Home Metamath Proof ExplorerTheorem List (p. 244 of 449) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28687) Hilbert Space Explorer (28688-30210) Users' Mathboxes (30211-44897)

Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremi1fsub 24301 The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.)
((𝐹 ∈ dom ∫1𝐺 ∈ dom ∫1) → (𝐹f𝐺) ∈ dom ∫1)

Theoremitg1sub 24302 The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.)
((𝐹 ∈ dom ∫1𝐺 ∈ dom ∫1) → (∫1‘(𝐹f𝐺)) = ((∫1𝐹) − (∫1𝐺)))

Theoremitg10a 24303* The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹𝑥) = 0)       (𝜑 → (∫1𝐹) = 0)

Theoremitg1ge0a 24304* The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ (𝐹𝑥))       (𝜑 → 0 ≤ (∫1𝐹))

Theoremitg1lea 24305* Approximate version of itg1le 24306. If 𝐹𝐺 for almost all 𝑥, then 1𝐹 ≤ ∫1𝐺. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   (𝜑𝐺 ∈ dom ∫1)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹𝑥) ≤ (𝐺𝑥))       (𝜑 → (∫1𝐹) ≤ (∫1𝐺))

Theoremitg1le 24306 If one simple function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
((𝐹 ∈ dom ∫1𝐺 ∈ dom ∫1𝐹r𝐺) → (∫1𝐹) ≤ (∫1𝐺))

Theoremitg1climres 24307* Restricting the simple function 𝐹 to the increasing sequence 𝐴(𝑛) of measurable sets whose union is yields a sequence of simple functions whose integrals approach the integral of 𝐹. (Contributed by Mario Carneiro, 15-Aug-2014.)
(𝜑𝐴:ℕ⟶dom vol)    &   ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ (𝐴‘(𝑛 + 1)))    &   (𝜑 ran 𝐴 = ℝ)    &   (𝜑𝐹 ∈ dom ∫1)    &   𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴𝑛), (𝐹𝑥), 0))       (𝜑 → (𝑛 ∈ ℕ ↦ (∫1𝐺)) ⇝ (∫1𝐹))

Theoremmbfi1fseqlem1 24308* Lemma for mbfi1fseq 24314. (Contributed by Mario Carneiro, 16-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))       (𝜑𝐽:(ℕ × ℝ)⟶(0[,)+∞))

Theoremmbfi1fseqlem2 24309* Lemma for mbfi1fseq 24314. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))       (𝐴 ∈ ℕ → (𝐺𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))

Theoremmbfi1fseqlem3 24310* Lemma for mbfi1fseq 24314. (Contributed by Mario Carneiro, 16-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))       ((𝜑𝐴 ∈ ℕ) → (𝐺𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))

Theoremmbfi1fseqlem4 24311* Lemma for mbfi1fseq 24314. This lemma is not as interesting as it is long - it is simply checking that 𝐺 is in fact a sequence of simple functions, by verifying that its range is in (0...𝑛2↑𝑛) / (2↑𝑛) (which is to say, the numbers from 0 to 𝑛 in increments of 1 / (2↑𝑛)), and also that the preimage of each point 𝑘 is measurable, because it is equal to (-𝑛[,]𝑛) ∩ (𝐹 “ (𝑘[,)𝑘 + 1 / (2↑𝑛))) for 𝑘 < 𝑛 and (-𝑛[,]𝑛) ∩ (𝐹 “ (𝑘[,)+∞)) for 𝑘 = 𝑛. (Contributed by Mario Carneiro, 16-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))       (𝜑𝐺:ℕ⟶dom ∫1)

Theoremmbfi1fseqlem5 24312* Lemma for mbfi1fseq 24314. Verify that 𝐺 describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))       ((𝜑𝐴 ∈ ℕ) → (0𝑝r ≤ (𝐺𝐴) ∧ (𝐺𝐴) ∘r ≤ (𝐺‘(𝐴 + 1))))

Theoremmbfi1fseqlem6 24313* Lemma for mbfi1fseq 24314. Verify that 𝐺 converges pointwise to 𝐹, and wrap up the existential quantifier. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))    &   𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))       (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))

Theoremmbfi1fseq 24314* A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function 𝐺 and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶(0[,)+∞))       (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝r ≤ (𝑔𝑛) ∧ (𝑔𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))

Theoremmbfi1flimlem 24315* Lemma for mbfi1flim 24316. (Contributed by Mario Carneiro, 5-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:ℝ⟶ℝ)       (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))

Theoremmbfi1flim 24316* Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐹:𝐴⟶ℝ)       (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥𝐴 (𝑛 ∈ ℕ ↦ ((𝑔𝑛)‘𝑥)) ⇝ (𝐹𝑥)))

Theoremmbfmullem2 24317* Lemma for mbfmul 24319. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐺:𝐴⟶ℝ)    &   (𝜑𝑃:ℕ⟶dom ∫1)    &   ((𝜑𝑥𝐴) → (𝑛 ∈ ℕ ↦ ((𝑃𝑛)‘𝑥)) ⇝ (𝐹𝑥))    &   (𝜑𝑄:ℕ⟶dom ∫1)    &   ((𝜑𝑥𝐴) → (𝑛 ∈ ℕ ↦ ((𝑄𝑛)‘𝑥)) ⇝ (𝐺𝑥))       (𝜑 → (𝐹f · 𝐺) ∈ MblFn)

Theoremmbfmullem 24318 Lemma for mbfmul 24319. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐺:𝐴⟶ℝ)       (𝜑 → (𝐹f · 𝐺) ∈ MblFn)

Theoremmbfmul 24319 The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)       (𝜑 → (𝐹f · 𝐺) ∈ MblFn)

Theoremitg2lcl 24320* The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))}       𝐿 ⊆ ℝ*

Theoremitg2val 24321* Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))}       (𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) = sup(𝐿, ℝ*, < ))

Theoremitg2l 24322* Elementhood in the set 𝐿 of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))}       (𝐴𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝐴 = (∫1𝑔)))

Theoremitg2lr 24323* Sufficient condition for elementhood in the set 𝐿. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔r𝐹𝑥 = (∫1𝑔))}       ((𝐺 ∈ dom ∫1𝐺r𝐹) → (∫1𝐺) ∈ 𝐿)

Theoremxrge0f 24324 A real function is a nonnegative extended real function if all its values are greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 28-Jul-2014.)
((𝐹:ℝ⟶ℝ ∧ 0𝑝r𝐹) → 𝐹:ℝ⟶(0[,]+∞))

Theoremitg2cl 24325 The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐹:ℝ⟶(0[,]+∞) → (∫2𝐹) ∈ ℝ*)

Theoremitg2ub 24326 The integral of a nonnegative real function 𝐹 is an upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐺 ∈ dom ∫1𝐺r𝐹) → (∫1𝐺) ≤ (∫2𝐹))

Theoremitg2leub 24327* Any upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹 is greater than (∫2𝐹), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2𝐹) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔r𝐹 → (∫1𝑔) ≤ 𝐴)))

Theoremitg2ge0 24328 The integral of a nonnegative real function is greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2𝐹))

Theoremitg2itg1 24329 The integral of a nonnegative simple function using 2 is the same as its value under 1. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹 ∈ dom ∫1 ∧ 0𝑝r𝐹) → (∫2𝐹) = (∫1𝐹))

Theoremitg20 24330 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
(∫2‘(ℝ × {0})) = 0

Theoremitg2lecl 24331 If an 2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2𝐹) ≤ 𝐴) → (∫2𝐹) ∈ ℝ)

Theoremitg2le 24332 If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)
((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ 𝐹r𝐺) → (∫2𝐹) ≤ (∫2𝐺))

Theoremitg2const 24333* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ (0[,)+∞)) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, 𝐵, 0))) = (𝐵 · (vol‘𝐴)))

Theoremitg2const2 24334* When the base set of a constant function has infinite volume, the integral is also infinite and vice-versa. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ+) → ((vol‘𝐴) ∈ ℝ ↔ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥𝐴, 𝐵, 0))) ∈ ℝ))

Theoremitg2seq 24335* Definitional property of the 2 integral: for any function 𝐹 there is a countable sequence 𝑔 of simple functions less than 𝐹 whose integrals converge to the integral of 𝐹. (This theorem is for the most part unnecessary in lieu of itg2i1fseq 24348, but unlike that theorem this one doesn't require 𝐹 to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.)
(𝐹:ℝ⟶(0[,]+∞) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (𝑔𝑛) ∘r𝐹 ∧ (∫2𝐹) = sup(ran (𝑛 ∈ ℕ ↦ (∫1‘(𝑔𝑛))), ℝ*, < )))

Theoremitg2uba 24336* Approximate version of itg2ub 24326. If 𝐹 approximately dominates 𝐺, then 1𝐺 ≤ ∫2𝐹. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,]+∞))    &   (𝜑𝐺 ∈ dom ∫1)    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐺𝑥) ≤ (𝐹𝑥))       (𝜑 → (∫1𝐺) ≤ (∫2𝐹))

Theoremitg2lea 24337* Approximate version of itg2le 24332. If 𝐹𝐺 for almost all 𝑥, then 2𝐹 ≤ ∫2𝐺. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,]+∞))    &   (𝜑𝐺:ℝ⟶(0[,]+∞))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹𝑥) ≤ (𝐺𝑥))       (𝜑 → (∫2𝐹) ≤ (∫2𝐺))

Theoremitg2eqa 24338* Approximate equality of integrals. If 𝐹 = 𝐺 for almost all 𝑥, then 2𝐹 = ∫2𝐺. (Contributed by Mario Carneiro, 12-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,]+∞))    &   (𝜑𝐺:ℝ⟶(0[,]+∞))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑 → (vol*‘𝐴) = 0)    &   ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹𝑥) = (𝐺𝑥))       (𝜑 → (∫2𝐹) = (∫2𝐺))

Theoremitg2mulclem 24339 Lemma for itg2mulc 24340. (Contributed by Mario Carneiro, 8-Jul-2014.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ+)       (𝜑 → (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ≤ (𝐴 · (∫2𝐹)))

Theoremitg2mulc 24340 The integral of a nonnegative constant times a function is the constant times the integral of the original function. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐴 ∈ (0[,)+∞))       (𝜑 → (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫2𝐹)))

Theoremitg2splitlem 24341* Lemma for itg2split 24342. (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐴 ∈ dom vol)    &   (𝜑𝐵 ∈ dom vol)    &   (𝜑 → (vol*‘(𝐴𝐵)) = 0)    &   (𝜑𝑈 = (𝐴𝐵))    &   ((𝜑𝑥𝑈) → 𝐶 ∈ (0[,]+∞))    &   𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, 𝐶, 0))    &   𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥𝐵, 𝐶, 0))    &   𝐻 = (𝑥 ∈ ℝ ↦ if(𝑥𝑈, 𝐶, 0))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑 → (∫2𝐺) ∈ ℝ)       (𝜑 → (∫2𝐻) ≤ ((∫2𝐹) + (∫2𝐺)))

Theoremitg2split 24342* The 2 integral splits under an almost disjoint union. (The proof avoids the use of itg2add 24352 which requires CC.) (Contributed by Mario Carneiro, 11-Aug-2014.)
(𝜑𝐴 ∈ dom vol)    &   (𝜑𝐵 ∈ dom vol)    &   (𝜑 → (vol*‘(𝐴𝐵)) = 0)    &   (𝜑𝑈 = (𝐴𝐵))    &   ((𝜑𝑥𝑈) → 𝐶 ∈ (0[,]+∞))    &   𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, 𝐶, 0))    &   𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥𝐵, 𝐶, 0))    &   𝐻 = (𝑥 ∈ ℝ ↦ if(𝑥𝑈, 𝐶, 0))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑 → (∫2𝐺) ∈ ℝ)       (𝜑 → (∫2𝐻) = ((∫2𝐹) + (∫2𝐺)))

Theoremitg2monolem1 24343* Lemma for itg2mono 24346. We show that for any constant 𝑡 less than one, 𝑡 · ∫1𝐻 is less than 𝑆, and so 1𝐻𝑆, which is one half of the equality in itg2mono 24346. Consider the sequence 𝐴(𝑛) = {𝑥𝑡 · 𝐻𝐹(𝑛)}. This is an increasing sequence of measurable sets whose union is , and so 𝐻𝐴(𝑛) has an integral which equals 1𝐻 in the limit, by itg1climres 24307. Then by taking the limit in (𝑡 · 𝐻) ↾ 𝐴(𝑛) ≤ 𝐹(𝑛), we get 𝑡 · ∫1𝐻𝑆 as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-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*, < )    &   (𝜑𝑇 ∈ (0(,)1))    &   (𝜑𝐻 ∈ dom ∫1)    &   (𝜑𝐻r𝐺)    &   (𝜑𝑆 ∈ ℝ)    &   𝐴 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻𝑥)) ≤ ((𝐹𝑛)‘𝑥)})       (𝜑 → (𝑇 · (∫1𝐻)) ≤ 𝑆)

Theoremitg2monolem2 24344* Lemma for itg2mono 24346. (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*, < )    &   (𝜑𝑃 ∈ dom ∫1)    &   (𝜑𝑃r𝐺)    &   (𝜑 → ¬ (∫1𝑃) ≤ 𝑆)       (𝜑𝑆 ∈ ℝ)

Theoremitg2monolem3 24345* Lemma for itg2mono 24346. (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*, < )    &   (𝜑𝑃 ∈ dom ∫1)    &   (𝜑𝑃r𝐺)    &   (𝜑 → ¬ (∫1𝑃) ≤ 𝑆)       (𝜑 → (∫1𝑃) ≤ 𝑆)

Theoremitg2mono 24346* 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 24347* Subject to the conditions coming from mbfi1fseq 24314, 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 24348* Subject to the conditions coming from mbfi1fseq 24314, 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 24349* In an extension to the results of itg2i1fseq 24348, 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 24350* Special case of itg2i1fseq2 24349: 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 24352 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 24353* 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 24354* Lemma for itgcn 24435. (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 24355* Lemma for itgcn 24435. (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 24356* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 24626 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 24357 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 24358* 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 24359* 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 24360 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
(𝐹 ∈ 𝐿1𝐹 ∈ MblFn)

Theoremiblitg 24361* 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 24362* Evaluate the class substitution in df-itg 24216. (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 24363 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐴𝐵 d𝑥 ∈ V

Theoremitgeq1f 24364 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Theoremitgeq1 24365* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Theoremnfitg1 24366 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝑥𝐴𝐵 d𝑥

Theoremnfitg 24367* 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 24368* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝑦𝐵    &   𝑥𝐶       𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦

Theoremcbvitgv 24369* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝑥 = 𝑦𝐵 = 𝐶)       𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦

Theoremitgeq2 24370 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(∀𝑥𝐴 𝐵 = 𝐶 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)

Theoremitgresr 24371 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 24372 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)
∫∅𝐴 d𝑥 = 0

Theoremitgz 24373 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 24374* Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)

Theoremitgmpt 24375* Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴((𝑥𝐴𝐵)‘𝑦) d𝑦)

Theoremitgcl 24376* 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 24377* 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 24378* Lemma for itgposval 24388 and itgreval 24389. (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 24379 The zero function is integrable on any measurable set. (Unlike iblconst 24410, this does not require 𝐴 to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)
(𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1)

Theoremiblcnlem1 24380* Lemma for iblcnlem 24381. (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 24381* Expand out the forall in isibl2 24359. (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 24382* Expand out the sum in dfitg 24362. (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 24383* 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 24384* Lemma for iblpos 24385. (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 24385* 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 24386* 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 24387* Lemma for itgposval 24388 and itgreval 24389. (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 24388* 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 24389* 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 24390* Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℝ)

Theoremiblcn 24391* 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 24392* 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 24393* 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 24394* 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 24395* 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 24396* Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝜑 → (𝑥𝐴𝐵) ∈ 𝐿1)       (𝜑 → -∫𝐴𝐵 d𝑥 = ∫𝐴-𝐵 d𝑥)

Theoremiblss 24397* 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 24398* 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 24399* 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 24400 A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)
(𝐹 ∈ dom ∫1𝐹 ∈ 𝐿1)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44897
 Copyright terms: Public domain < Previous  Next >