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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiblabs 23401* The absolute value of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (𝑥𝐴 ↦ (abs‘𝐵)) ∈ 𝐿1)

Theoremiblabsr 23402* A measurable function is integrable iff its absolute value is integrable. (See iblabs 23401 for the forward implication.) (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)    &   (𝜑 → (𝑥𝐴 ↦ (abs‘𝐵)) ∈ 𝐿1)       (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)

Theoremiblmulc2 23403* Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿1)

Theoremitgmulc2lem1 23404* Lemma for itgmulc2 23406: positive real case. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐶)    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐵)       (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)

Theoremitgmulc2lem2 23405* Lemma for itgmulc2 23406: real case. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)

Theoremitgmulc2 23406* Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)

Theoremitgabs 23407* The triangle inequality for integrals. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)

Theoremitgsplit 23408* The integral splits under an almost disjoint union. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑 → (vol*‘(𝐴𝐵)) = 0)    &   (𝜑𝑈 = (𝐴𝐵))    &   ((𝜑𝑥𝑈) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)    &   (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)       (𝜑 → ∫𝑈𝐶 d𝑥 = (∫𝐴𝐶 d𝑥 + ∫𝐵𝐶 d𝑥))

Theoremitgspliticc 23409* The integral splits on closed intervals with matching endpoints. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 ∈ (𝐴[,]𝐶))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐶)) → 𝐷𝑉)    &   (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐷) ∈ 𝐿1)    &   (𝜑 → (𝑥 ∈ (𝐵[,]𝐶) ↦ 𝐷) ∈ 𝐿1)       (𝜑 → ∫(𝐴[,]𝐶)𝐷 d𝑥 = (∫(𝐴[,]𝐵)𝐷 d𝑥 + ∫(𝐵[,]𝐶)𝐷 d𝑥))

Theoremitgsplitioo 23410* The integral splits on open intervals with matching endpoints. (Contributed by Mario Carneiro, 2-Sep-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 ∈ (𝐴[,]𝐶))    &   ((𝜑𝑥 ∈ (𝐴(,)𝐶)) → 𝐷 ∈ ℂ)    &   (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿1)    &   (𝜑 → (𝑥 ∈ (𝐵(,)𝐶) ↦ 𝐷) ∈ 𝐿1)       (𝜑 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))

Theorembddmulibl 23411* A bounded function times an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹𝑦)) ≤ 𝑥) → (𝐹𝑓 · 𝐺) ∈ 𝐿1)

Theorembddibl 23412* A bounded function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹𝑦)) ≤ 𝑥) → 𝐹 ∈ 𝐿1)

Theoremcniccibl 23413 A continuous function on a closed bounded interval is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) → 𝐹 ∈ 𝐿1)

Theoremitggt0 23414* The integral of a strictly positive function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)
(𝜑 → 0 < (vol‘𝐴))    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ+)       (𝜑 → 0 < ∫𝐴𝐵 d𝑥)

Theoremitgcn 23415* Transfer itg2cn 23336 to the full Lebesgue integral. (Contributed by Mario Carneiro, 1-Sep-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ∃𝑑 ∈ ℝ+𝑢 ∈ dom vol((𝑢𝐴 ∧ (vol‘𝑢) < 𝑑) → ∫𝑢(abs‘𝐵) d𝑥 < 𝐶))

13.2.2.2  Lesbesgue directed integral

Syntaxcdit 23416 Extend class notation with the directed integral.
class ⨜[𝐴𝐵]𝐶 d𝑥

Definitiondf-ditg 23417 Define the directed integral, which is just a regular integral but with a sign change when the limits are interchanged. The 𝐴 and 𝐵 here are the lower and upper limits of the integral, usually written as a subscript and superscript next to the integral sign. We define the region of integration to be an open interval instead of closed so that we can use +∞, -∞ for limits and also integrate up to a singularity at an endpoint. (Contributed by Mario Carneiro, 13-Aug-2014.)
⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)

Theoremditgeq1 23418* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝐴 = 𝐵 → ⨜[𝐴𝐶]𝐷 d𝑥 = ⨜[𝐵𝐶]𝐷 d𝑥)

Theoremditgeq2 23419* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝐴 = 𝐵 → ⨜[𝐶𝐴]𝐷 d𝑥 = ⨜[𝐶𝐵]𝐷 d𝑥)

Theoremditgeq3 23420* Equality theorem for the directed integral. (The domain of the equality here is very rough; for more precise bounds one should decompose it with ditgpos 23426 first and use the equality theorems for df-itg 23198.) (Contributed by Mario Carneiro, 13-Aug-2014.)
(∀𝑥 ∈ ℝ 𝐷 = 𝐸 → ⨜[𝐴𝐵]𝐷 d𝑥 = ⨜[𝐴𝐵]𝐸 d𝑥)

Theoremditgeq3dv 23421* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
((𝜑𝑥 ∈ ℝ) → 𝐷 = 𝐸)       (𝜑 → ⨜[𝐴𝐵]𝐷 d𝑥 = ⨜[𝐴𝐵]𝐸 d𝑥)

Theoremditgex 23422 A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
⨜[𝐴𝐵]𝐶 d𝑥 ∈ V

Theoremditg0 23423* Value of the directed integral from a point to itself. (Contributed by Mario Carneiro, 13-Aug-2014.)
⨜[𝐴𝐴]𝐵 d𝑥 = 0

Theoremcbvditg 23424* Change bound variable in a directed integral. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝑥 = 𝑦𝐶 = 𝐷)    &   𝑦𝐶    &   𝑥𝐷       ⨜[𝐴𝐵]𝐶 d𝑥 = ⨜[𝐴𝐵]𝐷 d𝑦

Theoremcbvditgv 23425* Change bound variable in a directed integral. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝑥 = 𝑦𝐶 = 𝐷)       ⨜[𝐴𝐵]𝐶 d𝑥 = ⨜[𝐴𝐵]𝐷 d𝑦

Theoremditgpos 23426* Value of the directed integral in the forward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝐴𝐵)       (𝜑 → ⨜[𝐴𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)

Theoremditgneg 23427* Value of the directed integral in the backward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ⨜[𝐵𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)

Theoremditgcl 23428* Closure of a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐶𝑉)    &   (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿1)       (𝜑 → ⨜[𝐴𝐵]𝐶 d𝑥 ∈ ℂ)

Theoremditgswap 23429* Reverse a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐶𝑉)    &   (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿1)       (𝜑 → ⨜[𝐵𝐴]𝐶 d𝑥 = -⨜[𝐴𝐵]𝐶 d𝑥)

Theoremditgsplitlem 23430* Lemma for ditgsplit 23431. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   (𝜑𝐶 ∈ (𝑋[,]𝑌))    &   ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐷𝑉)    &   (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ 𝐿1)    &   ((𝜓𝜃) ↔ (𝐴𝐵𝐵𝐶))       (((𝜑𝜓) ∧ 𝜃) → ⨜[𝐴𝐶]𝐷 d𝑥 = (⨜[𝐴𝐵]𝐷 d𝑥 + ⨜[𝐵𝐶]𝐷 d𝑥))

Theoremditgsplit 23431* This theorem is the raison d'être for the directed integral, because unlike itgspliticc 23409, there is no constraint on the ordering of the points 𝐴, 𝐵, 𝐶 in the domain. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   (𝜑𝐶 ∈ (𝑋[,]𝑌))    &   ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐷𝑉)    &   (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ 𝐿1)       (𝜑 → ⨜[𝐴𝐶]𝐷 d𝑥 = (⨜[𝐴𝐵]𝐷 d𝑥 + ⨜[𝐵𝐶]𝐷 d𝑥))

13.3  Derivatives

13.3.1  Real and complex differentiation

13.3.1.1  Derivatives of functions of one complex or real variable

Syntaxclimc 23432 The limit operator.
class lim

Syntaxcdv 23433 The derivative operator.
class D

Syntaxcdvn 23434 The 𝑛-th derivative operator.
class D𝑛

Syntaxccpn 23435 The set of 𝑛-times continuously differentiable functions.
class Cn

Definitiondf-limc 23436* Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.)
lim = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦[(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)})

Definitiondf-dv 23437* Define the derivative operator on functions on the reals. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set 𝑠 here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of and is well-behaved when 𝑠 contains no isolated points, we will restrict our attention to the cases 𝑠 = ℝ or 𝑠 = ℂ for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.)
D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)))

Definitiondf-dvn 23438* Define the 𝑛-th derivative operator on functions on the complex numbers. This just iterates the derivative operation according to the last argument. (Contributed by Mario Carneiro, 11-Feb-2015.)
D𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})))

Definitiondf-cpn 23439* Define the set of 𝑛-times continuously differentiable functions. (Contributed by Stefan O'Rear, 15-Nov-2014.)
Cn = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓cn→ℂ)}))

Theoremreldv 23440 The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 24-Dec-2016.)
Rel (𝑆 D 𝐹)

Theoremlimcvallem 23441* Lemma for ellimc 23443. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝐽 = (𝐾t (𝐴 ∪ {𝐵}))    &   𝐾 = (TopOpen‘ℂfld)    &   𝐺 = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))       ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐶 ∈ ℂ))

Theoremlimcfval 23442* Value and set bounds on the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝐽 = (𝐾t (𝐴 ∪ {𝐵}))    &   𝐾 = (TopOpen‘ℂfld)       ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 lim 𝐵) = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∧ (𝐹 lim 𝐵) ⊆ ℂ))

Theoremellimc 23443* Value of the limit predicate. 𝐶 is the limit of the function 𝐹 at 𝐵 if the function 𝐺, formed by adding 𝐵 to the domain of 𝐹 and setting it to 𝐶, is continuous at 𝐵. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝐽 = (𝐾t (𝐴 ∪ {𝐵}))    &   𝐾 = (TopOpen‘ℂfld)    &   𝐺 = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)))

Theoremlimcrcl 23444 Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.)
(𝐶 ∈ (𝐹 lim 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))

Theoremlimccl 23445 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝐹 lim 𝐵) ⊆ ℂ

Theoremlimcdif 23446 It suffices to consider functions which are not defined at 𝐵 to define the limit of a function. In particular, the value of the original function 𝐹 at 𝐵 does not affect the limit of 𝐹. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝜑𝐹:𝐴⟶ℂ)       (𝜑 → (𝐹 lim 𝐵) = ((𝐹 ↾ (𝐴 ∖ {𝐵})) lim 𝐵))

Theoremellimc2 23447* Write the definition of a limit directly in terms of open sets of the topology on the complex numbers. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   𝐾 = (TopOpen‘ℂfld)       (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑢𝐾 (𝐶𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))

Theoremlimcnlp 23448 If 𝐵 is not a limit point of the domain of the function 𝐹, then every point is a limit of 𝐹 at 𝐵. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑 → ¬ 𝐵 ∈ ((limPt‘𝐾)‘𝐴))       (𝜑 → (𝐹 lim 𝐵) = ℂ)

Theoremellimc3 23449* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ+𝑧𝐴 ((𝑧𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑦) → (abs‘((𝐹𝑧) − 𝐶)) < 𝑥))))

Theoremlimcflflem 23450 Lemma for limcflf 23451. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ((limPt‘𝐾)‘𝐴))    &   𝐾 = (TopOpen‘ℂfld)    &   𝐶 = (𝐴 ∖ {𝐵})    &   𝐿 = (((nei‘𝐾)‘{𝐵}) ↾t 𝐶)       (𝜑𝐿 ∈ (Fil‘𝐶))

Theoremlimcflf 23451 The limit operator can be expressed as a filter limit, from the filter of neighborhoods of 𝐵 restricted to 𝐴 ∖ {𝐵}, to the topology of the complex numbers. (If 𝐵 is not a limit point of 𝐴, then it is still formally a filter limit, but the neighborhood filter is not a proper filter in this case.) (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ((limPt‘𝐾)‘𝐴))    &   𝐾 = (TopOpen‘ℂfld)    &   𝐶 = (𝐴 ∖ {𝐵})    &   𝐿 = (((nei‘𝐾)‘{𝐵}) ↾t 𝐶)       (𝜑 → (𝐹 lim 𝐵) = ((𝐾 fLimf 𝐿)‘(𝐹𝐶)))

Theoremlimcmo 23452* If 𝐵 is a limit point of the domain of the function 𝐹, then there is at most one limit value of 𝐹 at 𝐵. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ((limPt‘𝐾)‘𝐴))    &   𝐾 = (TopOpen‘ℂfld)       (𝜑 → ∃*𝑥 𝑥 ∈ (𝐹 lim 𝐵))

Theoremlimcmpt 23453* Express the limit operator for a function defined by a mapping. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   ((𝜑𝑧𝐴) → 𝐷 ∈ ℂ)    &   𝐽 = (𝐾t (𝐴 ∪ {𝐵}))    &   𝐾 = (TopOpen‘ℂfld)       (𝜑 → (𝐶 ∈ ((𝑧𝐴𝐷) lim 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, 𝐷)) ∈ ((𝐽 CnP 𝐾)‘𝐵)))

Theoremlimcmpt2 23454* Express the limit operator for a function defined by a mapping. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵𝐴)    &   ((𝜑 ∧ (𝑧𝐴𝑧𝐵)) → 𝐷 ∈ ℂ)    &   𝐽 = (𝐾t 𝐴)    &   𝐾 = (TopOpen‘ℂfld)       (𝜑 → (𝐶 ∈ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ 𝐷) lim 𝐵) ↔ (𝑧𝐴 ↦ if(𝑧 = 𝐵, 𝐶, 𝐷)) ∈ ((𝐽 CnP 𝐾)‘𝐵)))

Theoremlimcresi 23455 Any limit of 𝐹 is also a limit of the restriction of 𝐹. (Contributed by Mario Carneiro, 28-Dec-2016.)
(𝐹 lim 𝐵) ⊆ ((𝐹𝐶) lim 𝐵)

Theoremlimcres 23456 If 𝐵 is an interior point of 𝐶 ∪ {𝐵} relative to the domain 𝐴, then a limit point of 𝐹𝐶 extends to a limit of 𝐹. (Contributed by Mario Carneiro, 27-Dec-2016.)
(𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐶𝐴)    &   (𝜑𝐴 ⊆ ℂ)    &   𝐾 = (TopOpen‘ℂfld)    &   𝐽 = (𝐾t (𝐴 ∪ {𝐵}))    &   (𝜑𝐵 ∈ ((int‘𝐽)‘(𝐶 ∪ {𝐵})))       (𝜑 → ((𝐹𝐶) lim 𝐵) = (𝐹 lim 𝐵))

Theoremcnplimc 23457 A function is continuous at 𝐵 iff its limit at 𝐵 equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.)
𝐾 = (TopOpen‘ℂfld)    &   𝐽 = (𝐾t 𝐴)       ((𝐴 ⊆ ℂ ∧ 𝐵𝐴) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵) ↔ (𝐹:𝐴⟶ℂ ∧ (𝐹𝐵) ∈ (𝐹 lim 𝐵))))

Theoremcnlimc 23458* 𝐹 is a continuous function iff the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.)
(𝐴 ⊆ ℂ → (𝐹 ∈ (𝐴cn→ℂ) ↔ (𝐹:𝐴⟶ℂ ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ (𝐹 lim 𝑥))))

Theoremcnlimci 23459 If 𝐹 is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹 ∈ (𝐴cn𝐷))    &   (𝜑𝐵𝐴)       (𝜑 → (𝐹𝐵) ∈ (𝐹 lim 𝐵))

Theoremcnmptlimc 23460* If 𝐹 is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
(𝜑 → (𝑥𝐴𝑋) ∈ (𝐴cn𝐷))    &   (𝜑𝐵𝐴)    &   (𝑥 = 𝐵𝑋 = 𝑌)       (𝜑𝑌 ∈ ((𝑥𝐴𝑋) lim 𝐵))

Theoremlimccnp 23461 If the limit of 𝐹 at 𝐵 is 𝐶 and 𝐺 is continuous at 𝐶, then the limit of 𝐺𝐹 at 𝐵 is 𝐺(𝐶). (Contributed by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹:𝐴𝐷)    &   (𝜑𝐷 ⊆ ℂ)    &   𝐾 = (TopOpen‘ℂfld)    &   𝐽 = (𝐾t 𝐷)    &   (𝜑𝐶 ∈ (𝐹 lim 𝐵))    &   (𝜑𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐶))       (𝜑 → (𝐺𝐶) ∈ ((𝐺𝐹) lim 𝐵))

Theoremlimccnp2 23462* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.)
((𝜑𝑥𝐴) → 𝑅𝑋)    &   ((𝜑𝑥𝐴) → 𝑆𝑌)    &   (𝜑𝑋 ⊆ ℂ)    &   (𝜑𝑌 ⊆ ℂ)    &   𝐾 = (TopOpen‘ℂfld)    &   𝐽 = ((𝐾 ×t 𝐾) ↾t (𝑋 × 𝑌))    &   (𝜑𝐶 ∈ ((𝑥𝐴𝑅) lim 𝐵))    &   (𝜑𝐷 ∈ ((𝑥𝐴𝑆) lim 𝐵))    &   (𝜑𝐻 ∈ ((𝐽 CnP 𝐾)‘⟨𝐶, 𝐷⟩))       (𝜑 → (𝐶𝐻𝐷) ∈ ((𝑥𝐴 ↦ (𝑅𝐻𝑆)) lim 𝐵))

Theoremlimcco 23463* Composition of two limits. (Contributed by Mario Carneiro, 29-Dec-2016.)
((𝜑 ∧ (𝑥𝐴𝑅𝐶)) → 𝑅𝐵)    &   ((𝜑𝑦𝐵) → 𝑆 ∈ ℂ)    &   (𝜑𝐶 ∈ ((𝑥𝐴𝑅) lim 𝑋))    &   (𝜑𝐷 ∈ ((𝑦𝐵𝑆) lim 𝐶))    &   (𝑦 = 𝑅𝑆 = 𝑇)    &   ((𝜑 ∧ (𝑥𝐴𝑅 = 𝐶)) → 𝑇 = 𝐷)       (𝜑𝐷 ∈ ((𝑥𝐴𝑇) lim 𝑋))

Theoremlimciun 23464* A point is a limit of 𝐹 on the finite union 𝑥𝐴𝐵(𝑥) iff it is the limit of the restriction of 𝐹 to each 𝐵(𝑥). (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝐴 ∈ Fin)    &   (𝜑 → ∀𝑥𝐴 𝐵 ⊆ ℂ)    &   (𝜑𝐹: 𝑥𝐴 𝐵⟶ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐹 lim 𝐶) = (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))

Theoremlimcun 23465 A point is a limit of 𝐹 on 𝐴𝐵 iff it is the limit of the restriction of 𝐹 to 𝐴 and to 𝐵. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ⊆ ℂ)    &   (𝜑𝐹:(𝐴𝐵)⟶ℂ)       (𝜑 → (𝐹 lim 𝐶) = (((𝐹𝐴) lim 𝐶) ∩ ((𝐹𝐵) lim 𝐶)))

Theoremdvlem 23466 Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
(𝜑𝐹:𝐷⟶ℂ)    &   (𝜑𝐷 ⊆ ℂ)    &   (𝜑𝐵𝐷)       ((𝜑𝐴 ∈ (𝐷 ∖ {𝐵})) → (((𝐹𝐴) − (𝐹𝐵)) / (𝐴𝐵)) ∈ ℂ)

Theoremdvfval 23467* Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 25-Dec-2016.)
𝑇 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)       ((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴𝑆) → ((𝑆 D 𝐹) = 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥))) lim 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝑇)‘𝐴) × ℂ)))

Theoremeldv 23468* The differentiable predicate. A function 𝐹 is differentiable at 𝐵 with derivative 𝐶 iff 𝐹 is defined in a neighborhood of 𝐵 and the difference quotient has limit 𝐶 at 𝐵. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 25-Dec-2016.)
𝑇 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   𝐺 = (𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹𝑧) − (𝐹𝐵)) / (𝑧𝐵)))    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)       (𝜑 → (𝐵(𝑆 D 𝐹)𝐶 ↔ (𝐵 ∈ ((int‘𝑇)‘𝐴) ∧ 𝐶 ∈ (𝐺 lim 𝐵))))

Theoremdvcl 23469 The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)       ((𝜑𝐵(𝑆 D 𝐹)𝐶) → 𝐶 ∈ ℂ)

Theoremdvbssntr 23470 The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)       (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘𝐽)‘𝐴))

Theoremdvbss 23471 The set of differentiable points is a subset of the domain of the function. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)       (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝐴)

Theoremdvbsss 23472 The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.)
dom (𝑆 D 𝐹) ⊆ 𝑆

Theoremperfdvf 23473 The derivative is a function, whenever it is defined relative to a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝐾 = (TopOpen‘ℂfld)       ((𝐾t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)

Theoremrecnprss 23474 Both and are subsets of . (Contributed by Mario Carneiro, 10-Feb-2015.)
(𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)

Theoremrecnperf 23475 Both and are perfect subsets of . (Contributed by Mario Carneiro, 28-Dec-2016.)
𝐾 = (TopOpen‘ℂfld)       (𝑆 ∈ {ℝ, ℂ} → (𝐾t 𝑆) ∈ Perf)

Theoremdvfg 23476 Explicitly write out the functionality condition on derivative for 𝑆 = ℝ and . (Contributed by Mario Carneiro, 9-Feb-2015.)
(𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)

Theoremdvf 23477 The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
(ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ

Theoremdvfcn 23478 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)
(ℂ D 𝐹):dom (ℂ D 𝐹)⟶ℂ

Theoremdvreslem 23479* Lemma for dvres 23481. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
𝐾 = (TopOpen‘ℂfld)    &   𝑇 = (𝐾t 𝑆)    &   𝐺 = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥)))    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝑦 ∈ ℂ)       (𝜑 → (𝑥(𝑆 D (𝐹𝐵))𝑦 ↔ (𝑥(𝑆 D 𝐹)𝑦𝑥 ∈ ((int‘𝑇)‘𝐵))))

Theoremdvres2lem 23480* Lemma for dvres2 23482. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
𝐾 = (TopOpen‘ℂfld)    &   𝑇 = (𝐾t 𝑆)    &   𝐺 = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥)))    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑𝑦 ∈ ℂ)    &   (𝜑𝑥(𝑆 D 𝐹)𝑦)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐵 D (𝐹𝐵))𝑦)

Theoremdvres 23481 Restriction of a derivative. Note that our definition of derivative df-dv 23437 would still make sense if we demanded that 𝑥 be an element of the domain instead of an interior point of the domain, but then it is possible for a non-differentiable function to have two different derivatives at a single point 𝑥 when restricted to different subsets containing 𝑥; a classic example is the absolute value function restricted to [0, +∞) and (-∞, 0]. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
𝐾 = (TopOpen‘ℂfld)    &   𝑇 = (𝐾t 𝑆)       (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴𝑆𝐵𝑆)) → (𝑆 D (𝐹𝐵)) = ((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵)))

Theoremdvres2 23482 Restriction of the base set of a derivative. The primary application of this theorem says that if a function is complex differentiable then it is also real differentiable. Unlike dvres 23481, there is no simple reverse relation relating real differentiable functions to complex differentiability, and indeed there are functions like ℜ(𝑥) which are everywhere real-differentiable but nowhere complex-differentiable.) (Contributed by Mario Carneiro, 9-Feb-2015.)
(((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴𝑆𝐵𝑆)) → ((𝑆 D 𝐹) ↾ 𝐵) ⊆ (𝐵 D (𝐹𝐵)))

Theoremdvres3 23483 Restriction of a complex differentiable function to the reals. (Contributed by Mario Carneiro, 10-Feb-2015.)
(((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ ℂ ∧ 𝑆 ⊆ dom (ℂ D 𝐹))) → (𝑆 D (𝐹𝑆)) = ((ℂ D 𝐹) ↾ 𝑆))

Theoremdvres3a 23484 Restriction of a complex differentiable function to the reals. This version of dvres3 23483 assumes that 𝐹 is differentiable on its domain, but does not require 𝐹 to be differentiable on the whole real line. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)       (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝑆 D (𝐹𝑆)) = ((ℂ D 𝐹) ↾ 𝑆))

Theoremdvidlem 23485* Lemma for dvid 23487 and dvconst 23486. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
(𝜑𝐹:ℂ⟶ℂ)    &   ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧𝑥)) → (((𝐹𝑧) − (𝐹𝑥)) / (𝑧𝑥)) = 𝐵)    &   𝐵 ∈ ℂ       (𝜑 → (ℂ D 𝐹) = (ℂ × {𝐵}))

Theoremdvconst 23486 Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
(𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0}))

Theoremdvid 23487 Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
(ℂ D ( I ↾ ℂ)) = (ℂ × {1})

Theoremdvcnp 23488* The difference quotient is continuous at 𝐵 when the original function is differentiable at 𝐵. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
𝐽 = (𝐾t 𝐴)    &   𝐾 = (TopOpen‘ℂfld)    &   𝐺 = (𝑧𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹𝑧) − (𝐹𝐵)) / (𝑧𝐵))))       (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵))

Theoremdvcnp2 23489 A function is continuous at each point for which it is differentiable. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
𝐽 = (𝐾t 𝐴)    &   𝐾 = (TopOpen‘ℂfld)       (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))

Theoremdvcn 23490 A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)
(((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴𝑆) ∧ dom (𝑆 D 𝐹) = 𝐴) → 𝐹 ∈ (𝐴cn→ℂ))

Theoremdvnfval 23491* Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐺 = (𝑥 ∈ V ↦ (𝑆 D 𝑥))       ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹) = seq0((𝐺 ∘ 1st ), (ℕ0 × {𝐹})))

Theoremdvnff 23492 The iterated derivative is a function. (Contributed by Mario Carneiro, 11-Feb-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D𝑛 𝐹):ℕ0⟶(ℂ ↑pm dom 𝐹))

Theoremdvn0 23493 Zero times iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘0) = 𝐹)

Theoremdvnp1 23494 Successor iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑁 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑁)))

Theoremdvn1 23495 One times iterated derivative. (Contributed by Mario Carneiro, 1-Jan-2017.)
((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘1) = (𝑆 D 𝐹))

Theoremdvnf 23496 The N-times derivative is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑁):dom ((𝑆 D𝑛 𝐹)‘𝑁)⟶ℂ)

Theoremdvnbss 23497 The set of N-times differentiable points is a subset of the domain of the function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)

Theoremdvnadd 23498 The 𝑁-th derivative of the 𝑀-th derivative of 𝐹 is the same as the 𝑀 + 𝑁-th derivative of 𝐹. (Contributed by Mario Carneiro, 11-Feb-2015.)
(((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘𝑁) = ((𝑆 D𝑛 𝐹)‘(𝑀 + 𝑁)))

Theoremdvn2bss 23499 An N-times differentiable point is an M-times differentiable point, if 𝑀𝑁. (Contributed by Mario Carneiro, 30-Dec-2016.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑀))

Theoremdvnres 23500 Multiple derivative version of dvres3a 23484. (Contributed by Mario Carneiro, 11-Feb-2015.)
(((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑁 ∈ ℕ0) ∧ dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹) → ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))

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