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Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvsinax 38701* Derivative exercise: the derivative with respect to y of sin(Ay), given a constant 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
 
Theoremdvsubf 38702 The subtraction rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → dom (𝑆 D 𝐺) = 𝑋)       (𝜑 → (𝑆 D (𝐹𝑓𝐺)) = ((𝑆 D 𝐹) ∘𝑓 − (𝑆 D 𝐺)))
 
Theoremdvmptconst 38703* Function-builder for derivative: derivative of a constant. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐴 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝑆 D (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ 0))
 
Theoremdvcnre 38704 From compex differentiation to real differentiation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐹:ℂ⟶ℂ ∧ ℝ ⊆ dom (ℂ D 𝐹)) → (ℝ D (𝐹 ↾ ℝ)) = ((ℂ D 𝐹) ↾ ℝ))
 
Theoremdvmptidg 38705* Function-builder for derivative: derivative of the identity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐴 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))       (𝜑 → (𝑆 D (𝑥𝐴𝑥)) = (𝑥𝐴 ↦ 1))
 
Theoremdvresntr 38706 Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐹:𝑋⟶ℂ)    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌)       (𝜑 → (𝑆 D 𝐹) = (𝑆 D (𝐹𝑌)))
 
Theoremdvmptdiv 38707* Function-builder for derivative, quotient rule. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   ((𝜑𝑥𝑋) → 𝐶 ∈ (ℂ ∖ {0}))    &   ((𝜑𝑥𝑋) → 𝐷 ∈ ℂ)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐶)) = (𝑥𝑋𝐷))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 / 𝐶))) = (𝑥𝑋 ↦ (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2))))
 
Theoremfperdvper 38708* The derivative of a periodic function is periodic. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℂ)    &   (𝜑𝑇 ∈ ℝ)    &   ((𝜑𝑥 ∈ ℝ) → (𝐹‘(𝑥 + 𝑇)) = (𝐹𝑥))    &   𝐺 = (ℝ D 𝐹)       ((𝜑𝑥 ∈ dom 𝐺) → ((𝑥 + 𝑇) ∈ dom 𝐺 ∧ (𝐺‘(𝑥 + 𝑇)) = (𝐺𝑥)))
 
Theoremdvmptresicc 38709* Derivative of a function restricted to a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐹 = (𝑥 ∈ ℂ ↦ 𝐴)    &   ((𝜑𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)    &   (𝜑 → (ℂ D 𝐹) = (𝑥 ∈ ℂ ↦ 𝐵))    &   ((𝜑𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)       (𝜑 → (ℝ D (𝑥 ∈ (𝐶[,]𝐷) ↦ 𝐴)) = (𝑥 ∈ (𝐶(,)𝐷) ↦ 𝐵))
 
Theoremdvasinbx 38710* Derivative exercise: the derivative with respect to y of A x sin(By), given two constants 𝐴 and 𝐵. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · (sin‘(𝐵 · 𝑦))))) = (𝑦 ∈ ℂ ↦ ((𝐴 · 𝐵) · (cos‘(𝐵 · 𝑦)))))
 
Theoremdvresioo 38711 Restriction of a derivative to an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ⊆ ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℝ D (𝐹 ↾ (𝐵(,)𝐶))) = ((ℝ D 𝐹) ↾ (𝐵(,)𝐶)))
 
Theoremdvdivf 38712 The quotient rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶(ℂ ∖ {0}))    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → dom (𝑆 D 𝐺) = 𝑋)       (𝜑 → (𝑆 D (𝐹𝑓 / 𝐺)) = ((((𝑆 D 𝐹) ∘𝑓 · 𝐺) ∘𝑓 − ((𝑆 D 𝐺) ∘𝑓 · 𝐹)) ∘𝑓 / (𝐺𝑓 · 𝐺)))
 
Theoremdvdivbd 38713* A sufficient condition for the derivative to be bounded, for the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐶))    &   ((𝜑𝑥𝑋) → 𝐶 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵 ∈ ℂ)    &   (𝜑𝑈 ∈ ℝ)    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑𝑇 ∈ ℝ)    &   (𝜑𝑄 ∈ ℝ)    &   ((𝜑𝑥𝑋) → (abs‘𝐶) ≤ 𝑈)    &   ((𝜑𝑥𝑋) → (abs‘𝐵) ≤ 𝑅)    &   ((𝜑𝑥𝑋) → (abs‘𝐷) ≤ 𝑇)    &   ((𝜑𝑥𝑋) → (abs‘𝐴) ≤ 𝑄)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐵)) = (𝑥𝑋𝐷))    &   ((𝜑𝑥𝑋) → 𝐷 ∈ ℂ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → ∀𝑥𝑋 𝐸 ≤ (abs‘𝐵))    &   𝐹 = (𝑆 D (𝑥𝑋 ↦ (𝐴 / 𝐵)))       (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥𝑋 (abs‘(𝐹𝑥)) ≤ 𝑏)
 
Theoremdvsubcncf 38714 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑 → (𝑆 D 𝐹) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑆 D 𝐺) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑆 D (𝐹𝑓𝐺)) ∈ (𝑋cn→ℂ))
 
Theoremdvmulcncf 38715 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑 → (𝑆 D 𝐹) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑆 D 𝐺) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑆 D (𝐹𝑓 · 𝐺)) ∈ (𝑋cn→ℂ))
 
Theoremdvcosax 38716* Derivative exercise: the derivative with respect to x of cos(Ax), given a constant 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘(𝐴 · 𝑥)))) = (𝑥 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑥)))))
 
Theoremdvdivcncf 38717 A sufficient condition for the derivative of a quotient to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶(ℂ ∖ {0}))    &   (𝜑 → (𝑆 D 𝐹) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑆 D 𝐺) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑆 D (𝐹𝑓 / 𝐺)) ∈ (𝑋cn→ℂ))
 
Theoremdvbdfbdioolem1 38718* Given a function with bounded derivative, on an open interval, here is an absolute bound to the difference of the image of two points in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑𝐾 ∈ ℝ)    &   (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)    &   (𝜑𝐶 ∈ (𝐴(,)𝐵))    &   (𝜑𝐷 ∈ (𝐶(,)𝐵))       (𝜑 → ((abs‘((𝐹𝐷) − (𝐹𝐶))) ≤ (𝐾 · (𝐷𝐶)) ∧ (abs‘((𝐹𝐷) − (𝐹𝐶))) ≤ (𝐾 · (𝐵𝐴))))
 
Theoremdvbdfbdioolem2 38719* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑𝐾 ∈ ℝ)    &   (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝐾)    &   𝑀 = ((abs‘(𝐹‘((𝐴 + 𝐵) / 2))) + (𝐾 · (𝐵𝐴)))       (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹𝑥)) ≤ 𝑀)
 
Theoremdvbdfbdioo 38720* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ∃𝑎 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑎)       (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹𝑥)) ≤ 𝑏)
 
Theoremioodvbdlimc1lem1 38721* If 𝐹 has bounded derivative on (𝐴(,)𝐵) then a sequence of points in its image converges to its lim sup. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑅:(ℤ𝑀)⟶(𝐴(,)𝐵))    &   𝑆 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐹‘(𝑅𝑗)))    &   (𝜑𝑅 ∈ dom ⇝ )    &   𝐾 = inf({𝑘 ∈ (ℤ𝑀) ∣ ∀𝑖 ∈ (ℤ𝑘)(abs‘((𝑅𝑖) − (𝑅𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}, ℝ, < )       (𝜑𝑆 ⇝ (lim sup‘𝑆))
 
Theoremioodvbdlimc1lem2 38722* Limit at the lower 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‘(𝐴 + (1 / 𝑗))))    &   𝑅 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐴 + (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‘(𝑧𝐴)) < (1 / 𝑗)))       (𝜑 → (lim sup‘𝑆) ∈ (𝐹 lim 𝐴))
 
Theoremioodvbdlimc1lem1OLD 38723* If 𝐹 has bounded derivative on (𝐴(,)𝐵) then a sequence of points in its image converges to its lim sup. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete version of ioodvbdlimc1lem1 38721 as of 3-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑅:(ℤ𝑀)⟶(𝐴(,)𝐵))    &   𝑆 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐹‘(𝑅𝑗)))    &   (𝜑𝑅 ∈ dom ⇝ )    &   𝐾 = sup({𝑘 ∈ (ℤ𝑀) ∣ ∀𝑖 ∈ (ℤ𝑘)(abs‘((𝑅𝑖) − (𝑅𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}, ℝ, < )       (𝜑𝑆 ⇝ (lim sup‘𝑆))
 
Theoremioodvbdlimc1lem2OLD 38724* Limit at the lower bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete version of ioodvbdlimc1lem2 38722 as of 3-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(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‘(𝐴 + (1 / 𝑗))))    &   𝑅 = (𝑗 ∈ (ℤ𝑀) ↦ (𝐴 + (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‘(𝑧𝐴)) < (1 / 𝑗)))       (𝜑 → (lim sup‘𝑆) ∈ (𝐹 lim 𝐴))
 
Theoremioodvbdlimc1 38725* 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 38726* 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))
 
Theoremioodvbdlimc2lemOLD 38727* Limit at the upper bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete version of ioodvbdlimc2lem 38726 as of 3-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(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 38728* 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 38729 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 38730* Function-builder for derivative, product rule. A version of dvmptmul 23405 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 38731* 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 38732 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 38733* 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 38734* 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 38735* 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 38736* 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 38737* Induction step for dvmptfprod 38738. (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 38738* 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 38739* 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 38740* Induction step for dvnprodlem2 38740. (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 38741* 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 38742* 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))))
 
20.31.11  Integrals
 
Theoremvolioo 38743 The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵𝐴))
 
Theoremitgsin0pilem1 38744* 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 38745* 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 38746 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 38747* 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 38748* 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 38749* 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 38750* A constant function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥𝐴𝐵) ∈ 𝐿1)
 
Theoremitgeq1d 38751* Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
 
Theoremmbf0 38752 The empty set is a measurable function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
∅ ∈ MblFn
 
Theoremmbfres2cn 38753 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 23093 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 38754 The measure of the empty set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(vol‘∅) = 0
 
Theoremditgeqiooicc 38755* 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 38756 The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ dom vol → 0 ≤ (vol‘𝐴))
 
Theoremcnbdibl 38757* 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 38758 A singleton is measurable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → {𝐴} ∈ dom vol)
 
Theoremditgeq3d 38759* 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 38760 The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
∅ ∈ 𝐿1
 
Theoremiblsplit 38761* 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 38762 A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → (vol‘{𝐴}) = 0)
 
Theoremitgvol0 38763* 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 38764* 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 38765* A version of iblsplit 38761 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 38766* 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 38767 The measure of left open, right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol‘(𝐴(,]𝐵)) = (𝐵𝐴))
 
Theoremiblspltprt 38768* 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 38769* 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 38770* lemma for itgsubsticc 38771. (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 38771* Integration by u-substitution. The main difference with respect to itgsubst 23491 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 38772* 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 38773 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 38774* 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 38775* 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 38776* 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 38777* 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)
 
Theoremitgeq2d 38778* Equality theorem for an integral. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)
 
Theoremvolico 38779 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 38780 The Lebesgue measure of a left-closed, right-open interval is greater 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 38781 Lebesgue measurable sets are subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
dom vol ⊆ 𝒫 ℝ
 
Theoremismbl3 38782* The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl2 22977, 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 38783 The function that assigns the Lebesgue measure to open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)
 
Theoremovolsplit 38784 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 38785 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 38786 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 38787 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 38788 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 38789* The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 22976, 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 38790* ((vol ∘ (,)) ∘ 𝐹) expressed in map-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))       (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥𝐴 ↦ (vol‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))))
 
Theoremvolicoff 38791 ((vol ∘ [,)) ∘ 𝐹) expressed in map-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴⟶(ℝ × ℝ*))       (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞))
 
Theoremvoliooicof 38792 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 38793* ((vol ∘ [,)) ∘ 𝐹) expressed in map-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶(ℝ × ℝ*))       (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑥𝐴 ↦ (vol‘((1st ‘(𝐹𝑥))[,)(2nd ‘(𝐹𝑥))))))
 
Theoremvolicc 38794 The Lebesgue measure of a closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol‘(𝐴[,]𝐵)) = (𝐵𝐴))
 
Theoremvoliccico 38795 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 38796 The domain of a measurable function is a subset of the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ)
 
20.31.12  Stone Weierstrass theorem - real version
 
Theoremstoweidlem1 38797 Lemma for stoweid 38859. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 12720. (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 38798* lemma for stoweid 38859: 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 38799* Lemma for stoweid 38859: 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 38800* Lemma for stoweid 38859: 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)
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