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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlimcrcl 24401 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 24402 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝐹 lim 𝐵) ⊆ ℂ
 
Theoremlimcdif 24403 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 24404* 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 24405 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 24406* 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 24407 Lemma for limcflf 24408. (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 24408 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 24409* 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 24410* 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 24411* 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 24412 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 24413 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 24414 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 24415* 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 24416 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 24417* 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 24418 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 24419* 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 24420* 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 24421* 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 24422 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 24423 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 24424* 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 24425* 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 24426 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 24427 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 24428 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 24429 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 24430 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 24431 Both and are subsets of . (Contributed by Mario Carneiro, 10-Feb-2015.)
(𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
 
Theoremrecnperf 24432 Both and are perfect subsets of . (Contributed by Mario Carneiro, 28-Dec-2016.)
𝐾 = (TopOpen‘ℂfld)       (𝑆 ∈ {ℝ, ℂ} → (𝐾t 𝑆) ∈ Perf)
 
Theoremdvfg 24433 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 24434 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 24435 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)
(ℂ D 𝐹):dom (ℂ D 𝐹)⟶ℂ
 
Theoremdvreslem 24436* Lemma for dvres 24438. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Commute the consequent and shorten proof. (Revised by Peter Mazsa, 2-Oct-2022.)
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 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦)))
 
Theoremdvres2lem 24437* Lemma for dvres2 24439. (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 24438 Restriction of a derivative. Note that our definition of derivative df-dv 24394 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 24439 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 24438, 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 24440 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 24441 Restriction of a complex differentiable function to the reals. This version of dvres3 24440 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 24442* Lemma for dvid 24444 and dvconst 24443. (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 24443 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 24444 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 24445* 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 24446 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 24447 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 24448* 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 24449 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 24450 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 24451 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 24452 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 24453 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 24454 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 24455 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 24456 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 24457 Multiple derivative version of dvres3a 24441. (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))
 
Theoremcpnfval 24458* Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝑆 ⊆ ℂ → (𝓑C𝑛𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓cn→ℂ)}))
 
Theoremfncpn 24459 The 𝓑C𝑛 object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝑆 ⊆ ℂ → (𝓑C𝑛𝑆) Fn ℕ0)
 
Theoremelcpn 24460 Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹 ∈ ((𝓑C𝑛𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹cn→ℂ))))
 
Theoremcpnord 24461 𝓑C𝑛 conditions are ordered by strength. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) → ((𝓑C𝑛𝑆)‘𝑁) ⊆ ((𝓑C𝑛𝑆)‘𝑀))
 
Theoremcpncn 24462 A 𝓑C𝑛 function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C𝑛𝑆)‘𝑁)) → 𝐹 ∈ (dom 𝐹cn→ℂ))
 
Theoremcpnres 24463 The restriction of a 𝓑C𝑛 function is 𝓑C𝑛. (Contributed by Mario Carneiro, 11-Feb-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C𝑛‘ℂ)‘𝑁)) → (𝐹𝑆) ∈ ((𝓑C𝑛𝑆)‘𝑁))
 
Theoremdvaddbr 24464 The sum rule for derivatives at a point. For the (simpler but more limited) function version, see dvadd 24466. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌⟶ℂ)    &   (𝜑𝑌𝑆)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑉)    &   (𝜑𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)       (𝜑𝐶(𝑆 D (𝐹f + 𝐺))(𝐾 + 𝐿))
 
Theoremdvmulbr 24465 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmul 24467. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌⟶ℂ)    &   (𝜑𝑌𝑆)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑉)    &   (𝜑𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)       (𝜑𝐶(𝑆 D (𝐹f · 𝐺))((𝐾 · (𝐺𝐶)) + (𝐿 · (𝐹𝐶))))
 
Theoremdvadd 24466 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddbr 24464. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌⟶ℂ)    &   (𝜑𝑌𝑆)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐹))    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐺))       (𝜑 → ((𝑆 D (𝐹f + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)))
 
Theoremdvmul 24467 The product rule for derivatives at a point. For the (more general) relation version, see dvmulbr 24465. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌⟶ℂ)    &   (𝜑𝑌𝑆)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐹))    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐺))       (𝜑 → ((𝑆 D (𝐹f · 𝐺))‘𝐶) = ((((𝑆 D 𝐹)‘𝐶) · (𝐺𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹𝐶))))
 
Theoremdvaddf 24468 The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → dom (𝑆 D 𝐺) = 𝑋)       (𝜑 → (𝑆 D (𝐹f + 𝐺)) = ((𝑆 D 𝐹) ∘f + (𝑆 D 𝐺)))
 
Theoremdvmulf 24469 The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑋⟶ℂ)    &   (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)))
 
Theoremdvcmul 24470 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐶 ∈ dom (𝑆 D 𝐹))       (𝜑 → ((𝑆 D ((𝑆 × {𝐴}) ∘f · 𝐹))‘𝐶) = (𝐴 · ((𝑆 D 𝐹)‘𝐶)))
 
Theoremdvcmulf 24471 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)       (𝜑 → (𝑆 D ((𝑆 × {𝐴}) ∘f · 𝐹)) = ((𝑆 × {𝐴}) ∘f · (𝑆 D 𝐹)))
 
Theoremdvcobr 24472 The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 24473. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌𝑋)    &   (𝜑𝑌𝑇)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑇 ⊆ ℂ)    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑉)    &   (𝜑 → (𝐺𝐶)(𝑆 D 𝐹)𝐾)    &   (𝜑𝐶(𝑇 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)       (𝜑𝐶(𝑇 D (𝐹𝐺))(𝐾 · 𝐿))
 
Theoremdvco 24473 The chain rule for derivatives at a point. For the (more general) relation version, see dvcobr 24472. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐺:𝑌𝑋)    &   (𝜑𝑌𝑇)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑇 ∈ {ℝ, ℂ})    &   (𝜑 → (𝐺𝐶) ∈ dom (𝑆 D 𝐹))    &   (𝜑𝐶 ∈ dom (𝑇 D 𝐺))       (𝜑 → ((𝑇 D (𝐹𝐺))‘𝐶) = (((𝑆 D 𝐹)‘(𝐺𝐶)) · ((𝑇 D 𝐺)‘𝐶)))
 
Theoremdvcof 24474 The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑇 ∈ {ℝ, ℂ})    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐺:𝑌𝑋)    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → dom (𝑇 D 𝐺) = 𝑌)       (𝜑 → (𝑇 D (𝐹𝐺)) = (((𝑆 D 𝐹) ∘ 𝐺) ∘f · (𝑇 D 𝐺)))
 
Theoremdvcjbr 24475 The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 24476. (This doesn't follow from dvcobr 24472 because is not a function on the reals, and even if we used complex derivatives, is not complex-differentiable.) (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝐶 ∈ dom (ℝ D 𝐹))       (𝜑𝐶(ℝ D (∗ ∘ 𝐹))(∗‘((ℝ D 𝐹)‘𝐶)))
 
Theoremdvcj 24476 The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 24475. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
((𝐹:𝑋⟶ℂ ∧ 𝑋 ⊆ ℝ) → (ℝ D (∗ ∘ 𝐹)) = (∗ ∘ (ℝ D 𝐹)))
 
Theoremdvfre 24477 The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)
((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
 
Theoremdvnfre 24478 The 𝑁-th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)
((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑁 ∈ ℕ0) → ((ℝ D𝑛 𝐹)‘𝑁):dom ((ℝ D𝑛 𝐹)‘𝑁)⟶ℝ)
 
Theoremdvexp 24479* Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
 
Theoremdvexp2 24480* Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
(𝑁 ∈ ℕ0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
 
Theoremdvrec 24481* Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
(𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ -(𝐴 / (𝑥↑2))))
 
Theoremdvmptres3 24482* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋𝐽)    &   (𝜑 → (𝑆𝑋) = 𝑌)    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (ℂ D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (𝑆 D (𝑥𝑌𝐴)) = (𝑥𝑌𝐵))
 
Theoremdvmptid 24483* Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})       (𝜑 → (𝑆 D (𝑥𝑆𝑥)) = (𝑥𝑆 ↦ 1))
 
Theoremdvmptc 24484* Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐴 ∈ ℂ)       (𝜑 → (𝑆 D (𝑥𝑆𝐴)) = (𝑥𝑆 ↦ 0))
 
Theoremdvmptcl 24485* Closure lemma for dvmptcmul 24490 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       ((𝜑𝑥𝑋) → 𝐵 ∈ ℂ)
 
Theoremdvmptadd 24486* Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(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 + 𝐷)))
 
Theoremdvmptmul 24487* Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(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))))
 
Theoremdvmptres2 24488* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝜑𝑍𝑋)    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑 → ((int‘𝐽)‘𝑍) = 𝑌)       (𝜑 → (𝑆 D (𝑥𝑍𝐴)) = (𝑥𝑌𝐵))
 
Theoremdvmptres 24489* Function-builder for derivative: restrict a derivative to an open subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝜑𝑌𝑋)    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝑌𝐽)       (𝜑 → (𝑆 D (𝑥𝑌𝐴)) = (𝑥𝑌𝐵))
 
Theoremdvmptcmul 24490* Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐶 · 𝐴))) = (𝑥𝑋 ↦ (𝐶 · 𝐵)))
 
Theoremdvmptdivc 24491* Function-builder for derivative, division rule for constant divisor. (Contributed by Mario Carneiro, 18-May-2016.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 / 𝐶))) = (𝑥𝑋 ↦ (𝐵 / 𝐶)))
 
Theoremdvmptneg 24492* Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ -𝐴)) = (𝑥𝑋 ↦ -𝐵))
 
Theoremdvmptsub 24493* Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(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 ↦ (𝐴𝐶))) = (𝑥𝑋 ↦ (𝐵𝐷)))
 
Theoremdvmptcj 24494* Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (ℝ D (𝑥𝑋 ↦ (∗‘𝐴))) = (𝑥𝑋 ↦ (∗‘𝐵)))
 
Theoremdvmptre 24495* Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.)
((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (ℝ D (𝑥𝑋 ↦ (ℜ‘𝐴))) = (𝑥𝑋 ↦ (ℜ‘𝐵)))
 
Theoremdvmptim 24496* Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.)
((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   (𝜑 → (ℝ D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (ℝ D (𝑥𝑋 ↦ (ℑ‘𝐴))) = (𝑥𝑋 ↦ (ℑ‘𝐵)))
 
Theoremdvmptntr 24497* Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑋𝑆)    &   ((𝜑𝑥𝑋) → 𝐴 ∈ ℂ)    &   𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌)       (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑆 D (𝑥𝑌𝐴)))
 
Theoremdvmptco 24498* Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑇 ∈ {ℝ, ℂ})    &   ((𝜑𝑥𝑋) → 𝐴𝑌)    &   ((𝜑𝑥𝑋) → 𝐵𝑉)    &   ((𝜑𝑦𝑌) → 𝐶 ∈ ℂ)    &   ((𝜑𝑦𝑌) → 𝐷𝑊)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))    &   (𝜑 → (𝑇 D (𝑦𝑌𝐶)) = (𝑦𝑌𝐷))    &   (𝑦 = 𝐴𝐶 = 𝐸)    &   (𝑦 = 𝐴𝐷 = 𝐹)       (𝜑 → (𝑆 D (𝑥𝑋𝐸)) = (𝑥𝑋 ↦ (𝐹 · 𝐵)))
 
Theoremdvrecg 24499* Derivative of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝐴 ∈ ℂ)    &   ((𝜑𝑥𝑋) → 𝐵 ∈ (ℂ ∖ {0}))    &   ((𝜑𝑥𝑋) → 𝐶𝑉)    &   (𝜑 → (𝑆 D (𝑥𝑋𝐵)) = (𝑥𝑋𝐶))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ (𝐴 / 𝐵))) = (𝑥𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2))))
 
Theoremdvmptdiv 24500* 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))))
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