Home Metamath Proof ExplorerTheorem List (p. 392 of 425) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26947) Hilbert Space Explorer (26948-28472) Users' Mathboxes (28473-42426)

Theorem List for Metamath Proof Explorer - 39101-39200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremqndenserrnopn 39101* n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝐼))    &   (𝜑𝑉𝐽)    &   (𝜑𝑉 ≠ ∅)       (𝜑 → ∃𝑦 ∈ (ℚ ↑𝑚 𝐼)𝑦𝑉)

Theoremqndenserrn 39102 n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝐼))       (𝜑 → ((cls‘𝐽)‘(ℚ ↑𝑚 𝐼)) = (ℝ ↑𝑚 𝐼))

Theoremrrxsnicc 39103* A multidimensional singleton expressed as a multidimensional closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ (ℝ ↑𝑚 𝑋))       (𝜑X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) = {𝐴})

Theoremrrnprjdstle 39104 The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝐺:𝑋⟶ℝ)    &   (𝜑𝐼𝑋)    &   𝐷 = (dist‘(ℝ^‘𝑋))       (𝜑 → (abs‘((𝐹𝐼) − (𝐺𝐼))) ≤ (𝐹𝐷𝐺))

Theoremrrndsmet 39105* 𝐷 is a metric for the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   𝐷 = (𝑓 ∈ (ℝ ↑𝑚 𝑋), 𝑔 ∈ (ℝ ↑𝑚 𝑋) ↦ (√‘Σ𝑘𝑋 (((𝑓𝑘) − (𝑔𝑘))↑2)))       (𝜑𝐷 ∈ (Met‘(ℝ ↑𝑚 𝑋)))

Theoremrrndsxmet 39106* 𝐷 is an extended metric for the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   𝐷 = (𝑓 ∈ (ℝ ↑𝑚 𝑋), 𝑔 ∈ (ℝ ↑𝑚 𝑋) ↦ (√‘Σ𝑘𝑋 (((𝑓𝑘) − (𝑔𝑘))↑2)))       (𝜑𝐷 ∈ (∞Met‘(ℝ ↑𝑚 𝑋)))

Theoremioorrnopnlem 39107* The a point in an indexed product of open intervals is contained in an open ball that is contained in the indexed product of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝐹X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)))    &   𝐻 = ran (𝑖𝑋 ↦ if(((𝐵𝑖) − (𝐹𝑖)) ≤ ((𝐹𝑖) − (𝐴𝑖)), ((𝐵𝑖) − (𝐹𝑖)), ((𝐹𝑖) − (𝐴𝑖))))    &   𝐸 = inf(𝐻, ℝ, < )    &   𝑉 = (𝐹(ball‘𝐷)𝐸)    &   𝐷 = (𝑓 ∈ (ℝ ↑𝑚 𝑋), 𝑔 ∈ (ℝ ↑𝑚 𝑋) ↦ (√‘Σ𝑘𝑋 (((𝑓𝑘) − (𝑔𝑘))↑2)))       (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹𝑣𝑣X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖))))

Theoremioorrnopn 39108* The indexed product of open intervals is an open set in (ℝ^‘𝑋). (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))

Theoremioorrnopnxrlem 39109* Given a point 𝐹 that belongs to an indexed product of (possibly unbounded) open intervals, then 𝐹 belongs to an open product of bounded open intervals that's a subset of the original indexed product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ*)    &   (𝜑𝐵:𝑋⟶ℝ*)    &   (𝜑𝐹X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)))    &   𝐿 = (𝑖𝑋 ↦ if((𝐴𝑖) = -∞, ((𝐹𝑖) − 1), (𝐴𝑖)))    &   𝑅 = (𝑖𝑋 ↦ if((𝐵𝑖) = +∞, ((𝐹𝑖) + 1), (𝐵𝑖)))    &   𝑉 = X𝑖𝑋 ((𝐿𝑖)(,)(𝑅𝑖))       (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹𝑣𝑣X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖))))

Theoremioorrnopnxr 39110* The indexed product of open intervals is an open set in (ℝ^‘𝑋). Similar to ioorrnopn 39108 but here unbounded intervals are allowed. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ*)    &   (𝜑𝐵:𝑋⟶ℝ*)       (𝜑X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))

20.31.19  Basic measure theory

20.31.19.1  σ-Algebras

Proofs for most of the theorems in section 111 of [Fremlin1]

Syntaxcsalg 39111 Extend class notation with the class of all sigma-algebras.
class SAlg

Definitiondf-salg 39112* Define the class of sigma-algebras. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥))}

Syntaxcsalon 39113 Extend class notation with the class of sigma-algebras on a set.
class SalOn

Definitiondf-salon 39114* Define the set of sigma-algebra on a given set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
SalOn = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ 𝑠 = 𝑥})

Syntaxcsalgen 39115 Extend class notation with the class of sigma-algebra generator.
class SalGen

Definitiondf-salgen 39116* Define the sigma-algebra generated by a given set. Definition 111G (b) of [Fremlin1] p. 13. The sigma-algebra generated by a set is the smallest sigma-algebra, on the same base set, that includes the set, see dfsalgen2 39142. The base set of the sigma-algebras used for the intersection needs to be the same, otherwise the resulting set is not guaranteed to be a sigma-algebra, as shown in the counterexample salgencntex 39144. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Revised by Glauco Siliprandi, 1-Jan-2021.)
SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)})

Theoremissal 39117* Express the predicate "𝑆 is a sigma-algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))

Theorempwsal 39118 The power set of a given set is a sigma-algebra (the so called discrete sigma-algebra). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)

Theoremsalunicl 39119 SAlg sigma-algebra is closed under countable union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝑇 ∈ 𝒫 𝑆)    &   (𝜑𝑇 ≼ ω)       (𝜑 𝑇𝑆)

Theoremsaluncl 39120 The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆𝐹𝑆) → (𝐸𝐹) ∈ 𝑆)

Theoremprsal 39121 The pair of the empty set and the whole base is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑋𝑉 → {∅, 𝑋} ∈ SAlg)

Theoremsaldifcl 39122 The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)

Theorem0sal 39123 The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆 ∈ SAlg → ∅ ∈ 𝑆)

Theoremsalgenval 39124* The sigma-algebra generated by a set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})

Theoremsaliuncl 39125* SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐾 ≼ ω)    &   ((𝜑𝑘𝐾) → 𝐸𝑆)       (𝜑 𝑘𝐾 𝐸𝑆)

Theoremsalincl 39126 The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆𝐹𝑆) → (𝐸𝐹) ∈ 𝑆)

Theoremsaluni 39127 A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆 ∈ SAlg → 𝑆𝑆)

Theoremsaliincl 39128* SAlg sigma-algebra is closed under countable indexed intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐾 ≼ ω)    &   (𝜑𝐾 ≠ ∅)    &   ((𝜑𝑘𝐾) → 𝐸𝑆)       (𝜑 𝑘𝐾 𝐸𝑆)

Theoremsaldifcl2 39129 The difference of two elements of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆𝐹𝑆) → (𝐸𝐹) ∈ 𝑆)

Theoremintsaluni 39130* The union of an arbitrary intersection of sigma-algebras on the same set 𝑋, is 𝑋. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐺 ⊆ SAlg)    &   (𝜑𝐺 ≠ ∅)    &   ((𝜑𝑠𝐺) → 𝑠 = 𝑋)       (𝜑 𝐺 = 𝑋)

Theoremintsal 39131* The arbitrary intersection of sigma-algebra (on the same set 𝑋) is a sigma-algebra ( on the same set 𝑋, see intsaluni 39130). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐺 ⊆ SAlg)    &   (𝜑𝐺 ≠ ∅)    &   ((𝜑𝑠𝐺) → 𝑠 = 𝑋)       (𝜑 𝐺 ∈ SAlg)

Theoremsalgenn0 39132* The set used in the definition of the generated sigma-algebra, is not empty. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)

Theoremsalgencl 39133 SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)

Theoremissald 39134* Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑆𝑉)    &   (𝜑 → ∅ ∈ 𝑆)    &   𝑋 = 𝑆    &   ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)    &   ((𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω) → 𝑦𝑆)       (𝜑𝑆 ∈ SAlg)

Theoremsalexct 39135* An example of non trivial sigma-algebra: the collection of all subsets which either are countable or have countable complement. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴𝑉)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}       (𝜑𝑆 ∈ SAlg)

Theoremsssalgen 39136 A set is a subset of the sigma-algebra it generates. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑆 = (SalGen‘𝑋)       (𝑋𝑉𝑋𝑆)

Theoremsalgenss 39137 The sigma-algebra generated by a set is the smallest sigma-algebra, on the same base set, that includes the set. Proposition 111G (b) of [Fremlin1] p. 13. Notice that the condition "on the same base set" is needed, see the counterexample salgensscntex 39145, where a sigma-algebra is shown that includes a set, but does not include the sigma-algebra generated (the key is that its base set is larger than the base set of the generating set). (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)    &   𝐺 = (SalGen‘𝑋)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝑋𝑆)    &   (𝜑 𝑆 = 𝑋)       (𝜑𝐺𝑆)

Theoremsalgenuni 39138 The base set of the sigma-algebra generated by a set is the union of the set itself. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)    &   𝑆 = (SalGen‘𝑋)    &   𝑈 = 𝑋       (𝜑 𝑆 = 𝑈)

Theoremissalgend 39139* One side of dfsalgen2 39142. If a sigma-algebra on 𝑋 includes 𝑋 and it is included in all the sigma-algebras with such two properties, then it is the sigma-algebra generated by 𝑋. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑 𝑆 = 𝑋)    &   (𝜑𝑋𝑆)    &   ((𝜑 ∧ (𝑦 ∈ SAlg ∧ 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)       (𝜑 → (SalGen‘𝑋) = 𝑆)

Theoremsalexct2 39140* An example of a subset that does not belong to a non trivial sigma-algebra, see salexct 39135. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (0[,]2)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}    &   𝐵 = (0[,]1)        ¬ 𝐵𝑆

Theoremunisalgen 39141 The union of a set belongs to the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)    &   𝑆 = (SalGen‘𝑋)    &   𝑈 = 𝑋       (𝜑𝑈𝑆)

Theoremdfsalgen2 39142* Alternate characterization of the sigma-algebra generated by a set. It is the smallest sigma-algebra, on the same base set, that includes the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)       (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))))

Theoremsalexct3 39143* An example of a sigma-algebra that's not closed under uncountable union. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (0[,]2)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}    &   𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦})       (𝑆 ∈ SAlg ∧ 𝑋𝑆 ∧ ¬ 𝑋𝑆)

Theoremsalgencntex 39144* This counterexample shows that df-salgen 39116 needs to require that all containing sigma-algebra have the same base set. Otherwise, the intersection could lead to a set that is not a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (0[,]2)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}    &   𝐵 = (0[,]1)    &   𝑇 = 𝒫 𝐵    &   𝐶 = (𝑆𝑇)    &   𝑍 = {𝑠 ∈ SAlg ∣ 𝐶𝑠}        ¬ 𝑍 ∈ SAlg

Theoremsalgensscntex 39145* This counterexample shows that the sigma-algebra generated by a set is not the smallest sigma-algebra containing the set, if we consider also sigma-algebras with a larger base set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (0[,]2)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}    &   𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦})    &   𝐺 = (SalGen‘𝑋)       (𝑋𝑆𝑆 ∈ SAlg ∧ ¬ 𝐺𝑆)

Theoremissalnnd 39146* Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑆𝑉)    &   (𝜑 → ∅ ∈ 𝑆)    &   𝑋 = 𝑆    &   ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)    &   ((𝜑𝑒:ℕ⟶𝑆) → 𝑛 ∈ ℕ (𝑒𝑛) ∈ 𝑆)       (𝜑𝑆 ∈ SAlg)

Theoremdmvolsal 39147 Lebesgue measurable sets form a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
dom vol ∈ SAlg

Theoremsaldifcld 39148 The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐸𝑆)       (𝜑 → ( 𝑆𝐸) ∈ 𝑆)

Theoremsaluncld 39149 The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐸𝑆)    &   (𝜑𝐹𝑆)       (𝜑 → (𝐸𝐹) ∈ 𝑆)

Theoremsalgencld 39150 SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋𝑉)    &   𝑆 = (SalGen‘𝑋)       (𝜑𝑆 ∈ SAlg)

Theorem0sald 39151 The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)       (𝜑 → ∅ ∈ 𝑆)

Theoremiooborel 39152 An open interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝐴(,)𝐶) ∈ 𝐵

Theoremsalincld 39153 The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐸𝑆)    &   (𝜑𝐹𝑆)       (𝜑 → (𝐸𝐹) ∈ 𝑆)

Theoremsalunid 39154 A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)       (𝜑 𝑆𝑆)

Theoremunisalgen2 39155 The union of a set belongs is equal to the union of the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   𝑆 = (SalGen‘𝐴)       (𝜑 𝑆 = 𝐴)

Theorembor1sal 39156 The Borel sigma-algebra on the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       𝐵 ∈ SAlg

Theoremiocborel 39157 A left-open, right-closed interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ)    &   𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝜑 → (𝐴(,]𝐶) ∈ 𝐵)

Theoremsubsaliuncllem 39158* A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑦𝜑    &   (𝜑𝑆𝑉)    &   𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})    &   𝐸 = (𝐻𝐺)    &   (𝜑𝐻 Fn ran 𝐺)    &   (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)       (𝜑 → ∃𝑒 ∈ (𝑆𝑚 ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))

Theoremsubsaliuncl 39159* A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷𝑉)    &   𝑇 = (𝑆t 𝐷)    &   (𝜑𝐹:ℕ⟶𝑇)       (𝜑 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇)

Theoremsubsalsal 39160 A subspace sigma-algebra is a sigma algebra. This is Lemma 121A of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐷𝑉)    &   𝑇 = (𝑆t 𝐷)       (𝜑𝑇 ∈ SAlg)

Theoremsubsaluni 39161 A set belongs to the subspace sigma-algebra it induces. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴 𝑆)       (𝜑𝐴 ∈ (𝑆t 𝐴))

20.31.19.2  Sum of nonnegative extended reals

Syntaxcsumge0 39162 Extend class notation to include the sum of nonnegative extended reals.
class Σ^

Definitiondf-sumge0 39163* Define the arbitrary sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) \$.
Σ^ = (𝑥 ∈ V ↦ if(+∞ ∈ ran 𝑥, +∞, sup(ran (𝑦 ∈ (𝒫 dom 𝑥 ∩ Fin) ↦ Σ𝑤𝑦 (𝑥𝑤)), ℝ*, < )))

Theoremsge0rnre 39164* When Σ^ is applied to nonnegative real numbers the range used in its definition is a subset of the reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹:𝑋⟶(0[,)+∞))       (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ⊆ ℝ)

Theoremfge0icoicc 39165 If 𝐹 maps to nonnegative reals, then 𝐹 maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹:𝑋⟶(0[,)+∞))       (𝜑𝐹:𝑋⟶(0[,]+∞))

Theoremsge0val 39166* The value of the sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑋𝑉𝐹:𝑋⟶(0[,]+∞)) → (Σ^𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤𝑦 (𝐹𝑤)), ℝ*, < )))

Theoremfge0npnf 39167 If 𝐹 maps to nonnegative reals, then +∞ is not in its range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹:𝑋⟶(0[,)+∞))       (𝜑 → ¬ +∞ ∈ ran 𝐹)

Theoremsge0rnn0 39168* The range used in the definition of Σ^ is not empty. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) ≠ ∅

Theoremsge0vald 39169* The value of the sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))       (𝜑 → (Σ^𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < )))

Theoremfge0iccico 39170 A range of nonnegative extended reals without plus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑 → ¬ +∞ ∈ ran 𝐹)       (𝜑𝐹:𝑋⟶(0[,)+∞))

Theoremgsumge0cl 39171 Closure of group sum, for finitely supported nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑𝐹 finSupp 0)       (𝜑 → (𝐺 Σg 𝐹) ∈ (0[,]+∞))

Theoremsge0reval 39172* Value of the sum of nonnegative extended reals, when all terms in the sum are reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,)+∞))       (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))

Theoremsge0pnfval 39173 If a term in the sum of nonnegative extended reals is +∞, then the value of the sum is +∞. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑 → +∞ ∈ ran 𝐹)       (𝜑 → (Σ^𝐹) = +∞)

Theoremfge0iccre 39174 A range of nonnegative extended reals without plus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑 → ¬ +∞ ∈ ran 𝐹)       (𝜑𝐹:𝑋⟶ℝ)

Theoremsge0z 39175* Any nonnegative extended sum of zero is zero. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)       (𝜑 → (Σ^‘(𝑘𝐴 ↦ 0)) = 0)

Theoremsge00 39176 The sum of nonnegative extended reals is zero when applied to the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
^‘∅) = 0

Theoremfsumlesge0 39177* Every finite subsum of nonnegative reals is less than or equal to the extended sum over the whole (possibly infinite) domain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,)+∞))    &   (𝜑𝑌𝑋)    &   (𝜑𝑌 ∈ Fin)       (𝜑 → Σ𝑥𝑌 (𝐹𝑥) ≤ (Σ^𝐹))

Theoremsge0revalmpt 39178* Value of the sum of nonnegative extended reals, when all terms in the sum are reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,)+∞))       (𝜑 → (Σ^‘(𝑥𝐴𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑥𝑦 𝐵), ℝ*, < ))

Theoremsge0sn 39179 A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:{𝐴}⟶(0[,]+∞))       (𝜑 → (Σ^𝐹) = (𝐹𝐴))

Theoremsge0tsms 39180 Σ^ applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))       (𝜑 → (Σ^𝐹) ∈ (𝐺 tsums 𝐹))

Theoremsge0cl 39181 The arbitrary sum of nonnegative extended reals is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))       (𝜑 → (Σ^𝐹) ∈ (0[,]+∞))

Theoremsge0f1o 39182* Re-index a nonnegative extended sum using a bijection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   𝑛𝜑    &   (𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶𝑉)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))

Theoremsge0snmpt 39183* A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐶 ∈ (0[,]+∞))    &   (𝑘 = 𝐴𝐵 = 𝐶)       (𝜑 → (Σ^‘(𝑘 ∈ {𝐴} ↦ 𝐵)) = 𝐶)

Theoremsge0ge0 39184 The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))       (𝜑 → 0 ≤ (Σ^𝐹))

Theoremsge0xrcl 39185 The arbitrary sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))       (𝜑 → (Σ^𝐹) ∈ ℝ*)

Theoremsge0repnf 39186 The of nonnegative extended reals is a real number if and only if it is not +∞. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))       (𝜑 → ((Σ^𝐹) ∈ ℝ ↔ ¬ (Σ^𝐹) = +∞))

Theoremsge0fsum 39187* The arbitrary sum of a finite set of nonnegative extended real numbers is equal to the sum of those numbers, when none of them is +∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐹:𝑋⟶(0[,)+∞))       (𝜑 → (Σ^𝐹) = Σ𝑥𝑋 (𝐹𝑥))

Theoremsge0rern 39188 If the sum of nonnegative extended reals is not +∞ then no terms is +∞. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑 → (Σ^𝐹) ∈ ℝ)       (𝜑 → ¬ +∞ ∈ ran 𝐹)

Theoremsge0supre 39189* If the arbitrary sum of nonnegative extended reals is real, then it is the supremum (in the real numbers) of finite subsums. Similar to sge0sup 39191, but here we can use sup with respect to instead of * (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑 → (Σ^𝐹) ∈ ℝ)       (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ, < ))

Theoremsge0fsummpt 39190* The arbitrary sum of a finite set of nonnegative extended real numbers is equal to the sum of those numbers, when none of them is +∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))       (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = Σ𝑘𝐴 𝐵)

Theoremsge0sup 39191* The arbitrary sum of nonnegative extended reals is the supremum of finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))       (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹𝑥))), ℝ*, < ))

Theoremsge0less 39192 A shorter sum of nonnegative extended reals is smaller than a longer one. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))       (𝜑 → (Σ^‘(𝐹𝑌)) ≤ (Σ^𝐹))

Theoremsge0rnbnd 39193* The range used in the definition of Σ^ is bounded, when the whole sum is a real number. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑 → (Σ^𝐹) ∈ ℝ)       (𝜑 → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦))𝑤𝑧)

Theoremsge0pr 39194* Sum of a pair of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷 ∈ (0[,]+∞))    &   (𝜑𝐸 ∈ (0[,]+∞))    &   (𝑘 = 𝐴𝐶 = 𝐷)    &   (𝑘 = 𝐵𝐶 = 𝐸)    &   (𝜑𝐴𝐵)       (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))

Theoremsge0gerp 39195* The arbitrary sum of nonnegative extended reals is larger or equal to a given extended real number if this number can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑𝐴 ∈ ℝ*)    &   ((𝜑𝑥 ∈ ℝ+) → ∃𝑧 ∈ (𝒫 𝑋 ∩ Fin)𝐴 ≤ ((Σ^‘(𝐹𝑧)) +𝑒 𝑥))       (𝜑𝐴 ≤ (Σ^𝐹))

Theoremsge0pnffigt 39196* If the sum of nonnegative extended reals is +∞, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑 → (Σ^𝐹) = +∞)    &   (𝜑𝑌 ∈ ℝ)       (𝜑 → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑌 < (Σ^‘(𝐹𝑥)))

Theoremsge0ssre 39197 If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑 → (Σ^𝐹) ∈ ℝ)       (𝜑 → (Σ^‘(𝐹𝑌)) ∈ ℝ)

Theoremsge0lefi 39198* A sum of nonnegative extended reals is smaller than a given extended real if and only if every finite subsum is smaller than it. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑𝐴 ∈ ℝ*)       (𝜑 → ((Σ^𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹𝑥)) ≤ 𝐴))

Theoremsge0lessmpt 39199* A shorter sum of nonnegative extended reals is smaller than a longer one. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶𝐴)       (𝜑 → (Σ^‘(𝑥𝐶𝐵)) ≤ (Σ^‘(𝑥𝐴𝐵)))

Theoremsge0ltfirp 39200* If the sum of nonnegative extended reals is real, then it can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑𝑌 ∈ ℝ+)    &   (𝜑 → (Σ^𝐹) ∈ ℝ)       (𝜑 → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^𝐹) < ((Σ^‘(𝐹𝑥)) + 𝑌))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 42401-42426
 Copyright terms: Public domain < Previous  Next >