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Theorem List for Metamath Proof Explorer - 31301-31400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremindf1o 31301 The bijection between a power set and the set of indicator functions. (Contributed by Thierry Arnoux, 14-Aug-2017.)
(𝑂𝑉 → (𝟭‘𝑂):𝒫 𝑂1-1-onto→({0, 1} ↑m 𝑂))

Theoremindpreima 31302 A function with range {0, 1} as an indicator of the preimage of {1}. (Contributed by Thierry Arnoux, 23-Aug-2017.)
((𝑂𝑉𝐹:𝑂⟶{0, 1}) → 𝐹 = ((𝟭‘𝑂)‘(𝐹 “ {1})))

Theoremindf1ofs 31303* The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
(𝑂𝑉 → ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin})

20.3.15.3  Extended sum

Syntaxcesum 31304 Extend class notation to include infinite summations.
class Σ*𝑘𝐴𝐵

Definitiondf-esum 31305 Define a short-hand for the possibly infinite sum over the extended nonnegative reals. Σ* is relying on the properties of the tsums, developped by Mario Carneiro. (Contributed by Thierry Arnoux, 21-Sep-2016.)
Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))

Theoremesumex 31306 An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
Σ*𝑘𝐴𝐵 ∈ V

Theoremesumcl 31307* Closure for extended sum in the extended positive reals. (Contributed by Thierry Arnoux, 2-Jan-2017.)
𝑘𝐴       ((𝐴𝑉 ∧ ∀𝑘𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘𝐴𝐵 ∈ (0[,]+∞))

Theoremesumeq12dvaf 31308 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)
𝑘𝜑    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)

Theoremesumeq12dva 31309* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) (Revised by Thierry Arnoux, 29-Jun-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)

Theoremesumeq12d 31310* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)

Theoremesumeq1 31311* Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
(𝐴 = 𝐵 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)

Theoremesumeq1d 31312 Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)

Theoremesumeq2 31313* Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
(∀𝑘𝐴 𝐵 = 𝐶 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Theoremesumeq2d 31314 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)
𝑘𝜑    &   (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Theoremesumeq2dv 31315* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
((𝜑𝑘𝐴) → 𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Theoremesumeq2sdv 31316* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Theoremnfesum1 31317 Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝐴       𝑘Σ*𝑘𝐴𝐵

Theoremnfesum2 31318* Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 2-May-2020.)
𝑥𝐴    &   𝑥𝐵       𝑥Σ*𝑘𝐴𝐵

Theoremcbvesum 31319* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
(𝑗 = 𝑘𝐵 = 𝐶)    &   𝑘𝐴    &   𝑗𝐴    &   𝑘𝐵    &   𝑗𝐶       Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶

Theoremcbvesumv 31320* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
(𝑗 = 𝑘𝐵 = 𝐶)       Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶

Theoremesumid 31321 Identify the extended sum as any limit points of the infinite sum. (Contributed by Thierry Arnoux, 9-May-2017.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)))       (𝜑 → Σ*𝑘𝐴𝐵 = 𝐶)

Theoremesumgsum 31322 A finite extended sum is the group sum over the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 24-Apr-2020.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝐴𝐵)))

Theoremesumval 31323* Develop the value of the extended sum. (Contributed by Thierry Arnoux, 4-Jan-2017.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑥𝐵)) = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < ))

Theoremesumel 31324* The extended sum is a limit point of the corresponding infinite group sum. (Contributed by Thierry Arnoux, 24-Mar-2017.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴𝐵 ∈ ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)))

Theoremesumnul 31325 Extended sum over the empty set. (Contributed by Thierry Arnoux, 19-Feb-2017.)
Σ*𝑥 ∈ ∅𝐴 = 0

Theoremesum0 31326* Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.)
𝑘𝐴       (𝐴𝑉 → Σ*𝑘𝐴0 = 0)

Theoremesumf1o 31327* Re-index an extended sum using a bijection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑛𝜑    &   𝑛𝐵    &   𝑘𝐷    &   𝑛𝐴    &   𝑛𝐶    &   𝑛𝐹    &   (𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑛𝐶𝐷)

Theoremesumc 31328* Convert from the collection form to the class-variable form of a sum. (Contributed by Thierry Arnoux, 10-May-2017.)
𝑘𝐷    &   𝑘𝜑    &   𝑘𝐴    &   (𝑦 = 𝐶𝐷 = 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑 → Fun (𝑘𝐴𝐶))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶𝑊)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐶}𝐷)

Theoremesumrnmpt 31329* Rewrite an extended sum into a sum on the range of a mapping function. (Contributed by Thierry Arnoux, 27-May-2020.)
𝑘𝐴    &   (𝑦 = 𝐵𝐶 = 𝐷)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐷 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (𝑊 ∖ {∅}))    &   (𝜑Disj 𝑘𝐴 𝐵)       (𝜑 → Σ*𝑦 ∈ ran (𝑘𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐷)

Theoremesumsplit 31330 Split an extended sum into two parts. (Contributed by Thierry Arnoux, 9-May-2017.)
𝑘𝜑    &   𝑘𝐴    &   𝑘𝐵    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑 → (𝐴𝐵) = ∅)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐵) → 𝐶 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘 ∈ (𝐴𝐵)𝐶 = (Σ*𝑘𝐴𝐶 +𝑒 Σ*𝑘𝐵𝐶))

Theoremesummono 31331* Extended sum is monotonic. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝜑    &   (𝜑𝐶𝑉)    &   ((𝜑𝑘𝐶) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐴𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 ≤ Σ*𝑘𝐶𝐵)

Theoremesumpad 31332* Extend an extended sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 31-May-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐵) → 𝐶 = 0)       (𝜑 → Σ*𝑘 ∈ (𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐶)

Theoremesumpad2 31333* Remove zeroes from an extended sum. (Contributed by Thierry Arnoux, 5-Jun-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐵) → 𝐶 = 0)       (𝜑 → Σ*𝑘 ∈ (𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐶)

(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴(𝐵 +𝑒 𝐶) = (Σ*𝑘𝐴𝐵 +𝑒 Σ*𝑘𝐴𝐶))

Theoremesumle 31335* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐵𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 ≤ Σ*𝑘𝐴𝐶)

Theoremgsumesum 31336* Relate a group sum on (ℝ*𝑠s (0[,]+∞)) to a finite extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.) (Proof shortened by AV, 12-Dec-2019.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝐴𝐵)) = Σ*𝑘𝐴𝐵)

Theoremesumlub 31337* The extended sum is the lowest upper bound for the partial sums. (Contributed by Thierry Arnoux, 19-Oct-2017.) (Proof shortened by AV, 12-Dec-2019.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝑋 ∈ ℝ*)    &   (𝜑𝑋 < Σ*𝑘𝐴𝐵)       (𝜑 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑋 < Σ*𝑘𝑎𝐵)

𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴(𝐵 +𝑒 𝐶) = (Σ*𝑘𝐴𝐵 +𝑒 Σ*𝑘𝐴𝐶))

Theoremesumlef 31339* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐵𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 ≤ Σ*𝑘𝐴𝐶)

Theoremesumcst 31340* The extended sum of a constant. (Contributed by Thierry Arnoux, 3-Mar-2017.) (Revised by Thierry Arnoux, 5-Jul-2017.)
𝑘𝐴    &   𝑘𝐵       ((𝐴𝑉𝐵 ∈ (0[,]+∞)) → Σ*𝑘𝐴𝐵 = ((♯‘𝐴) ·e 𝐵))

Theoremesumsnf 31341* The extended sum of a singleton is the term. (Contributed by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 2-May-2020.)
𝑘𝐵    &   ((𝜑𝑘 = 𝑀) → 𝐴 = 𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘 ∈ {𝑀}𝐴 = 𝐵)

Theoremesumsn 31342* The extended sum of a singleton is the term. (Contributed by Thierry Arnoux, 2-Jan-2017.) (Shortened by Thierry Arnoux, 2-May-2020.)
((𝜑𝑘 = 𝑀) → 𝐴 = 𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘 ∈ {𝑀}𝐴 = 𝐵)

Theoremesumpr 31343* Extended sum over a pair. (Contributed by Thierry Arnoux, 2-Jan-2017.)
((𝜑𝑘 = 𝐴) → 𝐶 = 𝐷)    &   ((𝜑𝑘 = 𝐵) → 𝐶 = 𝐸)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷 ∈ (0[,]+∞))    &   (𝜑𝐸 ∈ (0[,]+∞))    &   (𝜑𝐴𝐵)       (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸))

Theoremesumpr2 31344* Extended sum over a pair, with a relaxed condition compared to esumpr 31343. (Contributed by Thierry Arnoux, 2-Jan-2017.)
((𝜑𝑘 = 𝐴) → 𝐶 = 𝐷)    &   ((𝜑𝑘 = 𝐵) → 𝐶 = 𝐸)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷 ∈ (0[,]+∞))    &   (𝜑𝐸 ∈ (0[,]+∞))    &   (𝜑 → (𝐴 = 𝐵 → (𝐷 = 0 ∨ 𝐷 = +∞)))       (𝜑 → Σ*𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 +𝑒 𝐸))

Theoremesumrnmpt2 31345* Rewrite an extended sum into a sum on the range of a mapping function. (Contributed by Thierry Arnoux, 30-May-2020.)
(𝑦 = 𝐵𝐶 = 𝐷)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐷 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐵𝑊)    &   (((𝜑𝑘𝐴) ∧ 𝐵 = ∅) → 𝐷 = 0)    &   (𝜑Disj 𝑘𝐴 𝐵)       (𝜑 → Σ*𝑦 ∈ ran (𝑘𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐷)

Theoremesumfzf 31346* Formulating a partial extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)
𝑘𝐹       ((𝐹:ℕ⟶(0[,]+∞) ∧ 𝑁 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑁)(𝐹𝑘) = (seq1( +𝑒 , 𝐹)‘𝑁))

Theoremesumfsup 31347 Formulating an extended sum over integers using the recursive sequence builder. (Contributed by Thierry Arnoux, 18-Oct-2017.)
𝑘𝐹       (𝐹:ℕ⟶(0[,]+∞) → Σ*𝑘 ∈ ℕ(𝐹𝑘) = sup(ran seq1( +𝑒 , 𝐹), ℝ*, < ))

Theoremesumfsupre 31348 Formulating an extended sum over integers using the recursive sequence builder. This version is limited to real-valued functions. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝐹       (𝐹:ℕ⟶(0[,)+∞) → Σ*𝑘 ∈ ℕ(𝐹𝑘) = sup(ran seq1( + , 𝐹), ℝ*, < ))

Theoremesumss 31349 Change the index set to a subset by adding zeroes. (Contributed by Thierry Arnoux, 19-Jun-2017.)
𝑘𝜑    &   𝑘𝐴    &   𝑘𝐵    &   (𝜑𝐴𝐵)    &   (𝜑𝐵𝑉)    &   ((𝜑𝑘𝐵) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)

Theoremesumpinfval 31350* The value of the extended sum of nonnegative terms, with at least one infinite term. (Contributed by Thierry Arnoux, 19-Jun-2017.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑 → ∃𝑘𝐴 𝐵 = +∞)       (𝜑 → Σ*𝑘𝐴𝐵 = +∞)

Theoremesumpfinvallem 31351 Lemma for esumpfinval 31352. (Contributed by Thierry Arnoux, 28-Jun-2017.)
((𝐴𝑉𝐹:𝐴⟶(0[,)+∞)) → (ℂfld Σg 𝐹) = ((ℝ*𝑠s (0[,]+∞)) Σg 𝐹))

Theoremesumpfinval 31352* The value of the extended sum of a finite set of nonnegative finite terms. (Contributed by Thierry Arnoux, 28-Jun-2017.) (Proof shortened by AV, 25-Jul-2019.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))       (𝜑 → Σ*𝑘𝐴𝐵 = Σ𝑘𝐴 𝐵)

Theoremesumpfinvalf 31353 Same as esumpfinval 31352, minus distinct variable restrictions. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Proof shortened by AV, 25-Jul-2019.)
𝑘𝐴    &   𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))       (𝜑 → Σ*𝑘𝐴𝐵 = Σ𝑘𝐴 𝐵)

Theoremesumpinfsum 31354* The value of the extended sum of infinitely many terms greater than one. (Contributed by Thierry Arnoux, 29-Jun-2017.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   (𝜑 → ¬ 𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝑀𝐵)    &   (𝜑𝑀 ∈ ℝ*)    &   (𝜑 → 0 < 𝑀)       (𝜑 → Σ*𝑘𝐴𝐵 = +∞)

Theoremesumpcvgval 31355* The value of the extended sum when the corresponding series sum is convergent. (Contributed by Thierry Arnoux, 31-Jul-2017.)
((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))    &   (𝑘 = 𝑙𝐴 = 𝐵)    &   (𝜑 → (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)𝐴) ∈ dom ⇝ )       (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Theoremesumpmono 31356* The partial sums in an extended sum form a monotonic sequence. (Contributed by Thierry Arnoux, 31-Aug-2017.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))       (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ≤ Σ*𝑘 ∈ (1...𝑁)𝐴)

Theoremesumcocn 31357* Lemma for esummulc2 31359 and co. Composing with a continuous function preserves extended sums. (Contributed by Thierry Arnoux, 29-Jun-2017.)
𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ (𝐽 Cn 𝐽))    &   (𝜑 → (𝐶‘0) = 0)    &   ((𝜑𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶𝑥) +𝑒 (𝐶𝑦)))       (𝜑 → (𝐶‘Σ*𝑘𝐴𝐵) = Σ*𝑘𝐴(𝐶𝐵))

Theoremesummulc1 31358* An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ (0[,)+∞))       (𝜑 → (Σ*𝑘𝐴𝐵 ·e 𝐶) = Σ*𝑘𝐴(𝐵 ·e 𝐶))

Theoremesummulc2 31359* An extended sum multiplied by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ (0[,)+∞))       (𝜑 → (𝐶 ·e Σ*𝑘𝐴𝐵) = Σ*𝑘𝐴(𝐶 ·e 𝐵))

Theoremesumdivc 31360* An extended sum divided by a constant. (Contributed by Thierry Arnoux, 6-Jul-2017.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → (Σ*𝑘𝐴𝐵 /𝑒 𝐶) = Σ*𝑘𝐴(𝐵 /𝑒 𝐶))

Theoremhashf2 31361 Lemma for hasheuni 31362. (Contributed by Thierry Arnoux, 19-Nov-2016.)
♯:V⟶(0[,]+∞)

Theoremhasheuni 31362* The cardinality of a disjoint union, not necessarily finite. cf. hashuni 15170. (Contributed by Thierry Arnoux, 19-Nov-2016.) (Revised by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 20-Jun-2017.)
((𝐴𝑉Disj 𝑥𝐴 𝑥) → (♯‘ 𝐴) = Σ*𝑥𝐴(♯‘𝑥))

Theoremesumcvg 31363* The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 15073. (Contributed by Thierry Arnoux, 5-Sep-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))    &   𝐹 = (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)    &   ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))    &   (𝑘 = 𝑚𝐴 = 𝐵)       (𝜑𝐹(⇝𝑡𝐽*𝑘 ∈ ℕ𝐴)

Theoremesumcvg2 31364* Simpler version of esumcvg 31363. (Contributed by Thierry Arnoux, 5-Sep-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))    &   ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))    &   (𝑘 = 𝑙𝐴 = 𝐵)    &   (𝑘 = 𝑚𝐴 = 𝐶)       (𝜑 → (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴)(⇝𝑡𝐽*𝑘 ∈ ℕ𝐴)

Theoremesumcvgsum 31365* The value of the extended sum when the corresponding sum is convergent. (Contributed by Thierry Arnoux, 29-Oct-2019.)
(𝑘 = 𝑖𝐴 = 𝐵)    &   ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = 𝐴)    &   (𝜑 → seq1( + , 𝐹) ⇝ 𝐿)    &   (𝜑𝐿 ∈ ℝ)       (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = Σ𝑘 ∈ ℕ 𝐴)

Theoremesumsup 31366* Express an extended sum as a supremum of extended sums. (Contributed by Thierry Arnoux, 24-May-2020.)
(𝜑𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘 ∈ ℕ𝐴 = sup(ran (𝑛 ∈ ℕ ↦ Σ*𝑘 ∈ (1...𝑛)𝐴), ℝ*, < ))

Theoremesumgect 31367* "Send 𝑛 to +∞ " in an inequality with an extended sum. (Contributed by Thierry Arnoux, 24-May-2020.)
(𝜑𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))    &   ((𝜑𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)𝐴𝐵)       (𝜑 → Σ*𝑘 ∈ ℕ𝐴𝐵)

Theoremesumcvgre 31368* All terms of a converging extended sum shall be finite. (Contributed by Thierry Arnoux, 23-Sep-2019.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑 → Σ*𝑘𝐴𝐵 ∈ ℝ)       ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)

Theoremesum2dlem 31369* Lemma for esum2d 31370 (finite case). (Contributed by Thierry Arnoux, 17-May-2020.) (Proof shortened by AV, 17-Sep-2021.)
𝑘𝐹    &   (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ (0[,]+∞))    &   (𝜑𝐴 ∈ Fin)       (𝜑 → Σ*𝑗𝐴Σ*𝑘𝐵𝐶 = Σ*𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐹)

Theoremesum2d 31370* Write a double extended sum as a sum over a two-dimensional region. Note that 𝐵(𝑗) is a function of 𝑗. This can be seen as "slicing" the relation 𝐴. (Contributed by Thierry Arnoux, 17-May-2020.)
𝑘𝐹    &   (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ (0[,]+∞))       (𝜑 → Σ*𝑗𝐴Σ*𝑘𝐵𝐶 = Σ*𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐹)

Theoremesumiun 31371* Sum over a nonnecessarily disjoint indexed union. The inequality is strict in the case where the sets B(x) overlap. (Contributed by Thierry Arnoux, 21-Sep-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)    &   (((𝜑𝑗𝐴) ∧ 𝑘𝐵) → 𝐶 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘 𝑗𝐴 𝐵𝐶 ≤ Σ*𝑗𝐴Σ*𝑘𝐵𝐶)

20.3.16  Mixed Function/Constant operation

Syntaxcofc 31372 Extend class notation to include mapping of an operation to an operation for a function and a constant.
class f/c 𝑅

Definitiondf-ofc 31373* Define the function/constant operation map. The definition is designed so that if 𝑅 is a binary operation, then f/c 𝑅 is the analogous operation on functions and constants. (Contributed by Thierry Arnoux, 21-Jan-2017.)
f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))

Theoremofceq 31374 Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆)

Theoremofcfval 31375* Value of an operation applied to a function and a constant. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)       (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))

Theoremofcval 31376 Evaluate a function/constant operation at a point. (Contributed by Thierry Arnoux, 31-Jan-2017.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐵)       ((𝜑𝑋𝐴) → ((𝐹f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))

Theoremofcfn 31377 The function operation produces a function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → (𝐹f/c 𝑅𝐶) Fn 𝐴)

Theoremofcfeqd2 31378* Equality theorem for function/constant operation value. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)    &   ((𝜑𝑦𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶))    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f/c 𝑃𝐶))

Theoremofcfval3 31379* General value of (𝐹f/c 𝑅𝐶) with no assumptions on functionality of 𝐹. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐹𝑉𝐶𝑊) → (𝐹f/c 𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))

Theoremofcf 31380* The function/constant operation produces a function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑇)       (𝜑 → (𝐹f/c 𝑅𝐶):𝐴𝑈)

Theoremofcfval2 31381* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017.)
(𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   ((𝜑𝑥𝐴) → 𝐵𝑋)    &   (𝜑𝐹 = (𝑥𝐴𝐵))       (𝜑 → (𝐹f/c 𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))

Theoremofcfval4 31382* The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → (𝐹f/c 𝑅𝐶) = ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))

Theoremofcc 31383 Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)       (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Theoremofcof 31384 Relate function operation with operation with a constant. (Contributed by Thierry Arnoux, 3-Oct-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → (𝐹f/c 𝑅𝐶) = (𝐹f 𝑅(𝐴 × {𝐶})))

20.3.17  Abstract measure

20.3.17.1  Sigma-Algebra

Syntaxcsiga 31385 Extend class notation to include the function giving the sigma-algebras on a given base set.
class sigAlgebra

Definitiondf-siga 31386* Define a sigma-algebra, i.e. a set closed under complement and countable union. Literature usually uses capital greek sigma and omega letters for the algebra set, and the base set respectively. We are using 𝑆 and 𝑂 as a parallel. (Contributed by Thierry Arnoux, 3-Sep-2016.)
sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})

Theoremsigaex 31387* Lemma for issiga 31389 and isrnsiga 31390. The class of sigma-algebras with base set 𝑜 is a set. Note: a more generic version with (𝑂 ∈ V → ...) could be useful for sigaval 31388. (Contributed by Thierry Arnoux, 24-Oct-2016.)
{𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V

Theoremsigaval 31388* The set of sigma-algebra with a given base set. (Contributed by Thierry Arnoux, 23-Sep-2016.)
(𝑂 ∈ V → (sigAlgebra‘𝑂) = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})

Theoremissiga 31389* An alternative definition of the sigma-algebra, for a given base set. (Contributed by Thierry Arnoux, 19-Sep-2016.)
(𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ⊆ 𝒫 𝑂 ∧ (𝑂𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))

Theoremisrnsiga 31390* The property of being a sigma-algebra on an indefinite base set. (Contributed by Thierry Arnoux, 3-Sep-2016.) (Proof shortened by Thierry Arnoux, 23-Oct-2016.)
(𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))

Theorem0elsiga 31391 A sigma-algebra contains the empty set. (Contributed by Thierry Arnoux, 4-Sep-2016.)
(𝑆 ran sigAlgebra → ∅ ∈ 𝑆)

Theorembaselsiga 31392 A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
(𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴𝑆)

Theoremsigasspw 31393 A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.)
(𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ⊆ 𝒫 𝐴)

Theoremsigaclcu 31394 A sigma-algebra is closed under countable union. (Contributed by Thierry Arnoux, 26-Dec-2016.)
((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴𝑆)

Theoremsigaclcuni 31395* A sigma-algebra is closed under countable union: indexed union version. (Contributed by Thierry Arnoux, 8-Jun-2017.)
𝑘𝐴       ((𝑆 ran sigAlgebra ∧ ∀𝑘𝐴 𝐵𝑆𝐴 ≼ ω) → 𝑘𝐴 𝐵𝑆)

Theoremsigaclfu 31396 A sigma-algebra is closed under finite union. (Contributed by Thierry Arnoux, 28-Dec-2016.)
((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ∈ Fin) → 𝐴𝑆)

Theoremsigaclcu2 31397* A sigma-algebra is closed under countable union - indexing on (Contributed by Thierry Arnoux, 29-Dec-2016.)
((𝑆 ran sigAlgebra ∧ ∀𝑘 ∈ ℕ 𝐴𝑆) → 𝑘 ∈ ℕ 𝐴𝑆)

Theoremsigaclfu2 31398* A sigma-algebra is closed under finite union - indexing on (1..^𝑁). (Contributed by Thierry Arnoux, 28-Dec-2016.)
((𝑆 ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑁)𝐴𝑆) → 𝑘 ∈ (1..^𝑁)𝐴𝑆)

Theoremsigaclcu3 31399* A sigma-algebra is closed under countable or finite union. (Contributed by Thierry Arnoux, 6-Mar-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))    &   ((𝜑𝑘𝑁) → 𝐴𝑆)       (𝜑 𝑘𝑁 𝐴𝑆)

Theoremissgon 31400 Property of being a sigma-algebra with a given base set, noting that the base set of a sigma-algebra is actually its union set. (Contributed by Thierry Arnoux, 24-Sep-2016.) (Revised by Thierry Arnoux, 23-Oct-2016.)
(𝑆 ∈ (sigAlgebra‘𝑂) ↔ (𝑆 ran sigAlgebra ∧ 𝑂 = 𝑆))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45163
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