HomeHome Metamath Proof Explorer
Theorem List (p. 394 of 425)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26947)
  Hilbert Space Explorer  Hilbert Space Explorer
(26948-28472)
  Users' Mathboxes  Users' Mathboxes
(28473-42426)
 

Theorem List for Metamath Proof Explorer - 39301-39400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaragenss 39301 The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the domain of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑆 = (CaraGen‘𝑂)       (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
 
Theoremomeunile 39302 The outer measure of the union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝑌 ⊆ 𝒫 𝑋)    &   (𝜑𝑌 ≼ ω)       (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))
 
Theoremcaragen0 39303 The empty set belongs to any Caratheodory's construction. First part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑 → ∅ ∈ 𝑆)
 
Theoremomexrcl 39304 The outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)       (𝜑 → (𝑂𝐴) ∈ ℝ*)
 
Theoremcaragenunidm 39305 The base set of an outer measure belongs to the sigma-algebra generated by the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)       (𝜑𝑋𝑆)
 
Theoremcaragensspw 39306 The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the power set of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)       (𝜑𝑆 ⊆ 𝒫 𝑋)
 
Theoremomessre 39307 If the outer measure of a set is real, then the outer measure of any of its subset is real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) ∈ ℝ)    &   (𝜑𝐵𝐴)       (𝜑 → (𝑂𝐵) ∈ ℝ)
 
Theoremcaragenuni 39308 The base set of the sigma-algebra generated by the Caratheodory's construction is the whole base set of the original outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑 𝑆 = dom 𝑂)
 
Theoremcaragenuncllem 39309 The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸𝑆)    &   (𝜑𝐹𝑆)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)       (𝜑 → ((𝑂‘(𝐴 ∩ (𝐸𝐹))) +𝑒 (𝑂‘(𝐴 ∖ (𝐸𝐹)))) = (𝑂𝐴))
 
Theoremcaragenuncl 39310 The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸𝑆)    &   (𝜑𝐹𝑆)       (𝜑 → (𝐸𝐹) ∈ 𝑆)
 
Theoremcaragendifcl 39311 The Caratheodory's construction is closed under the complement operation. Second part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸𝑆)       (𝜑 → ( 𝑆𝐸) ∈ 𝑆)
 
Theoremcaragenfiiuncl 39312* The Caratheodory's construction is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝜑    &   (𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)       (𝜑 𝑘𝐴 𝐵𝑆)
 
Theoremomeunle 39313 The outer measure of the union of two sets is less or equal to the sum of the measures, Remark 113B (c) of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → (𝑂‘(𝐴𝐵)) ≤ ((𝑂𝐴) +𝑒 (𝑂𝐵)))
 
Theoremomeiunle 39314* The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑛𝜑    &   𝑛𝐸    &   (𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶𝒫 𝑋)       (𝜑 → (𝑂 𝑛𝑍 (𝐸𝑛)) ≤ (Σ^‘(𝑛𝑍 ↦ (𝑂‘(𝐸𝑛)))))
 
Theoremomelesplit 39315 The outer measure of a set 𝐴 is less than or equal to the extended addition of the outer measures of the decomposition induced on 𝐴 by any 𝐸. Step (a) in the proof of Caratheodory's Method, Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)       (𝜑 → (𝑂𝐴) ≤ ((𝑂‘(𝐴𝐸)) +𝑒 (𝑂‘(𝐴𝐸))))
 
Theoremomeiunltfirp 39316* If the outer measure of a countable union is not +∞, then it can be arbitrarily approximated by finite sums of outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑍 = (ℤ𝑁)    &   (𝜑𝐸:𝑍⟶𝒫 𝑋)    &   (𝜑 → (𝑂 𝑛𝑍 (𝐸𝑛)) ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ+)       (𝜑 → ∃𝑧 ∈ (𝒫 𝑍 ∩ Fin)(𝑂 𝑛𝑍 (𝐸𝑛)) < (Σ𝑛𝑧 (𝑂‘(𝐸𝑛)) + 𝑌))
 
Theoremomeiunlempt 39317* The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑛𝜑    &   (𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑍 = (ℤ𝑁)    &   ((𝜑𝑛𝑍) → 𝐸𝑋)       (𝜑 → (𝑂 𝑛𝑍 𝐸) ≤ (Σ^‘(𝑛𝑍 ↦ (𝑂𝐸))))
 
Theoremcarageniuncllem1 39318* The outer measure of 𝐴 ∩ (𝐺𝑛) is the sum of the outer measures of 𝐴 ∩ (𝐹𝑚). These are lines 7 to 10 of Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) ∈ ℝ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐸:𝑍𝑆)    &   𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)))    &   (𝜑𝐾𝑍)       (𝜑 → Σ𝑛 ∈ (𝑀...𝐾)(𝑂‘(𝐴 ∩ (𝐹𝑛))) = (𝑂‘(𝐴 ∩ (𝐺𝐾))))
 
Theoremcarageniuncllem2 39319* The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) ∈ ℝ)    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐸:𝑍𝑆)    &   (𝜑𝑌 ∈ ℝ+)    &   𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))    &   𝐹 = (𝑛𝑍 ↦ ((𝐸𝑛) ∖ 𝑖 ∈ (𝑀..^𝑛)(𝐸𝑖)))       (𝜑 → ((𝑂‘(𝐴 𝑛𝑍 (𝐸𝑛))) +𝑒 (𝑂‘(𝐴 𝑛𝑍 (𝐸𝑛)))) ≤ ((𝑂𝐴) + 𝑌))
 
Theoremcarageniuncl 39320* The Caratheodory's construction is closed under indexed countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐸:𝑍𝑆)       (𝜑 𝑛𝑍 (𝐸𝑛) ∈ 𝑆)
 
Theoremcaragenunicl 39321 The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝑋𝑆)    &   (𝜑𝑋 ≼ ω)       (𝜑 𝑋𝑆)
 
Theoremcaragensal 39322 Caratheodory's method generates a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑𝑆 ∈ SAlg)
 
Theoremcaratheodorylem1 39323* Lemma used to prove that Caratheodory's construction is sigma-additive. This is the proof of the statement in the middle of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐸:𝑍𝑆)    &   (𝜑Disj 𝑛𝑍 (𝐸𝑛))    &   𝐺 = (𝑛𝑍 𝑖 ∈ (𝑀...𝑛)(𝐸𝑖))    &   (𝜑𝑁 ∈ (ℤ𝑀))       (𝜑 → (𝑂‘(𝐺𝑁)) = (Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸𝑛)))))
 
Theoremcaratheodorylem2 39324* Caratheodory's construction is sigma-additive. Main part of Step (e) in the proof of Theorem 113C of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸:ℕ⟶𝑆)    &   (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))    &   𝐺 = (𝑘 ∈ ℕ ↦ 𝑛 ∈ (1...𝑘)(𝐸𝑛))       (𝜑 → (𝑂 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝐸𝑛)))))
 
Theoremcaratheodory 39325 Caratheodory's construction of a measure given an outer measure. Proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑆 = (CaraGen‘𝑂)       (𝜑 → (𝑂𝑆) ∈ Meas)
 
Theorem0ome 39326* The map that assigns 0 to every subset, is an outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋𝑉)    &   𝑂 = (𝑥 ∈ 𝒫 𝑋 ↦ 0)       (𝜑𝑂 ∈ OutMeas)
 
Theoremisomenndlem 39327* 𝑂 is sub-additive w.r.t. countable indexed union, implies that 𝑂 is sub-additive w.r.t. countable union. Thus, the definition of Outer Measure can be given using an indexed union. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))    &   (𝜑 → (𝑂‘∅) = 0)    &   (𝜑𝑌 ⊆ 𝒫 𝑋)    &   ((𝜑𝑎:ℕ⟶𝒫 𝑋) → (𝑂 𝑛 ∈ ℕ (𝑎𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎𝑛)))))    &   (𝜑𝐵 ⊆ ℕ)    &   (𝜑𝐹:𝐵1-1-onto𝑌)    &   𝐴 = (𝑛 ∈ ℕ ↦ if(𝑛𝐵, (𝐹𝑛), ∅))       (𝜑 → (𝑂 𝑌) ≤ (Σ^‘(𝑂𝑌)))
 
Theoremisomennd 39328* Sufficient condition to prove that 𝑂 is an outer measure. Definition 113A of [Fremlin1] p. 19 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋𝑉)    &   (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))    &   (𝜑 → (𝑂‘∅) = 0)    &   ((𝜑𝑥𝑋𝑦𝑥) → (𝑂𝑦) ≤ (𝑂𝑥))    &   ((𝜑𝑎:ℕ⟶𝒫 𝑋) → (𝑂 𝑛 ∈ ℕ (𝑎𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (𝑂‘(𝑎𝑛)))))       (𝜑𝑂 ∈ OutMeas)
 
Theoremcaragenel2d 39329* Membership in the Caratheodory's construction. Similar to carageneld 39299, but here "less then or equal" is used, instead of equality. This is Remark 113D of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   𝑆 = (CaraGen‘𝑂)    &   (𝜑𝐸 ∈ 𝒫 𝑋)    &   ((𝜑𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎𝐸)) +𝑒 (𝑂‘(𝑎𝐸))) ≤ (𝑂𝑎))       (𝜑𝐸𝑆)
 
Theoremomege0 39330 If the outer measure of a set is greater than or equal to 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)       (𝜑 → 0 ≤ (𝑂𝐴))
 
Theoremomess0 39331 If the outer measure of a set is 0, then the outer measure of its subsets is 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) = 0)    &   (𝜑𝐵𝐴)       (𝜑 → (𝑂𝐵) = 0)
 
Theoremcaragencmpl 39332 A measure built with the Caratheodory's construction is complete. See Definition 112Df of [Fremlin1] p. 19. This is Exercise 113Xa of [Fremlin1] p. 21 (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝑂 ∈ OutMeas)    &   𝑋 = dom 𝑂    &   (𝜑𝐸𝑋)    &   (𝜑 → (𝑂𝐸) = 0)    &   𝑆 = (CaraGen‘𝑂)       (𝜑𝐸𝑆)
 
20.31.19.5  Lebesgue measure on n-dimensional Real numbers

Proofs for most of the theorems in section 115 of [Fremlin1]

 
Syntaxcovoln 39333 Extend class notation with the class of Lebesgue outer measure for the space of multidimensional real numbers.
class voln*
 
Definitiondf-ovoln 39334* Define the outer measure for the space of multidimensional real numbers. The cardinality of 𝑥 is the dimension of the space modeled. Definition 115C of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
voln* = (𝑥 ∈ Fin ↦ (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑥 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑥 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))))
 
Syntaxcvoln 39335 Extend class notation with the class of Lebesgue measure for the space of multidimensional real numbers.
class voln
 
Definitiondf-voln 39336 Define the Lebesgue measure for the space of multidimensional real numbers. The cardinality of 𝑥 is the dimension of the space modeled. Definition 115C of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
 
Theoremvonval 39337 Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
 
Theoremovnval 39338* Value of the Lebesgue outer measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (voln*‘𝑋) = (𝑦 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))))
 
Theoremelhoi 39339* Membership in a multidimensional half-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋𝑉)       (𝜑 → (𝑌 ∈ ((𝐴[,)𝐵) ↑𝑚 𝑋) ↔ (𝑌:𝑋⟶ℝ* ∧ ∀𝑥𝑋 (𝑌𝑥) ∈ (𝐴[,)𝐵))))
 
Theoremicoresmbl 39340 A closed-below, open-above real interval is measurable, when the bounds are real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
ran ([,) ↾ (ℝ × ℝ)) ⊆ dom vol
 
Theoremhoissre 39341* The projection of a half-open interval onto a single dimension is a subset of . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐼:𝑋⟶(ℝ × ℝ))       ((𝜑𝑘𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ)
 
Theoremovnval2 39342* Value of the Lebesgue outer measure of a subset 𝐴 of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
 
Theoremvolicorecl 39343 The Lebesgue measure of a left-closed, right-open interval with real bounds, is real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ)
 
Theoremhoiprodcl 39344* The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐼:𝑋⟶(ℝ × ℝ))       (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ 𝐼)‘𝑘)) ∈ (0[,)+∞))
 
Theoremhoicvr 39345* 𝐼 is a countable set of half-open intervals that covers the whole multidimensional reals. See Definition 1135 (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝐼 = (𝑗 ∈ ℕ ↦ (𝑥𝑋 ↦ ⟨-𝑗, 𝑗⟩))    &   (𝜑𝑋 ∈ Fin)       (𝜑 → (ℝ ↑𝑚 𝑋) ⊆ 𝑗 ∈ ℕ X𝑖𝑋 (([,) ∘ (𝐼𝑗))‘𝑖))
 
Theoremhoissrrn 39346* A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐼:𝑋⟶(ℝ × ℝ))       (𝜑X𝑘𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ (ℝ ↑𝑚 𝑋))
 
Theoremovn0val 39347 The Lebesgue outer measure (for the zero dimensional space of reals) of every subset is zero. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴 ⊆ (ℝ ↑𝑚 ∅))       (𝜑 → ((voln*‘∅)‘𝐴) = 0)
 
Theoremovnn0val 39348* The value of a (multidimensional) Lebesgue outer measure, defined on a nonzero-dimensional space of reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑 → ((voln*‘𝑋)‘𝐴) = inf(𝑀, ℝ*, < ))
 
Theoremovnval2b 39349* Value of the Lebesgue outer measure of a subset 𝐴 of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   𝐿 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})       (𝜑 → ((voln*‘𝑋)‘𝐴) = if(𝑋 = ∅, 0, inf((𝐿𝐴), ℝ*, < )))
 
Theoremvolicorescl 39350 The Lebesgue measure of a left-closed, right-open interval with real bounds, is real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) → (vol‘𝐴) ∈ ℝ)
 
Theoremovnprodcl 39351* The product used in the definition of the outer Lebesgue measure in R^n is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐹:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))    &   (𝜑𝐼 ∈ ℕ)       (𝜑 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐹𝐼))‘𝑘)) ∈ (0[,)+∞))
 
Theoremhoiprodcl2 39352* The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))    &   (𝜑𝐼:𝑋⟶(ℝ × ℝ))       (𝜑 → (𝐿𝐼) ∈ (0[,)+∞))
 
Theoremhoicvrrex 39353* Any subset of the multidimensional reals can be covered by a countable set of half-open intervals, see Definition 115A (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ⊆ (ℝ ↑𝑚 𝑋))       (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑌 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
 
Theoremovnsupge0 39354* The set used in the definition of the Lebesgue outer measure is a subset of the nonnegative extended reals. This is a substep for (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑𝑀 ⊆ (0[,]+∞))
 
Theoremovnlecvr 39355* Given a subset of multidimensional reals and a set of half-open intervals that covers it, the Lebesgue outer measure of the set is bounded by the generalized sum of the pre-measure of the half-open intervals. The statement would also be true with 𝑋 the empty set, but covers are not used for the zero-dimensional case. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))    &   (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))    &   (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))       (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
 
Theoremovnpnfelsup 39356* +∞ is an element of the set used in the definition of the Lebesgue outer measure. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑 → +∞ ∈ 𝑀)
 
Theoremovnsslelem 39357* The (multidimensional, nonzero-dimensional) Lebesgue outer measure of a subset is less than the L.o.m. of the whole set. This is step (iii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ⊆ (ℝ ↑𝑚 𝑋))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}    &   𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐵 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵))
 
Theoremovnssle 39358 The (multidimensional) Lebesgue outer measure of a subset is less than the L.o.m. of the whole set. This is step (iii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴𝐵)    &   (𝜑𝐵 ⊆ (ℝ ↑𝑚 𝑋))       (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ ((voln*‘𝑋)‘𝐵))
 
Theoremovnlerp 39359* The Lebesgue outer measure of a subset of multidimensional real numbers can always be approximated by the total outer measure of a cover of half-open (multidimensional) intervals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}       (𝜑 → ∃𝑧𝑀 𝑧 ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
 
Theoremovnf 39360 The Lebesgue outer measure is a function that maps sets to nonnegative extended reals. This is step (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (voln*‘𝑋):𝒫 (ℝ ↑𝑚 𝑋)⟶(0[,]+∞))
 
Theoremovncvrrp 39361* The Lebesgue outer measure of a subset of multidimensional real numbers can always be approximated by the total outer measure of a cover of half-open (multidimensional) intervals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   (𝜑𝐸 ∈ ℝ+)    &   𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})    &   𝐿 = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))    &   𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))       (𝜑 → ∃𝑖 𝑖 ∈ ((𝐷𝐴)‘𝐸))
 
Theoremovn0lem 39362* For any finite dimension, the Lebesgue outer measure of the empty set is zero. This is step (a)(ii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))}    &   (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞))    &   𝐼 = (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨1, 0⟩))       (𝜑 → inf(𝑀, ℝ*, < ) = 0)
 
Theoremovn0 39363 For any finite dimension, the Lebesgue outer measure of the empty set is zero. This is step (ii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → ((voln*‘𝑋)‘∅) = 0)
 
Theoremovncl 39364 The Lebesgue outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))       (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ (0[,]+∞))
 
Theoremovn02 39365 For the zero-dimensional space, voln* assigns zero to every subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(voln*‘∅) = (𝑥 ∈ 𝒫 {∅} ↦ 0)
 
Theoremovnxrcl 39366 The Lebesgue outer measure of a set is an extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))       (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ*)
 
Theoremovnsubaddlem1 39367* The Lebesgue outer measure is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑍 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})    &   𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ { ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑗))‘𝑘)})    &   𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))    &   𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))    &   ((𝜑𝑛 ∈ ℕ) → (𝐼𝑛) ∈ ((𝐷‘(𝐴𝑛))‘(𝐸 / (2↑𝑛))))    &   (𝜑𝐹:ℕ–1-1-onto→(ℕ × ℕ))    &   𝐺 = (𝑚 ∈ ℕ ↦ ((𝐼‘(1st ‘(𝐹𝑚)))‘(2nd ‘(𝐹𝑚))))       (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))) +𝑒 𝐸))
 
Theoremovnsubaddlem2 39368* (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑍 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})    &   𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})    &   𝐿 = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))    &   𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑒 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑒)}))       (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))) +𝑒 𝐸))
 
Theoremovnsubadd 39369* (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:ℕ⟶𝒫 (ℝ ↑𝑚 𝑋))       (𝜑 → ((voln*‘𝑋)‘ 𝑛 ∈ ℕ (𝐴𝑛)) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ ((voln*‘𝑋)‘(𝐴𝑛)))))
 
Theoremovnome 39370 (voln*‘𝑋) is an outer measure on the space of multidimensional real numbers with dimension equal to the cardinality of the finite set 𝑋. Proposition 115D (a) of [Fremlin1] p. 30 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (voln*‘𝑋) ∈ OutMeas)
 
Theoremvonmea 39371 (voln‘𝑋) is a measure on the space of multidimensional real numbers with dimension equal to the cardinality of the finite set 𝑋. Comments in Definition 115E of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝑋 ∈ Fin)       (𝜑 → (voln‘𝑋) ∈ Meas)
 
Theoremvolicon0 39372 The measure of a nonempty left-closed, right-open interval. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (vol‘(𝐴[,)𝐵)) = (𝐵𝐴))
 
Theoremhsphoif 39373* 𝐻 is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑋𝑉)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑 → ((𝐻𝐴)‘𝐵):𝑋⟶ℝ)
 
Theoremhoidmvval 39374* The dimensional volume of a multidimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝑋 ∈ Fin)       (𝜑 → (𝐴(𝐿𝑋)𝐵) = if(𝑋 = ∅, 0, ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘)))))
 
Theoremhoissrrn2 39375* A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ*)       (𝜑X𝑘𝑋 (𝐴[,)𝐵) ⊆ (ℝ ↑𝑚 𝑋))
 
Theoremhsphoival 39376* 𝐻 is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐻 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑎𝑗), if((𝑎𝑗) ≤ 𝑥, (𝑎𝑗), 𝑥)))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑋𝑉)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝐾𝑋)       (𝜑 → (((𝐻𝐴)‘𝐵)‘𝐾) = if(𝐾𝑌, (𝐵𝐾), if((𝐵𝐾) ≤ 𝐴, (𝐵𝐾), 𝐴)))
 
Theoremhoiprodcl3 39377* The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑘𝜑    &   (𝜑𝑋 ∈ Fin)    &   ((𝜑𝑘𝑋) → 𝐴 ∈ ℝ)    &   ((𝜑𝑘𝑋) → 𝐵 ∈ ℝ)       (𝜑 → ∏𝑘𝑋 (vol‘(𝐴[,)𝐵)) ∈ (0[,)+∞))
 
Theoremvolicore 39378 The Lebesgue measure of a left-closed right-open interval is a real number. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ ℝ)
 
Theoremhoidmvcl 39379* The dimensional volume of a multidimensional half-open interval is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑 → (𝐴(𝐿𝑋)𝐵) ∈ (0[,)+∞))
 
Theoremhoidmv0val 39380* The dimensional volume of a 0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝐴:∅⟶ℝ)    &   (𝜑𝐵:∅⟶ℝ)       (𝜑 → (𝐴(𝐿‘∅)𝐵) = 0)
 
Theoremhoidmvn0val 39381* The dimensional volume of a non 0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑 → (𝐴(𝐿𝑋)𝐵) = ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
 
Theoremhsphoidmvle2 39382* The dimensional volume of a half-open interval intersected with a two half-spaces. Used in the last inequality of step (c) of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑋 = (𝑌 ∪ {𝑍})    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶𝐷)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)((𝐻𝐷)‘𝐵)))
 
Theoremhsphoidmvle 39383* The dimensional volume of a half-open interval intersected with a half-space, is less than or equal to the dimensional volume of the original half-open interval. Used in the last inequality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑋 = (𝑌 ∪ {𝑍})    &   (𝜑𝐶 ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑗𝑋 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑 → (𝐴(𝐿𝑋)((𝐻𝐶)‘𝐵)) ≤ (𝐴(𝐿𝑋)𝐵))
 
Theoremhoidmvval0 39384* The dimensional volume of the (half-open interval) empty set. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝑗𝜑    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑 → ∃𝑗𝑋 (𝐵𝑗) ≤ (𝐴𝑗))       (𝜑 → (𝐴(𝐿𝑋)𝐵) = 0)
 
Theoremhoiprodp1 39385* The dimensional volume of a half-open interval with dimension 𝑛 + 1. Used in the first equality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑌 ∈ Fin)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑍𝑌)    &   𝑋 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐺 = ∏𝑘𝑌 (vol‘((𝐴𝑘)[,)(𝐵𝑘)))       (𝜑 → (𝐴(𝐿𝑋)𝐵) = (𝐺 · (vol‘((𝐴𝑍)[,)(𝐵𝑍)))))
 
Theoremsge0hsphoire 39386* If the generalized sum of dimensional volumes of n-dimensional half-open intervals is finite, then the sum stays finite if every half-open interval is intersected with a half-space. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑌 ∈ Fin)    &   (𝜑𝑍 ∈ (𝑊𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   (𝜑𝑆 ∈ ℝ)       (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑆)‘(𝐷𝑗))))) ∈ ℝ)
 
Theoremhoidmvval0b 39387* The dimensional volume of the (half-open interval) empty set. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)       (𝜑 → (𝐴(𝐿𝑋)𝐴) = 0)
 
Theoremhoidmv1lelem1 39388* The supremum of 𝑈 belongs to 𝑈. This is the last part of step (a) and the whole step (b) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐶:ℕ⟶ℝ)    &   (𝜑𝐷:ℕ⟶ℝ)    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)    &   𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}    &   𝑆 = sup(𝑈, ℝ, < )       (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
 
Theoremhoidmv1lelem2 39389* This is the contradiction proven in step (c) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶:ℕ⟶ℝ)    &   (𝜑𝐷:ℕ⟶ℝ)    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)    &   𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}    &   (𝜑𝑆𝑈)    &   (𝜑𝐴𝑆)    &   (𝜑𝑆 < 𝐵)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑆 ∈ ((𝐶𝐾)[,)(𝐷𝐾)))    &   𝑀 = if((𝐷𝐾) ≤ 𝐵, (𝐷𝐾), 𝐵)       (𝜑 → ∃𝑢𝑈 𝑆 < 𝑢)
 
Theoremhoidmv1lelem3 39390* The dimensional volume of a 1-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is the non-empty, finite generalized sum, sub case in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐶:ℕ⟶ℝ)    &   (𝜑𝐷:ℕ⟶ℝ)    &   (𝜑 → (𝐴[,)𝐵) ⊆ 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)    &   𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}    &   𝑆 = sup(𝑈, ℝ, < )       (𝜑 → (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
 
Theoremhoidmv1le 39391* The dimensional volume of a 1-dimensional half-open interval is less than or equal to the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is one of the two base cases of the induction of Lemma 115B of [Fremlin1] p. 29 (the other base case is the 0-dimensional case). This proof of the 1-dimensional case is given in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑍𝑉)    &   𝑋 = {𝑍}    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑋))    &   (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑋))    &   (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))       (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
 
Theoremhoidmvlelem1 39392* The supremum of 𝑈 belongs to 𝑈. Step (c) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑊⟶ℝ)    &   (𝜑𝐵:𝑊⟶ℝ)    &   (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}    &   𝑆 = sup(𝑈, ℝ, < )    &   (𝜑 → (𝐴𝑍) < (𝐵𝑍))       (𝜑𝑆𝑈)
 
Theoremhoidmvlelem2 39393* This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑊⟶ℝ)    &   (𝜑𝐵:𝑊⟶ℝ)    &   (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   𝐹 = (𝑦𝑌 ↦ 0)    &   𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))    &   (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}    &   (𝜑𝑆𝑈)    &   (𝜑𝑆 < (𝐵𝑍))    &   𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃𝑗)))    &   𝑂 = ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍))} ↦ ((𝐷𝑖)‘𝑍))    &   𝑉 = ({(𝐵𝑍)} ∪ 𝑂)    &   𝑄 = inf(𝑉, ℝ, < )       (𝜑 → ∃𝑢𝑈 𝑆 < 𝑢)
 
Theoremhoidmvlelem3 39394* This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑊⟶ℝ)    &   (𝜑𝐵:𝑊⟶ℝ)    &   ((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))    &   𝐹 = (𝑦𝑌 ↦ 0)    &   (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐶𝑗) ↾ 𝑌), 𝐹))    &   (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶𝑗)‘𝑍)[,)((𝐷𝑗)‘𝑍)), ((𝐷𝑗) ↾ 𝑌), 𝐹))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}    &   (𝜑𝑆𝑈)    &   (𝜑𝑆 < (𝐵𝑍))    &   𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽𝑗)(𝐿𝑌)(𝐾𝑗)))    &   (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘𝑌 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑌 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑌)(𝑗))))))    &   (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))    &   𝑂 = (𝑥X𝑘𝑌 ((𝐴𝑘)[,)(𝐵𝑘)) ↦ (𝑘𝑊 ↦ if(𝑘𝑌, (𝑥𝑘), 𝑆)))       (𝜑 → ∃𝑢𝑈 𝑆 < 𝑢)
 
Theoremhoidmvlelem4 39395* The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29, case nonempty interval and dimension of the space greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌𝑋)    &   (𝜑𝑌 ≠ ∅)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑊⟶ℝ)    &   (𝜑𝐵:𝑊⟶ℝ)    &   ((𝜑𝑘𝑊) → (𝐴𝑘) < (𝐵𝑘))    &   (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))) ∈ ℝ)    &   𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗𝑊 ↦ if(𝑗𝑌, (𝑐𝑗), if((𝑐𝑗) ≤ 𝑥, (𝑐𝑗), 𝑥)))))    &   𝐺 = ((𝐴𝑌)(𝐿𝑌)(𝐵𝑌))    &   (𝜑𝐸 ∈ ℝ+)    &   𝑈 = {𝑧 ∈ ((𝐴𝑍)[,](𝐵𝑍)) ∣ (𝐺 · (𝑧 − (𝐴𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)((𝐻𝑧)‘(𝐷𝑗))))))}    &   𝑆 = sup(𝑈, ℝ, < )    &   (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘𝑌 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑌 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑌)(𝑗))))))    &   (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))       (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗))))))
 
Theoremhoidmvlelem5 39396* The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍 ∈ (𝑋𝑌))    &   𝑊 = (𝑌 ∪ {𝑍})    &   (𝜑𝐴:𝑊⟶ℝ)    &   (𝜑𝐵:𝑊⟶ℝ)    &   (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘𝑌 ((𝑒𝑘)[,)(𝑓𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑌 (((𝑔𝑗)‘𝑘)[,)((𝑗)‘𝑘)) → (𝑒(𝐿𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔𝑗)(𝐿𝑌)(𝑗))))))    &   (𝜑X𝑘𝑊 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑊 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))    &   (𝜑𝑌 ≠ ∅)       (𝜑 → (𝐴(𝐿𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑊)(𝐷𝑗)))))
 
Theoremhoidmvle 39397* The dimensional volume of a n-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝐶:ℕ⟶(ℝ ↑𝑚 𝑋))    &   (𝜑𝐷:ℕ⟶(ℝ ↑𝑚 𝑋))    &   (𝜑X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘)) ⊆ 𝑗 ∈ ℕ X𝑘𝑋 (((𝐶𝑗)‘𝑘)[,)((𝐷𝑗)‘𝑘)))       (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶𝑗)(𝐿𝑋)(𝐷𝑗)))))
 
Theoremovnhoilem1 39398* The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. First part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}    &   𝐻 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ if(𝑗 = 1, ⟨(𝐴𝑘), (𝐵𝑘)⟩, ⟨0, 0⟩)))       (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘𝑋 (vol‘((𝐴𝑘)[,)(𝐵𝑘))))
 
Theoremovnhoilem2 39399* The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. Second part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))    &   𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}    &   𝐹 = (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (1st ‘((𝑖𝑛)‘𝑙)))))    &   𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙𝑋 ↦ (2nd ‘((𝑖𝑛)‘𝑙)))))       (𝜑 → (𝐴(𝐿𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
 
Theoremovnhoi 39400* The Lebesgue outer measure of a multidimensional half-open interval is its dimensional volume (the product of its length in each dimension, when the dimension is nonzero). Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   𝐼 = X𝑘𝑋 ((𝐴𝑘)[,)(𝐵𝑘))    &   𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘𝑥 (vol‘((𝑎𝑘)[,)(𝑏𝑘))))))       (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿𝑋)𝐵))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
  Copyright terms: Public domain < Previous  Next >