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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dvbss 24501 | 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.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) ⇒ ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝐴) | ||
Theorem | dvbsss 24502 | The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) |
⊢ dom (𝑆 D 𝐹) ⊆ 𝑆 | ||
Theorem | perfdvf 24503 | 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) ⇒ ⊢ ((𝐾 ↾t 𝑆) ∈ Perf → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | ||
Theorem | recnprss 24504 | Both ℝ and ℂ are subsets of ℂ. (Contributed by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | ||
Theorem | recnperf 24505 | Both ℝ and ℂ are perfect subsets of ℂ. (Contributed by Mario Carneiro, 28-Dec-2016.) |
⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝐾 ↾t 𝑆) ∈ Perf) | ||
Theorem | dvfg 24506 | Explicitly write out the functionality condition on derivative for 𝑆 = ℝ and ℂ. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | ||
Theorem | dvf 24507 | 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 | ||
Theorem | dvfcn 24508 | The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ (ℂ D 𝐹):dom (ℂ D 𝐹)⟶ℂ | ||
Theorem | dvreslem 24509* | Lemma for dvres 24511. (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 = (𝐾 ↾t 𝑆) & ⊢ 𝐺 = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) & ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → 𝐵 ⊆ 𝑆) & ⊢ (𝜑 → 𝑦 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝑥(𝑆 D (𝐹 ↾ 𝐵))𝑦 ↔ (𝑥 ∈ ((int‘𝑇)‘𝐵) ∧ 𝑥(𝑆 D 𝐹)𝑦))) | ||
Theorem | dvres2lem 24510* | Lemma for dvres2 24512. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.) |
⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ 𝑇 = (𝐾 ↾t 𝑆) & ⊢ 𝐺 = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) & ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → 𝐵 ⊆ 𝑆) & ⊢ (𝜑 → 𝑦 ∈ ℂ) & ⊢ (𝜑 → 𝑥(𝑆 D 𝐹)𝑦) & ⊢ (𝜑 → 𝑥 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑥(𝐵 D (𝐹 ↾ 𝐵))𝑦) | ||
Theorem | dvres 24511 | Restriction of a derivative. Note that our definition of derivative df-dv 24467 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 = (𝐾 ↾t 𝑆) ⇒ ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑆 D (𝐹 ↾ 𝐵)) = ((𝑆 D 𝐹) ↾ ((int‘𝑇)‘𝐵))) | ||
Theorem | dvres2 24512 | 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 24511, 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) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝑆 D 𝐹) ↾ 𝐵) ⊆ (𝐵 D (𝐹 ↾ 𝐵))) | ||
Theorem | dvres3 24513 | 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 (𝐹 ↾ 𝑆)) = ((ℂ D 𝐹) ↾ 𝑆)) | ||
Theorem | dvres3a 24514 | Restriction of a complex differentiable function to the reals. This version of dvres3 24513 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) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝑆 D (𝐹 ↾ 𝑆)) = ((ℂ D 𝐹) ↾ 𝑆)) | ||
Theorem | dvidlem 24515* | Lemma for dvid 24517 and dvconst 24516. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.) |
⊢ (𝜑 → 𝐹:ℂ⟶ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑧 ≠ 𝑥)) → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = 𝐵) & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝜑 → (ℂ D 𝐹) = (ℂ × {𝐵})) | ||
Theorem | dvconst 24516 | 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})) | ||
Theorem | dvid 24517 | 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}) | ||
Theorem | dvcnp 24518* | 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 = (𝐾 ↾t 𝐴) & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))) ⇒ ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)) | ||
Theorem | dvcnp2 24519 | 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 = (𝐾 ↾t 𝐴) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵)) | ||
Theorem | dvcn 24520 | A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.) |
⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ dom (𝑆 D 𝐹) = 𝐴) → 𝐹 ∈ (𝐴–cn→ℂ)) | ||
Theorem | dvnfval 24521* | 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}))) | ||
Theorem | dvnff 24522 | 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)) | ||
Theorem | dvn0 24523 | 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) | ||
Theorem | dvnp1 24524 | 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))) | ||
Theorem | dvn1 24525 | One times iterated derivative. (Contributed by Mario Carneiro, 1-Jan-2017.) |
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 𝐹)‘1) = (𝑆 D 𝐹)) | ||
Theorem | dvnf 24526 | 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) | ||
Theorem | dvnbss 24527 | 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) | ||
Theorem | dvnadd 24528 | 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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑀))‘𝑁) = ((𝑆 D𝑛 𝐹)‘(𝑀 + 𝑁))) | ||
Theorem | dvn2bss 24529 | An N-times differentiable point is an M-times differentiable point, if 𝑀 ≤ 𝑁. (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)) | ||
Theorem | dvnres 24530 | Multiple derivative version of dvres3a 24514. (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 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆)) | ||
Theorem | cpnfval 24531* | Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) | ||
Theorem | fncpn 24532 | The 𝓑C𝑛 object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘𝑆) Fn ℕ0) | ||
Theorem | elcpn 24533 | 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 ∈ ((𝓑C𝑛‘𝑆)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑pm 𝑆) ∧ ((𝑆 D𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ)))) | ||
Theorem | cpnord 24534 | 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 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((𝓑C𝑛‘𝑆)‘𝑁) ⊆ ((𝓑C𝑛‘𝑆)‘𝑀)) | ||
Theorem | cpncn 24535 | A 𝓑C𝑛 function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C𝑛‘𝑆)‘𝑁)) → 𝐹 ∈ (dom 𝐹–cn→ℂ)) | ||
Theorem | cpnres 24536 | The restriction of a 𝓑C𝑛 function is 𝓑C𝑛. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C𝑛‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ ((𝓑C𝑛‘𝑆)‘𝑁)) | ||
Theorem | dvaddbr 24537 | The sum rule for derivatives at a point. For the (simpler but more limited) function version, see dvadd 24539. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶ℂ) & ⊢ (𝜑 → 𝑌 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑉) & ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)𝐾) & ⊢ (𝜑 → 𝐶(𝑆 D 𝐺)𝐿) & ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘f + 𝐺))(𝐾 + 𝐿)) | ||
Theorem | dvmulbr 24538 | The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmul 24540. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶ℂ) & ⊢ (𝜑 → 𝑌 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑉) & ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)𝐾) & ⊢ (𝜑 → 𝐶(𝑆 D 𝐺)𝐿) & ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘f · 𝐺))((𝐾 · (𝐺‘𝐶)) + (𝐿 · (𝐹‘𝐶)))) | ||
Theorem | dvadd 24539 | The sum rule for derivatives at a point. For the (more general) relation version, see dvaddbr 24537. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶ℂ) & ⊢ (𝜑 → 𝑌 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺)) ⇒ ⊢ (𝜑 → ((𝑆 D (𝐹 ∘f + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶))) | ||
Theorem | dvmul 24540 | The product rule for derivatives at a point. For the (more general) relation version, see dvmulbr 24538. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶ℂ) & ⊢ (𝜑 → 𝑌 ⊆ 𝑆) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺)) ⇒ ⊢ (𝜑 → ((𝑆 D (𝐹 ∘f · 𝐺))‘𝐶) = ((((𝑆 D 𝐹)‘𝐶) · (𝐺‘𝐶)) + (((𝑆 D 𝐺)‘𝐶) · (𝐹‘𝐶)))) | ||
Theorem | dvaddf 24541 | The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) & ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) & ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) ⇒ ⊢ (𝜑 → (𝑆 D (𝐹 ∘f + 𝐺)) = ((𝑆 D 𝐹) ∘f + (𝑆 D 𝐺))) | ||
Theorem | dvmulf 24542 | The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) & ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) & ⊢ (𝜑 → dom (𝑆 D 𝐺) = 𝑋) ⇒ ⊢ (𝜑 → (𝑆 D (𝐹 ∘f · 𝐺)) = (((𝑆 D 𝐹) ∘f · 𝐺) ∘f + ((𝑆 D 𝐺) ∘f · 𝐹))) | ||
Theorem | dvcmul 24543 | The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) ⇒ ⊢ (𝜑 → ((𝑆 D ((𝑆 × {𝐴}) ∘f · 𝐹))‘𝐶) = (𝐴 · ((𝑆 D 𝐹)‘𝐶))) | ||
Theorem | dvcmulf 24544 | The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) ⇒ ⊢ (𝜑 → (𝑆 D ((𝑆 × {𝐴}) ∘f · 𝐹)) = ((𝑆 × {𝐴}) ∘f · (𝑆 D 𝐹))) | ||
Theorem | dvcobr 24545 | The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 24546. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶𝑋) & ⊢ (𝜑 → 𝑌 ⊆ 𝑇) & ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝑇 ⊆ ℂ) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑉) & ⊢ (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾) & ⊢ (𝜑 → 𝐶(𝑇 D 𝐺)𝐿) & ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿)) | ||
Theorem | dvco 24546 | The chain rule for derivatives at a point. For the (more general) relation version, see dvcobr 24545. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝑌⟶𝑋) & ⊢ (𝜑 → 𝑌 ⊆ 𝑇) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → (𝐺‘𝐶) ∈ dom (𝑆 D 𝐹)) & ⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺)) ⇒ ⊢ (𝜑 → ((𝑇 D (𝐹 ∘ 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘(𝐺‘𝐶)) · ((𝑇 D 𝐺)‘𝐶))) | ||
Theorem | dvcof 24547 | The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐺:𝑌⟶𝑋) & ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) & ⊢ (𝜑 → dom (𝑇 D 𝐺) = 𝑌) ⇒ ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (((𝑆 D 𝐹) ∘ 𝐺) ∘f · (𝑇 D 𝐺))) | ||
Theorem | dvcjbr 24548 | The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 24549. (This doesn't follow from dvcobr 24545 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → 𝐶 ∈ dom (ℝ D 𝐹)) ⇒ ⊢ (𝜑 → 𝐶(ℝ D (∗ ∘ 𝐹))(∗‘((ℝ D 𝐹)‘𝐶))) | ||
Theorem | dvcj 24549 | The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 24548. (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))) | ||
Theorem | dvfre 24550 | 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) | ||
Theorem | dvnfre 24551 | 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) | ||
Theorem | dvexp 24552* | 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 ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1))))) | ||
Theorem | dvexp2 24553* | 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 ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))))) | ||
Theorem | dvrec 24554* | 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)))) | ||
Theorem | dvmptres3 24555* | Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑋 ∈ 𝐽) & ⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵)) | ||
Theorem | dvmptid 24556* | Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 𝑥)) = (𝑥 ∈ 𝑆 ↦ 1)) | ||
Theorem | dvmptc 24557* | Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 0)) | ||
Theorem | dvmptcl 24558* | Closure lemma for dvmptcmul 24563 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) | ||
Theorem | dvmptadd 24559* | Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 + 𝐷))) | ||
Theorem | dvmptmul 24560* | Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐶))) = (𝑥 ∈ 𝑋 ↦ ((𝐵 · 𝐶) + (𝐷 · 𝐴)))) | ||
Theorem | dvmptres2 24561* | Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ (𝜑 → 𝑍 ⊆ 𝑋) & ⊢ 𝐽 = (𝐾 ↾t 𝑆) & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ (𝜑 → ((int‘𝐽)‘𝑍) = 𝑌) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑍 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵)) | ||
Theorem | dvmptres 24562* | 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.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ (𝜑 → 𝑌 ⊆ 𝑋) & ⊢ 𝐽 = (𝐾 ↾t 𝑆) & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝑌 ∈ 𝐽) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵)) | ||
Theorem | dvmptcmul 24563* | Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐶 · 𝐴))) = (𝑥 ∈ 𝑋 ↦ (𝐶 · 𝐵))) | ||
Theorem | dvmptdivc 24564* | Function-builder for derivative, division rule for constant divisor. (Contributed by Mario Carneiro, 18-May-2016.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 / 𝐶))) | ||
Theorem | dvmptneg 24565* | Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ -𝐴)) = (𝑥 ∈ 𝑋 ↦ -𝐵)) | ||
Theorem | dvmptsub 24566* | Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 − 𝐶))) = (𝑥 ∈ 𝑋 ↦ (𝐵 − 𝐷))) | ||
Theorem | dvmptcj 24567* | Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ⇒ ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵))) | ||
Theorem | dvmptre 24568* | Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ⇒ ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (ℜ‘𝐵))) | ||
Theorem | dvmptim 24569* | Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ⇒ ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐵))) | ||
Theorem | dvmptntr 24570* | 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.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ 𝐽 = (𝐾 ↾t 𝑆) & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴))) | ||
Theorem | dvmptco 24571* | Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ (𝜑 → (𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) = (𝑦 ∈ 𝑌 ↦ 𝐷)) & ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐸) & ⊢ (𝑦 = 𝐴 → 𝐷 = 𝐹) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐸)) = (𝑥 ∈ 𝑋 ↦ (𝐹 · 𝐵))) | ||
Theorem | dvrecg 24572* | Derivative of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (ℂ ∖ {0})) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐵)) = (𝑥 ∈ 𝑋 ↦ 𝐶)) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐵))) = (𝑥 ∈ 𝑋 ↦ -((𝐴 · 𝐶) / (𝐵↑2)))) | ||
Theorem | dvmptdiv 24573* | Function-builder for derivative, quotient rule. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ (ℂ ∖ {0})) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ ℂ) & ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐶)) = (𝑥 ∈ 𝑋 ↦ 𝐷)) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ (𝐴 / 𝐶))) = (𝑥 ∈ 𝑋 ↦ (((𝐵 · 𝐶) − (𝐷 · 𝐴)) / (𝐶↑2)))) | ||
Theorem | dvmptfsum 24574* | Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.) |
⊢ 𝐽 = (𝐾 ↾t 𝑆) & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑋 ∈ 𝐽) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ⇒ ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵)) | ||
Theorem | dvcnvlem 24575 | Lemma for dvcnvre 24618. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t 𝑆) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑌 ∈ 𝐾) & ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌) & ⊢ (𝜑 → ◡𝐹 ∈ (𝑌–cn→𝑋)) & ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) & ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹)) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹‘𝐶)(𝑆 D ◡𝐹)(1 / ((𝑆 D 𝐹)‘𝐶))) | ||
Theorem | dvcnv 24576* | A weak version of dvcnvre 24618, 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 = (𝐽 ↾t 𝑆) & ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ (𝜑 → 𝑌 ∈ 𝐾) & ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌) & ⊢ (𝜑 → ◡𝐹 ∈ (𝑌–cn→𝑋)) & ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) & ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹)) ⇒ ⊢ (𝜑 → (𝑆 D ◡𝐹) = (𝑥 ∈ 𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(◡𝐹‘𝑥))))) | ||
Theorem | dvexp3 24577* | Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.) |
⊢ (𝑁 ∈ ℤ → (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑥↑𝑁))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝑁 · (𝑥↑(𝑁 − 1))))) | ||
Theorem | dveflem 24578 | Derivative of the exponential function at 0. The key step in the proof is eftlub 15464, 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 | ||
Theorem | dvef 24579 | Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.) |
⊢ (ℂ D exp) = exp | ||
Theorem | dvsincos 24580 | 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))) | ||
Theorem | dvsin 24581 | Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (ℂ D sin) = cos | ||
Theorem | dvcos 24582 | Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.) |
⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)) | ||
Theorem | dvferm1lem 24583* | Lemma for dvferm 24587. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℝ) & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋) & ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹)) & ⊢ (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) & ⊢ (𝜑 → 0 < ((ℝ D 𝐹)‘𝑈)) & ⊢ (𝜑 → 𝑇 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑋 ∖ {𝑈})((𝑧 ≠ 𝑈 ∧ (abs‘(𝑧 − 𝑈)) < 𝑇) → (abs‘((((𝐹‘𝑧) − (𝐹‘𝑈)) / (𝑧 − 𝑈)) − ((ℝ D 𝐹)‘𝑈))) < ((ℝ D 𝐹)‘𝑈))) & ⊢ 𝑆 = ((𝑈 + if(𝐵 ≤ (𝑈 + 𝑇), 𝐵, (𝑈 + 𝑇))) / 2) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | dvferm1 24584* | One-sided version of dvferm 24587. 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℝ) & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋) & ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹)) & ⊢ (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) ⇒ ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0) | ||
Theorem | dvferm2lem 24585* | Lemma for dvferm 24587. (Contributed by Mario Carneiro, 24-Feb-2015.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℝ) & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋) & ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹)) & ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) & ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) < 0) & ⊢ (𝜑 → 𝑇 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑧 ∈ (𝑋 ∖ {𝑈})((𝑧 ≠ 𝑈 ∧ (abs‘(𝑧 − 𝑈)) < 𝑇) → (abs‘((((𝐹‘𝑧) − (𝐹‘𝑈)) / (𝑧 − 𝑈)) − ((ℝ D 𝐹)‘𝑈))) < -((ℝ D 𝐹)‘𝑈))) & ⊢ 𝑆 = ((if(𝐴 ≤ (𝑈 − 𝑇), (𝑈 − 𝑇), 𝐴) + 𝑈) / 2) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | dvferm2 24586* | One-sided version of dvferm 24587. 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℝ) & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋) & ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹)) & ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) ⇒ ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈)) | ||
Theorem | dvferm 24587* | 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.) |
⊢ (𝜑 → 𝐹:𝑋⟶ℝ) & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋) & ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹)) & ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) ⇒ ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) | ||
Theorem | rollelem 24588* | Lemma for rolle 24589. (Contributed by Mario Carneiro, 1-Sep-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) & ⊢ (𝜑 → 𝑈 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → ¬ 𝑈 ∈ {𝐴, 𝐵}) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) | ||
Theorem | rolle 24589* | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = 0) | ||
Theorem | cmvth 24590* | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)(((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D 𝐺)‘𝑥)) = (((𝐺‘𝐵) − (𝐺‘𝐴)) · ((ℝ D 𝐹)‘𝑥))) | ||
Theorem | mvth 24591* | 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.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) = (((𝐹‘𝐵) − (𝐹‘𝐴)) / (𝐵 − 𝐴))) | ||
Theorem | dvlip 24592* | A function with derivative bounded by 𝑀 is 𝑀-Lipschitz continuous. (Contributed by Mario Carneiro, 3-Mar-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) & ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑀) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ (𝐴[,]𝐵) ∧ 𝑌 ∈ (𝐴[,]𝐵))) → (abs‘((𝐹‘𝑋) − (𝐹‘𝑌))) ≤ (𝑀 · (abs‘(𝑋 − 𝑌)))) | ||
Theorem | dvlipcn 24593* | A complex function with derivative bounded by 𝑀 on an open ball is 𝑀-Lipschitz continuous. (Contributed by Mario Carneiro, 18-Mar-2015.) |
⊢ (𝜑 → 𝑋 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝐵 = (𝐴(ball‘(abs ∘ − ))𝑅) & ⊢ (𝜑 → 𝐵 ⊆ dom (ℂ D 𝐹)) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (abs‘((ℂ D 𝐹)‘𝑥)) ≤ 𝑀) ⇒ ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (abs‘((𝐹‘𝑌) − (𝐹‘𝑍))) ≤ (𝑀 · (abs‘(𝑌 − 𝑍)))) | ||
Theorem | dvlip2 24594* | Combine the results of dvlip 24592 and dvlipcn 24593 into one. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Mario Carneiro, 8-Sep-2015.) |
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) & ⊢ 𝐽 = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ 𝐵 = (𝐴(ball‘𝐽)𝑅) & ⊢ (𝜑 → 𝐵 ⊆ dom (𝑆 D 𝐹)) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (abs‘((𝑆 D 𝐹)‘𝑥)) ≤ 𝑀) ⇒ ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (abs‘((𝐹‘𝑌) − (𝐹‘𝑍))) ≤ (𝑀 · (abs‘(𝑌 − 𝑍)))) | ||
Theorem | c1liplem1 24595* | Lemma for c1lip1 24596. (Contributed by Stefan O'Rear, 15-Nov-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℝ)) & ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ 𝐾 = sup((abs “ ((ℝ D 𝐹) “ (𝐴[,]𝐵))), ℝ, < ) ⇒ ⊢ (𝜑 → (𝐾 ∈ ℝ ∧ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝐾 · (abs‘(𝑦 − 𝑥)))))) | ||
Theorem | c1lip1 24596* | C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℝ)) & ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) & ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) | ||
Theorem | c1lip2 24597* | C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝓑C𝑛‘ℝ)‘1)) & ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) | ||
Theorem | c1lip3 24598* | C^1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝐹 ↾ ℝ) ∈ ((𝓑C𝑛‘ℝ)‘1)) & ⊢ (𝜑 → (𝐹 “ ℝ) ⊆ ℝ) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) ≤ (𝑘 · (abs‘(𝑦 − 𝑥)))) | ||
Theorem | dveq0 24599 | If a continuous function has zero derivative at all points on the interior of a closed interval, then it must be a constant function. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by Mario Carneiro, 3-Mar-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) & ⊢ (𝜑 → (ℝ D 𝐹) = ((𝐴(,)𝐵) × {0})) ⇒ ⊢ (𝜑 → 𝐹 = ((𝐴[,]𝐵) × {(𝐹‘𝐴)})) | ||
Theorem | dv11cn 24600 | Two functions defined on a ball whose derivatives are the same and which are equal at any given point 𝐶 in the ball must be equal everywhere. (Contributed by Mario Carneiro, 31-Mar-2015.) |
⊢ 𝑋 = (𝐴(ball‘(abs ∘ − ))𝑅) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑅 ∈ ℝ*) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) & ⊢ (𝜑 → dom (ℂ D 𝐹) = 𝑋) & ⊢ (𝜑 → (ℂ D 𝐹) = (ℂ D 𝐺)) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → (𝐹‘𝐶) = (𝐺‘𝐶)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) |
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