P. 410 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	sge0val 40901*	The value of the sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹:𝑋⟶(0[,]+∞)) → (Σ^‘𝐹) = if(+∞ ∈ ran 𝐹, +∞, sup(ran (𝑦 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑤 ∈ 𝑦 (𝐹‘𝑤)), ℝ*, < )))
Theorem	fge0npnf 40902	If 𝐹 maps to nonnegative reals, then +∞ is not in its range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞))    ⇒   ⊢ (𝜑 → ¬ +∞ ∈ ran 𝐹)
Theorem	sge0rnn0 40903*	The range used in the definition of Σ^ is not empty. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ≠ ∅
Theorem	sge0vald 40904*	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 40905	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 40906	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 40907*	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 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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → +∞ ∈ ran 𝐹)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = +∞)
Theorem	fge0iccre 40909	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 40910*	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 40911	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 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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞))    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ Fin)    ⇒   ⊢ (𝜑 → Σ𝑥 ∈ 𝑌 (𝐹‘𝑥) ≤ (Σ^‘𝐹))
Theorem	sge0revalmpt 40913*	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 40914	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 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*𝑠 ↾s (0[,]+∞))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) ∈ (𝐺 tsums 𝐹))
Theorem	sge0cl 40916	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 40917*	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 40918*	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 40919	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 40920	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 40921	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 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.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = Σ𝑥 ∈ 𝑋 (𝐹‘𝑥))
Theorem	sge0rern 40923	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 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 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 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.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
Theorem	sge0sup 40926*	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 40927	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 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.)
⊢ (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 40929*	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 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.)
⊢ (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 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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → (Σ^‘𝐹) = +∞)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)𝑌 < (Σ^‘(𝐹 ↾ 𝑥)))
Theorem	sge0ssre 40932	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 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.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((Σ^‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑥)) ≤ 𝐴))
Theorem	sge0lessmpt 40934*	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 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.)
⊢ (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 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.)
⊢ (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 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.)
⊢ 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 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.)
⊢ (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 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.)
⊢ (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 40940*	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 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.)
⊢ (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 40942*	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 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.)
⊢ 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 40944	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 40945*	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 40946*	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 40947*	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 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.)
⊢ (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 40949*	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 40950*	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 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.)
⊢ 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 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.)
⊢ 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 40953*	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 40954*	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 40955	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 40956*	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 40957*	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 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.)
⊢ 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 40959	Nonnegative integers are nonnegative reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ℕ0 ⊆ (0[,)+∞)
Theorem	sge0clmpt 40960*	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 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.)
⊢ 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 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.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶(0[,)+∞))    &   ⊢ 𝐺 = seq𝑀( + , 𝐹)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    ⇒   ⊢ (𝜑 → (Σ^‘𝐹) = 𝐵)
Theorem	sge0xrclmpt 40963*	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 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.)
⊢ 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 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.)
⊢ 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 40966*	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 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.)
⊢ 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 40968*	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 40969*	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 40970*	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 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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵)
Theorem	sge0snmptf 40972*	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 40973*	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 40974*	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 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.)
⊢ 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 40976*	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 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.)
⊢ 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 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.)
⊢ 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 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.)
⊢ 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 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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = +∞)
Theorem	sge0seq 40981	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 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.)
⊢ 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 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.)
⊢ 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, < ))
20.32.19.3  Measures Proofs for most of the theorems in section 112 of [Fremlin1]
Syntax	cmea 40984	Extend class notation with the class of measures.
class Meas
Definition	df-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 𝑤 ∈ 𝑦 𝑤) → (𝑥‘∪ 𝑦) = (Σ^‘(𝑥 ↾ 𝑦))))}
Theorem	ismea 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 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
Theorem	dmmeasal 40987	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 40988	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 40989	The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ Meas)    ⇒   ⊢ (𝜑 → (𝑀‘∅) = 0)
Theorem	nnfoctbdjlem 40990*	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 40991*	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 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.)
⊢ (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 40993	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 40994*	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 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.)
⊢ 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 40996	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 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.)
⊢ (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 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.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
Theorem	meaunle 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.)
⊢ (𝜑 → 𝑀 ∈ Meas)    &   ⊢ 𝑆 = dom 𝑀    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑀‘(𝐴 ∪ 𝐵)) ≤ ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵)))
Theorem	meadjiunlem 41000*	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)))