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Theorem List for Metamath Proof Explorer - 29201-29300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremindpi1 29201 Preimage of the singleton {1} by the indicator function. See i1f1lem 23140. (Contributed by Thierry Arnoux, 21-Aug-2017.)
((𝑂𝑉𝐴𝑂) → (((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴)
 
Theoremindsum 29202* Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 14-Aug-2017.)
(𝜑𝑂 ∈ Fin)    &   (𝜑𝐴𝑂)    &   ((𝜑𝑥𝑂) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑥𝑂 ((((𝟭‘𝑂)‘𝐴)‘𝑥) · 𝐵) = Σ𝑥𝐴 𝐵)
 
Theoremindf1o 29203 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 29204 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 29205* 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})
 
20.3.13.3  Extended sum
 
Syntaxcesum 29206 Extend class notation to include infinite summations.
class Σ*𝑘𝐴𝐵
 
Definitiondf-esum 29207 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 29208 An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
Σ*𝑘𝐴𝐵 ∈ V
 
Theoremesumcl 29209* 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 29210 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)
𝑘𝜑    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
 
Theoremesumeq12dva 29211* 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 29212* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
 
Theoremesumeq1 29213* Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
(𝐴 = 𝐵 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
 
Theoremesumeq1d 29214 Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
 
Theoremesumeq2 29215* Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
(∀𝑘𝐴 𝐵 = 𝐶 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
 
Theoremesumeq2d 29216 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)
𝑘𝜑    &   (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
 
Theoremesumeq2dv 29217* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
((𝜑𝑘𝐴) → 𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
 
Theoremesumeq2sdv 29218* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
 
Theoremnfesum1 29219 Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝐴       𝑘Σ*𝑘𝐴𝐵
 
Theoremnfesum2 29220* Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 2-May-2020.)
𝑥𝐴    &   𝑥𝐵       𝑥Σ*𝑘𝐴𝐵
 
Theoremcbvesum 29221* 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 29222* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
(𝑗 = 𝑘𝐵 = 𝐶)       Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶
 
Theoremesumid 29223 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 29224 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 29225* 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 29226* 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 29227 Extended sum over the empty set. (Contributed by Thierry Arnoux, 19-Feb-2017.)
Σ*𝑥 ∈ ∅𝐴 = 0
 
Theoremesum0 29228* Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.)
𝑘𝐴       (𝐴𝑉 → Σ*𝑘𝐴0 = 0)
 
Theoremesumf1o 29229* 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 29230* 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 29231* 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 29232 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 29233* Extended sum is monotonic. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝜑    &   (𝜑𝐶𝑉)    &   ((𝜑𝑘𝐶) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐴𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 ≤ Σ*𝑘𝐶𝐵)
 
Theoremesumpad 29234* 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 29235* 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)
 
Theoremesumadd 29236* Addition of infinite sums. (Contributed by Thierry Arnoux, 24-Mar-2017.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴(𝐵 +𝑒 𝐶) = (Σ*𝑘𝐴𝐵 +𝑒 Σ*𝑘𝐴𝐶))
 
Theoremesumle 29237* 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 29238* 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 29239* 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)
 
Theoremesumaddf 29240* Addition of infinite sums. (Contributed by Thierry Arnoux, 22-Jun-2017.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴(𝐵 +𝑒 𝐶) = (Σ*𝑘𝐴𝐵 +𝑒 Σ*𝑘𝐴𝐶))
 
Theoremesumlef 29241* 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 29242* 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 29243* 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 29244* 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 29245* 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 29246* Extended sum over a pair, with a relaxed condition compared to esumpr 29245. (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 29247* 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 29248* 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 29249 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 29250 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 29251 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 29252* 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 29253 Lemma for esumpfinval 29254. (Contributed by Thierry Arnoux, 28-Jun-2017.)
((𝐴𝑉𝐹:𝐴⟶(0[,)+∞)) → (ℂfld Σg 𝐹) = ((ℝ*𝑠s (0[,]+∞)) Σg 𝐹))
 
Theoremesumpfinval 29254* 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 29255 Same as esumpfinval 29254, 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 29256* 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 29257* 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 29258* 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 29259* Lemma for esummulc2 29261 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 29260* 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 29261* 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 29262* 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 29263 Lemma for hasheuni 29264. (Contributed by Thierry Arnoux, 19-Nov-2016.)
#:V⟶(0[,]+∞)
 
Theoremhasheuni 29264* The cardinality of a disjoint union, not necessarily finite. cf. hashuni 14269. (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 29265* The sequence of partial sums of an extended sum converges to the whole sum. cf. fsumcvg2 14177. (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 29266* Simpler version of esumcvg 29265. (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 29267* 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 29268* 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 29269* "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 29270* 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 29271* Lemma for esum2d 29272 (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 29272* 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 29273* Sum over a non necessarily disjoint indexed union. The inegality 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)
 
20.3.14  Mixed Function/Constant operation
 
Syntaxcofc 29274 Extend class notation to include mapping of an operation to an operation for a function and a constant.
class 𝑓/𝑐𝑅
 
Definitiondf-ofc 29275* Define the function/constant operation map. The definition is designed so that if 𝑅 is a binary operation, then 𝑓/𝑐𝑅 is the analogous operation on functions and constants. (Contributed by Thierry Arnoux, 21-Jan-2017.)
𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
 
Theoremofceq 29276 Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝑅 = 𝑆 → ∘𝑓/𝑐𝑅 = ∘𝑓/𝑐𝑆)
 
Theoremofcfval 29277* 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/𝑐𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
 
Theoremofcval 29278 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/𝑐𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))
 
Theoremofcfn 29279 The function operation produces a function. (Contributed by Thierry Arnoux, 31-Jan-2017.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) Fn 𝐴)
 
Theoremofcfeqd2 29280* 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/𝑐𝑅𝐶) = (𝐹𝑓/𝑐𝑃𝐶))
 
Theoremofcfval3 29281* General value of (𝐹𝑓/𝑐𝑅𝐶) with no assumptions on functionality of 𝐹. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐹𝑉𝐶𝑊) → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥 ∈ dom 𝐹 ↦ ((𝐹𝑥)𝑅𝐶)))
 
Theoremofcf 29282* The function/constant operation produces a function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑇)       (𝜑 → (𝐹𝑓/𝑐𝑅𝐶):𝐴𝑈)
 
Theoremofcfval2 29283* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017.)
(𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   ((𝜑𝑥𝐴) → 𝐵𝑋)    &   (𝜑𝐹 = (𝑥𝐴𝐵))       (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
 
Theoremofcfval4 29284* The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = ((𝑥𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))
 
Theoremofcc 29285 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/𝑐𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))
 
Theoremofcof 29286 Relate function operation with operation with a constant. (Contributed by Thierry Arnoux, 3-Oct-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)       (𝜑 → (𝐹𝑓/𝑐𝑅𝐶) = (𝐹𝑓 𝑅(𝐴 × {𝐶})))
 
20.3.15  Abstract measure
 
20.3.15.1  Sigma-Algebra
 
Syntaxcsiga 29287 Extend class notation to include the function giving the sigma-algebras on a given base set.
class sigAlgebra
 
Definitiondf-siga 29288* 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 29289* Lemma for issiga 29291 and isrnsiga 29293. 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 29290. (Contributed by Thierry Arnoux, 24-Oct-2016.)
{𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V
 
Theoremsigaval 29290* 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 29291* 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)))))
 
TheoremisrnsigaOLD 29292* The property of being a sigma-algebra on an indefinite base set. (Contributed by Thierry Arnoux, 3-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
 
Theoremisrnsiga 29293* 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 29294 A sigma-algebra contains the empty set. (Contributed by Thierry Arnoux, 4-Sep-2016.)
(𝑆 ran sigAlgebra → ∅ ∈ 𝑆)
 
Theorembaselsiga 29295 A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
(𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴𝑆)
 
Theoremsigasspw 29296 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 29297 A sigma-algebra is closed under countable union. (Contributed by Thierry Arnoux, 26-Dec-2016.)
((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴𝑆)
 
Theoremsigaclcuni 29298* 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 29299 A sigma-algebra is closed under finite union. (Contributed by Thierry Arnoux, 28-Dec-2016.)
((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ∈ Fin) → 𝐴𝑆)
 
Theoremsigaclcu2 29300* 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)
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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 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