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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremitgeq1f 24301 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
 
Theoremitgeq1 24302* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
 
Theoremnfitg1 24303 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝑥𝐴𝐵 d𝑥
 
Theoremnfitg 24304* Bound-variable hypothesis builder for an integral: if 𝑦 is (effectively) not free in 𝐴 and 𝐵, it is not free in 𝐴𝐵 d𝑥. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝑦𝐴    &   𝑦𝐵       𝑦𝐴𝐵 d𝑥
 
Theoremcbvitg 24305* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝑦𝐵    &   𝑥𝐶       𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦
 
Theoremcbvitgv 24306* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝑥 = 𝑦𝐵 = 𝐶)       𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦
 
Theoremitgeq2 24307 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(∀𝑥𝐴 𝐵 = 𝐶 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)
 
Theoremitgresr 24308 The domain of an integral only matters in its intersection with . (Contributed by Mario Carneiro, 29-Jun-2014.)
𝐴𝐵 d𝑥 = ∫(𝐴 ∩ ℝ)𝐵 d𝑥
 
Theoremitg0 24309 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)
∫∅𝐴 d𝑥 = 0
 
Theoremitgz 24310 The integral of zero on any set is zero. (Contributed by Mario Carneiro, 29-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐴0 d𝑥 = 0
 
Theoremitgeq2dv 24311* Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)
 
Theoremitgmpt 24312* Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴((𝑥𝐴𝐵)‘𝑦) d𝑦)
 
Theoremitgcl 24313* The integral of an integrable function is a complex number. This is Metamath 100 proof #86. (Contributed by Mario Carneiro, 29-Jun-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ)
 
Theoremitgvallem 24314* Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(i↑𝐾) = 𝑇       (𝑘 = 𝐾 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0))))
 
Theoremitgvallem3 24315* Lemma for itgposval 24325 and itgreval 24326. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 = 0)       (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) = 0)
 
Theoremibl0 24316 The zero function is integrable on any measurable set. (Unlike iblconst 24347, this does not require 𝐴 to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)
(𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1)
 
Theoremiblcnlem1 24317* Lemma for iblcnlem 24318. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))    &   𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))    &   𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))    &   𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
 
Theoremiblcnlem 24318* Expand out the forall in isibl2 24296. (Contributed by Mario Carneiro, 6-Aug-2014.)
𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))    &   𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))    &   𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))    &   𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
 
Theoremitgcnlem 24319* Expand out the sum in dfitg 24299. (Contributed by Mario Carneiro, 1-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))    &   𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))    &   𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))    &   𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → ∫𝐴𝐵 d𝑥 = ((𝑅𝑆) + (i · (𝑇𝑈))))
 
Theoremiblrelem 24320* Integrability of a real function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)))
 
Theoremiblposlem 24321* Lemma for iblpos 24322. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐵)       (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) = 0)
 
Theoremiblpos 24322* Integrability of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐵)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, 𝐵, 0))) ∈ ℝ)))
 
Theoremiblre 24323* Integrability of a real function. (Contributed by Mario Carneiro, 11-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
 
Theoremitgrevallem1 24324* Lemma for itgposval 24325 and itgreval 24326. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → ∫𝐴𝐵 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))))
 
Theoremitgposval 24325* The integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐵)       (𝜑 → ∫𝐴𝐵 d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, 𝐵, 0))))
 
Theoremitgreval 24326* Decompose the integral of a real function into positive and negative parts. (Contributed by Mario Carneiro, 31-Jul-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
 
Theoremitgrecl 24327* Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℝ)
 
Theoremiblcn 24328* Integrability of a complex function. (Contributed by Mario Carneiro, 6-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1)))
 
Theoremitgcnval 24329* Decompose the integral of a complex function into real and imaginary parts. (Contributed by Mario Carneiro, 6-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
 
Theoremitgre 24330* Real part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (ℜ‘∫𝐴𝐵 d𝑥) = ∫𝐴(ℜ‘𝐵) d𝑥)
 
Theoremitgim 24331* Imaginary part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (ℑ‘∫𝐴𝐵 d𝑥) = ∫𝐴(ℑ‘𝐵) d𝑥)
 
Theoremiblneg 24332* The negative of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (𝑥𝐴 ↦ -𝐵) ∈ 𝐿1)
 
Theoremitgneg 24333* Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)
 
Theoremiblss 24334* A subset of an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 ∈ dom vol)    &   ((𝜑𝑥𝐵) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)       (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)
 
Theoremiblss2 24335* Change the domain of an integrability predicate. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ dom vol)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   ((𝜑𝑥 ∈ (𝐵𝐴)) → 𝐶 = 0)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)       (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)
 
Theoremitgitg2 24336* Transfer an integral using 2 to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)
((𝜑𝑥 ∈ ℝ) → 𝐴 ∈ ℝ)    &   ((𝜑𝑥 ∈ ℝ) → 0 ≤ 𝐴)    &   (𝜑 → (𝑥 ∈ ℝ ↦ 𝐴) ∈ 𝐿1)       (𝜑 → ∫ℝ𝐴 d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ 𝐴)))
 
Theoremi1fibl 24337 A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)
(𝐹 ∈ dom ∫1𝐹 ∈ 𝐿1)
 
Theoremitgitg1 24338* Transfer an integral using 1 to an equivalent integral using . (Contributed by Mario Carneiro, 6-Aug-2014.)
(𝐹 ∈ dom ∫1 → ∫ℝ(𝐹𝑥) d𝑥 = (∫1𝐹))
 
Theoremitgle 24339* Monotonicity of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → ∫𝐴𝐵 d𝑥 ≤ ∫𝐴𝐶 d𝑥)
 
Theoremitgge0 24340* The integral of a positive function is positive. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐵)       (𝜑 → 0 ≤ ∫𝐴𝐵 d𝑥)
 
Theoremitgss 24341* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐴𝐵)    &   ((𝜑𝑥 ∈ (𝐵𝐴)) → 𝐶 = 0)       (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
 
Theoremitgss2 24342* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝐴𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵if(𝑥𝐴, 𝐶, 0) d𝑥)
 
Theoremitgeqa 24343* Approximate equality of integrals. If 𝐶(𝑥) = 𝐷(𝑥) for almost all 𝑥, then 𝐵𝐶(𝑥) d𝑥 = ∫𝐵𝐷(𝑥) d𝑥 and one is integrable iff the other is. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
((𝜑𝑥𝐵) → 𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐵) → 𝐷 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (𝐵𝐴)) → 𝐶 = 𝐷)       (𝜑 → (((𝑥𝐵𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐷) ∈ 𝐿1) ∧ ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥))
 
Theoremitgss3 24344* Expand the set of an integral by a nullset. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ⊆ ℝ)    &   (𝜑 → (vol*‘(𝐵𝐴)) = 0)    &   ((𝜑𝑥𝐵) → 𝐶 ∈ ℂ)       (𝜑 → (((𝑥𝐴𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐶) ∈ 𝐿1) ∧ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥))
 
Theoremitgioo 24345* Equality of integrals on open and closed intervals. (Contributed by Mario Carneiro, 2-Sep-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → ∫(𝐴(,)𝐵)𝐶 d𝑥 = ∫(𝐴[,]𝐵)𝐶 d𝑥)
 
Theoremitgless 24346* Expand the integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Aug-2014.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 ∈ dom vol)    &   ((𝜑𝑥𝐵) → 𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐵) → 0 ≤ 𝐶)    &   (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)       (𝜑 → ∫𝐴𝐶 d𝑥 ≤ ∫𝐵𝐶 d𝑥)
 
Theoremiblconst 24347 A constant function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ 𝐿1)
 
Theoremitgconst 24348* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.)
((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → ∫𝐴𝐵 d𝑥 = (𝐵 · (vol‘𝐴)))
 
Theoremibladdlem 24349* Lemma for ibladd 24350. (Contributed by Mario Carneiro, 17-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐷 = (𝐵 + 𝐶))    &   (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)    &   (𝜑 → (𝑥𝐴𝐶) ∈ MblFn)    &   (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)    &   (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)       (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ∈ ℝ)
 
Theoremibladd 24350* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)       (𝜑 → (𝑥𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿1)
 
Theoremiblsub 24351* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)       (𝜑 → (𝑥𝐴 ↦ (𝐵𝐶)) ∈ 𝐿1)
 
Theoremitgaddlem1 24352* Lemma for itgadd 24354. (Contributed by Mario Carneiro, 17-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐵)    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐶)       (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
 
Theoremitgaddlem2 24353* Lemma for itgadd 24354. (Contributed by Mario Carneiro, 17-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ)       (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
 
Theoremitgadd 24354* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)       (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
 
Theoremitgsub 24355* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)       (𝜑 → ∫𝐴(𝐵𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 − ∫𝐴𝐶 d𝑥))
 
Theoremitgfsum 24356* Take a finite sum of integrals over the same domain. (Contributed by Mario Carneiro, 24-Aug-2014.)
(𝜑𝐴 ∈ dom vol)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑥𝐴𝑘𝐵)) → 𝐶𝑉)    &   ((𝜑𝑘𝐵) → (𝑥𝐴𝐶) ∈ 𝐿1)       (𝜑 → ((𝑥𝐴 ↦ Σ𝑘𝐵 𝐶) ∈ 𝐿1 ∧ ∫𝐴Σ𝑘𝐵 𝐶 d𝑥 = Σ𝑘𝐵𝐴𝐶 d𝑥))
 
Theoremiblabslem 24357* Lemma for iblabs 24358. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (abs‘(𝐹𝐵)), 0))    &   (𝜑 → (𝑥𝐴 ↦ (𝐹𝐵)) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → (𝐹𝐵) ∈ ℝ)       (𝜑 → (𝐺 ∈ MblFn ∧ (∫2𝐺) ∈ ℝ))
 
Theoremiblabs 24358* The absolute value of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (𝑥𝐴 ↦ (abs‘𝐵)) ∈ 𝐿1)
 
Theoremiblabsr 24359* A measurable function is integrable iff its absolute value is integrable. (See iblabs 24358 for the forward implication.) (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ MblFn)    &   (𝜑 → (𝑥𝐴 ↦ (abs‘𝐵)) ∈ 𝐿1)       (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)
 
Theoremiblmulc2 24360* Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (𝑥𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿1)
 
Theoremitgmulc2lem1 24361* Lemma for itgmulc2 24363: positive real case. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐶)    &   ((𝜑𝑥𝐴) → 0 ≤ 𝐵)       (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
 
Theoremitgmulc2lem2 24362* Lemma for itgmulc2 24363: real case. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   (𝜑𝐶 ∈ ℝ)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)       (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
 
Theoremitgmulc2 24363* Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝜑𝐶 ∈ ℂ)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
 
Theoremitgabs 24364* The triangle inequality for integrals. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)
 
Theoremitgsplit 24365* The integral splits under an almost disjoint union. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑 → (vol*‘(𝐴𝐵)) = 0)    &   (𝜑𝑈 = (𝐴𝐵))    &   ((𝜑𝑥𝑈) → 𝐶𝑉)    &   (𝜑 → (𝑥𝐴𝐶) ∈ 𝐿1)    &   (𝜑 → (𝑥𝐵𝐶) ∈ 𝐿1)       (𝜑 → ∫𝑈𝐶 d𝑥 = (∫𝐴𝐶 d𝑥 + ∫𝐵𝐶 d𝑥))
 
Theoremitgspliticc 24366* The integral splits on closed intervals with matching endpoints. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 ∈ (𝐴[,]𝐶))    &   ((𝜑𝑥 ∈ (𝐴[,]𝐶)) → 𝐷𝑉)    &   (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐷) ∈ 𝐿1)    &   (𝜑 → (𝑥 ∈ (𝐵[,]𝐶) ↦ 𝐷) ∈ 𝐿1)       (𝜑 → ∫(𝐴[,]𝐶)𝐷 d𝑥 = (∫(𝐴[,]𝐵)𝐷 d𝑥 + ∫(𝐵[,]𝐶)𝐷 d𝑥))
 
Theoremitgsplitioo 24367* The integral splits on open intervals with matching endpoints. (Contributed by Mario Carneiro, 2-Sep-2014.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 ∈ (𝐴[,]𝐶))    &   ((𝜑𝑥 ∈ (𝐴(,)𝐶)) → 𝐷 ∈ ℂ)    &   (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐷) ∈ 𝐿1)    &   (𝜑 → (𝑥 ∈ (𝐵(,)𝐶) ↦ 𝐷) ∈ 𝐿1)       (𝜑 → ∫(𝐴(,)𝐶)𝐷 d𝑥 = (∫(𝐴(,)𝐵)𝐷 d𝑥 + ∫(𝐵(,)𝐶)𝐷 d𝑥))
 
Theorembddmulibl 24368* A bounded function times an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹𝑦)) ≤ 𝑥) → (𝐹f · 𝐺) ∈ 𝐿1)
 
Theorembddibl 24369* A bounded function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹𝑦)) ≤ 𝑥) → 𝐹 ∈ 𝐿1)
 
Theoremcniccibl 24370 A continuous function on a closed bounded interval is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) → 𝐹 ∈ 𝐿1)
 
Theoremitggt0 24371* The integral of a strictly positive function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)
(𝜑 → 0 < (vol‘𝐴))    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ+)       (𝜑 → 0 < ∫𝐴𝐵 d𝑥)
 
Theoremitgcn 24372* Transfer itg2cn 24293 to the full Lebesgue integral. (Contributed by Mario Carneiro, 1-Sep-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ∃𝑑 ∈ ℝ+𝑢 ∈ dom vol((𝑢𝐴 ∧ (vol‘𝑢) < 𝑑) → ∫𝑢(abs‘𝐵) d𝑥 < 𝐶))
 
13.2.2.2  Lesbesgue directed integral
 
Syntaxcdit 24373 Extend class notation with the directed integral.
class ⨜[𝐴𝐵]𝐶 d𝑥
 
Definitiondf-ditg 24374 Define the directed integral, which is just a regular integral but with a sign change when the limits are interchanged. The 𝐴 and 𝐵 here are the lower and upper limits of the integral, usually written as a subscript and superscript next to the integral sign. We define the region of integration to be an open interval instead of closed so that we can use +∞, -∞ for limits and also integrate up to a singularity at an endpoint. (Contributed by Mario Carneiro, 13-Aug-2014.)
⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
 
Theoremditgeq1 24375* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝐴 = 𝐵 → ⨜[𝐴𝐶]𝐷 d𝑥 = ⨜[𝐵𝐶]𝐷 d𝑥)
 
Theoremditgeq2 24376* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝐴 = 𝐵 → ⨜[𝐶𝐴]𝐷 d𝑥 = ⨜[𝐶𝐵]𝐷 d𝑥)
 
Theoremditgeq3 24377* Equality theorem for the directed integral. (The domain of the equality here is very rough; for more precise bounds one should decompose it with ditgpos 24383 first and use the equality theorems for df-itg 24153.) (Contributed by Mario Carneiro, 13-Aug-2014.)
(∀𝑥 ∈ ℝ 𝐷 = 𝐸 → ⨜[𝐴𝐵]𝐷 d𝑥 = ⨜[𝐴𝐵]𝐸 d𝑥)
 
Theoremditgeq3dv 24378* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
((𝜑𝑥 ∈ ℝ) → 𝐷 = 𝐸)       (𝜑 → ⨜[𝐴𝐵]𝐷 d𝑥 = ⨜[𝐴𝐵]𝐸 d𝑥)
 
Theoremditgex 24379 A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
⨜[𝐴𝐵]𝐶 d𝑥 ∈ V
 
Theoremditg0 24380* Value of the directed integral from a point to itself. (Contributed by Mario Carneiro, 13-Aug-2014.)
⨜[𝐴𝐴]𝐵 d𝑥 = 0
 
Theoremcbvditg 24381* Change bound variable in a directed integral. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝑥 = 𝑦𝐶 = 𝐷)    &   𝑦𝐶    &   𝑥𝐷       ⨜[𝐴𝐵]𝐶 d𝑥 = ⨜[𝐴𝐵]𝐷 d𝑦
 
Theoremcbvditgv 24382* Change bound variable in a directed integral. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝑥 = 𝑦𝐶 = 𝐷)       ⨜[𝐴𝐵]𝐶 d𝑥 = ⨜[𝐴𝐵]𝐷 d𝑦
 
Theoremditgpos 24383* Value of the directed integral in the forward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝐴𝐵)       (𝜑 → ⨜[𝐴𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)
 
Theoremditgneg 24384* Value of the directed integral in the backward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ⨜[𝐵𝐴]𝐶 d𝑥 = -∫(𝐴(,)𝐵)𝐶 d𝑥)
 
Theoremditgcl 24385* Closure of a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐶𝑉)    &   (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿1)       (𝜑 → ⨜[𝐴𝐵]𝐶 d𝑥 ∈ ℂ)
 
Theoremditgswap 24386* Reverse a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐶𝑉)    &   (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐶) ∈ 𝐿1)       (𝜑 → ⨜[𝐵𝐴]𝐶 d𝑥 = -⨜[𝐴𝐵]𝐶 d𝑥)
 
Theoremditgsplitlem 24387* Lemma for ditgsplit 24388. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   (𝜑𝐶 ∈ (𝑋[,]𝑌))    &   ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐷𝑉)    &   (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ 𝐿1)    &   ((𝜓𝜃) ↔ (𝐴𝐵𝐵𝐶))       (((𝜑𝜓) ∧ 𝜃) → ⨜[𝐴𝐶]𝐷 d𝑥 = (⨜[𝐴𝐵]𝐷 d𝑥 + ⨜[𝐵𝐶]𝐷 d𝑥))
 
Theoremditgsplit 24388* This theorem is the raison d'être for the directed integral, because unlike itgspliticc 24366, there is no constraint on the ordering of the points 𝐴, 𝐵, 𝐶 in the domain. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐴 ∈ (𝑋[,]𝑌))    &   (𝜑𝐵 ∈ (𝑋[,]𝑌))    &   (𝜑𝐶 ∈ (𝑋[,]𝑌))    &   ((𝜑𝑥 ∈ (𝑋(,)𝑌)) → 𝐷𝑉)    &   (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐷) ∈ 𝐿1)       (𝜑 → ⨜[𝐴𝐶]𝐷 d𝑥 = (⨜[𝐴𝐵]𝐷 d𝑥 + ⨜[𝐵𝐶]𝐷 d𝑥))
 
13.3  Derivatives
 
13.3.1  Real and complex differentiation
 
13.3.1.1  Derivatives of functions of one complex or real variable
 
Syntaxclimc 24389 The limit operator.
class lim
 
Syntaxcdv 24390 The derivative operator.
class D
 
Syntaxcdvn 24391 The 𝑛-th derivative operator.
class D𝑛
 
Syntaxccpn 24392 The set of 𝑛-times continuously differentiable functions.
class 𝓑C𝑛
 
Definitiondf-limc 24393* Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.)
lim = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦[(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓𝑧))) ∈ (((𝑗t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)})
 
Definitiondf-dv 24394* Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set 𝑠 here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of and is well-behaved when 𝑠 contains no isolated points, we will restrict our attention to the cases 𝑠 = ℝ or 𝑠 = ℂ for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.)
D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)))
 
Definitiondf-dvn 24395* Define the 𝑛-th derivative operator on functions on the complex numbers. This just iterates the derivative operation according to the last argument. (Contributed by Mario Carneiro, 11-Feb-2015.)
D𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ), (ℕ0 × {𝑓})))
 
Definitiondf-cpn 24396* Define the set of 𝑛-times continuously differentiable functions. (Contributed by Stefan O'Rear, 15-Nov-2014.)
𝓑C𝑛 = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓cn→ℂ)}))
 
Theoremreldv 24397 The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 24-Dec-2016.)
Rel (𝑆 D 𝐹)
 
Theoremlimcvallem 24398* Lemma for ellimc 24400. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝐽 = (𝐾t (𝐴 ∪ {𝐵}))    &   𝐾 = (TopOpen‘ℂfld)    &   𝐺 = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))       ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵) → 𝐶 ∈ ℂ))
 
Theoremlimcfval 24399* Value and set bounds on the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝐽 = (𝐾t (𝐴 ∪ {𝐵}))    &   𝐾 = (TopOpen‘ℂfld)       ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 lim 𝐵) = {𝑦 ∣ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑦, (𝐹𝑧))) ∈ ((𝐽 CnP 𝐾)‘𝐵)} ∧ (𝐹 lim 𝐵) ⊆ ℂ))
 
Theoremellimc 24400* Value of the limit predicate. 𝐶 is the limit of the function 𝐹 at 𝐵 if the function 𝐺, formed by adding 𝐵 to the domain of 𝐹 and setting it to 𝐶, is continuous at 𝐵. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝐽 = (𝐾t (𝐴 ∪ {𝐵}))    &   𝐾 = (TopOpen‘ℂfld)    &   𝐺 = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)))
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