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Theorem List for Metamath Proof Explorer - 40901-41000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsge0val 40901* 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 40902 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 40903* 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 40904* 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 40905 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 40906 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 40907* 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 40908 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 40909 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 40910* Any nonnegative extended sum of zero is zero. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)       (𝜑 → (Σ^‘(𝑘𝐴 ↦ 0)) = 0)

Theoremsge00 40911 The sum of nonnegative extended reals is zero when applied to the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
^‘∅) = 0

Theoremfsumlesge0 40912* 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 40913* 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 40914 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 40915 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 40916 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 40917* 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 40918* 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 40919 The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝐹:𝑋⟶(0[,]+∞))       (𝜑 → 0 ≤ (Σ^𝐹))

Theoremsge0xrcl 40920 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 40921 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 40922* 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 40923 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 40924* 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 40926, 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 40925* 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 40926* 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 40927 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 40928* 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 40929* 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 40930* 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 40931* 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 40932 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 40933* 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 40934* 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 40935* 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 40936* 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 40929. (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 40937* 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.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ ℝ*)    &   ((𝜑𝑦 ∈ ℝ+) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐶 ≤ ((Σ^‘(𝑥𝑧𝐵)) +𝑒 𝑦))       (𝜑𝐶 ≤ (Σ^‘(𝑥𝐴𝐵)))

Theoremsge0resrnlem 40938 The sum of nonnegative extended reals restricted to the range of a function is less 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 40939 The sum of nonnegative extended reals restricted to the range of a function is less 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 40940* 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 40941 Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 40944. (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 40942* 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 40943* 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 40944 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 40945* 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 40946* 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 40947* 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 40948* 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 40949* 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 40950* 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 40951* 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 40952* 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 40953* 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 40954* 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)))))

Theoremsge0nemnf 40955 The generalized sum of nonnegative extended reals is not minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))       (𝜑 → (Σ^𝐹) ≠ -∞)

Theoremsge0rpcpnf 40956* The sum of an infinite number of a positive constant, is +∞ (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   (𝜑 → ¬ 𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (Σ^‘(𝑥𝐴𝐵)) = +∞)

Theoremsge0rernmpt 40957* If the sum of nonnegative extended reals is not +∞ then no term is +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ ℝ)       ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,)+∞))

Theoremsge0lefimpt 40958* 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, 11-Oct-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → ((Σ^‘(𝑥𝐴𝐵)) ≤ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥𝑦𝐵)) ≤ 𝐶))

Theoremnn0ssge0 40959 Nonnegative integers are nonnegative reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
0 ⊆ (0[,)+∞)

Theoremsge0clmpt 40960* The generalized sum of nonnegative extended reals is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ (0[,]+∞))

Theoremsge0ltfirpmpt2 40961* 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)) < (Σ𝑥𝑦 𝐵 + 𝑌))

Theoremsge0isum 40962 If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶(0[,)+∞))    &   𝐺 = seq𝑀( + , 𝐹)    &   (𝜑𝐺𝐵)       (𝜑 → (Σ^𝐹) = 𝐵)

Theoremsge0xrclmpt 40963* The generalized sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑥𝐴𝐵)) ∈ ℝ*)

Theoremsge0xp 40964* Combine two generalized sums of nonnegative extended reals into a single generalized sum over the cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑗𝐴𝑘𝐵) → 𝐶 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑗𝐴 ↦ (Σ^‘(𝑘𝐵𝐶)))) = (Σ^‘(𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐷)))

Theoremsge0isummpt 40965* If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   ((𝜑𝑘𝑍) → 𝐴 ∈ (0[,)+∞))    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑 → seq𝑀( + , (𝑘𝑍𝐴)) ⇝ 𝐵)       (𝜑 → (Σ^‘(𝑘𝑍𝐴)) = 𝐵)

Theoremsge0ad2en 40966* The value of the infinite geometric series 2↑-1 + 2↑-2 +... , multiplied by a constant. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ∈ (0[,)+∞))       (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴)

Theoremsge0isummpt2 40967* If a series of nonnegative reals is convergent, then it agrees with the generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   ((𝜑𝑘𝑍) → 𝐴 ∈ (0[,)+∞))    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑 → seq𝑀( + , (𝑘𝑍𝐴)) ⇝ 𝐵)       (𝜑 → (Σ^‘(𝑘𝑍𝐴)) = Σ𝑘𝑍 𝐴)

Theoremsge0xaddlem1 40968* The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,)+∞))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝑈𝐴)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑𝑊𝐴)    &   (𝜑𝑊 ∈ Fin)    &   (𝜑 → (Σ^‘(𝑘𝐴𝐵)) < (Σ𝑘𝑈 𝐵 + (𝐸 / 2)))    &   (𝜑 → (Σ^‘(𝑘𝐴𝐶)) < (Σ𝑘𝑊 𝐶 + (𝐸 / 2)))    &   (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑥 (𝐵 + 𝐶)), ℝ*, < ) ∈ (0[,]+∞))    &   (𝜑 → (Σ^‘(𝑘𝐴𝐵)) ∈ ℝ)    &   (𝜑 → (Σ^‘(𝑘𝐴𝐶)) ∈ ℝ)       (𝜑 → ((Σ^‘(𝑘𝐴𝐵)) + (Σ^‘(𝑘𝐴𝐶))) ≤ (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘𝑥 (𝐵 + 𝐶)), ℝ*, < ) +𝑒 𝐸))

Theoremsge0xaddlem2 40969* The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,)+∞))    &   (𝜑 → (Σ^‘(𝑘𝐴𝐵)) ∈ ℝ)    &   (𝜑 → (Σ^‘(𝑘𝐴𝐶)) ∈ ℝ)       (𝜑 → (Σ^‘(𝑘𝐴 ↦ (𝐵 +𝑒 𝐶))) = ((Σ^‘(𝑘𝐴𝐵)) +𝑒^‘(𝑘𝐴𝐶))))

Theoremsge0xadd 40970* The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑘𝐴 ↦ (𝐵 +𝑒 𝐶))) = ((Σ^‘(𝑘𝐴𝐵)) +𝑒^‘(𝑘𝐴𝐶))))

Theoremsge0fsummptf 40971* The generalized 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, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))       (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = Σ𝑘𝐴 𝐵)

Theoremsge0snmptf 40972* A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝐶 ∈ (0[,]+∞))    &   (𝑘 = 𝐴𝐵 = 𝐶)       (𝜑 → (Σ^‘(𝑘 ∈ {𝐴} ↦ 𝐵)) = 𝐶)

Theoremsge0ge0mpt 40973* The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → 0 ≤ (Σ^‘(𝑘𝐴𝐵)))

Theoremsge0repnfmpt 40974* The of nonnegative extended reals is a real number if and only if it is not +∞. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → ((Σ^‘(𝑘𝐴𝐵)) ∈ ℝ ↔ ¬ (Σ^‘(𝑘𝐴𝐵)) = +∞))

Theoremsge0pnffigtmpt 40975* If the generalized sum of nonnegative reals is +∞, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = +∞)    &   (𝜑𝑌 ∈ ℝ)       (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘(𝑘𝑥𝐵)))

Theoremsge0splitsn 40976* Separate out a term in a generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑 → ¬ 𝐵𝐴)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   (𝑘 = 𝐵𝐶 = 𝐷)    &   (𝜑𝐷 ∈ (0[,]+∞))       (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∪ {𝐵}) ↦ 𝐶)) = ((Σ^‘(𝑘𝐴𝐶)) +𝑒 𝐷))

Theoremsge0pnffsumgt 40977* If the sum of nonnegative extended reals is +∞, then any real number can be dominated by finite subsums. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))    &   (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = +∞)    &   (𝜑𝑌 ∈ ℝ)       (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < Σ𝑘𝑥 𝐵)

Theoremsge0gtfsumgt 40978* If the generalized sum of nonnegative reals is larger than a given number, then that number can be dominated by a finite subsum. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 < (Σ^‘(𝑘𝐴𝐵)))       (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)𝐶 < Σ𝑘𝑦 𝐵)

Theoremsge0uzfsumgt 40979* If a real number is smaller than a generalized sum of nonnegative reals, then it is smaller than some finite subsum. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝐾 ∈ ℤ)    &   𝑍 = (ℤ𝐾)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ (0[,)+∞))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 < (Σ^‘(𝑘𝑍𝐵)))       (𝜑 → ∃𝑚𝑍 𝐶 < Σ𝑘 ∈ (𝐾...𝑚)𝐵)

Theoremsge0pnfmpt 40980* If a term in the sum of nonnegative extended reals is +∞, then the value of the sum is +∞ (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑 → ∃𝑘𝐴 𝐵 = +∞)       (𝜑 → (Σ^‘(𝑘𝐴𝐵)) = +∞)

Theoremsge0seq 40981 A series of nonnegative reals agrees with the generalized sum of nonnegative reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶(0[,)+∞))    &   𝐺 = seq𝑀( + , 𝐹)       (𝜑 → (Σ^𝐹) = sup(ran 𝐺, ℝ*, < ))

Theoremsge0reuz 40982* Value of the generalized sum of nonnegative reals, when the domain is a set of upper integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ (0[,)+∞))       (𝜑 → (Σ^‘(𝑘𝑍𝐵)) = sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))

Theoremsge0reuzb 40983* Value of the generalized sum of uniformly bounded nonnegative reals, when the domain is a set of upper integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑘𝜑    &   𝑥𝜑    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ (0[,)+∞))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵𝑥)       (𝜑 → (Σ^‘(𝑘𝑍𝐵)) = sup(ran (𝑛𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ, < ))

20.32.19.3  Measures

Proofs for most of the theorems in section 112 of [Fremlin1]

Syntaxcmea 40984 Extend class notation with the class of measures.
class Meas

Definitiondf-mea 40985* Define the class of measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Meas = {𝑥 ∣ (((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 ∈ SAlg) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥((𝑦 ≼ ω ∧ Disj 𝑤𝑦 𝑤) → (𝑥 𝑦) = (Σ^‘(𝑥𝑦))))}

Theoremismea 40986* Express the predicate "𝑀 is a measure." Definition 112A of [Fremlin1] p. 14. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))

Theoremdmmeasal 40987 The domain of a measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀       (𝜑𝑆 ∈ SAlg)

Theoremmeaf 40988 A measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀       (𝜑𝑀:𝑆⟶(0[,]+∞))

Theoremmea0 40989 The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)       (𝜑 → (𝑀‘∅) = 0)

Theoremnnfoctbdjlem 40990* There exists a mapping from onto any (nonempty) countable set of disjoint sets, such that elements in the range of the map are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝐺:𝐴1-1-onto𝑋)    &   (𝜑Disj 𝑦𝑋 𝑦)    &   𝐹 = (𝑛 ∈ ℕ ↦ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))))       (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓𝑛)))

Theoremnnfoctbdj 40991* There exists a mapping from onto any (nonempty) countable set of disjoint sets, such that elements in the range of the map are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑋 ≼ ω)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑Disj 𝑦𝑋 𝑦)       (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓𝑛)))

Theoremmeadjuni 40992* The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝑋𝑆)    &   (𝜑𝑋 ≼ ω)    &   (𝜑Disj 𝑥𝑋 𝑥)       (𝜑 → (𝑀 𝑋) = (Σ^‘(𝑀𝑋)))

Theoremmeacl 40993 The measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)       (𝜑 → (𝑀𝐴) ∈ (0[,]+∞))

Theoremiundjiunlem 40994* The sets in the sequence 𝐹 are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑍 = (ℤ𝑁)    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑁..^𝑛)(𝐸𝑖)))    &   (𝜑𝐽𝑍)    &   (𝜑𝐾𝑍)    &   (𝜑𝐽 < 𝐾)       (𝜑 → ((𝐹𝐽) ∩ (𝐹𝐾)) = ∅)

Theoremiundjiun 40995* Given a sequence 𝐸 of sets, a sequence 𝐹 of disjoint sets is built, such that the indexed union stays the same. As in the proof of Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑛𝜑    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍𝑉)    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑁..^𝑛)(𝐸𝑖)))       (𝜑 → ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)(𝐹𝑛) = 𝑛 ∈ (𝑁...𝑚)(𝐸𝑛) ∧ 𝑛𝑍 (𝐹𝑛) = 𝑛𝑍 (𝐸𝑛)) ∧ Disj 𝑛𝑍 (𝐹𝑛)))

Theoremmeaxrcl 40996 The measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)       (𝜑 → (𝑀𝐴) ∈ ℝ*)

Theoremmeadjun 40997 The measure of the union of two disjoint sets is the sum of the measures, Property 112C (a) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴𝐵) = ∅)       (𝜑 → (𝑀‘(𝐴𝐵)) = ((𝑀𝐴) +𝑒 (𝑀𝐵)))

Theoremmeassle 40998 The measure of a set is larger or equal to the measure of a subset, Property 112C (b) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑀𝐴) ≤ (𝑀𝐵))

Theoremmeaunle 40999 The measure of the union of two sets is less or equal to the sum of the measures, Property 112C (c) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)       (𝜑 → (𝑀‘(𝐴𝐵)) ≤ ((𝑀𝐴) +𝑒 (𝑀𝐵)))

Theoremmeadjiunlem 41000* The sum of nonnegative extended reals, restricted to the range of another function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑀 ∈ Meas)    &   𝑆 = dom 𝑀    &   (𝜑𝑋𝑉)    &   (𝜑𝐺:𝑋𝑆)    &   𝑌 = {𝑖𝑋 ∣ (𝐺𝑖) ≠ ∅}    &   (𝜑Disj 𝑖𝑋 (𝐺𝑖))       (𝜑 → (Σ^‘(𝑀 ↾ ran 𝐺)) = (Σ^‘(𝑀𝐺)))

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