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Type | Label | Description |
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Statement | ||
Theorem | dfsalgen2 42501* | Alternate characterization of the sigma-algebra generated by a set. It is the smallest sigma-algebra, on the same base set, that includes the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ ∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆) ∧ ∀𝑦 ∈ SAlg ((∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦) → 𝑆 ⊆ 𝑦)))) | ||
Theorem | salexct3 42502* | An example of a sigma-algebra that's not closed under uncountable union. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ 𝐴 = (0[,]2) & ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} & ⊢ 𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ⇒ ⊢ (𝑆 ∈ SAlg ∧ 𝑋 ⊆ 𝑆 ∧ ¬ ∪ 𝑋 ∈ 𝑆) | ||
Theorem | salgencntex 42503* | This counterexample shows that df-salgen 42475 needs to require that all containing sigma-algebra have the same base set. Otherwise, the intersection could lead to a set that is not a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ 𝐴 = (0[,]2) & ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} & ⊢ 𝐵 = (0[,]1) & ⊢ 𝑇 = 𝒫 𝐵 & ⊢ 𝐶 = (𝑆 ∩ 𝑇) & ⊢ 𝑍 = ∩ {𝑠 ∈ SAlg ∣ 𝐶 ⊆ 𝑠} ⇒ ⊢ ¬ 𝑍 ∈ SAlg | ||
Theorem | salgensscntex 42504* | This counterexample shows that the sigma-algebra generated by a set is not the smallest sigma-algebra containing the set, if we consider also sigma-algebras with a larger base set. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ 𝐴 = (0[,]2) & ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} & ⊢ 𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) & ⊢ 𝐺 = (SalGen‘𝑋) ⇒ ⊢ (𝑋 ⊆ 𝑆 ∧ 𝑆 ∈ SAlg ∧ ¬ 𝐺 ⊆ 𝑆) | ||
Theorem | issalnnd 42505* | Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → ∅ ∈ 𝑆) & ⊢ 𝑋 = ∪ 𝑆 & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑒:ℕ⟶𝑆) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝑆 ∈ SAlg) | ||
Theorem | dmvolsal 42506 | Lebesgue measurable sets form a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ dom vol ∈ SAlg | ||
Theorem | saldifcld 42507 | The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) ⇒ ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) | ||
Theorem | saluncld 42508 | The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ 𝑆) | ||
Theorem | salgencld 42509 | SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ 𝑆 = (SalGen‘𝑋) ⇒ ⊢ (𝜑 → 𝑆 ∈ SAlg) | ||
Theorem | 0sald 42510 | The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) ⇒ ⊢ (𝜑 → ∅ ∈ 𝑆) | ||
Theorem | iooborel 42511 | An open interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ (𝐴(,)𝐶) ∈ 𝐵 | ||
Theorem | salincld 42512 | The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐸 ∩ 𝐹) ∈ 𝑆) | ||
Theorem | salunid 42513 | A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) ⇒ ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆) | ||
Theorem | unisalgen2 42514 | The union of a set belongs is equal to the union of the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ 𝑆 = (SalGen‘𝐴) ⇒ ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐴) | ||
Theorem | bor1sal 42515 | The Borel sigma-algebra on the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ 𝐵 ∈ SAlg | ||
Theorem | iocborel 42516 | A left-open, right-closed interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) ⇒ ⊢ (𝜑 → (𝐴(,]𝐶) ∈ 𝐵) | ||
Theorem | subsaliuncllem 42517* | A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) & ⊢ 𝐸 = (𝐻 ∘ 𝐺) & ⊢ (𝜑 → 𝐻 Fn ran 𝐺) & ⊢ (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦) ⇒ ⊢ (𝜑 → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) | ||
Theorem | subsaliuncl 42518* | A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ 𝑇 = (𝑆 ↾t 𝐷) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑇) ⇒ ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇) | ||
Theorem | subsalsal 42519 | 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 42520 | 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 42521 | Extend class notation to include the sum of nonnegative extended reals. |
class Σ^ | ||
Definition | df-sumge0 42522* | 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 42523* | 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 42524 | 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 42525* | 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 42526 | 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 42527* | 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 42528* | 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 42529 | 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 42530 | 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 42531* | 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 42532 | 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 42533 | 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 42534* | 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 42535 | 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 42536* | 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 42537* | 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 42538 | 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 42539 | 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 42540 | 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 42541* | 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 42542* | 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 42543 | 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 42544 | 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 42545 | 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 42546* | 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 42547 | 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 42548* | 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 42550, 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 42549* | 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 42550* | 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 42551 | 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 42552* | 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 42553* | 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 42554* | 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 42555* | 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 42556 | 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 42557* | 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 42558* | 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 42559* | 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 42560* | 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 42553. (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 42561* | 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 42562 | 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 42563 | 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 42564* | 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 42565 | Σ^ splits into two parts, when it's a real number. This is a special case of sge0split 42568. (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 42566* | 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 42567* | 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 42568 | 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 42569* | 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 42570* | 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 42571* | 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 42572* | 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 42573* | 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 42574* | 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 42575* | 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 42576* | 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 42577* | 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 42578* | 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 42579 | 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 42580* | 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 42581* | 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 42582* | 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 42583 | Nonnegative integers are nonnegative reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ ℕ0 ⊆ (0[,)+∞) | ||
Theorem | sge0clmpt 42584* | 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 42585* | 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 42586 | 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 42587* | 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 42588* | 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 42589* | 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 42590* | 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 42591* | 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 42592* | 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 42593* | 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 42594* | 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 42595* | 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 42596* | 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 42597* | 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 42598* | 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 42599* | 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 42600* | 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)) |
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