P. 438 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43701-43800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	etransclem36 43701*	The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝑃 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   ⊢ (𝜑 → 𝐽 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    &   ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})    ⇒   ⊢ (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽) ∈ ℤ)
Theorem	etransclem37 43702*	(𝑃 − 1) factorial divides the 𝑁-th derivative of 𝐹 applied to 𝐽. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝑃 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ 𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})    &   ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))    &   ⊢ (𝜑 → 𝐽 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (!‘(𝑃 − 1)) ∥ (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽))
Theorem	etransclem38 43703*	𝑃 divides the I -th derivative of 𝐹 applied to 𝐽. if it is not the case that 𝐼 = 𝑃 − 1 and 𝐽 = 0. This is case 1 and the second part of case 2 proven in in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑃 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    &   ⊢ (𝜑 → 𝐼 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐽 ∈ (0...𝑀))    &   ⊢ (𝜑 → ¬ (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛})    ⇒   ⊢ (𝜑 → 𝑃 ∥ ((((ℝ D𝑛 𝐹)‘𝐼)‘𝐽) / (!‘(𝑃 − 1))))
Theorem	etransclem39 43704*	𝐺 is a function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑃 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    &   ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D𝑛 𝐹)‘𝑖)‘𝑥))    ⇒   ⊢ (𝜑 → 𝐺:ℝ⟶ℂ)
Theorem	etransclem40 43705*	The 𝑁-th derivative of 𝐹 is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝑃 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑘 ∈ (1...𝑀)((𝑥 − 𝑘)↑𝑃)))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (𝑋–cn→ℂ))
Theorem	etransclem41 43706*	𝑃 does not divide the P-1 -th derivative of 𝐹 applied to 0. This is the first part of case 2: proven in in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → (!‘𝑀) < 𝑃)    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    ⇒   ⊢ (𝜑 → ¬ 𝑃 ∥ ((((ℝ D𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))))
Theorem	etransclem42 43707*	The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝑃 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    ⇒   ⊢ (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽) ∈ ℤ)
Theorem	etransclem43 43708*	𝐺 is a continuous function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ⊢ (𝜑 → 𝑃 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    &   ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D𝑛 𝐹)‘𝑖)‘𝑥))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑋–cn→ℂ))
Theorem	etransclem44 43709*	The given finite sum is nonzero. This is the claim proved after equation (7) in [Juillerat] p. 12 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴:ℕ0⟶ℤ)    &   ⊢ (𝜑 → (𝐴‘0) ≠ 0)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)    &   ⊢ (𝜑 → (!‘𝑀) < 𝑃)    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    &   ⊢ 𝐾 = (Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1st ‘𝑘)) · (((ℝ D𝑛 𝐹)‘(2nd ‘𝑘))‘(1st ‘𝑘))) / (!‘(𝑃 − 1)))    ⇒   ⊢ (𝜑 → 𝐾 ≠ 0)
Theorem	etransclem45 43710*	𝐾 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑃 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    &   ⊢ (𝜑 → 𝐴:ℕ0⟶ℤ)    &   ⊢ 𝐾 = (Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1st ‘𝑘)) · (((ℝ D𝑛 𝐹)‘(2nd ‘𝑘))‘(1st ‘𝑘))) / (!‘(𝑃 − 1)))    ⇒   ⊢ (𝜑 → 𝐾 ∈ ℤ)
Theorem	etransclem46 43711*	This is the proof for equation *(7) in [Juillerat] p. 12. The proven equality will lead to a contradiction, because the left-hand side goes to 0 for large 𝑃, but the right-hand side is a nonzero integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑄 ∈ ((Poly‘ℤ) ∖ {0𝑝}))    &   ⊢ (𝜑 → (𝑄‘e) = 0)    &   ⊢ 𝐴 = (coeff‘𝑄)    &   ⊢ 𝑀 = (deg‘𝑄)    &   ⊢ (𝜑 → ℝ ⊆ ℝ)    &   ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})    &   ⊢ (𝜑 → ℝ ∈ ((TopOpen‘ℂfld) ↾t ℝ))    &   ⊢ (𝜑 → 𝑃 ∈ ℕ)    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    &   ⊢ 𝐿 = Σ𝑗 ∈ (0...𝑀)(((𝐴‘𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · (𝐹‘𝑥)) d𝑥)    &   ⊢ 𝑅 = ((𝑀 · 𝑃) + (𝑃 − 1))    &   ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D𝑛 𝐹)‘𝑖)‘𝑥))    &   ⊢ 𝑂 = (𝑥 ∈ (0[,]𝑗) ↦ -((e↑𝑐-𝑥) · (𝐺‘𝑥)))    ⇒   ⊢ (𝜑 → (𝐿 / (!‘(𝑃 − 1))) = (-Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1st ‘𝑘)) · (((ℝ D𝑛 𝐹)‘(2nd ‘𝑘))‘(1st ‘𝑘))) / (!‘(𝑃 − 1))))
Theorem	etransclem47 43712*	e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝑄 ∈ ((Poly‘ℤ) ∖ {0𝑝}))    &   ⊢ (𝜑 → (𝑄‘e) = 0)    &   ⊢ 𝐴 = (coeff‘𝑄)    &   ⊢ (𝜑 → (𝐴‘0) ≠ 0)    &   ⊢ 𝑀 = (deg‘𝑄)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → (abs‘(𝐴‘0)) < 𝑃)    &   ⊢ (𝜑 → (!‘𝑀) < 𝑃)    &   ⊢ (𝜑 → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑃 − 1)) / (!‘(𝑃 − 1)))) < 1)    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑃)))    &   ⊢ 𝐿 = Σ𝑗 ∈ (0...𝑀)(((𝐴‘𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · (𝐹‘𝑥)) d𝑥)    &   ⊢ 𝐾 = (𝐿 / (!‘(𝑃 − 1)))    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))
Theorem	etransclem48 43713*	e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. In this lemma, a large enough prime 𝑝 is chosen: it will be used by subsequent lemmas. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by AV, 28-Sep-2020.)
⊢ (𝜑 → 𝑄 ∈ ((Poly‘ℤ) ∖ {0𝑝}))    &   ⊢ (𝜑 → (𝑄‘e) = 0)    &   ⊢ 𝐴 = (coeff‘𝑄)    &   ⊢ (𝜑 → (𝐴‘0) ≠ 0)    &   ⊢ 𝑀 = (deg‘𝑄)    &   ⊢ 𝐶 = Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1))))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))))    &   ⊢ 𝐼 = inf({𝑖 ∈ ℕ0 ∣ ∀𝑛 ∈ (ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1}, ℝ, < )    &   ⊢ 𝑇 = sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, < )    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))
Theorem	etransc 43714	e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Proof shortened by AV, 28-Sep-2020.)
⊢ e ∈ (ℂ ∖ 𝔸)
20.37.18  n-dimensional Euclidean space
Theorem	rrxtopn 43715*	The topology of the generalized real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐼 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝfld Σg (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))))
Theorem	rrxngp 43716	Generalized Euclidean real spaces are normed groups. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) ∈ NrmGrp)
Theorem	rrxtps 43717	Generalized Euclidean real spaces are topological spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) ∈ TopSp)
Theorem	rrxtopnfi 43718*	The topology of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐼 ∈ Fin)    ⇒   ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))))
Theorem	rrxtopon 43719	The topology on generalized Euclidean real spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼))    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼))))
Theorem	rrxtop 43720	The topology on generalized Euclidean real spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼))    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐽 ∈ Top)
Theorem	rrndistlt 43721*	Given two points in the space of n-dimensional real numbers, if every component is closer than 𝐸 then the distance between the two points is less then ((√‘𝑛) · 𝐸). (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ≠ ∅)    &   ⊢ 𝑁 = (♯‘𝐼)    &   ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑m 𝐼))    &   ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑m 𝐼))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (abs‘((𝑋‘𝑖) − (𝑌‘𝑖))) < 𝐸)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ (𝜑 → (𝑋𝐷𝑌) < ((√‘𝑁) · 𝐸))
Theorem	rrxtoponfi 43722	The topology on n-dimensional Euclidean real spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼))    ⇒   ⊢ (𝐼 ∈ Fin → 𝐽 ∈ (TopOn‘(ℝ ↑m 𝐼)))
Theorem	rrxunitopnfi 43723	The base set of the standard topology on the space of n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝑋 ∈ Fin → ∪ (TopOpen‘(ℝ^‘𝑋)) = (ℝ ↑m 𝑋))
Theorem	rrxtopn0 43724	The topology of the zero-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (TopOpen‘(ℝ^‘∅)) = 𝒫 {∅}
Theorem	qndenserrnbllem 43725*	n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ≠ ∅)    &   ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑m 𝐼))    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
Theorem	qndenserrnbl 43726*	n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (ℝ ↑m 𝐼))    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
Theorem	rrxtopn0b 43727	The topology of the zero-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (TopOpen‘(ℝ^‘∅)) = {∅, {∅}}
Theorem	qndenserrnopnlem 43728*	n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼))    &   ⊢ (𝜑 → 𝑉 ∈ 𝐽)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑉)
Theorem	qndenserrnopn 43729*	n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼))    &   ⊢ (𝜑 → 𝑉 ∈ 𝐽)    &   ⊢ (𝜑 → 𝑉 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ 𝑉)
Theorem	qndenserrn 43730	n-dimensional rational numbers are dense in the space of n-dimensional real numbers, with respect to the n-dimensional standard topology. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ 𝐽 = (TopOpen‘(ℝ^‘𝐼))    ⇒   ⊢ (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) = (ℝ ↑m 𝐼))
Theorem	rrxsnicc 43731*	A multidimensional singleton expressed as a multidimensional closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑m 𝑋))    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐴‘𝑘)) = {𝐴})
Theorem	rrnprjdstle 43732	The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐺:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ 𝐷 = (dist‘(ℝ^‘𝑋))    ⇒   ⊢ (𝜑 → (abs‘((𝐹‘𝐼) − (𝐺‘𝐼))) ≤ (𝐹𝐷𝐺))
Theorem	rrndsmet 43733*	𝐷 is a metric for the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝐷 = (𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))    ⇒   ⊢ (𝜑 → 𝐷 ∈ (Met‘(ℝ ↑m 𝑋)))
Theorem	rrndsxmet 43734*	𝐷 is an extended metric for the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝐷 = (𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))    ⇒   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘(ℝ ↑m 𝑋)))
Theorem	ioorrnopnlem 43735*	The a point in an indexed product of open intervals is contained in an open ball that is contained in the indexed product of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))    &   ⊢ 𝐻 = ran (𝑖 ∈ 𝑋 ↦ if(((𝐵‘𝑖) − (𝐹‘𝑖)) ≤ ((𝐹‘𝑖) − (𝐴‘𝑖)), ((𝐵‘𝑖) − (𝐹‘𝑖)), ((𝐹‘𝑖) − (𝐴‘𝑖))))    &   ⊢ 𝐸 = inf(𝐻, ℝ, < )    &   ⊢ 𝑉 = (𝐹(ball‘𝐷)𝐸)    &   ⊢ 𝐷 = (𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦ (√‘Σ𝑘 ∈ 𝑋 (((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
Theorem	ioorrnopn 43736*	The indexed product of open intervals is an open set in (ℝ^‘𝑋). (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
Theorem	ioorrnopnxrlem 43737*	Given a point 𝐹 that belongs to an indexed product of (possibly unbounded) open intervals, then 𝐹 belongs to an open product of bounded open intervals that's a subset of the original indexed product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ*)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ*)    &   ⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))    &   ⊢ 𝐿 = (𝑖 ∈ 𝑋 ↦ if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))    &   ⊢ 𝑅 = (𝑖 ∈ 𝑋 ↦ if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))    &   ⊢ 𝑉 = X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖))    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
Theorem	ioorrnopnxr 43738*	The indexed product of open intervals is an open set in (ℝ^‘𝑋). Similar to ioorrnopn 43736 but here unbounded intervals are allowed. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ*)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ*)    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
20.37.19  Basic measure theory
20.37.19.1  σ-Algebras Proofs for most of the theorems in section 111 of [Fremlin1]
Syntax	csalg 43739	Extend class notation with the class of all sigma-algebras.
class SAlg
Definition	df-salg 43740*	Define the class of sigma-algebras. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
Syntax	csalon 43741	Extend class notation with the class of sigma-algebras on a set.
class SalOn
Definition	df-salon 43742*	Define the set of sigma-algebra on a given set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ SalOn = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ∪ 𝑠 = 𝑥})
Syntax	csalgen 43743	Extend class notation with the class of sigma-algebra generator.
class SalGen
Definition	df-salgen 43744*	Define the sigma-algebra generated by a given set. Definition 111G (b) of [Fremlin1] p. 13. The sigma-algebra generated by a set is the smallest sigma-algebra, on the same base set, that includes the set, see dfsalgen2 43770. The base set of the sigma-algebras used for the intersection needs to be the same, otherwise the resulting set is not guaranteed to be a sigma-algebra, as shown in the counterexample salgencntex 43772. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Revised by Glauco Siliprandi, 1-Jan-2021.)
⊢ SalGen = (𝑥 ∈ V ↦ ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)})
Theorem	issal 43745*	Express the predicate "𝑆 is a sigma-algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
Theorem	pwsal 43746	The power set of a given set is a sigma-algebra (the so called discrete sigma-algebra). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ SAlg)
Theorem	salunicl 43747	SAlg sigma-algebra is closed under countable union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑆 ∈ SAlg)    &   ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑆)    &   ⊢ (𝜑 → 𝑇 ≼ ω)    ⇒   ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑆)
Theorem	saluncl 43748	The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → (𝐸 ∪ 𝐹) ∈ 𝑆)
Theorem	prsal 43749	The pair of the empty set and the whole base is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝑋 ∈ 𝑉 → {∅, 𝑋} ∈ SAlg)
Theorem	saldifcl 43750	The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)
Theorem	0sal 43751	The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
Theorem	salgenval 43752*	The sigma-algebra generated by a set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)})
Theorem	saliuncl 43753*	SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑆 ∈ SAlg)    &   ⊢ (𝜑 → 𝐾 ≼ ω)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆)
Theorem	salincl 43754	The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → (𝐸 ∩ 𝐹) ∈ 𝑆)
Theorem	saluni 43755	A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆)
Theorem	saliincl 43756*	SAlg sigma-algebra is closed under countable indexed intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝑆 ∈ SAlg)    &   ⊢ (𝜑 → 𝐾 ≼ ω)    &   ⊢ (𝜑 → 𝐾 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆)
Theorem	saldifcl2 43757	The difference of two elements of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆 ∧ 𝐹 ∈ 𝑆) → (𝐸 ∖ 𝐹) ∈ 𝑆)
Theorem	intsaluni 43758*	The union of an arbitrary intersection of sigma-algebras on the same set 𝑋, is 𝑋. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐺 ⊆ SAlg)    &   ⊢ (𝜑 → 𝐺 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 = 𝑋)    ⇒   ⊢ (𝜑 → ∪ ∩ 𝐺 = 𝑋)
Theorem	intsal 43759*	The arbitrary intersection of sigma-algebra (on the same set 𝑋) is a sigma-algebra ( on the same set 𝑋, see intsaluni 43758). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐺 ⊆ SAlg)    &   ⊢ (𝜑 → 𝐺 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 = 𝑋)    ⇒   ⊢ (𝜑 → ∩ 𝐺 ∈ SAlg)
Theorem	salgenn0 43760*	The set used in the definition of the generated sigma-algebra, is not empty. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ≠ ∅)
Theorem	salgencl 43761	SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg)
Theorem	issald 43762*	Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → ∅ ∈ 𝑆)    &   ⊢ 𝑋 = ∪ 𝑆    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝑆 ∈ SAlg)
Theorem	salexct 43763*	An example of nontrivial sigma-algebra: the collection of all subsets which either are countable or have countable complement. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)}    ⇒   ⊢ (𝜑 → 𝑆 ∈ SAlg)
Theorem	sssalgen 43764	A set is a subset of the sigma-algebra it generates. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ 𝑆 = (SalGen‘𝑋)    ⇒   ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ 𝑆)
Theorem	salgenss 43765	The sigma-algebra generated by a set is the smallest sigma-algebra, on the same base set, that includes the set. Proposition 111G (b) of [Fremlin1] p. 13. Notice that the condition "on the same base set" is needed, see the counterexample salgensscntex 43773, where a sigma-algebra is shown that includes a set, but does not include the sigma-algebra generated (the key is that its base set is larger than the base set of the generating set). (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ 𝐺 = (SalGen‘𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ SAlg)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝑋)    ⇒   ⊢ (𝜑 → 𝐺 ⊆ 𝑆)
Theorem	salgenuni 43766	The base set of the sigma-algebra generated by a set is the union of the set itself. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ 𝑆 = (SalGen‘𝑋)    &   ⊢ 𝑈 = ∪ 𝑋    ⇒   ⊢ (𝜑 → ∪ 𝑆 = 𝑈)
Theorem	issalgend 43767*	One side of dfsalgen2 43770. If a sigma-algebra on ∪ 𝑋 includes 𝑋 and it is included in all the sigma-algebras with such two properties, then it is the sigma-algebra generated by 𝑋. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ SAlg)    &   ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝑋)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ SAlg ∧ ∪ 𝑦 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑦)) → 𝑆 ⊆ 𝑦)    ⇒   ⊢ (𝜑 → (SalGen‘𝑋) = 𝑆)
Theorem	salexct2 43768*	An example of a subset that does not belong to a nontrivial sigma-algebra, see salexct 43763. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ 𝐴 = (0[,]2)    &   ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)}    &   ⊢ 𝐵 = (0[,]1)    ⇒   ⊢ ¬ 𝐵 ∈ 𝑆
Theorem	unisalgen 43769	The union of a set belongs to the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ 𝑆 = (SalGen‘𝑋)    &   ⊢ 𝑈 = ∪ 𝑋    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝑆)
Theorem	dfsalgen2 43770*	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 43771*	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 43772*	This counterexample shows that df-salgen 43744 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 43773*	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 43774*	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 43775	Lebesgue measurable sets form a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ dom vol ∈ SAlg
Theorem	saldifcld 43776	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 43777	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 43778	SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ 𝑆 = (SalGen‘𝑋)    ⇒   ⊢ (𝜑 → 𝑆 ∈ SAlg)
Theorem	0sald 43779	The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝑆 ∈ SAlg)    ⇒   ⊢ (𝜑 → ∅ ∈ 𝑆)
Theorem	iooborel 43780	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 43781	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 43782	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 43783	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 43784	The Borel sigma-algebra on the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐵 = (SalGen‘𝐽)    ⇒   ⊢ 𝐵 ∈ SAlg
Theorem	iocborel 43785	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 43786*	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 43787*	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 43788	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 43789	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))
20.37.19.2  Sum of nonnegative extended reals
Syntax	csumge0 43790	Extend class notation to include the sum of nonnegative extended reals.
class Σ^
Definition	df-sumge0 43791*	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 43792*	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 43793	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 43794*	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 43795	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 43796*	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 43797*	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 43798	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 43799	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 43800*	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*, < ))