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Theorem List for Metamath Proof Explorer - 42801-42900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsaluncl 42801 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 42802 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 42803 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 42804 The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆 ∈ SAlg → ∅ ∈ 𝑆)

Theoremsalgenval 42805* The sigma-algebra generated by a set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})

Theoremsaliuncl 42806* 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 42807 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 42808 A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆 ∈ SAlg → 𝑆𝑆)

Theoremsaliincl 42809* 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 42810 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 42811* 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 42812* The arbitrary intersection of sigma-algebra (on the same set 𝑋) is a sigma-algebra ( on the same set 𝑋, see intsaluni 42811). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐺 ⊆ SAlg)    &   (𝜑𝐺 ≠ ∅)    &   ((𝜑𝑠𝐺) → 𝑠 = 𝑋)       (𝜑 𝐺 ∈ SAlg)

Theoremsalgenn0 42813* 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 42814 SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)

Theoremissald 42815* 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 42816* An example of nontrivial 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 42817 A set is a subset of the sigma-algebra it generates. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑆 = (SalGen‘𝑋)       (𝑋𝑉𝑋𝑆)

Theoremsalgenss 42818 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 42826, 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 42819 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 42820* One side of dfsalgen2 42823. 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 42821* An example of a subset that does not belong to a nontrivial sigma-algebra, see salexct 42816. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (0[,]2)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}    &   𝐵 = (0[,]1)        ¬ 𝐵𝑆

Theoremunisalgen 42822 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 42823* 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 42824* 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 42825* This counterexample shows that df-salgen 42797 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 42826* 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 42827* 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 42828 Lebesgue measurable sets form a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
dom vol ∈ SAlg

Theoremsaldifcld 42829 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 42830 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 42831 SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋𝑉)    &   𝑆 = (SalGen‘𝑋)       (𝜑𝑆 ∈ SAlg)

Theorem0sald 42832 The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)       (𝜑 → ∅ ∈ 𝑆)

Theoremiooborel 42833 An open interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝐴(,)𝐶) ∈ 𝐵

Theoremsalincld 42834 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 42835 A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)       (𝜑 𝑆𝑆)

Theoremunisalgen2 42836 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 42837 The Borel sigma-algebra on the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       𝐵 ∈ SAlg

Theoremiocborel 42838 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 42839* 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 42840* 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 42841 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 42842 A set belongs to the subspace sigma-algebra it induces. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐴 𝑆)       (𝜑𝐴 ∈ (𝑆t 𝐴))

20.37.19.2  Sum of nonnegative extended reals

Syntaxcsumge0 42843 Extend class notation to include the sum of nonnegative extended reals.
class Σ^

Definitiondf-sumge0 42844* 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 42845* 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 42846 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 42847* 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 42848 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 42849* 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 42850* 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 42851 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 42852 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 42853* 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 42854 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 42855 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 42856* Any nonnegative extended sum of zero is zero. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)       (𝜑 → (Σ^‘(𝑘𝐴 ↦ 0)) = 0)

Theoremsge00 42857 The sum of nonnegative extended reals is zero when applied to the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
^‘∅) = 0

Theoremfsumlesge0 42858* 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 42859* 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 42860 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 42861 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 42862 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 42863* 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 42864* 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 42865 The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))       (𝜑 → 0 ≤ (Σ^𝐹))

Theoremsge0xrcl 42866 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 42867 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 42868* 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 42869 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 42870* 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 42872, 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 42871* 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 42872* 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 42873 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 42874* 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 42875* 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 42876* The arbitrary sum of nonnegative extended reals is greater than 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 42877* 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 42878 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 42879* 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 42880* 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 42881* 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))

Theoremsge0prle 42882* The sum of a pair of nonnegative extended reals is less than or equal their extended addition. When it is a distinct pair, than equality holds, see sge0pr 42875. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷 ∈ (0[,]+∞))    &   (𝜑𝐸 ∈ (0[,]+∞))    &   (𝑘 = 𝐴𝐶 = 𝐷)    &   (𝑘 = 𝐵𝐶 = 𝐸)       (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸))

Theoremsge0gerpmpt 42883* The arbitrary sum of nonnegative extended reals is greater than 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.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ ℝ*)    &   ((𝜑𝑦 ∈ ℝ+) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐶 ≤ ((Σ^‘(𝑥𝑧𝐵)) +𝑒 𝑦))       (𝜑𝐶 ≤ (Σ^‘(𝑥𝐴𝐵)))

Theoremsge0resrnlem 42884 The sum of nonnegative extended reals restricted to the range of a function is less than or equal to the sum of the composition of the two functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐵⟶(0[,]+∞))    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝑋 ∈ 𝒫 𝐴)    &   (𝜑 → (𝐺𝑋):𝑋1-1-onto→ran 𝐺)       (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹𝐺)))

Theoremsge0resrn 42885 The sum of nonnegative extended reals restricted to the range of a function is less than or equal to the sum of the composition of the two functions (well-order hypothesis allows to avoid using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐵⟶(0[,]+∞))    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝑅 We 𝐴)       (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹𝐺)))

Theoremsge0ssrempt 42886* If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ ℝ)    &   (𝜑𝐶𝐴)       (𝜑 → (Σ^‘(𝑥𝐶𝐵)) ∈ ℝ)

Theoremsge0resplit 42887 Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 42890. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝑈 = (𝐴𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝐹:𝑈⟶(0[,]+∞))    &   (𝜑 → (Σ^𝐹) ∈ ℝ)       (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) + (Σ^‘(𝐹𝐵))))

Theoremsge0le 42888* If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑𝐺:𝑋⟶(0[,]+∞))    &   ((𝜑𝑥𝑋) → (𝐹𝑥) ≤ (𝐺𝑥))       (𝜑 → (Σ^𝐹) ≤ (Σ^𝐺))

Theoremsge0ltfirpmpt 42889* If the extended sum of nonnegative reals is not +∞, then it can be approximated from below by finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝑌 ∈ ℝ+)    &   (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ ℝ)       (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥𝐴𝐵)) < ((Σ^‘(𝑥𝑦𝐵)) + 𝑌))

Theoremsge0split 42890 Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   𝑈 = (𝐴𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝐹:𝑈⟶(0[,]+∞))       (𝜑 → (Σ^𝐹) = ((Σ^‘(𝐹𝐴)) +𝑒^‘(𝐹𝐵))))

Theoremsge0lempt 42891* If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑥𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ≤ (Σ^‘(𝑥𝐴𝐶)))

Theoremsge0splitmpt 42892* Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑 → (𝐴𝐵) = ∅)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑥𝐵) → 𝐶 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑥 ∈ (𝐴𝐵) ↦ 𝐶)) = ((Σ^‘(𝑥𝐴𝐶)) +𝑒^‘(𝑥𝐵𝐶))))

Theoremsge0ss 42893* Change the index set to a subset in a sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝐵𝑉)    &   (𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 0)       (𝜑 → (Σ^‘(𝑘𝐴𝐶)) = (Σ^‘(𝑘𝐵𝐶)))

Theoremsge0iunmptlemfi 42894* Sum of nonnegative extended reals over a disjoint indexed union (in this lemma, for a finite index set). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   ((𝜑𝑥𝐴𝑘𝐵) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑥𝐴) → (Σ^‘(𝑘𝐵𝐶)) ∈ ℝ)       (𝜑 → (Σ^‘(𝑘 𝑥𝐴 𝐵𝐶)) = (Σ^‘(𝑥𝐴 ↦ (Σ^‘(𝑘𝐵𝐶)))))

Theoremsge0p1 42895* The addition of the next term in a finite sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ (0[,]+∞))    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)       (𝜑 → (Σ^‘(𝑘 ∈ (𝑀...(𝑁 + 1)) ↦ 𝐴)) = ((Σ^‘(𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)) +𝑒 𝐵))

Theoremsge0iunmptlemre 42896* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   ((𝜑𝑥𝐴𝑘𝐵) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑥𝐴) → (Σ^‘(𝑘𝐵𝐶)) ∈ ℝ)    &   (𝜑 → (Σ^‘(𝑘 𝑥𝐴 𝐵𝐶)) ∈ ℝ*)    &   (𝜑 → (Σ^‘(𝑥𝐴 ↦ (Σ^‘(𝑘𝐵𝐶)))) ∈ ℝ*)    &   (𝜑 → (𝑘 𝑥𝐴 𝐵𝐶): 𝑥𝐴 𝐵⟶(0[,]+∞))    &   (𝜑 𝑥𝐴 𝐵 ∈ V)       (𝜑 → (Σ^‘(𝑘 𝑥𝐴 𝐵𝐶)) = (Σ^‘(𝑥𝐴 ↦ (Σ^‘(𝑘𝐵𝐶)))))

Theoremsge0fodjrnlem 42897* Re-index a nonnegative extended sum using an onto function with disjoint range, when the empty set is assigned 0 in the sum (this is true, for example, both for measures and outer measures). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   𝑛𝜑    &   (𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶𝑉)    &   (𝜑𝐹:𝐶onto𝐴)    &   (𝜑Disj 𝑛𝐶 (𝐹𝑛))    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘 = ∅) → 𝐵 = 0)    &   𝑍 = (𝐹 “ {∅})       (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))

Theoremsge0fodjrn 42898* Re-index a nonnegative extended sum using an onto function with disjoint range, when the empty set is assigned 0 in the sum (this is true, for example, both for measures and outer measures). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   𝑛𝜑    &   (𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶𝑉)    &   (𝜑𝐹:𝐶onto𝐴)    &   (𝜑Disj 𝑛𝐶 (𝐹𝑛))    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘 = ∅) → 𝐵 = 0)       (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = (Σ^‘(𝑛𝐶𝐷)))

Theoremsge0iunmpt 42899* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   (𝜑Disj 𝑥𝐴 𝐵)    &   ((𝜑𝑥𝐴𝑘𝐵) → 𝐶 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑘 𝑥𝐴 𝐵𝐶)) = (Σ^‘(𝑥𝐴 ↦ (Σ^‘(𝑘𝐵𝐶)))))

Theoremsge0iun 42900* Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)    &   𝑋 = 𝑥𝐴 𝐵    &   (𝜑𝐹:𝑋⟶(0[,]+∞))    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑 → (Σ^𝐹) = (Σ^‘(𝑥𝐴 ↦ (Σ^‘(𝐹𝐵)))))

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