P. 351 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35001-35100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mbfmco2 35001*	The pair building of two measurable functions is measurable. ( cf. cnmpt1t 24028). (Contributed by Thierry Arnoux, 6-Jun-2017.)
⊢ (𝜑 → 𝑅 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑅MblFnM𝑇))    &   ⊢ 𝐻 = (𝑥 ∈ ∪ 𝑅 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)    ⇒   ⊢ (𝜑 → 𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)))
Theorem	mbfmvolf 35002	Measurable functions with respect to the Lebesgue measure are real-valued functions on the real numbers. (Contributed by Thierry Arnoux, 27-Mar-2017.)
⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹:ℝ⟶ℝ)
Theorem	elmbfmvol2 35003	Measurable functions with respect to the Lebesgue measure. We only have the inclusion, since MblFn includes complex-valued functions. (Contributed by Thierry Arnoux, 26-Jan-2017.)
⊢ (𝐹 ∈ (dom volMblFnM𝔅ℝ) → 𝐹 ∈ MblFn)
Theorem	mbfmcnt 35004	All functions are measurable with respect to the counting measure. (Contributed by Thierry Arnoux, 24-Jan-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂MblFnM𝔅ℝ) = (ℝ ↑m 𝑂))
21.4.18.12  Borel Algebra on ` ( RR X. RR ) `
Theorem	br2base 35005*	The base set for the generator of the Borel sigma-algebra on (ℝ × ℝ) is indeed (ℝ × ℝ). (Contributed by Thierry Arnoux, 22-Sep-2017.)
⊢ ∪ ran (𝑥 ∈ 𝔅ℝ, 𝑦 ∈ 𝔅ℝ ↦ (𝑥 × 𝑦)) = (ℝ × ℝ)
Theorem	dya2ub 35006	An upper bound for a dyadic number. (Contributed by Thierry Arnoux, 19-Sep-2017.)
⊢ (𝑅 ∈ ℝ+ → (1 / (2↑(⌊‘(1 − (2 logb 𝑅))))) < 𝑅)
Theorem	sxbrsigalem0 35007*	The closed half-spaces of (ℝ × ℝ) cover (ℝ × ℝ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
Theorem	sxbrsigalem3 35008*	The sigma-algebra generated by the closed half-spaces of (ℝ × ℝ) is a subset of the sigma-algebra generated by the closed sets of (ℝ × ℝ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))) ⊆ (sigaGen‘(Clsd‘(𝐽 ×t 𝐽)))
Theorem	dya2iocival 35009*	The function 𝐼 returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 25965. (Contributed by Thierry Arnoux, 24-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    ⇒   ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) = ((𝑋 / (2↑𝑁))[,)((𝑋 + 1) / (2↑𝑁))))
Theorem	dya2iocress 35010*	Dyadic intervals are subsets of ℝ. (Contributed by Thierry Arnoux, 18-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    ⇒   ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ⊆ ℝ)
Theorem	dya2iocbrsiga 35011*	Dyadic intervals are Borel sets of ℝ. (Contributed by Thierry Arnoux, 22-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    ⇒   ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ ℤ) → (𝑋𝐼𝑁) ∈ 𝔅ℝ)
Theorem	dya2icobrsiga 35012*	Dyadic intervals are Borel sets of ℝ. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-Oct-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    ⇒   ⊢ ran 𝐼 ⊆ 𝔅ℝ
Theorem	dya2icoseg 35013*	For any point and any closed-below, open-above interval of ℝ centered on that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 19-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑁 = (⌊‘(1 − (2 logb 𝐷)))    ⇒   ⊢ ((𝑋 ∈ ℝ ∧ 𝐷 ∈ ℝ+) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ ((𝑋 − 𝐷)(,)(𝑋 + 𝐷))))
Theorem	dya2icoseg2 35014*	For any point and any open interval of ℝ containing that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 12-Oct-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    ⇒   ⊢ ((𝑋 ∈ ℝ ∧ 𝐸 ∈ ran (,) ∧ 𝑋 ∈ 𝐸) → ∃𝑏 ∈ ran 𝐼(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐸))
Theorem	dya2iocrfn 35015*	The function returning dyadic square covering for a given size has domain (ran 𝐼 × ran 𝐼). (Contributed by Thierry Arnoux, 19-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    ⇒   ⊢ 𝑅 Fn (ran 𝐼 × ran 𝐼)
Theorem	dya2iocct 35016*	The dyadic rectangle set is countable. (Contributed by Thierry Arnoux, 18-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    ⇒   ⊢ ran 𝑅 ≼ ω
Theorem	dya2iocnrect 35017*	For any point of an open rectangle in (ℝ × ℝ), there is a closed-below open-above dyadic rational square which contains that point and is included in the rectangle. (Contributed by Thierry Arnoux, 12-Oct-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    &   ⊢ 𝐵 = ran (𝑒 ∈ ran (,), 𝑓 ∈ ran (,) ↦ (𝑒 × 𝑓))    ⇒   ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
Theorem	dya2iocnei 35018*	For any point of an open set of the usual topology on (ℝ × ℝ) there is a closed-below open-above dyadic rational square which contains that point and is entirely in the open set. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    ⇒   ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑏 ∈ ran 𝑅(𝑋 ∈ 𝑏 ∧ 𝑏 ⊆ 𝐴))
Theorem	dya2iocuni 35019*	Every open set of (ℝ × ℝ) is a union of closed-below open-above dyadic rational rectangular subsets of (ℝ × ℝ). This union must be a countable union by dya2iocct 35016. (Contributed by Thierry Arnoux, 18-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    ⇒   ⊢ (𝐴 ∈ (𝐽 ×t 𝐽) → ∃𝑐 ∈ 𝒫 ran 𝑅∪ 𝑐 = 𝐴)
Theorem	dya2iocucvr 35020*	The dyadic rectangular set collection covers (ℝ × ℝ). (Contributed by Thierry Arnoux, 18-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    ⇒   ⊢ ∪ ran 𝑅 = (ℝ × ℝ)
Theorem	sxbrsigalem1 35021*	The Borel algebra on (ℝ × ℝ) is a subset of the sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ). This is a step of the proof of Proposition 1.1.5 of [Cohn] p. 4. (Contributed by Thierry Arnoux, 17-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    ⇒   ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (sigaGen‘ran 𝑅)
Theorem	sxbrsigalem2 35022*	The sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ) is a subset of the sigma-algebra generated by the closed half-spaces of (ℝ × ℝ). The proof goes by noting the fact that the dyadic rectangles are intersections of a 'vertical band' and an 'horizontal band', which themselves are differences of closed half-spaces. (Contributed by Thierry Arnoux, 17-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    ⇒   ⊢ (sigaGen‘ran 𝑅) ⊆ (sigaGen‘(ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
Theorem	sxbrsigalem4 35023*	The Borel algebra on (ℝ × ℝ) is generated by the dyadic closed-below, open-above rectangular subsets of (ℝ × ℝ). Proposition 1.1.5 of [Cohn] p. 4 . Note that the interval used in this formalization are closed-below, open-above instead of open-below, closed-above in the proof as they are ultimately generated by the floor function. (Contributed by Thierry Arnoux, 21-Sep-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    ⇒   ⊢ (sigaGen‘(𝐽 ×t 𝐽)) = (sigaGen‘ran 𝑅)
Theorem	sxbrsigalem5 35024*	First direction for sxbrsiga 35026. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 11-Oct-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐼 = (𝑥 ∈ ℤ, 𝑛 ∈ ℤ ↦ ((𝑥 / (2↑𝑛))[,)((𝑥 + 1) / (2↑𝑛))))    &   ⊢ 𝑅 = (𝑢 ∈ ran 𝐼, 𝑣 ∈ ran 𝐼 ↦ (𝑢 × 𝑣))    ⇒   ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅ℝ ×s 𝔅ℝ)
Theorem	sxbrsigalem6 35025	First direction for sxbrsiga 35026, same as sxbrsigalem6, dealing with the antecedents. (Contributed by Thierry Arnoux, 10-Oct-2017.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ (sigaGen‘(𝐽 ×t 𝐽)) ⊆ (𝔅ℝ ×s 𝔅ℝ)
Theorem	sxbrsiga 35026	The product sigma-algebra (𝔅ℝ ×s 𝔅ℝ) is the Borel algebra on (ℝ × ℝ) See example 5.1.1 of [Cohn] p. 143 . (Contributed by Thierry Arnoux, 10-Oct-2017.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ (𝔅ℝ ×s 𝔅ℝ) = (sigaGen‘(𝐽 ×t 𝐽))
21.4.18.13  Caratheodory's extension theorem In this section, we define a function toOMeas which constructs an outer measure, from a pre-measure 𝑅. An explicit generic definition of an outer measure is not given. It consists of the three following statements: - the outer measure of an empty set is zero (oms0 35033) - it is monotone (omsmon 35034) - it is countably sub-additive (omssubadd 35036) See Definition 1.11.1 of [Bogachev] p. 41.
Syntax	coms 35027	Class declaration for the outer measure construction function.
class toOMeas
Definition	df-oms 35028*	Define a function constructing an outer measure. See omsval 35029 for its value. Definition 1.5 of [Bogachev] p. 16. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
⊢ toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 ∪ dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑟‘𝑦)), (0[,]+∞), < )))
Theorem	omsval 35029*	Value of the function mapping a content function to the corresponding outer measure. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
⊢ (𝑅 ∈ V → (toOMeas‘𝑅) = (𝑎 ∈ 𝒫 ∪ dom 𝑅 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < )))
Theorem	omsfval 35030*	Value of the outer measure evaluated for a given set 𝐴. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ⊆ ∪ 𝑄) → ((toOMeas‘𝑅)‘𝐴) = inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)), (0[,]+∞), < ))
Theorem	omscl 35031*	A closure lemma for the constructed outer measure. (Contributed by Thierry Arnoux, 17-Sep-2019.)
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞) ∧ 𝐴 ∈ 𝒫 ∪ dom 𝑅) → ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦ Σ*𝑦 ∈ 𝑥(𝑅‘𝑦)) ⊆ (0[,]+∞))
Theorem	omsf 35032	A constructed outer measure is a function. (Contributed by Thierry Arnoux, 17-Sep-2019.) (Revised by AV, 4-Oct-2020.)
⊢ ((𝑄 ∈ 𝑉 ∧ 𝑅:𝑄⟶(0[,]+∞)) → (toOMeas‘𝑅):𝒫 ∪ dom 𝑅⟶(0[,]+∞))
Theorem	oms0 35033	A constructed outer measure evaluates to zero for the empty set. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
⊢ 𝑀 = (toOMeas‘𝑅)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ (𝜑 → ∅ ∈ dom 𝑅)    &   ⊢ (𝜑 → (𝑅‘∅) = 0)    ⇒   ⊢ (𝜑 → (𝑀‘∅) = 0)
Theorem	omsmon 35034	A constructed outer measure is monotone. Note in Example 1.5.2 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 15-Sep-2019.) (Revised by AV, 4-Oct-2020.)
⊢ 𝑀 = (toOMeas‘𝑅)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝑄)    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
Theorem	omssubaddlem 35035*	For any small margin 𝐸, we can find a covering approaching the outer measure of a set 𝐴 by that margin. (Contributed by Thierry Arnoux, 18-Sep-2019.) (Revised by AV, 4-Oct-2020.)
⊢ 𝑀 = (toOMeas‘𝑅)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑄)    &   ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}Σ*𝑤 ∈ 𝑥(𝑅‘𝑤) < ((𝑀‘𝐴) + 𝐸))
Theorem	omssubadd 35036*	A constructed outer measure is countably sub-additive. Lemma 1.5.4 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 21-Sep-2019.) (Revised by AV, 4-Oct-2020.) Remove dependency on ax-reg 9621. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
⊢ 𝑀 = (toOMeas‘𝑅)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐴 ⊆ ∪ 𝑄)    &   ⊢ (𝜑 → 𝑋 ≼ ω)    ⇒   ⊢ (𝜑 → (𝑀‘∪ 𝑦 ∈ 𝑋 𝐴) ≤ Σ*𝑦 ∈ 𝑋(𝑀‘𝐴))
Syntax	ccarsg 35037	Class declaration for the Caratheodory sigma-Algebra construction.
class toCaraSiga
Definition	df-carsg 35038*	Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 35039 for its value. Definition 1.11.2 of [Bogachev] p. 41. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +e (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)})
Theorem	carsgval 35039*	Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +e (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)})
Theorem	carsgcl 35040	Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
Theorem	elcarsg 35041*	Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +e (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒))))
Theorem	baselcarsg 35042	The universe set, 𝑂, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    ⇒   ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀))
Theorem	0elcarsg 35043	The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    ⇒   ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀))
Theorem	carsguni 35044	The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    ⇒   ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) = 𝑂)
Theorem	elcarsgss 35045	Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)
Theorem	difelcarsg 35046	The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝑂 ∖ 𝐴) ∈ (toCaraSiga‘𝑀))
Theorem	inelcarsg 35047*	The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +e (𝑀‘𝑏)))    &   ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ (toCaraSiga‘𝑀))
Theorem	unelcarsg 35048*	The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +e (𝑀‘𝑏)))    &   ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ (toCaraSiga‘𝑀))
Theorem	difelcarsg2 35049*	The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +e (𝑀‘𝑏)))    &   ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ (toCaraSiga‘𝑀))
Theorem	carsgmon 35050*	Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑂)    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    ⇒   ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
Theorem	carsgsigalem 35051*	Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    ⇒   ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +e (𝑀‘𝑓)))
Theorem	fiunelcarsg 35052*	The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀))
Theorem	carsgclctunlem1 35053*	Lemma for carsgclctun 35057. (Contributed by Thierry Arnoux, 23-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)    ⇒   ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))
Theorem	carsggect 35054*	The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ (𝜑 → ¬ ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    ⇒   ⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴))
Theorem	carsgclctunlem2 35055*	Lemma for carsgclctun 35057. (Contributed by Thierry Arnoux, 25-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    &   ⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)    &   ⊢ (𝜑 → (𝑀‘𝐸) ≠ +∞)    ⇒   ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +e (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸))
Theorem	carsgclctunlem3 35056*	Lemma for carsgclctun 35057. (Contributed by Thierry Arnoux, 24-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    &   ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)    ⇒   ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝐴)) +e (𝑀‘(𝐸 ∖ ∪ 𝐴))) ≤ (𝑀‘𝐸))
Theorem	carsgclctun 35057*	The Caratheodory measurable sets are closed under countable union. (Contributed by Thierry Arnoux, 21-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀))
Theorem	carsgsiga 35058*	The Caratheodory measurable sets constructed from outer measures form a Sigma-algebra. Statement (iii) of Theorem 1.11.4 of [Bogachev] p. 42. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑀‘∅) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))    &   ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))    ⇒   ⊢ (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂))
Theorem	omsmeas 35059	The restriction of a constructed outer measure to Caratheodory measurable sets is a measure. This theorem allows to construct measures from pre-measures with the required characteristics, as for the Lebesgue measure. (Contributed by Thierry Arnoux, 17-May-2020.)
⊢ 𝑀 = (toOMeas‘𝑅)    &   ⊢ 𝑆 = (toCaraSiga‘𝑀)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ⊢ (𝜑 → ∅ ∈ dom 𝑅)    &   ⊢ (𝜑 → (𝑅‘∅) = 0)    ⇒   ⊢ (𝜑 → (𝑀 ↾ 𝑆) ∈ (measures‘𝑆))
Theorem	pmeasmono 35060*	This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑃‘∅) = 0)    &   ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑅)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑅)    &   ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑅)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
Theorem	pmeasadd 35061*	A premeasure on a ring of sets is additive on disjoint countable collections. This is called sigma-additivity. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞))    &   ⊢ (𝜑 → (𝑃‘∅) = 0)    &   ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦))    &   ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}    &   ⊢ (𝜑 → 𝑅 ∈ 𝑄)    &   ⊢ (𝜑 → 𝐴 ≼ ω)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑅)    &   ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ*𝑘 ∈ 𝐴(𝑃‘𝐵))
21.4.19  Integration
21.4.19.1  Lebesgue integral - misc additions
Theorem	itgeq12dv 35062*	Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
21.4.19.2  Bochner integral
Syntax	citgm 35063	Extend class notation with the (measure) Bochner integral.
class itgm
Syntax	csitm 35064	Extend class notation with the integral metric for simple functions.
class sitm
Syntax	csitg 35065	Extend class notation with the integral of simple functions.
class sitg
Definition	df-sitg 35066*	Define the integral of simple functions from a measurable space dom 𝑚 to a generic space 𝑤 equipped with the right scalar product. 𝑤 will later be required to be a Banach space. These simple functions are required to take finitely many different values: this is expressed by ran 𝑔 ∈ Fin in the definition. Moreover, for each 𝑥, the pre-image (◡𝑔 “ {𝑥}) is requested to be measurable, of finite measure. In this definition, (sigaGen‘(TopOpen‘𝑤)) is the Borel sigma-algebra on 𝑤, and the functions 𝑔 range over the measurable functions over that Borel algebra. Definition 2.4.1 of [Bogachev] p. 118. (Contributed by Thierry Arnoux, 21-Oct-2017.)
⊢ sitg = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g‘𝑤)})(𝑚‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g‘𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(◡𝑓 “ {𝑥})))( ·𝑠 ‘𝑤)𝑥)))))
Definition	df-sitm 35067*	Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take nonnegative values in ℝ*, the range structure for this integral is (ℝ*𝑠 ↾s (0[,]+∞)). See definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑤)𝑔))))
Theorem	sitgval 35068*	Value of the simple function integral builder for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    ⇒   ⊢ (𝜑 → (𝑊sitg𝑀) = (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥)))))
Theorem	issibf 35069*	The predicate "𝐹 is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
Theorem	sibf0 35070	The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-Mar-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝑊 ∈ Mnd)    ⇒   ⊢ (𝜑 → (∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀))
Theorem	sibfmbl 35071	A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM𝑆))
Theorem	sibff 35072	A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → 𝐹:∪ dom 𝑀⟶∪ 𝐽)
Theorem	sibfrn 35073	A simple function has finite range. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → ran 𝐹 ∈ Fin)
Theorem	sibfima 35074	Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))
Theorem	sibfinima 35075	The measure of the intersection of any two preimages by simple functions is a real number. (Contributed by Thierry Arnoux, 21-Mar-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝐽 ∈ Fre)    ⇒   ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,)+∞))
Theorem	sibfof 35076	Applying function operations on simple functions results in simple functions with regard to the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-Mar-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ 𝐶 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶)    &   ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝐾 ∈ TopSp)    &   ⊢ (𝜑 → 𝐽 ∈ Fre)    &   ⊢ (𝜑 → ( 0 + 0 ) = (0g‘𝐾))    ⇒   ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom (𝐾sitg𝑀))
Theorem	sitgfval 35077*	Value of the Bochner integral for a simple function 𝐹. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥))))
Theorem	sitgclg 35078*	Closure of the Bochner integral on simple functions, generic version. See sitgclbn 35079 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.) (Proof shortened by AV, 12-Dec-2019.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ 𝐺 = (Scalar‘𝑊)    &   ⊢ 𝐷 = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝑊 ∈ CMnd)    &   ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt )    &   ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → (𝑚 · 𝑥) ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
Theorem	sitgclbn 35079	Closure of the Bochner integral on a simple function. This version is specific to Banach spaces, with additional conditions on its scalar field. (Contributed by Thierry Arnoux, 24-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝑊 ∈ Ban)    &   ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt )    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
Theorem	sitgclcn 35080	Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the complex numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝑊 ∈ Ban)    &   ⊢ (𝜑 → (Scalar‘𝑊) = ℂfld)    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
Theorem	sitgclre 35081	Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝑊 ∈ Ban)    &   ⊢ (𝜑 → (Scalar‘𝑊) = ℝfld)    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵)
Theorem	sitg0 35082	The integral of the constant zero function is zero. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝑊 ∈ Mnd)    ⇒   ⊢ (𝜑 → ((𝑊sitg𝑀)‘(∪ dom 𝑀 × { 0 })) = 0 )
Theorem	sitgf 35083*	The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵)
Theorem	sitgaddlemb 35084	Lemma for * sitgadd . (Contributed by Thierry Arnoux, 10-Mar-2019.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝑆 = (sigaGen‘𝐽)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → (𝑊 ↾v (𝐻 “ (0[,)+∞))) ∈ SLMod)    &   ⊢ (𝜑 → 𝐽 ∈ Fre)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt )    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝜑 ∧ 𝑝 ∈ ((ran 𝐹 × ran 𝐺) ∖ {⟨ 0 , 0 ⟩})) → ((𝐻‘(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))) · (2nd ‘𝑝)) ∈ 𝐵)
Theorem	sitmval 35085*	Value of the simple function integral metric for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐷 = (dist‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    ⇒   ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔))))
Theorem	sitmfval 35086	Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐷 = (dist‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘f 𝐷𝐺)))
Theorem	sitmcl 35087	Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.)
⊢ (𝜑 → 𝑊 ∈ Mnd)    &   ⊢ (𝜑 → 𝑊 ∈ ∞MetSp)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    &   ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))    &   ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀))    ⇒   ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) ∈ (0[,]+∞))
Theorem	sitmf 35088	The integral metric as a function. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ (𝜑 → 𝑊 ∈ Mnd)    &   ⊢ (𝜑 → 𝑊 ∈ ∞MetSp)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)    ⇒   ⊢ (𝜑 → (𝑊sitm𝑀):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞))
Definition	df-itgm 35089*	Define the Bochner integral as the extension by continuity of the Bochnel integral for simple functions. Bogachev first defines 'fundamental in the mean' sequences, in definition 2.3.1 of [Bogachev] p. 116, and notes that those are actually Cauchy sequences for the pseudometric (𝑤sitm𝑚). He then defines the Bochner integral in chapter 2.4.4 in [Bogachev] p. 118. The definition of the Lebesgue integral, df-itg 25988. (Contributed by Thierry Arnoux, 13-Feb-2018.)
⊢ itgm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚)))
21.4.20  Euler's partition theorem
Theorem	oddpwdc 35090*	Lemma for eulerpart 35118. The function 𝐹 that decomposes a number into its "odd" and "even" parts, which is to say the largest power of two and largest odd divisor of a number, is a bijection from pairs of a nonnegative integer and an odd number to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.)
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    ⇒   ⊢ 𝐹:(𝐽 × ℕ0)⟶1-1ontoℕ
Theorem	oddpwdcv 35091*	Lemma for eulerpart 35118: value of the 𝐹 function. (Contributed by Thierry Arnoux, 9-Sep-2017.)
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    ⇒   ⊢ (𝑊 ∈ (𝐽 × ℕ0) → (𝐹‘𝑊) = ((2↑(2nd ‘𝑊)) · (1st ‘𝑊)))
Theorem	eulerpartlemsv1 35092*	Lemma for eulerpart 35118. Value of the sum of a partition 𝐴. (Contributed by Thierry Arnoux, 26-Aug-2018.)
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘))
Theorem	eulerpartlemelr 35093*	Lemma for eulerpart 35118. (Contributed by Thierry Arnoux, 8-Aug-2018.)
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin))
Theorem	eulerpartlemsv2 35094*	Lemma for eulerpart 35118. Value of the sum of a finite partition 𝐴 (Contributed by Thierry Arnoux, 19-Aug-2018.)
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘))
Theorem	eulerpartlemsf 35095*	Lemma for eulerpart 35118. (Contributed by Thierry Arnoux, 8-Aug-2018.)
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ 𝑆:((ℕ0 ↑m ℕ) ∩ 𝑅)⟶ℕ0
Theorem	eulerpartlems 35096*	Lemma for eulerpart 35118. (Contributed by Thierry Arnoux, 6-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0)
Theorem	eulerpartlemsv3 35097*	Lemma for eulerpart 35118. Value of the sum of a finite partition 𝐴 (Contributed by Thierry Arnoux, 19-Aug-2018.)
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘))
Theorem	eulerpartlemgc 35098*	Lemma for eulerpart 35118. (Contributed by Thierry Arnoux, 9-Aug-2018.)
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆‘𝐴))
Theorem	eulerpartleme 35099*	Lemma for eulerpart 35118. (Contributed by Mario Carneiro, 26-Jan-2015.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    ⇒   ⊢ (𝐴 ∈ 𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = 𝑁))
Theorem	eulerpartlemv 35100*	Lemma for eulerpart 35118. (Contributed by Thierry Arnoux, 19-Aug-2018.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    ⇒   ⊢ (𝐴 ∈ 𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘) = 𝑁))