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

Theoremi1f1lem 23501* Lemma for i1f1 23502 and itg11 23503. (Contributed by Mario Carneiro, 18-Jun-2014.)
𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, 1, 0))       (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (𝐹 “ {1}) = 𝐴))

Theoremi1f1 23502* Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.)
𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, 1, 0))       ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1)

Theoremitg11 23503* The integral of an indicator function is the volume of the set. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, 1, 0))       ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (∫1𝐹) = (vol‘𝐴))

Theoremitg1addlem1 23504* Decompose a preimage, which is always a disjoint union. (Contributed by Mario Carneiro, 25-Jun-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
(𝜑𝐹:𝑋𝑌)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ⊆ (𝐹 “ {𝑘}))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ dom vol)    &   ((𝜑𝑘𝐴) → (vol‘𝐵) ∈ ℝ)       (𝜑 → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵))

Theoremi1faddlem 23505* Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)       ((𝜑𝐴 ∈ ℂ) → ((𝐹𝑓 + 𝐺) “ {𝐴}) = 𝑦 ∈ ran 𝐺((𝐹 “ {(𝐴𝑦)}) ∩ (𝐺 “ {𝑦})))

Theoremi1fmullem 23506* Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)       ((𝜑𝐴 ∈ (ℂ ∖ {0})) → ((𝐹𝑓 · 𝐺) “ {𝐴}) = 𝑦 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝐴 / 𝑦)}) ∩ (𝐺 “ {𝑦})))

Theoremi1fadd 23507 The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)       (𝜑 → (𝐹𝑓 + 𝐺) ∈ dom ∫1)

Theoremi1fmul 23508 The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)       (𝜑 → (𝐹𝑓 · 𝐺) ∈ dom ∫1)

Theoremitg1addlem2 23509* Lemma for itg1add 23513. The function 𝐼 represents the pieces into which we will break up the domain of the sum. Since it is infinite only when both 𝑖 and 𝑗 are zero, we arbitrarily define it to be zero there to simplify the sums that are manipulated in itg1addlem4 23511 and itg1addlem5 23512. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)    &   𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))       (𝜑𝐼:(ℝ × ℝ)⟶ℝ)

(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)    &   𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((𝐹 “ {𝐴}) ∩ (𝐺 “ {𝐵}))))

(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)    &   𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))    &   𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))       (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))

(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)    &   𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))    &   𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))       (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = ((∫1𝐹) + (∫1𝐺)))

Theoremitg1add 23513 The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐺 ∈ dom ∫1)       (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = ((∫1𝐹) + (∫1𝐺)))

Theoremi1fmulclem 23514 Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ∈ ℝ)       (((𝜑𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → (((ℝ × {𝐴}) ∘𝑓 · 𝐹) “ {𝐵}) = (𝐹 “ {(𝐵 / 𝐴)}))

Theoremi1fmulc 23515 A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → ((ℝ × {𝐴}) ∘𝑓 · 𝐹) ∈ dom ∫1)

Theoremitg1mulc 23516 The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)
(𝜑𝐹 ∈ dom ∫1)    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (∫1‘((ℝ × {𝐴}) ∘𝑓 · 𝐹)) = (𝐴 · (∫1𝐹)))

Theoremi1fres 23517* The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside 𝐴.) (Contributed by Mario Carneiro, 29-Jun-2014.)
𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥𝐴, (𝐹𝑥), 0))       ((𝐹 ∈ dom ∫1𝐴 ∈ dom vol) → 𝐺 ∈ dom ∫1)

Theoremi1fpos 23518* The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐺 = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹𝑥), (𝐹𝑥), 0))       (𝐹 ∈ dom ∫1𝐺 ∈ dom ∫1)

Theoremi1fposd 23519* Deduction form of i1fposd 23519. (Contributed by Mario Carneiro, 6-Aug-2014.)
(𝜑 → (𝑥 ∈ ℝ ↦ 𝐴) ∈ dom ∫1)       (𝜑 → (𝑥 ∈ ℝ ↦ if(0 ≤ 𝐴, 𝐴, 0)) ∈ dom ∫1)

Theoremi1fsub 23520 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 23521 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 23522* 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 23523* 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 23524* Approximate version of itg1le 23525. 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 23525 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 23526* 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 23527* Lemma for mbfi1fseq 23533. (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 23528* Lemma for mbfi1fseq 23533. (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 23529* Lemma for mbfi1fseq 23533. (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 23530* Lemma for mbfi1fseq 23533. 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 23531* Lemma for mbfi1fseq 23533. 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 23532* Lemma for mbfi1fseq 23533. Verify that 𝐺 converges pointwise to 𝐹, and wrap up the existence 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 23533* 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 23534* Lemma for mbfi1flim 23535. (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 23535* 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 23536* Lemma for mbfmul 23538. (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 23537 Lemma for mbfmul 23538. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)    &   (𝜑𝐹:𝐴⟶ℝ)    &   (𝜑𝐺:𝐴⟶ℝ)       (𝜑 → (𝐹𝑓 · 𝐺) ∈ MblFn)

Theoremmbfmul 23538 The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
(𝜑𝐹 ∈ MblFn)    &   (𝜑𝐺 ∈ MblFn)       (𝜑 → (𝐹𝑓 · 𝐺) ∈ MblFn)

Theoremitg2lcl 23539* 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 23540* 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 23541* 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 23542* 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 23543 A real function is a nonnegative extended real function if all its values are greater 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 23544 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 23545 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 23546* 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 23547 The integral of a nonnegative real function is greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2𝐹))

Theoremitg2itg1 23548 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 23549 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
(∫2‘(ℝ × {0})) = 0

Theoremitg2lecl 23550 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 23551 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 23552* 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 23553* 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 23554* 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 23567, 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 23555* Approximate version of itg2ub 23545. 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 23556* Approximate version of itg2le 23551. 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 23557* 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 23558 Lemma for itg2mulc 23559. (Contributed by Mario Carneiro, 8-Jul-2014.)
(𝜑𝐹:ℝ⟶(0[,)+∞))    &   (𝜑 → (∫2𝐹) ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ+)       (𝜑 → (∫2‘((ℝ × {𝐴}) ∘𝑓 · 𝐹)) ≤ (𝐴 · (∫2𝐹)))

Theoremitg2mulc 23559 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 23560* Lemma for itg2split 23561. (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 23561* The 2 integral splits under an almost disjoint union. (The proof avoids the use of itg2add 23571 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 23562* Lemma for itg2mono 23565. 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 23565. 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 23526. 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 23563* Lemma for itg2mono 23565. (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 23564* Lemma for itg2mono 23565. (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 23565* 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 23566* Subject to the conditions coming from mbfi1fseq 23533, 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 23567* Subject to the conditions coming from mbfi1fseq 23533, 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 23568* In an extension to the results of itg2i1fseq 23567, 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 23569* Special case of itg2i1fseq2 23568: 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 23571 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 23572* 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 23573* Lemma for itgcn 23654. (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 23574* Lemma for itgcn 23654. (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 23575* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 23845 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 23576 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 23577* 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 23578* 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 23579 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
(𝐹 ∈ 𝐿1𝐹 ∈ MblFn)

Theoremiblitg 23580* 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 23581* Evaluate the class substitution in df-itg 23437. (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 23582 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝐴𝐵 d𝑥 ∈ V

Theoremitgeq1f 23583 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Theoremitgeq1 23584* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Theoremnfitg1 23585 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
𝑥𝐴𝐵 d𝑥

Theoremnfitg 23586* 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 23587* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝑦𝐵    &   𝑥𝐶       𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦

Theoremcbvitgv 23588* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝑥 = 𝑦𝐵 = 𝐶)       𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦

Theoremitgeq2 23589 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
(∀𝑥𝐴 𝐵 = 𝐶 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)

Theoremitgresr 23590 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 23591 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)
∫∅𝐴 d𝑥 = 0

Theoremitgz 23592 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 23593* Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)

Theoremitgmpt 23594* Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)
((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴((𝑥𝐴𝐵)‘𝑦) d𝑦)

Theoremitgcl 23595* 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 23596* 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 23597* Lemma for itgposval 23607 and itgreval 23608. (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 23598 The zero function is integrable on any measurable set. (Unlike iblconst 23629, this does not require 𝐴 to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)
(𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1)

Theoremiblcnlem1 23599* Lemma for iblcnlem 23600. (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 23600* Expand out the forall in isibl2 23578. (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))))

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