P. 469 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46801-46900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sge0vald 46801*	The value of the sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < )))
Theorem	fge0iccico 46802	A range of nonnegative extended reals without plus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → ¬ +∞ ∈ ran 𝐹)    ⇒   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞))
Theorem	gsumge0cl 46803	Closure of group sum, for finitely supported nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐹 finSupp 0)    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (0[,]+∞))
Theorem	sge0reval 46804*	Value of the sum of nonnegative extended reals, when all terms in the sum are reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
Theorem	sge0pnfval 46805	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → +∞ ∈ ran 𝐹)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = +∞)
Theorem	fge0iccre 46806	A range of nonnegative extended reals without plus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → ¬ +∞ ∈ ran 𝐹)    ⇒   ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
Theorem	sge0z 46807*	Any nonnegative extended sum of zero is zero. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 0)) = 0)
Theorem	sge00 46808	The sum of nonnegative extended reals is zero when applied to the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (Σ^‘∅) = 0
Theorem	fsumlesge0 46809*	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞))    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ Fin)    ⇒   ⊢ (𝜑 → Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ≤ (Σ^‘𝐹))
Theorem	sge0revalmpt 46810*	Value of the sum of nonnegative extended reals, when all terms in the sum are reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = sup(ran (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑥 ∈ 𝑦 𝐵), ℝ*, < ))
Theorem	sge0sn 46811	A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:{𝐴}⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = (𝐹‘𝐴))
Theorem	sge0tsms 46812	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*𝑠 ↾s (0[,]+∞))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) ∈ (𝐺 tsums 𝐹))
Theorem	sge0cl 46813	The arbitrary sum of nonnegative extended reals is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) ∈ (0[,]+∞))
Theorem	sge0f1o 46814*	Re-index a nonnegative extended sum using a bijection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑛𝜑    &   ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
Theorem	sge0snmpt 46815*	A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ (0[,]+∞))    &   ⊢ (𝑘 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴} ↦ 𝐵)) = 𝐶)
Theorem	sge0ge0 46816	The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → 0 ≤ (Σ^‘𝐹))
Theorem	sge0xrcl 46817	The arbitrary sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
Theorem	sge0repnf 46818	The of nonnegative extended reals is a real number if and only if it is not +∞. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → ((Σ^‘𝐹) ∈ ℝ ↔ ¬ (Σ^‘𝐹) = +∞))
Theorem	sge0fsum 46819*	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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = Σ𝑥 ∈ 𝑋 (𝐹‘𝑥))
Theorem	sge0rern 46820	If the sum of nonnegative extended reals is not +∞ then no terms is +∞. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → ¬ +∞ ∈ ran 𝐹)
Theorem	sge0supre 46821*	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 46823, but here we can use sup with respect to ℝ instead of ℝ*. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ, < ))
Theorem	sge0fsummpt 46822*	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.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
Theorem	sge0sup 46823*	The arbitrary sum of nonnegative extended reals is the supremum of finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ))
Theorem	sge0less 46824	A shorter sum of nonnegative extended reals is smaller than a longer one. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ≤ (Σ^‘𝐹))
Theorem	sge0rnbnd 46825*	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))𝑤 ≤ 𝑧)
Theorem	sge0pr 46826*	Sum of a pair of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞))    &   ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸))
Theorem	sge0gerp 46827*	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑧 ∈ (𝒫 𝑋 ∩ Fin)𝐴 ≤ ((Σ^‘(𝐹 ↾ 𝑧)) +𝑒 𝑥))    ⇒   ⊢ (𝜑 → 𝐴 ≤ (Σ^‘𝐹))
Theorem	sge0pnffigt 46828*	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘𝐹) = +∞)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑌 < (Σ^‘(𝐹 ↾ 𝑥)))
Theorem	sge0ssre 46829	If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ^‘(𝐹 ↾ 𝑌)) ∈ ℝ)
Theorem	sge0lefi 46830*	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((Σ^‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑥)) ≤ 𝐴))
Theorem	sge0lessmpt 46831*	A shorter sum of nonnegative extended reals is smaller than a longer one. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐶 ↦ 𝐵)) ≤ (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
Theorem	sge0ltfirp 46832*	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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ+)    &   ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘𝐹) < ((Σ^‘(𝐹 ↾ 𝑥)) + 𝑌))
Theorem	sge0prle 46833*	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 46826. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐸 ∈ (0[,]+∞))    &   ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) ≤ (𝐷 +𝑒 𝐸))
Theorem	sge0gerpmpt 46834*	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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ+) → ∃𝑧 ∈ (𝒫 𝐴 ∩ Fin)𝐶 ≤ ((Σ^‘(𝑥 ∈ 𝑧 ↦ 𝐵)) +𝑒 𝑦))    ⇒   ⊢ (𝜑 → 𝐶 ≤ (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
Theorem	sge0resrnlem 46835	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐵⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝐴)    &   ⊢ (𝜑 → (𝐺 ↾ 𝑋):𝑋–1-1-onto→ran 𝐺)    ⇒   ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺)))
Theorem	sge0resrn 46836	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐵⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐴)    ⇒   ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺)))
Theorem	sge0ssrempt 46837*	If a sum of nonnegative extended reals is real, than any subsum is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐶 ↦ 𝐵)) ∈ ℝ)
Theorem	sge0resplit 46838	Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 46841. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝑈 = (𝐴 ∪ 𝐵)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝐹:𝑈⟶(0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) + (Σ^‘(𝐹 ↾ 𝐵))))
Theorem	sge0le 46839*	If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐺:𝑋⟶(0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) ≤ (Σ^‘𝐺))
Theorem	sge0ltfirpmpt 46840*	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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ+)    &   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < ((Σ^‘(𝑥 ∈ 𝑦 ↦ 𝐵)) + 𝑌))
Theorem	sge0split 46841	Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ 𝑈 = (𝐴 ∪ 𝐵)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝐹:𝑈⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = ((Σ^‘(𝐹 ↾ 𝐴)) +𝑒 (Σ^‘(𝐹 ↾ 𝐵))))
Theorem	sge0lempt 46842*	If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≤ (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐶)))
Theorem	sge0splitmpt 46843*	Split a sum of nonnegative extended reals into two parts. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶)) = ((Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐶)) +𝑒 (Σ^‘(𝑥 ∈ 𝐵 ↦ 𝐶))))
Theorem	sge0ss 46844*	Change the index set to a subset in a sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) = (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))
Theorem	sge0iunmptlemfi 46845*	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.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)) ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ 𝐶)) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))))
Theorem	sge0p1 46846*	The addition of the next term in a finite sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ (0[,]+∞))    &   ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝑀...(𝑁 + 1)) ↦ 𝐴)) = ((Σ^‘(𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)) +𝑒 𝐵))
Theorem	sge0iunmptlemre 46847*	Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)) ∈ ℝ)    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ 𝐶)) ∈ ℝ*)    &   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))) ∈ ℝ*)    &   ⊢ (𝜑 → (𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ 𝐶):∪ 𝑥 ∈ 𝐴 𝐵⟶(0[,]+∞))    &   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ 𝐶)) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))))
Theorem	sge0fodjrnlem 46848*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑛𝜑    &   ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐶–onto→𝐴)    &   ⊢ (𝜑 → Disj 𝑛 ∈ 𝐶 (𝐹‘𝑛))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0)    &   ⊢ 𝑍 = (◡𝐹 “ {∅})    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
Theorem	sge0fodjrn 46849*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑛𝜑    &   ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐶–onto→𝐴)    &   ⊢ (𝜑 → Disj 𝑛 ∈ 𝐶 (𝐹‘𝑛))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝐵 = 0)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = (Σ^‘(𝑛 ∈ 𝐶 ↦ 𝐷)))
Theorem	sge0iunmpt 46850*	Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ 𝐶)) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))))
Theorem	sge0iun 46851*	Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ 𝑋 = ∪ 𝑥 ∈ 𝐴 𝐵    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝐹 ↾ 𝐵)))))
Theorem	sge0nemnf 46852	The generalized sum of nonnegative extended reals is not minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) ≠ -∞)
Theorem	sge0rpcpnf 46853*	The sum of an infinite number of a positive constant, is +∞ (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)
Theorem	sge0rernmpt 46854*	If the sum of nonnegative extended reals is not +∞ then no term is +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))
Theorem	sge0lefimpt 46855*	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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≤ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ≤ 𝐶))
Theorem	nn0ssge0 46856	Nonnegative integers are nonnegative reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ℕ0 ⊆ (0[,)+∞)
Theorem	sge0clmpt 46857*	The generalized sum of nonnegative extended reals is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ (0[,]+∞))
Theorem	sge0ltfirpmpt2 46858*	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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ+)    &   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ𝑥 ∈ 𝑦 𝐵 + 𝑌))
Theorem	sge0isum 46859	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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶(0[,)+∞))    &   ⊢ 𝐺 = seq𝑀( + , 𝐹)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = 𝐵)
Theorem	sge0xrclmpt 46860*	The generalized sum of nonnegative extended reals is an extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ*)
Theorem	sge0xp 46861*	Combine two generalized sums of nonnegative extended reals into a single generalized sum over the cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ 𝐴 ↦ (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)))) = (Σ^‘(𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐷)))
Theorem	sge0isummpt 46862*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ (0[,)+∞))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐴)) = 𝐵)
Theorem	sge0ad2en 46863*	The value of the infinite geometric series 2↑-1 + 2↑-2 +... , multiplied by a constant. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴)
Theorem	sge0isummpt2 46864*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ (0[,)+∞))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐴)) = Σ𝑘 ∈ 𝑍 𝐴)
Theorem	sge0xaddlem1 46865*	The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,)+∞))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝑊 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ Fin)    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) < (Σ𝑘 ∈ 𝑈 𝐵 + (𝐸 / 2)))    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) < (Σ𝑘 ∈ 𝑊 𝐶 + (𝐸 / 2)))    &   ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 (𝐵 + 𝐶)), ℝ*, < ) ∈ (0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) ∈ ℝ)    ⇒   ⊢ (𝜑 → ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) + (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶))) ≤ (sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ Σ𝑘 ∈ 𝑥 (𝐵 + 𝐶)), ℝ*, < ) +𝑒 𝐸))
Theorem	sge0xaddlem2 46866*	The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,)+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ (𝐵 +𝑒 𝐶))) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) +𝑒 (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶))))
Theorem	sge0xadd 46867*	The extended addition of two generalized sums of nonnegative extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ (𝐵 +𝑒 𝐶))) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) +𝑒 (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶))))
Theorem	sge0fsummptf 46868*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
Theorem	sge0snmptf 46869*	A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ (0[,]+∞))    &   ⊢ (𝑘 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ {𝐴} ↦ 𝐵)) = 𝐶)
Theorem	sge0ge0mpt 46870*	The sum of nonnegative extended reals is nonnegative. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → 0 ≤ (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)))
Theorem	sge0repnfmpt 46871*	The of nonnegative extended reals is a real number if and only if it is not +∞. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ ¬ (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞))
Theorem	sge0pnffigtmpt 46872*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < (Σ^‘(𝑘 ∈ 𝑥 ↦ 𝐵)))
Theorem	sge0splitsn 46873*	Separate out a term in a generalized sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞))    &   ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∪ {𝐵}) ↦ 𝐶)) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 𝐷))
Theorem	sge0pnffsumgt 46874*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    &   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑌 < Σ𝑘 ∈ 𝑥 𝐵)
Theorem	sge0gtfsumgt 46875*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 < (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)𝐶 < Σ𝑘 ∈ 𝑦 𝐵)
Theorem	sge0uzfsumgt 46876*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝐾)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 < (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)))    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 𝐶 < Σ𝑘 ∈ (𝐾...𝑚)𝐵)
Theorem	sge0pnfmpt 46877*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞)
Theorem	sge0seq 46878	A series of nonnegative reals agrees with the generalized sum of nonnegative reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶(0[,)+∞))    &   ⊢ 𝐺 = seq𝑀( + , 𝐹)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran 𝐺, ℝ*, < ))
Theorem	sge0reuz 46879*	Value of the generalized sum of nonnegative reals, when the domain is a set of upper integers. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ*, < ))
Theorem	sge0reuzb 46880*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ (0[,)+∞))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 Σ𝑘 ∈ (𝑀...𝑛)𝐵 ≤ 𝑥)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = sup(ran (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐵), ℝ, < ))
21.43.19.3  Measures Proofs for most of the theorems in section 112 of [Fremlin1]
Syntax	cmea 46881	Extend class notation with the class of measures.
class Meas
Definition	df-mea 46882*	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 𝑤 ∈ 𝑦 𝑤) → (𝑥‘∪ 𝑦) = (Σ^‘(𝑥 ↾ 𝑦))))}
Theorem	ismea 46883*	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 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
Theorem	dmmeasal 46884	The domain of a measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    ⇒   ⊢ (𝜑 → 𝑆 ∈ SAlg)
Theorem	meaf 46885	A measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    ⇒   ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))
Theorem	mea0 46886	The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    ⇒   ⊢ (𝜑 → (𝑀‘∅) = 0)
Theorem	nnfoctbdjlem 46887*	There exists a mapping from ℕ 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.)
⊢ (𝜑 → 𝐴 ⊆ ℕ)    &   ⊢ (𝜑 → 𝐺:𝐴–1-1-onto→𝑋)    &   ⊢ (𝜑 → Disj 𝑦 ∈ 𝑋 𝑦)    &   ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if((𝑛 = 1 ∨ ¬ (𝑛 − 1) ∈ 𝐴), ∅, (𝐺‘(𝑛 − 1))))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛)))
Theorem	nnfoctbdj 46888*	There exists a mapping from ℕ 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.)
⊢ (𝜑 → 𝑋 ≼ ω)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → Disj 𝑦 ∈ 𝑋 𝑦)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:ℕ–onto→(𝑋 ∪ {∅}) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛)))
Theorem	meadjuni 46889*	The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝑋 ≼ ω)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝑥)    ⇒   ⊢ (𝜑 → (𝑀‘∪ 𝑋) = (Σ^‘(𝑀 ↾ 𝑋)))
Theorem	meacl 46890	The measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ (0[,]+∞))
Theorem	iundjiunlem 46891*	The sets in the sequence 𝐹 are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝑍 = (ℤ≥‘𝑁)    &   ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)))    &   ⊢ (𝜑 → 𝐽 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐽 < 𝐾)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐽) ∩ (𝐹‘𝐾)) = ∅)
Theorem	iundjiun 46892*	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.)
⊢ Ⅎ𝑛𝜑    &   ⊢ 𝑍 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝐸:𝑍⟶𝑉)    &   ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)))    ⇒   ⊢ (𝜑 → ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)(𝐹‘𝑛) = ∪ 𝑛 ∈ (𝑁...𝑚)(𝐸‘𝑛) ∧ ∪ 𝑛 ∈ 𝑍 (𝐹‘𝑛) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∧ Disj 𝑛 ∈ 𝑍 (𝐹‘𝑛)))
Theorem	meaxrcl 46893	The measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ*)
Theorem	meadjun 46894	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.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵)))
Theorem	meassle 46895	The measure of a set is greater than or equal to the measure of a subset, Property 112C (b) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
Theorem	meaunle 46896	The measure of the union of two sets is less than or equal to the sum of the measures, Property 112C (c) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵)))
Theorem	meadjiunlem 46897*	The sum of nonnegative extended reals, restricted to the range of another function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺:𝑋⟶𝑆)    &   ⊢ 𝑌 = {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅}    &   ⊢ (𝜑 → Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑀 ↾ ran 𝐺)) = (Σ^‘(𝑀 ∘ 𝐺)))
Theorem	meadjiun 46898*	The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ 𝐴 𝐵) = (Σ^‘(𝑘 ∈ 𝐴 ↦ (𝑀‘𝐵))))
Theorem	ismeannd 46899*	Sufficient condition to prove that 𝑀 is a measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑆 ∈ SAlg)    &   ⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑒:ℕ⟶𝑆 ∧ Disj 𝑛 ∈ ℕ (𝑒‘𝑛)) → (𝑀‘∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑀‘(𝑒‘𝑛)))))    ⇒   ⊢ (𝜑 → 𝑀 ∈ Meas)
Theorem	meaiunlelem 46900*	The measure of the union of countable sets is less than or equal to the sum of the measures, Property 112C (d) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑛𝜑    &   ⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ 𝑍 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝐸:𝑍⟶𝑆)    &   ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)))    ⇒   ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛)))))