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Theorem List for Metamath Proof Explorer - 23501-23600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremitg2mono 23501* The Monotone Convergence Theorem for nonnegative functions. If {(𝐹𝑛):𝑛 ∈ ℕ} is a monotone increasing sequence of positive, measurable, real-valued functions, and 𝐺 is the pointwise limit of the sequence, then (∫2𝐺) is the limit of the sequence {(∫2‘(𝐹𝑛)):𝑛 ∈ ℕ}. (Contributed by Mario Carneiro, 16-Aug-2014.)
𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹𝑛)‘𝑥)), ℝ, < ))    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ MblFn)    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛):ℝ⟶(0[,)+∞))    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))    &   ((𝜑𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹𝑛)‘𝑥) ≤ 𝑦)    &   𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹𝑛))), ℝ*, < )       (𝜑 → (∫2𝐺) = 𝑆)

Theoremitg2i1fseqle 23502* Subject to the conditions coming from mbfi1fseq 23469, the sequence of simple functions are all less than the target function 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))       ((𝜑𝑀 ∈ ℕ) → (𝑃𝑀) ∘𝑟𝐹)

Theoremitg2i1fseq 23503* Subject to the conditions coming from mbfi1fseq 23469, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))    &   𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))       (𝜑 → (∫2𝐹) = sup(ran 𝑆, ℝ*, < ))

Theoremitg2i1fseq2 23504* In an extension to the results of itg2i1fseq 23503, if there is an upper bound on the integrals of the simple functions approaching 𝐹, then 2𝐹 is real and the standard limit relation applies. (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))    &   𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))    &   (𝜑𝑀 ∈ ℝ)    &   ((𝜑𝑘 ∈ ℕ) → (∫1‘(𝑃𝑘)) ≤ 𝑀)       (𝜑𝑆 ⇝ (∫2𝐹))

Theoremitg2i1fseq3 23505* Special case of itg2i1fseq2 23504: if the integral of 𝐹 is a real number, then the standard limit relation holds on the integrals of simple functions approaching 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))    &   𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃𝑚)))    &   (𝜑 → (∫2𝐹) ∈ ℝ)       (𝜑𝑆 ⇝ (∫2𝐹))

(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐺:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐺) ∈ ℝ)    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑃𝑛) ∧ (𝑃𝑛) ∘𝑟 ≤ (𝑃‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))    &   (𝜑𝑄:ℕ⟶dom ∫1)    &   (𝜑 → ∀𝑛 ∈ ℕ (0𝑝𝑟 ≤ (𝑄𝑛) ∧ (𝑄𝑛) ∘𝑟 ≤ (𝑄‘(𝑛 + 1))))    &   (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑄𝑛)‘𝑥)) ⇝ (𝐺𝑥))       (𝜑 → (∫2‘(𝐹𝑓 + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))

Theoremitg2add 23507 The 2 integral is linear. (Measurability is an essential component of this theorem; otherwise consider the characteristic function of a nonmeasurable set and its complement.) (Contributed by Mario Carneiro, 17-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐺:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐺) ∈ ℝ)       (𝜑 → (∫2‘(𝐹𝑓 + 𝐺)) = ((∫2𝐹) + (∫2𝐺)))

Theoremitg2gt0 23508* If the function 𝐹 is strictly positive on a set of positive measure, then the integral of the function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)
(𝜑𝐴 ∈ dom vol)    &   (𝜑 → 0 < (vol‘𝐴))    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝐹 ∈ MblFn)    &   ((𝜑𝑥𝐴) → 0 < (𝐹𝑥))       (𝜑 → 0 < (∫2𝐹))

Theoremitg2cnlem1 23509* Lemma for itgcn 23590. (Contributed by Mario Carneiro, 30-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝐹 ∈ MblFn)    &   (𝜑 → (∫2𝐹) ∈ ℝ)       (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))), ℝ*, < ) = (∫2𝐹))

Theoremitg2cnlem2 23510* Lemma for itgcn 23590. (Contributed by Mario Carneiro, 31-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝐹 ∈ MblFn)    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → ¬ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑀, (𝐹𝑥), 0))) ≤ ((∫2𝐹) − (𝐶 / 2)))       (𝜑 → ∃𝑑 ∈ ℝ+𝑢 ∈ dom vol((vol‘𝑢) < 𝑑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝑢, (𝐹𝑥), 0))) < 𝐶))

Theoremitg2cn 23511* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 23781 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑𝐹 ∈ MblFn)    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)       (𝜑 → ∃𝑑 ∈ ℝ+𝑢 ∈ dom vol((vol‘𝑢) < 𝑑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝑢, (𝐹𝑥), 0))) < 𝐶))

Theoremibllem 23512 Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))

Theoremisibl 23513* The predicate "𝐹 is integrable". The "integrable" predicate corresponds roughly to the range of validity of 𝐴𝐵 d𝑥, which is to say that the expression 𝐴𝐵 d𝑥 doesn't make sense unless (𝑥𝐴𝐵) ∈ 𝐿1. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))    &   ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))    &   (𝜑 → dom 𝐹 = 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)       (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))

Theoremisibl2 23514* The predicate "𝐹 is integrable" when 𝐹 is a mapping operation. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))    &   ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2𝐺) ∈ ℝ)))

Theoremiblmbf 23515 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
(𝐹 ∈ 𝐿1𝐹 ∈ MblFn)

Theoremiblitg 23516* If a function is integrable, then the 2 integrals of the function's decompositions all exist. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))    &   ((𝜑𝑥𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝐾))))    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       ((𝜑𝐾 ∈ ℤ) → (∫2𝐺) ∈ ℝ)

Theoremdfitg 23517* Evaluate the class substitution in df-itg 23373. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝑇 = (ℜ‘(𝐵 / (i↑𝑘)))       𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))))

Theoremitgex 23518 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐴𝐵 d𝑥 ∈ V

Theoremitgeq1f 23519 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Theoremitgeq1 23520* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Theoremnfitg1 23521 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝑥𝐴𝐵 d𝑥

Theoremnfitg 23522* 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 23523* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝑦𝐵    &   𝑥𝐶       𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦

Theoremcbvitgv 23524* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝑥 = 𝑦𝐵 = 𝐶)       𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦

Theoremitgeq2 23525 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(∀𝑥𝐴 𝐵 = 𝐶 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)

Theoremitgresr 23526 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 23527 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)
∫∅𝐴 d𝑥 = 0

Theoremitgz 23528 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 23529* Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)

Theoremitgmpt 23530* Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴((𝑥𝐴𝐵)‘𝑦) d𝑦)

Theoremitgcl 23531* 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 23532* 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 23533* Lemma for itgposval 23543 and itgreval 23544. (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 23534 The zero function is integrable on any measurable set. (Unlike iblconst 23565, this does not require 𝐴 to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)
(𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1)

Theoremiblcnlem1 23535* Lemma for iblcnlem 23536. (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 23536* Expand out the forall in isibl2 23514. (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 23537* Expand out the sum in dfitg 23517. (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 23538* 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 23539* Lemma for iblpos 23540. (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 23540* 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 23541* 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 23542* Lemma for itgposval 23543 and itgreval 23544. (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 23543* 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 23544* 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 23545* Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℝ)

Theoremiblcn 23546* 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 23547* 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 23548* 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 23549* 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 23550* 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 23551* Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)

Theoremiblss 23552* 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 23553* 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 23554* 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 23555 A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)
(𝐹 ∈ dom ∫1𝐹 ∈ 𝐿1)

Theoremitgitg1 23556* 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 23557* 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 23558* 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 23559* 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 23560* 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 23561* 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 23562* 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 23563* 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 23564* 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 23565 A constant function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ 𝐿1)

Theoremitgconst 23566* 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)))

((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 23568* 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 23569* 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)

((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))

((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 23572* 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 23573* 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 23574* 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 23575* Lemma for iblabs 23576. (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 23576* 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 23577* A measurable function is integrable iff its absolute value is integrable. (See iblabs 23576 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 23578* 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 23579* Lemma for itgmulc2 23581: 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 23580* Lemma for itgmulc2 23581: 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 23581* 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 23582* 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 23583* 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 23584* 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 23585* 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 23586* 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 23587* 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 23588 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 23589* 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 23590* Transfer itg2cn 23511 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 23591 Extend class notation with the directed integral.
class ⨜[𝐴𝐵]𝐶 d𝑥

Definitiondf-ditg 23592 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 23593* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝐴 = 𝐵 → ⨜[𝐴𝐶]𝐷 d𝑥 = ⨜[𝐵𝐶]𝐷 d𝑥)

Theoremditgeq2 23594* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
(𝐴 = 𝐵 → ⨜[𝐶𝐴]𝐷 d𝑥 = ⨜[𝐶𝐵]𝐷 d𝑥)

Theoremditgeq3 23595* 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 23601 first and use the equality theorems for df-itg 23373.) (Contributed by Mario Carneiro, 13-Aug-2014.)
(∀𝑥 ∈ ℝ 𝐷 = 𝐸 → ⨜[𝐴𝐵]𝐷 d𝑥 = ⨜[𝐴𝐵]𝐸 d𝑥)

Theoremditgeq3dv 23596* 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 23597 A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
⨜[𝐴𝐵]𝐶 d𝑥 ∈ V

Theoremditg0 23598* Value of the directed integral from a point to itself. (Contributed by Mario Carneiro, 13-Aug-2014.)
⨜[𝐴𝐴]𝐵 d𝑥 = 0

Theoremcbvditg 23599* 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 23600* Change bound variable in a directed integral. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝑥 = 𝑦𝐶 = 𝐷)       ⨜[𝐴𝐵]𝐶 d𝑥 = ⨜[𝐴𝐵]𝐷 d𝑦

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