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Type | Label | Description |
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Statement | ||
Theorem | subsalsal 41501 | A subspace sigma-algebra is a sigma algebra. This is Lemma 121A of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑇 = (𝑆 ↾t 𝐷) ⇒ ⊢ (𝜑 → 𝑇 ∈ SAlg) | ||
Theorem | subsaluni 41502 | A set belongs to the subspace sigma-algebra it induces. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝑆 ↾t 𝐴)) | ||
Syntax | csumge0 41503 | Extend class notation to include the sum of nonnegative extended reals. |
class Σ^ | ||
Definition | df-sumge0 41504* | 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) ↦ Σ𝑤 ∈ 𝑦 (𝑥‘𝑤)), ℝ*, < ))) | ||
Theorem | sge0rnre 41505* | 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞)) ⇒ ⊢ (𝜑 → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) ⊆ ℝ) | ||
Theorem | fge0icoicc 41506 | If 𝐹 maps to nonnegative reals, then 𝐹 maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐹:𝑋⟶(0[,)+∞)) ⇒ ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) | ||
Theorem | sge0val 41507* | 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 41508 | 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 41509* | 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 41510* | 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 41511 | 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 41512 | 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 41513* | 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 41514 | 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 41515 | 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 41516* | 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 41517 | 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 41518* | 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 41519* | 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 41520 | 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 41521 | 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 41522 | 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 41523* | 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 41524* | 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 41525 | 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 41526 | 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 41527 | 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 41528* | 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 41529 | 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 41530* | 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 41532, 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 41531* | 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 41532* | 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 41533 | 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 41534* | 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 41535* | 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 41536* | 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 41537* | 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 41538 | 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 41539* | 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 41540* | 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 41541* | 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 41542* | 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 41535. (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 41543* | 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 41544 | 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 41545 | 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 41546* | 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 41547 | Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 41550. (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 41548* | 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 41549* | 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 41550 | 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 41551* | 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 41552* | 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 41553* | 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 41554* | 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 41555* | 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 41556* | 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 41557* | 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 41558* | 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 41559* | 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 41560* | 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 41561 | 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 41562* | 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 41563* | 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 41564* | 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 41565 | Nonnegative integers are nonnegative reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ ℕ0 ⊆ (0[,)+∞) | ||
Theorem | sge0clmpt 41566* | 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 41567* | 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 41568 | 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 41569* | 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 41570* | 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 41571* | 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 41572* | 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 41573* | 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 41574* | 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 41575* | 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 41576* | 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 41577* | 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 41578* | 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 41579* | 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 41580* | 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 41581* | 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 41582* | 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 41583* | 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 41584* | 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 41585* | 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 41586* | 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 41587 | 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 41588* | 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 41589* | 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, < )) | ||
Proofs for most of the theorems in section 112 of [Fremlin1] | ||
Syntax | cmea 41590 | Extend class notation with the class of measures. |
class Meas | ||
Definition | df-mea 41591* | 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 41592* | 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 41593 | 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 41594 | 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 41595 | The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝑀 ∈ Meas) ⇒ ⊢ (𝜑 → (𝑀‘∅) = 0) | ||
Theorem | nnfoctbdjlem 41596* | 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 41597* | 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 41598* | 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 41599 | 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 41600* | 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)) = ∅) |
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