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Theorem List for Metamath Proof Explorer - 24501-24600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlimciun 24501* 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 24502 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 24503 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 24504* 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 24505* 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 24506 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 24507 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 24508 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 24509 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 24510 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 24511 Both and are subsets of . (Contributed by Mario Carneiro, 10-Feb-2015.)
(𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
 
Theoremrecnperf 24512 Both and are perfect subsets of . (Contributed by Mario Carneiro, 28-Dec-2016.)
𝐾 = (TopOpen‘ℂfld)       (𝑆 ∈ {ℝ, ℂ} → (𝐾t 𝑆) ∈ Perf)
 
Theoremdvfg 24513 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 24514 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 24515 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)
(ℂ D 𝐹):dom (ℂ D 𝐹)⟶ℂ
 
Theoremdvreslem 24516* Lemma for dvres 24518. (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 24517* Lemma for dvres2 24519. (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 24518 Restriction of a derivative. Note that our definition of derivative df-dv 24474 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 24519 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 24518, 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 24520 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 24521 Restriction of a complex differentiable function to the reals. This version of dvres3 24520 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 24522* Lemma for dvid 24525 and dvconst 24524. (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}))
 
Theoremdvmptresicc 24523* 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))
 
Theoremdvconst 24524 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 24525 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 24526* 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 24527 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 24528 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 24529* 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 24530 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 24531 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 24532 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 24533 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 24534 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 24535 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 24536 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 24537 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 24538 Multiple derivative version of dvres3a 24521. (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 24539* 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 24540 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 24541 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 24542 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 24543 A 𝓑C𝑛 function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)
((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C𝑛𝑆)‘𝑁)) → 𝐹 ∈ (dom 𝐹cn→ℂ))
 
Theoremcpnres 24544 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 24545 The sum rule for derivatives at a point. For the (simpler but more limited) function version, see dvadd 24547. (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 24546 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmul 24548. (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 24547 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddbr 24545. (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 24548 The product rule for derivatives at a point. For the (more general) relation version, see dvmulbr 24546. (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 24549 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 24550 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 24551 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 24552 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 24553 The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 24554. (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 24554 The chain rule for derivatives at a point. For the (more general) relation version, see dvcobr 24553. (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 24555 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 24556 The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 24557. (This doesn't follow from dvcobr 24553 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 24557 The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 24556. (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 24558 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 24559 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 24560* 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 24561* 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 24562* 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 24563* 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 24564* 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 24565* 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 24566* Closure lemma for dvmptcmul 24571 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 24567* 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 24568* 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 24569* 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 24570* 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 24571* 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 24572* 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 24573* 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 24574* 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 24575* 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 24576* 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 24577* 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 24578* 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 24579* 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 24580* 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 24581* 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))))
 
Theoremdvmptfsum 24582* Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑋𝐽)    &   (𝜑𝐼 ∈ Fin)    &   ((𝜑𝑖𝐼𝑥𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ)    &   ((𝜑𝑖𝐼) → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))       (𝜑 → (𝑆 D (𝑥𝑋 ↦ Σ𝑖𝐼 𝐴)) = (𝑥𝑋 ↦ Σ𝑖𝐼 𝐵))
 
Theoremdvcnvlem 24583 Lemma for dvcnvre 24626. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝑆)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑌𝐾)    &   (𝜑𝐹:𝑋1-1-onto𝑌)    &   (𝜑𝐹 ∈ (𝑌cn𝑋))    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))    &   (𝜑𝐶𝑋)       (𝜑 → (𝐹𝐶)(𝑆 D 𝐹)(1 / ((𝑆 D 𝐹)‘𝐶)))
 
Theoremdvcnv 24584* A weak version of dvcnvre 24626, valid for both real and complex domains but under the hypothesis that the inverse function is already known to be continuous, and the image set is known to be open. A more advanced proof can show that these conditions are unnecessary. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝑆)    &   (𝜑𝑆 ∈ {ℝ, ℂ})    &   (𝜑𝑌𝐾)    &   (𝜑𝐹:𝑋1-1-onto𝑌)    &   (𝜑𝐹 ∈ (𝑌cn𝑋))    &   (𝜑 → dom (𝑆 D 𝐹) = 𝑋)    &   (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))       (𝜑 → (𝑆 D 𝐹) = (𝑥𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(𝐹𝑥)))))
 
Theoremdvexp3 24585* Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)
(𝑁 ∈ ℤ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
 
Theoremdveflem 24586 Derivative of the exponential function at 0. The key step in the proof is eftlub 15458, to show that abs(exp(𝑥) − 1 − 𝑥) ≤ abs(𝑥)↑2 · (3 / 4). (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
0(ℂ D exp)1
 
Theoremdvef 24587 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)
(ℂ D exp) = exp
 
Theoremdvsincos 24588 Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)
((ℂ D sin) = cos ∧ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
 
Theoremdvsin 24589 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)
(ℂ D sin) = cos
 
Theoremdvcos 24590 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)
(ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
 
13.3.1.2  Results on real differentiation
 
Theoremdvferm1lem 24591* Lemma for dvferm 24595. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝑈 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   (𝜑𝑈 ∈ dom (ℝ D 𝐹))    &   (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹𝑦) ≤ (𝐹𝑈))    &   (𝜑 → 0 < ((ℝ D 𝐹)‘𝑈))    &   (𝜑𝑇 ∈ ℝ+)    &   (𝜑 → ∀𝑧 ∈ (𝑋 ∖ {𝑈})((𝑧𝑈 ∧ (abs‘(𝑧𝑈)) < 𝑇) → (abs‘((((𝐹𝑧) − (𝐹𝑈)) / (𝑧𝑈)) − ((ℝ D 𝐹)‘𝑈))) < ((ℝ D 𝐹)‘𝑈)))    &   𝑆 = ((𝑈 + if(𝐵 ≤ (𝑈 + 𝑇), 𝐵, (𝑈 + 𝑇))) / 2)        ¬ 𝜑
 
Theoremdvferm1 24592* One-sided version of dvferm 24595. A point 𝑈 which is the local maximum of its right neighborhood has derivative at most zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝑈 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   (𝜑𝑈 ∈ dom (ℝ D 𝐹))    &   (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹𝑦) ≤ (𝐹𝑈))       (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0)
 
Theoremdvferm2lem 24593* Lemma for dvferm 24595. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝑈 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   (𝜑𝑈 ∈ dom (ℝ D 𝐹))    &   (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹𝑦) ≤ (𝐹𝑈))    &   (𝜑 → ((ℝ D 𝐹)‘𝑈) < 0)    &   (𝜑𝑇 ∈ ℝ+)    &   (𝜑 → ∀𝑧 ∈ (𝑋 ∖ {𝑈})((𝑧𝑈 ∧ (abs‘(𝑧𝑈)) < 𝑇) → (abs‘((((𝐹𝑧) − (𝐹𝑈)) / (𝑧𝑈)) − ((ℝ D 𝐹)‘𝑈))) < -((ℝ D 𝐹)‘𝑈)))    &   𝑆 = ((if(𝐴 ≤ (𝑈𝑇), (𝑈𝑇), 𝐴) + 𝑈) / 2)        ¬ 𝜑
 
Theoremdvferm2 24594* One-sided version of dvferm 24595. A point 𝑈 which is the local maximum of its left neighborhood has derivative at least zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
(𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝑈 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   (𝜑𝑈 ∈ dom (ℝ D 𝐹))    &   (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹𝑦) ≤ (𝐹𝑈))       (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈))
 
Theoremdvferm 24595* Fermat's theorem on stationary points. A point 𝑈 which is a local maximum has derivative equal to zero. (Contributed by Mario Carneiro, 1-Sep-2014.)
(𝜑𝐹:𝑋⟶ℝ)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝑈 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋)    &   (𝜑𝑈 ∈ dom (ℝ D 𝐹))    &   (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹𝑦) ≤ (𝐹𝑈))       (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)
 
Theoremrollelem 24596* Lemma for rolle 24597. (Contributed by Mario Carneiro, 1-Sep-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹𝑦) ≤ (𝐹𝑈))    &   (𝜑𝑈 ∈ (𝐴[,]𝐵))    &   (𝜑 → ¬ 𝑈 ∈ {𝐴, 𝐵})       (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
 
Theoremrolle 24597* Rolle's theorem. If 𝐹 is a real continuous function on [𝐴, 𝐵] which is differentiable on (𝐴, 𝐵), and 𝐹(𝐴) = 𝐹(𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that (ℝ D 𝐹)‘𝑥 = 0. (Contributed by Mario Carneiro, 1-Sep-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → (𝐹𝐴) = (𝐹𝐵))       (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0)
 
Theoremcmvth 24598* Cauchy's Mean Value Theorem. If 𝐹, 𝐺 are real continuous functions on [𝐴, 𝐵] differentiable on (𝐴, 𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that 𝐹' (𝑥) / 𝐺' (𝑥) = (𝐹(𝐴) − 𝐹(𝐵)) / (𝐺(𝐴) − 𝐺(𝐵)). (We express the condition without division, so that we need no nonzero constraints.) (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))       (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)(((𝐹𝐵) − (𝐹𝐴)) · ((ℝ D 𝐺)‘𝑥)) = (((𝐺𝐵) − (𝐺𝐴)) · ((ℝ D 𝐹)‘𝑥)))
 
Theoremmvth 24599* The Mean Value Theorem. If 𝐹 is a real continuous function on [𝐴, 𝐵] which is differentiable on (𝐴, 𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that (ℝ D 𝐹)‘𝑥 is equal to the average slope over [𝐴, 𝐵]. This is Metamath 100 proof #75. (Contributed by Mario Carneiro, 1-Sep-2014.) (Proof shortened by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))       (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = (((𝐹𝐵) − (𝐹𝐴)) / (𝐵𝐴)))
 
Theoremdvlip 24600* A function with derivative bounded by 𝑀 is 𝑀-Lipschitz continuous. (Contributed by Mario Carneiro, 3-Mar-2015.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑀)       ((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹𝑋) − (𝐹𝑌))) ≤ (𝑀 · (abs‘(𝑋𝑌))))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45330
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