HomeHome Metamath Proof Explorer
Theorem List (p. 431 of 460)
< Previous  Next >
Bad symbols? Try the
GIF version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-28845)
  Hilbert Space Explorer  Hilbert Space Explorer
(28846-30368)
  Users' Mathboxes  Users' Mathboxes
(30369-45946)
 

Theorem List for Metamath Proof Explorer - 43001-43100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremioodvbdlimc2lem 43001* Limit at the upper bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)    &   𝑌 = sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )    &   𝑀 = ((⌊‘(1 / (𝐵𝐴))) + 1)    &   𝑆 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐹‘(𝐵 − (1 / 𝑗))))    &   𝑅 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐵 − (1 / 𝑗)))    &   𝑁 = if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀)    &   (𝜒 ↔ (((((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ (ℤ𝑁)) ∧ (abs‘((𝑆𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧𝐵)) < (1 / 𝑗)))       (𝜑 → (lim sup‘𝑆) ∈ (𝐹 lim 𝐵))
 
Theoremioodvbdlimc2 43002* A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by AV, 3-Oct-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)       (𝜑 → (𝐹 lim 𝐵) ≠ ∅)
 
Theoremdvdmsscn 43003 𝑋 is a subset of . This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))       (𝜑𝑋 ⊆ ℂ)
 
Theoremdvmptmulf 43004* Function-builder for derivative, product rule. A version of dvmptmul 24705 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑥𝜑    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   ((𝜑𝑥𝑋) → 𝐶 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐷𝑊)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐶)) = (𝑥𝑋𝐷))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 · 𝐶))) = (𝑥𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴))))
 
Theoremdvnmptdivc 43005* Function-builder for iterated derivative, division rule for constant divisor. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋𝑆)    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋𝑛 ∈ (0...𝑀)) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑛) = (𝑥𝑋𝐵))    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)    &   (𝜑𝑀 ∈ ℕ0)       ((𝜑𝑛 ∈ (0...𝑀)) → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 / 𝐶)))‘𝑛) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))
 
Theoremdvdsn1add 43006 If 𝐾 divides 𝑁 but 𝐾 does not divide 𝑀, then 𝐾 does not divide (𝑀 + 𝑁). (Contributed by Glauco Siliprandi, 5-Apr-2020.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((¬ 𝐾𝑀𝐾𝑁) → ¬ 𝐾 ∥ (𝑀 + 𝑁)))
 
Theoremdvxpaek 43007* Derivative of the polynomial (𝑥 + 𝐴)↑𝐾. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ)       (𝜑 → (𝑆 D (𝑥𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))) = (𝑥𝑋 ↦ (𝐾 · ((𝑥 + 𝐴)↑(𝐾 − 1)))))
 
Theoremdvnmptconst 43008* The 𝑁-th derivative of a constant function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝑆 D𝑛 (𝑥𝑋𝐴))‘𝑁) = (𝑥𝑋 ↦ 0))
 
Theoremdvnxpaek 43009* The 𝑛-th derivative of the polynomial (𝑥 + 𝐴)↑𝐾. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ0)    &   𝐹 = (𝑥𝑋 ↦ ((𝑥 + 𝐴)↑𝐾))       ((𝜑𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ if(𝐾 < 𝑁, 0, (((!‘𝐾) / (!‘(𝐾𝑁))) · ((𝑥 + 𝐴)↑(𝐾𝑁))))))
 
Theoremdvnmul 43010* Function-builder for the 𝑁-th derivative, product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   𝐹 = (𝑥𝑋𝐴)    &   𝐺 = (𝑥𝑋𝐵)    &   ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):𝑋⟶ℂ)    &   ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐺)‘𝑘):𝑋⟶ℂ)    &   𝐶 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐹)‘𝑘))    &   𝐷 = (𝑘 ∈ (0...𝑁) ↦ ((𝑆 D𝑛 𝐺)‘𝑘))       (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ (𝐴 · 𝐵)))‘𝑁) = (𝑥𝑋 ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · (((𝐶𝑘)‘𝑥) · ((𝐷‘(𝑁𝑘))‘𝑥)))))
 
Theoremdvmptfprodlem 43011* Induction step for dvmptfprod 43012. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑥𝜑    &   𝑖𝜑    &   𝑗𝜑    &   𝑖𝐹    &   𝑗𝐺    &   ((𝜑𝑖𝐼𝑥𝑋) → 𝐴 ∈ ℂ)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝐸 ∈ V)    &   (𝜑 → ¬ 𝐸𝐷)    &   (𝜑 → (𝐷 ∪ {𝐸}) ⊆ 𝐼)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (((𝜑𝑥𝑋) ∧ 𝑗𝐷) → 𝐶 ∈ ℂ)    &   (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐷 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐷 (𝐶 · ∏𝑖 ∈ (𝐷 ∖ {𝑗})𝐴)))    &   ((𝜑𝑥𝑋) → 𝐺 ∈ ℂ)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐹)) = (𝑥𝑋𝐺))    &   (𝑖 = 𝐸𝐴 = 𝐹)    &   (𝑗 = 𝐸𝐶 = 𝐺)       (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝐷 ∪ {𝐸})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝐷 ∪ {𝐸})(𝐶 · ∏𝑖 ∈ ((𝐷 ∪ {𝐸}) ∖ {𝑗})𝐴)))
 
Theoremdvmptfprod 43012* Function-builder for derivative, finite product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑖𝜑    &   𝑗𝜑    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋𝐽)    &   (𝜑𝐼 ∈ Fin)    &   ((𝜑𝑖𝐼𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ)    &   ((𝜑𝑖𝐼) → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝑖 = 𝑗𝐵 = 𝐶)       (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
 
Theoremdvnprodlem1 43013* 𝐷 is bijective. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}))    &   (𝜑𝐽 ∈ ℕ0)    &   𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)    &   (𝜑𝑇 ∈ Fin)    &   (𝜑𝑍𝑇)    &   (𝜑 → ¬ 𝑍𝑅)    &   (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)       (𝜑𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶𝑅)‘𝑘)))
 
Theoremdvnprodlem2 43014* Induction step for dvnprodlem2 43014. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑇 ∈ Fin)    &   ((𝜑𝑡𝑇) → (𝐻𝑡):𝑋⟶ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑡𝑇𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻𝑡))‘𝑗):𝑋⟶ℂ)    &   𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}))    &   (𝜑𝑅𝑇)    &   (𝜑𝑍 ∈ (𝑇𝑅))    &   (𝜑 → ∀𝑘 ∈ (0...𝑁)((𝑆 D𝑛 (𝑥𝑋 ↦ ∏𝑡𝑅 ((𝐻𝑡)‘𝑥)))‘𝑘) = (𝑥𝑋 ↦ Σ𝑐 ∈ ((𝐶𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡𝑅 (!‘(𝑐𝑡))) · ∏𝑡𝑅 (((𝑆 D𝑛 (𝐻𝑡))‘(𝑐𝑡))‘𝑥))))    &   (𝜑𝐽 ∈ (0...𝑁))    &   𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐𝑍)), (𝑐𝑅)⟩)       (𝜑 → ((𝑆 D𝑛 (𝑥𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻𝑡)‘𝑥)))‘𝐽) = (𝑥𝑋 ↦ Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D𝑛 (𝐻𝑡))‘(𝑐𝑡))‘𝑥))))
 
Theoremdvnprodlem3 43015* The multinomial formula for the 𝑘-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑇 ∈ Fin)    &   ((𝜑𝑡𝑇) → (𝐻𝑡):𝑋⟶ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑡𝑇𝑗 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻𝑡))‘𝑗):𝑋⟶ℂ)    &   𝐹 = (𝑥𝑋 ↦ ∏𝑡𝑇 ((𝐻𝑡)‘𝑥))    &   𝐷 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡𝑠 (𝑐𝑡) = 𝑛}))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑇) ∣ Σ𝑡𝑇 (𝑐𝑡) = 𝑛})       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ Σ𝑐 ∈ (𝐶𝑁)(((!‘𝑁) / ∏𝑡𝑇 (!‘(𝑐𝑡))) · ∏𝑡𝑇 (((𝑆 D𝑛 (𝐻𝑡))‘(𝑐𝑡))‘𝑥))))
 
Theoremdvnprod 43016* The multinomial formula for the 𝑁-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝑇 ∈ Fin)    &   ((𝜑𝑡𝑇) → (𝐻𝑡):𝑋⟶ℂ)    &   (𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑡𝑇𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 (𝐻𝑡))‘𝑘):𝑋⟶ℂ)    &   𝐹 = (𝑥𝑋 ↦ ∏𝑡𝑇 ((𝐻𝑡)‘𝑥))    &   𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑇) ∣ Σ𝑡𝑇 (𝑐𝑡) = 𝑛})       (𝜑 → ((𝑆 D𝑛 𝐹)‘𝑁) = (𝑥𝑋 ↦ Σ𝑐 ∈ (𝐶𝑁)(((!‘𝑁) / ∏𝑡𝑇 (!‘(𝑐𝑡))) · ∏𝑡𝑇 (((𝑆 D𝑛 (𝐻𝑡))‘(𝑐𝑡))‘𝑥))))
 
20.37.11  Integrals
 
Theoremitgsin0pilem1 43017* Calculation of the integral for sine on the (0,π) interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐶 = (𝑡 ∈ (0[,]π) ↦ -(cos‘𝑡))       ∫(0(,)π)(sin‘𝑥) d𝑥 = 2
 
Theoremibliccsinexp 43018* sin^n on a closed interval is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿1)
 
Theoremitgsin0pi 43019 Calculation of the integral for sine on the (0,π) interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
∫(0(,)π)(sin‘𝑥) d𝑥 = 2
 
Theoremiblioosinexp 43020* sin^n on an open integral is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿1)
 
Theoremitgsinexplem1 43021* Integration by parts is applied to integrate sin^(N+1). (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐹 = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))    &   𝐺 = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))    &   𝐻 = (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))    &   𝐼 = (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))    &   𝐿 = (𝑥 ∈ ℂ ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)))    &   𝑀 = (𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑𝑁) · (sin‘𝑥)) d𝑥 = (𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))
 
Theoremitgsinexp 43022* A recursive formula for the integral of sin^N on the interval (0,π) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)    &   (𝜑𝑁 ∈ (ℤ‘2))       (𝜑 → (𝐼𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))
 
Theoremiblconstmpt 43023* A constant function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥𝐴𝐵) ∈ 𝐿1)
 
Theoremitgeq1d 43024* Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
 
Theoremmbfres2cn 43025 Measurability of a piecewise function: if 𝐹 is measurable on subsets 𝐵 and 𝐶 of its domain, and these pieces make up all of 𝐴, then 𝐹 is measurable on the whole domain. Similar to mbfres2 24390 but here the theorem is extended to complex-valued functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑 → (𝐹𝐵) ∈ MblFn)    &   (𝜑 → (𝐹𝐶) ∈ MblFn)    &   (𝜑 → (𝐵𝐶) = 𝐴)       (𝜑𝐹 ∈ MblFn)
 
Theoremvol0 43026 The measure of the empty set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(vol‘∅) = 0
 
Theoremditgeqiooicc 43027* A function 𝐹 on an open interval, has the same directed integral as its extension 𝐺 on the closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)       (𝜑 → ⨜[𝐴𝐵](𝐹𝑥) d𝑥 = ⨜[𝐴𝐵](𝐺𝑥) d𝑥)
 
Theoremvolge0 43028 The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ dom vol → 0 ≤ (vol‘𝐴))
 
Theoremcnbdibl 43029* A continuous bounded function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ dom vol)    &   (𝜑 → (vol‘𝐴) ∈ ℝ)    &   (𝜑𝐹 ∈ (𝐴cn→ℂ))    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹𝑦)) ≤ 𝑥)       (𝜑𝐹 ∈ 𝐿1)
 
Theoremsnmbl 43030 A singleton is measurable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → {𝐴} ∈ dom vol)
 
Theoremditgeq3d 43031* Equality theorem for the directed integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 𝐷 = 𝐸)       (𝜑 → ⨜[𝐴𝐵]𝐷 d𝑥 = ⨜[𝐴𝐵]𝐸 d𝑥)
 
Theoremiblempty 43032 The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
∅ ∈ 𝐿1
 
Theoremiblsplit 43033* The union of two integrable functions is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (vol*‘(𝐴𝐵)) = 0)    &   (𝜑𝑈 = (𝐴𝐵))    &   ((𝜑𝑥𝑈) → 𝐶 ∈ ℂ)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)    &   (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)       (𝜑 → (𝑥𝑈𝐶) ∈ 𝐿1)
 
Theoremvolsn 43034 A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → (vol‘{𝐴}) = 0)
 
Theoremitgvol0 43035* If the domani is negligible, the function is integrable and the integral is 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ∧ ∫𝐴𝐵 d𝑥 = 0))
 
Theoremitgcoscmulx 43036* Exercise: the integral of 𝑥 ↦ cos𝑎𝑥 on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)    &   (𝜑𝐴 ≠ 0)       (𝜑 → ∫(𝐵(,)𝐶)(cos‘(𝐴 · 𝑥)) d𝑥 = (((sin‘(𝐴 · 𝐶)) − (sin‘(𝐴 · 𝐵))) / 𝐴))
 
Theoremiblsplitf 43037* A version of iblsplit 43033 using bound-variable hypotheses instead of distinct variable conditions". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑥𝜑    &   (𝜑 → (vol*‘(𝐴𝐵)) = 0)    &   (𝜑𝑈 = (𝐴𝐵))    &   ((𝜑𝑥𝑈) → 𝐶 ∈ ℂ)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)    &   (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)       (𝜑 → (𝑥𝑈𝐶) ∈ 𝐿1)
 
Theoremibliooicc 43038* If a function is integrable on an open interval, it is integrable on the corresponding closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1)    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿1)
 
Theoremvolioc 43039 The measure of a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol‘(𝐴(,]𝐵)) = (𝐵𝐴))
 
Theoremiblspltprt 43040* If a function is integrable on any interval of a partition, then it is integrable on the whole interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑡𝜑    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   ((𝜑𝑖 ∈ (𝑀...𝑁)) → (𝑃𝑖) ∈ ℝ)    &   ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑃𝑖) < (𝑃‘(𝑖 + 1)))    &   ((𝜑𝑡 ∈ ((𝑃𝑀)[,](𝑃𝑁))) → 𝐴 ∈ ℂ)    &   ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑡 ∈ ((𝑃𝑖)[,](𝑃‘(𝑖 + 1))) ↦ 𝐴) ∈ 𝐿1)       (𝜑 → (𝑡 ∈ ((𝑃𝑀)[,](𝑃𝑁)) ↦ 𝐴) ∈ 𝐿1)
 
Theoremitgsincmulx 43041* Exercise: the integral of 𝑥 ↦ sin𝑎𝑥 on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵𝐶)       (𝜑 → ∫(𝐵(,)𝐶)(sin‘(𝐴 · 𝑥)) d𝑥 = (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴))
 
Theoremitgsubsticclem 43042* lemma for itgsubsticc 43043. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)    &   𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹𝑢), if(𝑢 < 𝐾, (𝐹𝐾), (𝐹𝐿))))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑌)    &   (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)))    &   (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))    &   (𝜑𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ))    &   (𝜑𝐾 ∈ ℝ)    &   (𝜑𝐿 ∈ ℝ)    &   (𝜑𝐾𝐿)    &   (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))    &   (𝑢 = 𝐴𝐶 = 𝐸)    &   (𝑥 = 𝑋𝐴 = 𝐾)    &   (𝑥 = 𝑌𝐴 = 𝐿)       (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
 
Theoremitgsubsticc 43043* Integration by u-substitution. The main difference with respect to itgsubst 24793 is that here we consider the range of 𝐴(𝑥) to be in the closed interval (𝐾[,]𝐿). If 𝐴(𝑥) is a continuous, differentiable function from [𝑋, 𝑌] to (𝑍, 𝑊), whose derivative is continuous and integrable, and 𝐶(𝑢) is a continuous function on (𝑍, 𝑊), then the integral of 𝐶(𝑢) from 𝐾 = 𝐴(𝑋) to 𝐿 = 𝐴(𝑌) is equal to the integral of 𝐶(𝐴(𝑥)) D 𝐴(𝑥) from 𝑋 to 𝑌. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑌)    &   (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)))    &   (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶) ∈ ((𝐾[,]𝐿)–cn→ℂ))    &   (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))    &   (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))    &   (𝑢 = 𝐴𝐶 = 𝐸)    &   (𝑥 = 𝑋𝐴 = 𝐾)    &   (𝑥 = 𝑌𝐴 = 𝐿)    &   (𝜑𝐾 ∈ ℝ)    &   (𝜑𝐿 ∈ ℝ)       (𝜑 → ⨜[𝐾𝐿]𝐶 d𝑢 = ⨜[𝑋𝑌](𝐸 · 𝐵) d𝑥)
 
Theoremitgioocnicc 43044* The integral of a piecewise continuous function 𝐹 on an open interval is equal to the integral of the continuous function 𝐺, in the corresponding closed interval. 𝐺 is equal to 𝐹 on the open interval, but it is continuous at the two boundaries, also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:dom 𝐹⟶ℂ)    &   (𝜑 → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹)    &   (𝜑𝑅 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim 𝐴))    &   (𝜑𝐿 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) lim 𝐵))    &   𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹𝑥))))       (𝜑 → (𝐺 ∈ 𝐿1 ∧ ∫(𝐴[,]𝐵)(𝐺𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹𝑥) d𝑥))
 
Theoremiblcncfioo 43045 A continuous function 𝐹 on an open interval (𝐴(,)𝐵) with a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵 is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑𝐿 ∈ (𝐹 lim 𝐵))    &   (𝜑𝑅 ∈ (𝐹 lim 𝐴))       (𝜑𝐹 ∈ 𝐿1)
 
Theoremitgspltprt 43046* The integral splits on a given partition 𝑃. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   (𝜑𝑃:(𝑀...𝑁)⟶ℝ)    &   ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑃𝑖) < (𝑃‘(𝑖 + 1)))    &   ((𝜑𝑡 ∈ ((𝑃𝑀)[,](𝑃𝑁))) → 𝐴 ∈ ℂ)    &   ((𝜑𝑖 ∈ (𝑀..^𝑁)) → (𝑡 ∈ ((𝑃𝑖)[,](𝑃‘(𝑖 + 1))) ↦ 𝐴) ∈ 𝐿1)       (𝜑 → ∫((𝑃𝑀)[,](𝑃𝑁))𝐴 d𝑡 = Σ𝑖 ∈ (𝑀..^𝑁)∫((𝑃𝑖)[,](𝑃‘(𝑖 + 1)))𝐴 d𝑡)
 
Theoremitgiccshift 43047* The integral of a function, 𝐹 stays the same if its closed interval domain is shifted by 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    &   (𝜑𝑇 ∈ ℝ+)    &   𝐺 = (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑥𝑇)))       (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐺𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹𝑥) d𝑥)
 
Theoremitgperiod 43048* The integral of a periodic function, with period 𝑇 stays the same if the domain of integration is shifted. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑇 ∈ ℝ+)    &   (𝜑𝐹:ℝ⟶ℂ)    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))       (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹𝑥) d𝑥)
 
Theoremitgsbtaddcnst 43049* Integral substitution, adding a constant to the function's argument. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))       (𝜑 → ⨜[(𝐴𝑋) → (𝐵𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴𝐵](𝐹𝑡) d𝑡)
 
Theoremvolico 43050 The measure of left-closed, right-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵𝐴), 0))
 
Theoremsublevolico 43051 The Lebesgue measure of a left-closed, right-open interval is greater than or equal to the difference of the two bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐵𝐴) ≤ (vol‘(𝐴[,)𝐵)))
 
Theoremdmvolss 43052 Lebesgue measurable sets are subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
dom vol ⊆ 𝒫 ℝ
 
Theoremismbl3 43053* The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl2 24272, but here +𝑒 is used, and the precondition (vol*‘𝑥) ∈ ℝ can be dropped. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴))) ≤ (vol*‘𝑥)))
 
Theoremvolioof 43054 The function that assigns the Lebesgue measure to open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)
 
Theoremovolsplit 43055 The Lebesgue outer measure function is finitely sub-additive: application to a set split in two parts, using addition for extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ⊆ ℝ)       (𝜑 → (vol*‘𝐴) ≤ ((vol*‘(𝐴𝐵)) +𝑒 (vol*‘(𝐴𝐵))))
 
Theoremfvvolioof 43056 The function value of the Lebesgue measure of an open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴⟶(ℝ* × ℝ*))    &   (𝜑𝑋𝐴)       (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
 
Theoremvolioore 43057 The measure of an open interval. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) = if(𝐴𝐵, (𝐵𝐴), 0))
 
Theoremfvvolicof 43058 The function value of the Lebesgue measure of a left-closed right-open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴⟶(ℝ* × ℝ*))    &   (𝜑𝑋𝐴)       (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))
 
Theoremvoliooico 43059 An open interval and a left-closed, right-open interval with the same real bounds, have the same Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))
 
Theoremismbl4 43060* The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 24271, but here +𝑒 is used, and the precondition (vol*‘𝑥) ∈ ℝ can be dropped. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))))
 
Theoremvolioofmpt 43061* ((vol ∘ (,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))       (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥𝐴 ↦ (vol‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))))
 
Theoremvolicoff 43062 ((vol ∘ [,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴⟶(ℝ × ℝ*))       (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞))
 
Theoremvoliooicof 43063 The Lebesgue measure of open intervals is the same as the Lebesgue measure of left-closed right-open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴⟶(ℝ × ℝ))       (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = ((vol ∘ [,)) ∘ 𝐹))
 
Theoremvolicofmpt 43064* ((vol ∘ [,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶(ℝ × ℝ*))       (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑥𝐴 ↦ (vol‘((1st ‘(𝐹𝑥))[,)(2nd ‘(𝐹𝑥))))))
 
Theoremvolicc 43065 The Lebesgue measure of a closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol‘(𝐴[,]𝐵)) = (𝐵𝐴))
 
Theoremvoliccico 43066 A closed interval and a left-closed, right-open interval with the same real bounds, have the same Lebesgue measure. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (vol‘(𝐴[,]𝐵)) = (vol‘(𝐴[,)𝐵)))
 
Theoremmbfdmssre 43067 The domain of a measurable function is a subset of the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ)
 
20.37.12  Stone Weierstrass theorem - real version
 
Theoremstoweidlem1 43068 Lemma for stoweid 43130. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 13675. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 ≤ 1)    &   (𝜑𝐷𝐴)       (𝜑 → ((1 − (𝐴𝑁))↑(𝐾𝑁)) ≤ (1 / ((𝐾 · 𝐷)↑𝑁)))
 
Theoremstoweidlem2 43069* lemma for stoweid 43130: here we prove that the subalgebra of continuous functions, which contains constant functions, is closed under scaling. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹𝐴)       (𝜑 → (𝑡𝑇 ↦ (𝐸 · (𝐹𝑡))) ∈ 𝐴)
 
Theoremstoweidlem3 43070* Lemma for stoweid 43130: if 𝐴 is positive and all 𝑀 terms of a finite product are larger than 𝐴, then the finite product is larger than 𝐴𝑀. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝐹    &   𝑖𝜑    &   𝑋 = seq1( · , 𝐹)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)⟶ℝ)    &   ((𝜑𝑖 ∈ (1...𝑀)) → 𝐴 < (𝐹𝑖))    &   (𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴𝑀) < (𝑋𝑀))
 
Theoremstoweidlem4 43071* Lemma for stoweid 43130: a class variable replaces a setvar variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)       ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)
 
Theoremstoweidlem5 43072* There exists a δ as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 0 < δ < 1 , p >= δ on 𝑇𝑈. Here 𝐷 is used to represent δ in the paper and 𝑄 to represent 𝑇𝑈 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝐷 = if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2))    &   (𝜑𝑃:𝑇⟶ℝ)    &   (𝜑𝑄𝑇)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑡𝑄 𝐶 ≤ (𝑃𝑡))       (𝜑 → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡𝑄 𝑑 ≤ (𝑃𝑡)))
 
Theoremstoweidlem6 43073* Lemma for stoweid 43130: two class variables replace two setvar variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡 𝑓 = 𝐹    &   𝑡 𝑔 = 𝐺    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)       ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴)
 
Theoremstoweidlem7 43074* This lemma is used to prove that qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91, (at the top of page 91), is such that qn < ε on 𝑇𝑈, and qn > 1 - ε on 𝑉. Here it is proven that, for 𝑛 large enough, 1-(k*δ/2)^n > 1 - ε , and 1/(k*δ)^n < ε. The variable 𝐴 is used to represent (k*δ) in the paper, and 𝐵 is used to represent (k*δ/2). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐹 = (𝑖 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑖))    &   𝐺 = (𝑖 ∈ ℕ0 ↦ (𝐵𝑖))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐵 < 1)    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑛 ∈ ℕ ((1 − 𝐸) < (1 − (𝐵𝑛)) ∧ (1 / (𝐴𝑛)) < 𝐸))
 
Theoremstoweidlem8 43075* Lemma for stoweid 43130: two class variables replace two setvar variables, for the sum of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   𝑡𝐹    &   𝑡𝐺       ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝐺𝑡))) ∈ 𝐴)
 
Theoremstoweidlem9 43076* Lemma for stoweid 43130: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑇 = ∅)    &   (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)       (𝜑 → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
 
Theoremstoweidlem10 43077 Lemma for stoweid 43130. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90, this lemma is an application of Bernoulli's inequality. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0𝐴 ≤ 1) → (1 − (𝑁 · 𝐴)) ≤ ((1 − 𝐴)↑𝑁))
 
Theoremstoweidlem11 43078* This lemma is used to prove that there is a function 𝑔 as in the proof of [BrosowskiDeutsh] p. 92 (at the top of page 92): this lemma proves that g(t) < ( j + 1 / 3 ) * ε. Here 𝐸 is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝑡𝑇)    &   (𝜑𝑗 ∈ (1...𝑁))    &   ((𝜑𝑖 ∈ (0...𝑁)) → (𝑋𝑖):𝑇⟶ℝ)    &   ((𝜑𝑖 ∈ (0...𝑁)) → ((𝑋𝑖)‘𝑡) ≤ 1)    &   ((𝜑𝑖 ∈ (𝑗...𝑁)) → ((𝑋𝑖)‘𝑡) < (𝐸 / 𝑁))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))       (𝜑 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸))
 
Theoremstoweidlem12 43079* Lemma for stoweid 43130. This Lemma is used by other three Lemmas. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))    &   (𝜑𝑃:𝑇⟶ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)       ((𝜑𝑡𝑇) → (𝑄𝑡) = ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
 
Theoremstoweidlem13 43080 Lemma for stoweid 43130. This lemma is used to prove the statement abs( f(t) - g(t) ) < 2 epsilon, in the last step of the proof in [BrosowskiDeutsh] p. 92. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝐸 ∈ ℝ+)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑗 ∈ ℝ)    &   (𝜑 → ((𝑗 − (4 / 3)) · 𝐸) < 𝑋)    &   (𝜑𝑋 ≤ ((𝑗 − (1 / 3)) · 𝐸))    &   (𝜑 → ((𝑗 − (4 / 3)) · 𝐸) < 𝑌)    &   (𝜑𝑌 < ((𝑗 + (1 / 3)) · 𝐸))       (𝜑 → (abs‘(𝑌𝑋)) < (2 · 𝐸))
 
Theoremstoweidlem14 43081* There exists a 𝑘 as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 𝑘 is an integer and 1 < k * δ < 2. 𝐷 is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐴 = {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗}    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐷 < 1)       (𝜑 → ∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))
 
Theoremstoweidlem15 43082* This lemma is used to prove the existence of a function 𝑝 as in Lemma 1 from [BrosowskiDeutsh] p. 90: 𝑝 is in the subalgebra, such that 0 ≤ p ≤ 1, p_(t0) = 0, and p > 0 on T - U. Here (𝐺𝐼) is used to represent p_(ti) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   (𝜑𝐺:(1...𝑀)⟶𝑄)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → (((𝐺𝐼)‘𝑆) ∈ ℝ ∧ 0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
 
Theoremstoweidlem16 43083* Lemma for stoweid 43130. The subset 𝑌 of functions in the algebra 𝐴, with values in [ 0 , 1 ], is closed under multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}    &   𝐻 = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)       ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝑌)
 
Theoremstoweidlem17 43084* This lemma proves that the function 𝑔 (as defined in [BrosowskiDeutsh] p. 91, at the end of page 91) belongs to the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑋:(0...𝑁)⟶𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝐸 ∈ ℝ)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
 
Theoremstoweidlem18 43085* This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 92 when A is empty, the trivial case. Here D is used to denote the set A of Lemma 2, because the variable A is used for the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐷    &   𝑡𝜑    &   𝐹 = (𝑡𝑇 ↦ 1)    &   𝑇 = 𝐽    &   ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)    &   (𝜑𝐵 ∈ (Clsd‘𝐽))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐷 = ∅)       (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
 
Theoremstoweidlem19 43086* If a set of real functions is closed under multiplication and it contains constants, then it is closed under finite exponentiation. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝜑    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝐹𝐴)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴)
 
Theoremstoweidlem20 43087* If a set A of real functions from a common domain T is closed under the sum of two functions, then it is closed under the sum of a finite number of functions, indexed by G. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝐹 = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑀)⟶𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       (𝜑𝐹𝐴)
 
Theoremstoweidlem21 43088* Once the Stone Weierstrass theorem has been proven for approximating nonnegative functions, then this lemma is used to extend the result to functions with (possibly) negative values. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐺    &   𝑡𝐻    &   𝑡𝑆    &   𝑡𝜑    &   𝐺 = (𝑡𝑇 ↦ ((𝐻𝑡) + 𝑆))    &   (𝜑𝐹:𝑇⟶ℝ)    &   (𝜑𝑆 ∈ ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑 → ∀𝑓𝐴 𝑓:𝑇⟶ℝ)    &   (𝜑𝐻𝐴)    &   (𝜑 → ∀𝑡𝑇 (abs‘((𝐻𝑡) − ((𝐹𝑡) − 𝑆))) < 𝐸)       (𝜑 → ∃𝑓𝐴𝑡𝑇 (abs‘((𝑓𝑡) − (𝐹𝑡))) < 𝐸)
 
Theoremstoweidlem22 43089* If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝑡𝐹    &   𝑡𝐺    &   𝐻 = (𝑡𝑇 ↦ ((𝐹𝑡) − (𝐺𝑡)))    &   𝐼 = (𝑡𝑇 ↦ -1)    &   𝐿 = (𝑡𝑇 ↦ ((𝐼𝑡) · (𝐺𝑡)))    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)       ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝐺𝑡))) ∈ 𝐴)
 
Theoremstoweidlem23 43090* This lemma is used to prove the existence of a function pt as in the beginning of Lemma 1 [BrosowskiDeutsh] p. 90: for all t in T - U, there exists a function p in the subalgebra, such that pt ( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝑡𝐺    &   𝐻 = (𝑡𝑇 ↦ ((𝐺𝑡) − (𝐺𝑍)))    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝑆𝑇)    &   (𝜑𝑍𝑇)    &   (𝜑𝐺𝐴)    &   (𝜑 → (𝐺𝑆) ≠ (𝐺𝑍))       (𝜑 → (𝐻𝐴 ∧ (𝐻𝑆) ≠ (𝐻𝑍) ∧ (𝐻𝑍) = 0))
 
Theoremstoweidlem24 43091* This lemma proves that for 𝑛 sufficiently large, qn( t ) > ( 1 - epsilon ), for all 𝑡 in 𝑉: see Lemma 1 [BrosowskiDeutsh] p. 90, (at the bottom of page 90). 𝑄 is used to represent qn in the paper, 𝑁 to represent 𝑛 in the paper, 𝐾 to represent 𝑘, 𝐷 to represent δ, and 𝐸 to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑉 = {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}    &   𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))    &   (𝜑𝑃:𝑇⟶ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (1 − 𝐸) < (1 − (((𝐾 · 𝐷) / 2)↑𝑁)))    &   (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))       ((𝜑𝑡𝑉) → (1 − 𝐸) < (𝑄𝑡))
 
Theoremstoweidlem25 43092* This lemma proves that for n sufficiently large, qn( t ) < ε, for all 𝑡 in 𝑇𝑈: see Lemma 1 [BrosowskiDeutsh] p. 91 (at the top of page 91). 𝑄 is used to represent qn in the paper, 𝑁 to represent n in the paper, 𝐾 to represent k, 𝐷 to represent δ, 𝑃 to represent p, and 𝐸 to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝑃:𝑇⟶ℝ)    &   (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))    &   (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (1 / ((𝐾 · 𝐷)↑𝑁)) < 𝐸)       ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑄𝑡) < 𝐸)
 
Theoremstoweidlem26 43093* This lemma is used to prove that there is a function 𝑔 as in the proof of [BrosowskiDeutsh] p. 92: this lemma proves that g(t) > ( j - 4 / 3 ) * ε. Here 𝐿 is used to represnt j in the paper, 𝐷 is used to represent A in the paper, 𝑆 is used to represent t, and 𝐸 is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑗𝜑    &   𝑡𝜑    &   𝐷 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})    &   𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇 ∈ V)    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝑆 ∈ ((𝐷𝐿) ∖ (𝐷‘(𝐿 − 1))))    &   (𝜑𝐹:𝑇⟶ℝ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))    &   ((𝜑𝑖 ∈ (0...𝑁)) → (𝑋𝑖):𝑇⟶ℝ)    &   ((𝜑𝑖 ∈ (0...𝑁) ∧ 𝑡𝑇) → 0 ≤ ((𝑋𝑖)‘𝑡))    &   ((𝜑𝑖 ∈ (0...𝑁) ∧ 𝑡 ∈ (𝐵𝑖)) → (1 − (𝐸 / 𝑁)) < ((𝑋𝑖)‘𝑡))       (𝜑 → ((𝐿 − (4 / 3)) · 𝐸) < ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑆))
 
Theoremstoweidlem27 43094* This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Here (𝑞𝑖) is used to represent p_(ti) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐺 = (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})    &   (𝜑𝑄 ∈ V)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑌 Fn ran 𝐺)    &   (𝜑 → ran 𝐺 ∈ V)    &   ((𝜑𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑙)    &   (𝜑𝐹:(1...𝑀)–1-1-onto→ran 𝐺)    &   (𝜑 → (𝑇𝑈) ⊆ 𝑋)    &   𝑡𝜑    &   𝑤𝜑    &   𝑄       (𝜑 → ∃𝑞(𝑀 ∈ ℕ ∧ (𝑞:(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡))))
 
Theoremstoweidlem28 43095* There exists a δ as in Lemma 1 [BrosowskiDeutsh] p. 90: 0 < delta < 1 and p >= delta on 𝑇𝑈. Here 𝑑 is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝑈    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑃 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))    &   (𝜑𝑈𝐽)       (𝜑 → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
 
Theoremstoweidlem29 43096* When the hypothesis for the extreme value theorem hold, then the inf of the range of the function belongs to the range, it is real and it a lower bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.) (Revised by AV, 13-Sep-2020.)
𝑡𝐹    &   𝑡𝜑    &   𝑇 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑇 ≠ ∅)       (𝜑 → (inf(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ inf(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡𝑇 inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑡)))
 
Theoremstoweidlem30 43097* This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, (𝐺𝑖) is used for p_(ti). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑃 = (𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑀)⟶𝑄)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       ((𝜑𝑆𝑇) → (𝑃𝑆) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑆)))
 
Theoremstoweidlem31 43098* This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that 𝑅 is a finite subset of 𝑉, 𝑥 indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all 𝑖 ranging in the finite indexing set, 0 ≤ xi ≤ 1, xi < ε / m on V(ti), and xi > 1 - ε / m on 𝐵. Here M is used to represent m in the paper, 𝐸 is used to represent ε in the paper, vi is used to represent V(ti). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝜑    &   𝑡𝜑    &   𝑤𝜑    &   𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}    &   𝑉 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}    &   𝐺 = (𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑣:(1...𝑀)–1-1-onto𝑅)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐵 ⊆ (𝑇𝑈))    &   (𝜑𝑉 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑 → ran 𝐺 ∈ Fin)       (𝜑 → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡))))
 
Theoremstoweidlem32 43099* If a set A of real functions from a common domain T is a subalgebra and it contains constants, then it is closed under the sum of a finite number of functions, indexed by G and finally scaled by a real Y. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝑃 = (𝑡𝑇 ↦ (𝑌 · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))    &   𝐹 = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))    &   𝐻 = (𝑡𝑇𝑌)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐺:(1...𝑀)⟶𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       (𝜑𝑃𝐴)
 
Theoremstoweidlem33 43100* If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝐺    &   𝑡𝜑    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)       ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝐺𝑡))) ∈ 𝐴)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-45946
  Copyright terms: Public domain < Previous  Next >