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Theorem List for Metamath Proof Explorer - 42501-42600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremioodvbdlimc1 42501* 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 𝐴) ≠ ∅)

Theoremioodvbdlimc2lem 42502* 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 42503* 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 42504 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 42505* Function-builder for derivative, product rule. A version of dvmptmul 24567 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 42506* 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 42507 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 42508* 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 42509* 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 42510* 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 42511* 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 42512* Induction step for dvmptfprod 42513. (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 42513* 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 42514* 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 42515* Induction step for dvnprodlem2 42515. (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 42516* 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 42517* 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 42518* 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 42519* 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 42520 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 42521* 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 42522* 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 42523* 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 42524* A constant function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥𝐴𝐵) ∈ 𝐿1)

Theoremitgeq1d 42525* Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Theoremmbfres2cn 42526 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 24252 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 42527 The measure of the empty set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(vol‘∅) = 0

Theoremditgeqiooicc 42528* 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 42529 The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ dom vol → 0 ≤ (vol‘𝐴))

Theoremcnbdibl 42530* 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 42531 A singleton is measurable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → {𝐴} ∈ dom vol)

Theoremditgeq3d 42532* 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 42533 The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
∅ ∈ 𝐿1

Theoremiblsplit 42534* 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 42535 A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → (vol‘{𝐴}) = 0)

Theoremitgvol0 42536* 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 42537* 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 42538* A version of iblsplit 42534 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 42539* 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 42540 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 42541* 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 42542* 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 42543* lemma for itgsubsticc 42544. (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 42544* Integration by u-substitution. The main difference with respect to itgsubst 24655 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 42545* 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 42546 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 42547* 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 42548* 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 42549* 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 42550* 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 42551 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 42552 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 42553 Lebesgue measurable sets are subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
dom vol ⊆ 𝒫 ℝ

Theoremismbl3 42554* The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl2 24134, 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 42555 The function that assigns the Lebesgue measure to open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)

Theoremovolsplit 42556 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 42557 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 42558 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 42559 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 42560 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 42561* The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 24133, 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 42562* ((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 42563 ((vol ∘ [,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴⟶(ℝ × ℝ*))       (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞))

Theoremvoliooicof 42564 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 42565* ((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 42566 The Lebesgue measure of a closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol‘(𝐴[,]𝐵)) = (𝐵𝐴))

Theoremvoliccico 42567 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 42568 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 42569 Lemma for stoweid 42631. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 13595. (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 42570* lemma for stoweid 42631: 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 42571* Lemma for stoweid 42631: if 𝐴 is positive and all 𝑀 terms of a finite product are larger than 𝐴, then the finite product is larger than A^M. (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 42572* Lemma for stoweid 42631: 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 42573* 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 42574* Lemma for stoweid 42631: 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 42575* 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 42576* Lemma for stoweid 42631: 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 42577* Lemma for stoweid 42631: 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 42578 Lemma for stoweid 42631. 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 42579* 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 42580* Lemma for stoweid 42631. 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 42581 Lemma for stoweid 42631. 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 42582* 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 42583* 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 42584* Lemma for stoweid 42631. 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 42585* 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 42586* 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 42587* 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 42588* 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 42589* 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 42590* 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 42591* 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 42592* 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 42593* 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 42594* 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 42595* 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 42596* 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 42597* 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 42598* 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 42599* 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 42600* 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)

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