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Theorem List for Metamath Proof Explorer - 24401-24500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremitgrevallem1 24401* Lemma for itgposval 24402 and itgreval 24403. (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 24402* 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 24403* 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 24404* Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℝ)
 
Theoremiblcn 24405* 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 24406* 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 24407* 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 24408* 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 24409* 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 24410* Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)
 
Theoremiblss 24411* 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 24412* 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 24413* 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 24414 A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)
(𝐹 ∈ dom ∫1𝐹 ∈ 𝐿1)
 
Theoremitgitg1 24415* 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 24416* 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 24417* 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 24418* 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 24419* 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 24420* 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 24421* 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 24422* 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 24423* 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 24424 A constant function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ 𝐿1)
 
Theoremitgconst 24425* 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 24426* Lemma for ibladd 24427. (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 24427* 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 24428* 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 24429* Lemma for itgadd 24431. (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 24430* Lemma for itgadd 24431. (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 24431* 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 24432* 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 24433* 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 24434* Lemma for iblabs 24435. (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 24435* 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 24436* A measurable function is integrable iff its absolute value is integrable. (See iblabs 24435 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 24437* 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 24438* Lemma for itgmulc2 24440: 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 24439* Lemma for itgmulc2 24440: 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 24440* 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 24441* 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 24442* 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 24443* 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 24444* 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 24445* 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 24446* 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 24447 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)
 
Theorembddiblnc 24448* Choice-free proof of bddibl 24446. (Contributed by Brendan Leahy, 2-Nov-2017.) (Revised by Brendan Leahy, 6-Nov-2017.)
((𝐹 ∈ MblFn ∧ (vol‘dom 𝐹) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹𝑦)) ≤ 𝑥) → 𝐹 ∈ 𝐿1)
 
Theoremcnicciblnc 24449 Choice-free proof of cniccibl 24447. (Contributed by Brendan Leahy, 2-Nov-2017.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) → 𝐹 ∈ 𝐿1)
 
Theoremitggt0 24450* 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 24451* Transfer itg2cn 24370 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 24452 Extend class notation with the directed integral.
class ⨜[𝐴𝐵]𝐶 d𝑥
 
Definitiondf-ditg 24453 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 24454* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝐴 = 𝐵 → ⨜[𝐴𝐶]𝐷 d𝑥 = ⨜[𝐵𝐶]𝐷 d𝑥)
 
Theoremditgeq2 24455* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝐴 = 𝐵 → ⨜[𝐶𝐴]𝐷 d𝑥 = ⨜[𝐶𝐵]𝐷 d𝑥)
 
Theoremditgeq3 24456* 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 24462 first and use the equality theorems for df-itg 24230.) (Contributed by Mario Carneiro, 13-Aug-2014.)
(∀𝑥 ∈ ℝ 𝐷 = 𝐸 → ⨜[𝐴𝐵]𝐷 d𝑥 = ⨜[𝐴𝐵]𝐸 d𝑥)
 
Theoremditgeq3dv 24457* 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 24458 A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
⨜[𝐴𝐵]𝐶 d𝑥 ∈ V
 
Theoremditg0 24459* Value of the directed integral from a point to itself. (Contributed by Mario Carneiro, 13-Aug-2014.)
⨜[𝐴𝐴]𝐵 d𝑥 = 0
 
Theoremcbvditg 24460* 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 24461* Change bound variable in a directed integral. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝑥 = 𝑦𝐶 = 𝐷)       ⨜[𝐴𝐵]𝐶 d𝑥 = ⨜[𝐴𝐵]𝐷 d𝑦
 
Theoremditgpos 24462* 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 24463* 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 24464* 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 24465* 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 24466* Lemma for ditgsplit 24467. (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 24467* This theorem is the raison d'être for the directed integral, because unlike itgspliticc 24443, 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 24468 The limit operator.
class lim
 
Syntaxcdv 24469 The derivative operator.
class D
 
Syntaxcdvn 24470 The 𝑛-th derivative operator.
class D𝑛
 
Syntaxccpn 24471 The set of 𝑛-times continuously differentiable functions.
class 𝓑C𝑛
 
Definitiondf-limc 24472* 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 24473* 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 24474* 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 24475* 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 24476 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 24477* Lemma for ellimc 24479. (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 24478* 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 24479* Value of the limit predicate. 𝐶 is the limit of the function 𝐹 at 𝐵 if the function 𝐺, formed by adding 𝐵 to the domain of 𝐹 and setting it to 𝐶, is continuous at 𝐵. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝐽 = (𝐾t (𝐴 ∪ {𝐵}))    &   𝐾 = (TopOpen‘ℂfld)    &   𝐺 = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝐶, (𝐹𝑧)))    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (𝐶 ∈ (𝐹 lim 𝐵) ↔ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵)))
 
Theoremlimcrcl 24480 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 24481 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
(𝐹 lim 𝐵) ⊆ ℂ
 
Theoremlimcdif 24482 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 24483* 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 24484 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 24485* 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 24486 Lemma for limcflf 24487. (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 24487 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 24488* 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 24489* 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 24490* 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 24491 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 24492 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 24493 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 24494* 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 24495 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 24496* 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 24497 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 24498* 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 24499* 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 24500* 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)))
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