnoux, 26-Jul-2017.)
(TopOpen‘(ℝ*𝑠s (0[,]+∞))) ∈ Haus

Theoremxrge0tmdOLD 30589 The extended nonnegative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(ℝ*𝑠s (0[,]+∞)) ∈ TopMnd

Theoremxrge0tmd 30590 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

Theoremlmlim 30591 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 30592 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 30593 Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 16029. (Contributed by Thierry Arnoux, 28-Jul-2017.)
((𝐹:ℕ⟶(0[,)+∞) ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → sup(ran seq1( + , 𝐹), ℝ, < ) ∈ ℝ)

Theoremfsumcvg4 30594 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 30595* A neighborhood of +∞ contains an unbounded interval based at a real number. See pnfnei 21432. (Contributed by Thierry Arnoux, 31-Jul-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))       ((𝐴𝐽 ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)

Theoremlmxrge0 30596* 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 30597* 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 30598* 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)+∞)

20.3.11.15  Univariate polynomials

Theorempl1cn 30599 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))

20.3.12  Uniform Stuctures and Spaces

20.3.12.1  Hausdorff uniform completion

Syntaxchcmp 30600 Extend class notation with the Hausdorff uniform completion relation.
class HCmp

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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 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