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Theorem List for Metamath Proof Explorer - 43801-43900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremetransclem19 43801* The 𝑁-th derivative of 𝐻 is 0 if 𝑁 is large enough. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   (𝜑𝐽 ∈ (0...𝑀))    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑 → if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁)       (𝜑 → ((𝑆 D𝑛 (𝐻𝐽))‘𝑁) = (𝑥𝑋 ↦ 0))
 
Theoremetransclem20 43802* 𝐻 is smooth. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   (𝜑𝐽 ∈ (0...𝑀))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝑆 D𝑛 (𝐻𝐽))‘𝑁):𝑋⟶ℂ)
 
Theoremetransclem21 43803* The 𝑁-th derivative of 𝐻 applied to 𝑌. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   (𝜑𝐽 ∈ (0...𝑀))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑌𝑋)       (𝜑 → (((𝑆 D𝑛 (𝐻𝐽))‘𝑁)‘𝑌) = if(if(𝐽 = 0, (𝑃 − 1), 𝑃) < 𝑁, 0, (((!‘if(𝐽 = 0, (𝑃 − 1), 𝑃)) / (!‘(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁))) · ((𝑌𝐽)↑(if(𝐽 = 0, (𝑃 − 1), 𝑃) − 𝑁)))))
 
Theoremetransclem22 43804* The 𝑁-th derivative of 𝐻 is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   (𝜑𝐽 ∈ (0...𝑀))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝑆 D𝑛 (𝐻𝐽))‘𝑁) ∈ (𝑋cn→ℂ))
 
Theoremetransclem23 43805* This is the claim proof in [Juillerat] p. 14 (but in our proof, Stirling's approximation is not used). (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝐴:ℕ0⟶ℤ)    &   𝐿 = Σ𝑗 ∈ (0...𝑀)(((𝐴𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · (𝐹𝑥)) d𝑥)    &   𝐾 = (𝐿 / (!‘(𝑃 − 1)))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑 → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑃 − 1)) / (!‘(𝑃 − 1)))) < 1)       (𝜑 → (abs‘𝐾) < 1)
 
Theoremetransclem24 43806* 𝑃 divides the I -th derivative of 𝐹 applied to 𝐽. when 𝐽 = 0 and 𝐼 is not equal to 𝑃 − 1. This is the second part of case 2 proven in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝐼 ∈ ℕ0)    &   (𝜑𝐼 ≠ (𝑃 − 1))    &   (𝜑𝐽 = 0)    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})    &   (𝜑𝐷 ∈ (𝐶𝐼))       (𝜑𝑃 ∥ ((((!‘𝐼) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷𝑗)))) · ((𝐽𝑗)↑(𝑃 − (𝐷𝑗))))))) / (!‘(𝑃 − 1))))
 
Theoremetransclem25 43807* 𝑃 factorial divides the 𝑁-th derivative of 𝐹 applied to 𝐽. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐶:(0...𝑀)⟶(0...𝑁))    &   (𝜑 → Σ𝑗 ∈ (0...𝑀)(𝐶𝑗) = 𝑁)    &   𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐶𝑗))) · (if((𝑃 − 1) < (𝐶‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐶‘0)))) · (𝐽↑((𝑃 − 1) − (𝐶‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐶𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐶𝑗)))) · ((𝐽𝑗)↑(𝑃 − (𝐶𝑗)))))))    &   (𝜑𝐽 ∈ (1...𝑀))       (𝜑 → (!‘𝑃) ∥ 𝑇)
 
Theoremetransclem26 43808* Every term in the sum of the 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐽 ∈ ℤ)    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})    &   (𝜑𝐷 ∈ (𝐶𝑁))       (𝜑 → (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷𝑗)))) · ((𝐽𝑗)↑(𝑃 − (𝐷𝑗))))))) ∈ ℤ)
 
Theoremetransclem27 43809* The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   (𝜑𝐶 ∈ Fin)    &   (𝜑𝐶:dom 𝐶⟶(ℕ0m (0...𝑀)))    &   𝐺 = (𝑥𝑋 ↦ Σ𝑙 ∈ dom 𝐶𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘((𝐶𝑙)‘𝑗))‘𝑥))    &   (𝜑𝐽𝑋)    &   (𝜑𝐽 ∈ ℤ)       (𝜑 → (𝐺𝐽) ∈ ℤ)
 
Theoremetransclem28 43810* (𝑃 − 1) factorial divides the 𝑁-th derivative of 𝐹 applied to 𝐽. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})    &   (𝜑𝐷 ∈ (𝐶𝑁))    &   (𝜑𝐽 ∈ (0...𝑀))    &   𝑇 = (((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝐷𝑗))) · (if((𝑃 − 1) < (𝐷‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝐷‘0)))) · (𝐽↑((𝑃 − 1) − (𝐷‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝐷𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝐷𝑗)))) · ((𝐽𝑗)↑(𝑃 − (𝐷𝑗)))))))       (𝜑 → (!‘(𝑃 − 1)) ∥ 𝑇)
 
Theoremetransclem29 43811* The 𝑁-th derivative of 𝐹. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})    &   𝐸 = (𝑥𝑋 ↦ ∏𝑗 ∈ (0...𝑀)((𝐻𝑗)‘𝑥))       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ Σ𝑐 ∈ (𝐶𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥))))
 
Theoremetransclem30 43812* The 𝑁-th derivative of 𝐹. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ Σ𝑐 ∈ (𝐶𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · ∏𝑗 ∈ (0...𝑀)(((𝑆 D𝑛 (𝐻𝑗))‘(𝑐𝑗))‘𝑥))))
 
Theoremetransclem31 43813* The 𝑁-th derivative of 𝐻 applied to 𝑌. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})    &   (𝜑𝑌𝑋)       (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁)‘𝑌) = Σ𝑐 ∈ (𝐶𝑁)(((!‘𝑁) / ∏𝑗 ∈ (0...𝑀)(!‘(𝑐𝑗))) · (if((𝑃 − 1) < (𝑐‘0), 0, (((!‘(𝑃 − 1)) / (!‘((𝑃 − 1) − (𝑐‘0)))) · (𝑌↑((𝑃 − 1) − (𝑐‘0))))) · ∏𝑗 ∈ (1...𝑀)if(𝑃 < (𝑐𝑗), 0, (((!‘𝑃) / (!‘(𝑃 − (𝑐𝑗)))) · ((𝑌𝑗)↑(𝑃 − (𝑐𝑗))))))))
 
Theoremetransclem32 43814* This is the proof for the last equation in the proof of the derivative calculated in [Juillerat] p. 12, just after equation *(6) . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ((𝑀 · 𝑃) + (𝑃 − 1)) < 𝑁)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ 0))
 
Theoremetransclem33 43815* 𝐹 is smooth. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁):𝑋⟶ℂ)
 
Theoremetransclem34 43816* The 𝑁-th derivative of 𝐹 is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑘 ∈ (1...𝑀)((𝑥𝑘)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   𝐻 = (𝑘 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑘 ∈ (0...𝑀)(𝑐𝑘) = 𝑛})       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (𝑋cn→ℂ))
 
Theoremetransclem35 43817* 𝑃 does not divide the P-1 -th derivative of 𝐹 applied to 0. This is case 2 of the proof in [Juillerat] p. 13 . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})    &   𝐷 = (𝑗 ∈ (0...𝑀) ↦ if(𝑗 = 0, (𝑃 − 1), 0))       (𝜑 → (((ℝ D𝑛 𝐹)‘(𝑃 − 1))‘0) = ((!‘(𝑃 − 1)) · (∏𝑗 ∈ (1...𝑀)-𝑗𝑃)))
 
Theoremetransclem36 43818* The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   (𝜑𝐽𝑋)    &   (𝜑𝐽 ∈ ℤ)    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})       (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽) ∈ ℤ)
 
Theoremetransclem37 43819* (𝑃 − 1) factorial divides the 𝑁-th derivative of 𝐹 applied to 𝐽. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   𝐻 = (𝑗 ∈ (0...𝑀) ↦ (𝑥𝑋 ↦ ((𝑥𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})    &   (𝜑𝐽 ∈ (0...𝑀))    &   (𝜑𝐽𝑋)       (𝜑 → (!‘(𝑃 − 1)) ∥ (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽))
 
Theoremetransclem38 43820* 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.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝐼 ∈ ℕ0)    &   (𝜑𝐽 ∈ (0...𝑀))    &   (𝜑 → ¬ (𝐼 = (𝑃 − 1) ∧ 𝐽 = 0))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐𝑗) = 𝑛})       (𝜑𝑃 ∥ ((((ℝ D𝑛 𝐹)‘𝐼)‘𝐽) / (!‘(𝑃 − 1))))
 
Theoremetransclem39 43821* 𝐺 is a function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐺 = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D𝑛 𝐹)‘𝑖)‘𝑥))       (𝜑𝐺:ℝ⟶ℂ)
 
Theoremetransclem40 43822* The 𝑁-th derivative of 𝐹 is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑘 ∈ (1...𝑀)((𝑥𝑘)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (𝑋cn→ℂ))
 
Theoremetransclem41 43823* 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.)
(𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (!‘𝑀) < 𝑃)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))       (𝜑 → ¬ 𝑃 ∥ ((((ℝ D𝑛 𝐹)‘(𝑃 − 1))‘0) / (!‘(𝑃 − 1))))
 
Theoremetransclem42 43824* The 𝑁-th derivative of 𝐹 applied to 𝐽 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐽𝑋)    &   (𝜑𝐽 ∈ ℤ)       (𝜑 → (((𝑆 D𝑛 𝐹)‘𝑁)‘𝐽) ∈ ℤ)
 
Theoremetransclem43 43825* 𝐺 is a continuous function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐺 = (𝑥𝑋 ↦ Σ𝑖 ∈ (0...𝑅)(((𝑆 D𝑛 𝐹)‘𝑖)‘𝑥))       (𝜑𝐺 ∈ (𝑋cn→ℂ))
 
Theoremetransclem44 43826* 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.)
(𝜑𝐴:ℕ0⟶ℤ)    &   (𝜑 → (𝐴‘0) ≠ 0)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (abs‘(𝐴‘0)) < 𝑃)    &   (𝜑 → (!‘𝑀) < 𝑃)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐾 = (Σ𝑘 ∈ ((0...𝑀) × (0...((𝑀 · 𝑃) + (𝑃 − 1))))((𝐴‘(1st𝑘)) · (((ℝ D𝑛 𝐹)‘(2nd𝑘))‘(1st𝑘))) / (!‘(𝑃 − 1)))       (𝜑𝐾 ≠ 0)
 
Theoremetransclem45 43827* 𝐾 is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑃 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   (𝜑𝐴:ℕ0⟶ℤ)    &   𝐾 = (Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1st𝑘)) · (((ℝ D𝑛 𝐹)‘(2nd𝑘))‘(1st𝑘))) / (!‘(𝑃 − 1)))       (𝜑𝐾 ∈ ℤ)
 
Theoremetransclem46 43828* 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.)
(𝜑𝑄 ∈ ((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))    &   (𝜑𝑃 ∈ ℕ)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐿 = Σ𝑗 ∈ (0...𝑀)(((𝐴𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · (𝐹𝑥)) d𝑥)    &   𝑅 = ((𝑀 · 𝑃) + (𝑃 − 1))    &   𝐺 = (𝑥 ∈ ℝ ↦ Σ𝑖 ∈ (0...𝑅)(((ℝ D𝑛 𝐹)‘𝑖)‘𝑥))    &   𝑂 = (𝑥 ∈ (0[,]𝑗) ↦ -((e↑𝑐-𝑥) · (𝐺𝑥)))       (𝜑 → (𝐿 / (!‘(𝑃 − 1))) = (-Σ𝑘 ∈ ((0...𝑀) × (0...𝑅))((𝐴‘(1st𝑘)) · (((ℝ D𝑛 𝐹)‘(2nd𝑘))‘(1st𝑘))) / (!‘(𝑃 − 1))))
 
Theoremetransclem47 43829* 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.)
(𝜑𝑄 ∈ ((Poly‘ℤ) ∖ {0𝑝}))    &   (𝜑 → (𝑄‘e) = 0)    &   𝐴 = (coeff‘𝑄)    &   (𝜑 → (𝐴‘0) ≠ 0)    &   𝑀 = (deg‘𝑄)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑 → (abs‘(𝐴‘0)) < 𝑃)    &   (𝜑 → (!‘𝑀) < 𝑃)    &   (𝜑 → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑃 − 1)) / (!‘(𝑃 − 1)))) < 1)    &   𝐹 = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑃 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥𝑗)↑𝑃)))    &   𝐿 = Σ𝑗 ∈ (0...𝑀)(((𝐴𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · (𝐹𝑥)) d𝑥)    &   𝐾 = (𝐿 / (!‘(𝑃 − 1)))       (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))
 
Theoremetransclem48 43830* 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.)
(𝜑𝑄 ∈ ((Poly‘ℤ) ∖ {0𝑝}))    &   (𝜑 → (𝑄‘e) = 0)    &   𝐴 = (coeff‘𝑄)    &   (𝜑 → (𝐴‘0) ≠ 0)    &   𝑀 = (deg‘𝑄)    &   𝐶 = Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1))))    &   𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))))    &   𝐼 = inf({𝑖 ∈ ℕ0 ∣ ∀𝑛 ∈ (ℤ𝑖)(abs‘(𝑆𝑛)) < 1}, ℝ, < )    &   𝑇 = sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, < )       (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))
 
Theoremetransc 43831 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
 
Theoremrrxtopn 43832* The topology of the generalized real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼𝑉)       (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝfld Σg (𝑥𝐼 ↦ (((𝑓𝑥) − (𝑔𝑥))↑2)))))))
 
Theoremrrxngp 43833 Generalized Euclidean real spaces are normed groups. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝐼𝑉 → (ℝ^‘𝐼) ∈ NrmGrp)
 
Theoremrrxtps 43834 Generalized Euclidean real spaces are topological spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝐼𝑉 → (ℝ^‘𝐼) ∈ TopSp)
 
Theoremrrxtopnfi 43835* The topology of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐼 ∈ Fin)       (𝜑 → (TopOpen‘(ℝ^‘𝐼)) = (MetOpen‘(𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (√‘Σ𝑘𝐼 (((𝑓𝑘) − (𝑔𝑘))↑2)))))
 
Theoremrrxtopon 43836 The topology on generalized Euclidean real spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐽 = (TopOpen‘(ℝ^‘𝐼))       (𝐼𝑉𝐽 ∈ (TopOn‘(Base‘(ℝ^‘𝐼))))
 
Theoremrrxtop 43837 The topology on generalized Euclidean real spaces. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝐽 = (TopOpen‘(ℝ^‘𝐼))       (𝐼𝑉𝐽 ∈ Top)
 
Theoremrrndistlt 43838* 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.)
(𝜑𝐼 ∈ Fin)    &   (𝜑𝐼 ≠ ∅)    &   𝑁 = (♯‘𝐼)    &   (𝜑𝑋 ∈ (ℝ ↑m 𝐼))    &   (𝜑𝑌 ∈ (ℝ ↑m 𝐼))    &   ((𝜑𝑖𝐼) → (abs‘((𝑋𝑖) − (𝑌𝑖))) < 𝐸)    &   (𝜑𝐸 ∈ ℝ+)    &   𝐷 = (dist‘(ℝ^‘𝐼))       (𝜑 → (𝑋𝐷𝑌) < ((√‘𝑁) · 𝐸))
 
Theoremrrxtoponfi 43839 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)))
 
Theoremrrxunitopnfi 43840 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))
 
Theoremrrxtopn0 43841 The topology of the zero-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(TopOpen‘(ℝ^‘∅)) = 𝒫 {∅}
 
Theoremqndenserrnbllem 43842* 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.)
(𝜑𝐼 ∈ Fin)    &   (𝜑𝐼 ≠ ∅)    &   (𝜑𝑋 ∈ (ℝ ↑m 𝐼))    &   𝐷 = (dist‘(ℝ^‘𝐼))    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
 
Theoremqndenserrnbl 43843* 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.)
(𝜑𝐼 ∈ Fin)    &   (𝜑𝑋 ∈ (ℝ ↑m 𝐼))    &   𝐷 = (dist‘(ℝ^‘𝐼))    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦 ∈ (𝑋(ball‘𝐷)𝐸))
 
Theoremrrxtopn0b 43844 The topology of the zero-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(TopOpen‘(ℝ^‘∅)) = {∅, {∅}}
 
Theoremqndenserrnopnlem 43845* 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.)
(𝜑𝐼 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝐼))    &   (𝜑𝑉𝐽)    &   (𝜑𝑋𝑉)    &   𝐷 = (dist‘(ℝ^‘𝐼))       (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦𝑉)
 
Theoremqndenserrnopn 43846* 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.)
(𝜑𝐼 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝐼))    &   (𝜑𝑉𝐽)    &   (𝜑𝑉 ≠ ∅)       (𝜑 → ∃𝑦 ∈ (ℚ ↑m 𝐼)𝑦𝑉)
 
Theoremqndenserrn 43847 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.)
(𝜑𝐼 ∈ Fin)    &   𝐽 = (TopOpen‘(ℝ^‘𝐼))       (𝜑 → ((cls‘𝐽)‘(ℚ ↑m 𝐼)) = (ℝ ↑m 𝐼))
 
Theoremrrxsnicc 43848* A multidimensional singleton expressed as a multidimensional closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ (ℝ ↑m 𝑋))       (𝜑X𝑘𝑋 ((𝐴𝑘)[,](𝐴𝑘)) = {𝐴})
 
Theoremrrnprjdstle 43849 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝐺:𝑋⟶ℝ)    &   (𝜑𝐼𝑋)    &   𝐷 = (dist‘(ℝ^‘𝑋))       (𝜑 → (abs‘((𝐹𝐼) − (𝐺𝐼))) ≤ (𝐹𝐷𝐺))
 
Theoremrrndsmet 43850* 𝐷 is a metric for the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   𝐷 = (𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦ (√‘Σ𝑘𝑋 (((𝑓𝑘) − (𝑔𝑘))↑2)))       (𝜑𝐷 ∈ (Met‘(ℝ ↑m 𝑋)))
 
Theoremrrndsxmet 43851* 𝐷 is an extended metric for the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   𝐷 = (𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦ (√‘Σ𝑘𝑋 (((𝑓𝑘) − (𝑔𝑘))↑2)))       (𝜑𝐷 ∈ (∞Met‘(ℝ ↑m 𝑋)))
 
Theoremioorrnopnlem 43852* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)    &   (𝜑𝐹X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)))    &   𝐻 = ran (𝑖𝑋 ↦ if(((𝐵𝑖) − (𝐹𝑖)) ≤ ((𝐹𝑖) − (𝐴𝑖)), ((𝐵𝑖) − (𝐹𝑖)), ((𝐹𝑖) − (𝐴𝑖))))    &   𝐸 = inf(𝐻, ℝ, < )    &   𝑉 = (𝐹(ball‘𝐷)𝐸)    &   𝐷 = (𝑓 ∈ (ℝ ↑m 𝑋), 𝑔 ∈ (ℝ ↑m 𝑋) ↦ (√‘Σ𝑘𝑋 (((𝑓𝑘) − (𝑔𝑘))↑2)))       (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹𝑣𝑣X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖))))
 
Theoremioorrnopn 43853* The indexed product of open intervals is an open set in (ℝ^‘𝑋). (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ)    &   (𝜑𝐵:𝑋⟶ℝ)       (𝜑X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
 
Theoremioorrnopnxrlem 43854* 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.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ*)    &   (𝜑𝐵:𝑋⟶ℝ*)    &   (𝜑𝐹X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)))    &   𝐿 = (𝑖𝑋 ↦ if((𝐴𝑖) = -∞, ((𝐹𝑖) − 1), (𝐴𝑖)))    &   𝑅 = (𝑖𝑋 ↦ if((𝐵𝑖) = +∞, ((𝐹𝑖) + 1), (𝐵𝑖)))    &   𝑉 = X𝑖𝑋 ((𝐿𝑖)(,)(𝑅𝑖))       (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹𝑣𝑣X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖))))
 
Theoremioorrnopnxr 43855* The indexed product of open intervals is an open set in (ℝ^‘𝑋). Similar to ioorrnopn 43853 but here unbounded intervals are allowed. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑋 ∈ Fin)    &   (𝜑𝐴:𝑋⟶ℝ*)    &   (𝜑𝐵:𝑋⟶ℝ*)       (𝜑X𝑖𝑋 ((𝐴𝑖)(,)(𝐵𝑖)) ∈ (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]

 
Syntaxcsalg 43856 Extend class notation with the class of all sigma-algebras.
class SAlg
 
Definitiondf-salg 43857* Define the class of sigma-algebras. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥))}
 
Syntaxcsalon 43858 Extend class notation with the class of sigma-algebras on a set.
class SalOn
 
Definitiondf-salon 43859* Define the set of sigma-algebra on a given set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
SalOn = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ 𝑠 = 𝑥})
 
Syntaxcsalgen 43860 Extend class notation with the class of sigma-algebra generator.
class SalGen
 
Definitiondf-salgen 43861* 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 43887. 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 43889. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Revised by Glauco Siliprandi, 1-Jan-2021.)
SalGen = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑥𝑥𝑠)})
 
Theoremissal 43862* Express the predicate "𝑆 is a sigma-algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
 
Theorempwsal 43863 The power set of a given set is a sigma-algebra (the so called discrete sigma-algebra). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑋𝑉 → 𝒫 𝑋 ∈ SAlg)
 
Theoremsalunicl 43864 SAlg sigma-algebra is closed under countable union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝑇 ∈ 𝒫 𝑆)    &   (𝜑𝑇 ≼ ω)       (𝜑 𝑇𝑆)
 
Theoremsaluncl 43865 The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆𝐹𝑆) → (𝐸𝐹) ∈ 𝑆)
 
Theoremprsal 43866 The pair of the empty set and the whole base is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑋𝑉 → {∅, 𝑋} ∈ SAlg)
 
Theoremsaldifcl 43867 The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)
 
Theorem0sal 43868 The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆 ∈ SAlg → ∅ ∈ 𝑆)
 
Theoremsalgenval 43869* The sigma-algebra generated by a set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
 
Theoremsaliuncl 43870* SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐾 ≼ ω)    &   ((𝜑𝑘𝐾) → 𝐸𝑆)       (𝜑 𝑘𝐾 𝐸𝑆)
 
Theoremsalincl 43871 The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆𝐹𝑆) → (𝐸𝐹) ∈ 𝑆)
 
Theoremsaluni 43872 A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑆 ∈ SAlg → 𝑆𝑆)
 
Theoremsaliincl 43873* SAlg sigma-algebra is closed under countable indexed intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐾 ≼ ω)    &   (𝜑𝐾 ≠ ∅)    &   ((𝜑𝑘𝐾) → 𝐸𝑆)       (𝜑 𝑘𝐾 𝐸𝑆)
 
Theoremsaldifcl2 43874 The difference of two elements of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝑆 ∈ SAlg ∧ 𝐸𝑆𝐹𝑆) → (𝐸𝐹) ∈ 𝑆)
 
Theoremintsaluni 43875* The union of an arbitrary intersection of sigma-algebras on the same set 𝑋, is 𝑋. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐺 ⊆ SAlg)    &   (𝜑𝐺 ≠ ∅)    &   ((𝜑𝑠𝐺) → 𝑠 = 𝑋)       (𝜑 𝐺 = 𝑋)
 
Theoremintsal 43876* The arbitrary intersection of sigma-algebra (on the same set 𝑋) is a sigma-algebra ( on the same set 𝑋, see intsaluni 43875). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐺 ⊆ SAlg)    &   (𝜑𝐺 ≠ ∅)    &   ((𝜑𝑠𝐺) → 𝑠 = 𝑋)       (𝜑 𝐺 ∈ SAlg)
 
Theoremsalgenn0 43877* The set used in the definition of the generated sigma-algebra, is not empty. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
 
Theoremsalgencl 43878 SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
 
Theoremissald 43879* Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑆𝑉)    &   (𝜑 → ∅ ∈ 𝑆)    &   𝑋 = 𝑆    &   ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)    &   ((𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω) → 𝑦𝑆)       (𝜑𝑆 ∈ SAlg)
 
Theoremsalexct 43880* 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.)
(𝜑𝐴𝑉)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}       (𝜑𝑆 ∈ SAlg)
 
Theoremsssalgen 43881 A set is a subset of the sigma-algebra it generates. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑆 = (SalGen‘𝑋)       (𝑋𝑉𝑋𝑆)
 
Theoremsalgenss 43882 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 43890, 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.)
(𝜑𝑋𝑉)    &   𝐺 = (SalGen‘𝑋)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑𝑋𝑆)    &   (𝜑 𝑆 = 𝑋)       (𝜑𝐺𝑆)
 
Theoremsalgenuni 43883 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.)
(𝜑𝑋𝑉)    &   𝑆 = (SalGen‘𝑋)    &   𝑈 = 𝑋       (𝜑 𝑆 = 𝑈)
 
Theoremissalgend 43884* One side of dfsalgen2 43887. 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.)
(𝜑𝑋𝑉)    &   (𝜑𝑆 ∈ SAlg)    &   (𝜑 𝑆 = 𝑋)    &   (𝜑𝑋𝑆)    &   ((𝜑 ∧ (𝑦 ∈ SAlg ∧ 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)       (𝜑 → (SalGen‘𝑋) = 𝑆)
 
Theoremsalexct2 43885* An example of a subset that does not belong to a nontrivial sigma-algebra, see salexct 43880. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = (0[,]2)    &   𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴𝑥) ≼ ω)}    &   𝐵 = (0[,]1)        ¬ 𝐵𝑆
 
Theoremunisalgen 43886 The union of a set belongs to the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝑋𝑉)    &   𝑆 = (SalGen‘𝑋)    &   𝑈 = 𝑋       (𝜑𝑈𝑆)
 
Theoremdfsalgen2 43887* 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.)
(𝜑𝑋𝑉)       (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))))
 
Theoremsalexct3 43888* 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 ∨ (𝐴𝑥) ≼ ω)}    &   𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦})       (𝑆 ∈ SAlg ∧ 𝑋𝑆 ∧ ¬ 𝑋𝑆)
 
Theoremsalgencntex 43889* This counterexample shows that df-salgen 43861 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 ∨ (𝐴𝑥) ≼ ω)}    &   𝐵 = (0[,]1)    &   𝑇 = 𝒫 𝐵    &   𝐶 = (𝑆𝑇)    &   𝑍 = {𝑠 ∈ SAlg ∣ 𝐶𝑠}        ¬ 𝑍 ∈ SAlg
 
Theoremsalgensscntex 43890* 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 ∨ (𝐴𝑥) ≼ ω)}    &   𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦})    &   𝐺 = (SalGen‘𝑋)       (𝑋𝑆𝑆 ∈ SAlg ∧ ¬ 𝐺𝑆)
 
Theoremissalnnd 43891* Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝑆𝑉)    &   (𝜑 → ∅ ∈ 𝑆)    &   𝑋 = 𝑆    &   ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)    &   ((𝜑𝑒:ℕ⟶𝑆) → 𝑛 ∈ ℕ (𝑒𝑛) ∈ 𝑆)       (𝜑𝑆 ∈ SAlg)
 
Theoremdmvolsal 43892 Lebesgue measurable sets form a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
dom vol ∈ SAlg
 
Theoremsaldifcld 43893 The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐸𝑆)       (𝜑 → ( 𝑆𝐸) ∈ 𝑆)
 
Theoremsaluncld 43894 The union of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐸𝑆)    &   (𝜑𝐹𝑆)       (𝜑 → (𝐸𝐹) ∈ 𝑆)
 
Theoremsalgencld 43895 SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑋𝑉)    &   𝑆 = (SalGen‘𝑋)       (𝜑𝑆 ∈ SAlg)
 
Theorem0sald 43896 The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)       (𝜑 → ∅ ∈ 𝑆)
 
Theoremiooborel 43897 An open interval is a Borel set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐽 = (topGen‘ran (,))    &   𝐵 = (SalGen‘𝐽)       (𝐴(,)𝐶) ∈ 𝐵
 
Theoremsalincld 43898 The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)    &   (𝜑𝐸𝑆)    &   (𝜑𝐹𝑆)       (𝜑 → (𝐸𝐹) ∈ 𝑆)
 
Theoremsalunid 43899 A set is an element of any sigma-algebra on it . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝑆 ∈ SAlg)       (𝜑 𝑆𝑆)
 
Theoremunisalgen2 43900 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.)
(𝜑𝐴𝑉)    &   𝑆 = (SalGen‘𝐴)       (𝜑 𝑆 = 𝐴)
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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 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