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Type | Label | Description |
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Statement | ||
Theorem | itg1addlem2 24301* | Lemma for itg1add 24305. 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 24303 and itg1addlem5 24304. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) & ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))) ⇒ ⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ) | ||
Theorem | itg1addlem3 24302* | Lemma for itg1add 24305. (Contributed by Mario Carneiro, 26-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ 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})))) | ||
Theorem | itg1addlem4 24303* | Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ 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‘(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧))) | ||
Theorem | itg1addlem5 24304* | Lemma for itg1add . (Contributed by Mario Carneiro, 27-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ 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‘(𝐹 ∘f + 𝐺)) = ((∫1‘𝐹) + (∫1‘𝐺))) | ||
Theorem | itg1add 24305 | The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) ⇒ ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = ((∫1‘𝐹) + (∫1‘𝐺))) | ||
Theorem | i1fmulclem 24306 | Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → (◡((ℝ × {𝐴}) ∘f · 𝐹) “ {𝐵}) = (◡𝐹 “ {(𝐵 / 𝐴)})) | ||
Theorem | i1fmulc 24307 | A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ((ℝ × {𝐴}) ∘f · 𝐹) ∈ dom ∫1) | ||
Theorem | itg1mulc 24308 | The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (∫1‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫1‘𝐹))) | ||
Theorem | i1fres 24309* | 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(𝑥 ∈ 𝐴, (𝐹‘𝑥), 0)) ⇒ ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐴 ∈ dom vol) → 𝐺 ∈ dom ∫1) | ||
Theorem | i1fpos 24310* | The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0)) ⇒ ⊢ (𝐹 ∈ dom ∫1 → 𝐺 ∈ dom ∫1) | ||
Theorem | i1fposd 24311* | Deduction form of i1fposd 24311. (Contributed by Mario Carneiro, 6-Aug-2014.) |
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ 𝐴) ∈ dom ∫1) ⇒ ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(0 ≤ 𝐴, 𝐴, 0)) ∈ dom ∫1) | ||
Theorem | i1fsub 24312 | The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.) |
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (𝐹 ∘f − 𝐺) ∈ dom ∫1) | ||
Theorem | itg1sub 24313 | The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.) |
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘(𝐹 ∘f − 𝐺)) = ((∫1‘𝐹) − (∫1‘𝐺))) | ||
Theorem | itg10a 24314* | The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 0) ⇒ ⊢ (𝜑 → (∫1‘𝐹) = 0) | ||
Theorem | itg1ge0a 24315* | The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → 0 ≤ (∫1‘𝐹)) | ||
Theorem | itg1lea 24316* | Approximate version of itg1le 24317. If 𝐹 ≤ 𝐺 for almost all 𝑥, then ∫1𝐹 ≤ ∫1𝐺. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (∫1‘𝐹) ≤ (∫1‘𝐺)) | ||
Theorem | itg1le 24317 | 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 ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) → (∫1‘𝐹) ≤ (∫1‘𝐺)) | ||
Theorem | itg1climres 24318* | Restricting the simple function 𝐹 to the increasing sequence 𝐴(𝑛) of measurable sets whose union is ℝ yields a sequence of simple functions whose integrals approach the integral of 𝐹. (Contributed by Mario Carneiro, 15-Aug-2014.) |
⊢ (𝜑 → 𝐴:ℕ⟶dom vol) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1))) & ⊢ (𝜑 → ∪ ran 𝐴 = ℝ) & ⊢ (𝜑 → 𝐹 ∈ dom ∫1) & ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0)) ⇒ ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ (∫1‘𝐹)) | ||
Theorem | mbfi1fseqlem1 24319* | Lemma for mbfi1fseq 24325. (Contributed by Mario Carneiro, 16-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) ⇒ ⊢ (𝜑 → 𝐽:(ℕ × ℝ)⟶(0[,)+∞)) | ||
Theorem | mbfi1fseqlem2 24320* | Lemma for mbfi1fseq 24325. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ⇒ ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))) | ||
Theorem | mbfi1fseqlem3 24321* | Lemma for mbfi1fseq 24325. (Contributed by Mario Carneiro, 16-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))) | ||
Theorem | mbfi1fseqlem4 24322* | Lemma for mbfi1fseq 24325. 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.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ⇒ ⊢ (𝜑 → 𝐺:ℕ⟶dom ∫1) | ||
Theorem | mbfi1fseqlem5 24323* | Lemma for mbfi1fseq 24325. Verify that 𝐺 describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) & ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (0𝑝 ∘r ≤ (𝐺‘𝐴) ∧ (𝐺‘𝐴) ∘r ≤ (𝐺‘(𝐴 + 1)))) | ||
Theorem | mbfi1fseqlem6 24324* | Lemma for mbfi1fseq 24325. 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.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (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 (0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))) | ||
Theorem | mbfi1fseq 24325* | 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.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))) | ||
Theorem | mbfi1flimlem 24326* | Lemma for mbfi1flim 24327. (Contributed by Mario Carneiro, 5-Sep-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶ℝ) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))) | ||
Theorem | mbfi1flim 24327* | Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ 𝐴 (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))) | ||
Theorem | mbfmullem2 24328* | Lemma for mbfmul 24330. (Contributed by Mario Carneiro, 7-Sep-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐺 ∈ MblFn) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐺:𝐴⟶ℝ) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) & ⊢ (𝜑 → 𝑄:ℕ⟶dom ∫1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ ℕ ↦ ((𝑄‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ MblFn) | ||
Theorem | mbfmullem 24329 | Lemma for mbfmul 24330. (Contributed by Mario Carneiro, 7-Sep-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐺 ∈ MblFn) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐺:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ MblFn) | ||
Theorem | mbfmul 24330 | The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐺 ∈ MblFn) ⇒ ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ MblFn) | ||
Theorem | itg2lcl 24331* | The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⇒ ⊢ 𝐿 ⊆ ℝ* | ||
Theorem | itg2val 24332* | Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⇒ ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < )) | ||
Theorem | itg2l 24333* | Elementhood in the set 𝐿 of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⇒ ⊢ (𝐴 ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔))) | ||
Theorem | itg2lr 24334* | Sufficient condition for elementhood in the set 𝐿. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))} ⇒ ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹) → (∫1‘𝐺) ∈ 𝐿) | ||
Theorem | xrge0f 24335 | 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 ∧ 0𝑝 ∘r ≤ 𝐹) → 𝐹:ℝ⟶(0[,]+∞)) | ||
Theorem | itg2cl 24336 | The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) ∈ ℝ*) | ||
Theorem | itg2ub 24337 | 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 ∫1 ∧ 𝐺 ∘r ≤ 𝐹) → (∫1‘𝐺) ≤ (∫2‘𝐹)) | ||
Theorem | itg2leub 24338* | Any upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹 is greater than (∫2‘𝐹), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2‘𝐹) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴))) | ||
Theorem | itg2ge0 24339 | 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 ≤ (∫2‘𝐹)) | ||
Theorem | itg2itg1 24340 | 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 ∧ 0𝑝 ∘r ≤ 𝐹) → (∫2‘𝐹) = (∫1‘𝐹)) | ||
Theorem | itg20 24341 | The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ (∫2‘(ℝ × {0})) = 0 | ||
Theorem | itg2lecl 24342 | If an ∫2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2‘𝐹) ≤ 𝐴) → (∫2‘𝐹) ∈ ℝ) | ||
Theorem | itg2le 24343 | 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[,]+∞) ∧ 𝐹 ∘r ≤ 𝐺) → (∫2‘𝐹) ≤ (∫2‘𝐺)) | ||
Theorem | itg2const 24344* | 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(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝐵 · (vol‘𝐴))) | ||
Theorem | itg2const2 24345* | 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(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ)) | ||
Theorem | itg2seq 24346* | 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 24359, 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 (𝑔‘𝑛) ∘r ≤ 𝐹 ∧ (∫2‘𝐹) = sup(ran (𝑛 ∈ ℕ ↦ (∫1‘(𝑔‘𝑛))), ℝ*, < ))) | ||
Theorem | itg2uba 24347* | Approximate version of itg2ub 24337. If 𝐹 approximately dominates 𝐺, then ∫1𝐺 ≤ ∫2𝐹. (Contributed by Mario Carneiro, 11-Aug-2014.) |
⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐺 ∈ dom ∫1) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐺‘𝑥) ≤ (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → (∫1‘𝐺) ≤ (∫2‘𝐹)) | ||
Theorem | itg2lea 24348* | Approximate version of itg2le 24343. If 𝐹 ≤ 𝐺 for almost all 𝑥, then ∫2𝐹 ≤ ∫2𝐺. (Contributed by Mario Carneiro, 11-Aug-2014.) |
⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (∫2‘𝐹) ≤ (∫2‘𝐺)) | ||
Theorem | itg2eqa 24349* | Approximate equality of integrals. If 𝐹 = 𝐺 for almost all 𝑥, then ∫2𝐹 = ∫2𝐺. (Contributed by Mario Carneiro, 12-Aug-2014.) |
⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → (vol*‘𝐴) = 0) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (∫2‘𝐹) = (∫2‘𝐺)) | ||
Theorem | itg2mulclem 24350 | Lemma for itg2mulc 24351. (Contributed by Mario Carneiro, 8-Jul-2014.) |
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ≤ (𝐴 · (∫2‘𝐹))) | ||
Theorem | itg2mulc 24351 | 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.) |
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ (0[,)+∞)) ⇒ ⊢ (𝜑 → (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫2‘𝐹))) | ||
Theorem | itg2splitlem 24352* | Lemma for itg2split 24353. (Contributed by Mario Carneiro, 11-Aug-2014.) |
⊢ (𝜑 → 𝐴 ∈ dom vol) & ⊢ (𝜑 → 𝐵 ∈ dom vol) & ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0) & ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (0[,]+∞)) & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐶, 0)) & ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, 𝐶, 0)) & ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑈, 𝐶, 0)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ) ⇒ ⊢ (𝜑 → (∫2‘𝐻) ≤ ((∫2‘𝐹) + (∫2‘𝐺))) | ||
Theorem | itg2split 24353* | The ∫2 integral splits under an almost disjoint union. The proof avoids the use of itg2add 24363, which requires countable choice. (Contributed by Mario Carneiro, 11-Aug-2014.) |
⊢ (𝜑 → 𝐴 ∈ dom vol) & ⊢ (𝜑 → 𝐵 ∈ dom vol) & ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0) & ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (0[,]+∞)) & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐶, 0)) & ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, 𝐶, 0)) & ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑈, 𝐶, 0)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ) ⇒ ⊢ (𝜑 → (∫2‘𝐻) = ((∫2‘𝐹) + (∫2‘𝐺))) | ||
Theorem | itg2monolem1 24354* | Lemma for itg2mono 24357. We show that for any constant 𝑡 less than one, 𝑡 · ∫1𝐻 is less than 𝑆, and so ∫1𝐻 ≤ 𝑆, which is one half of the equality in itg2mono 24357. Consider the sequence 𝐴(𝑛) = {𝑥 ∣ 𝑡 · 𝐻 ≤ 𝐹(𝑛)}. This is an increasing sequence of measurable sets whose union is ℝ, and so 𝐻 ↾ 𝐴(𝑛) has an integral which equals ∫1𝐻 in the limit, by itg1climres 24318. Then by taking the limit in (𝑡 · 𝐻) ↾ 𝐴(𝑛) ≤ 𝐹(𝑛), we get 𝑡 · ∫1𝐻 ≤ 𝑆 as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) & ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) & ⊢ (𝜑 → 𝑇 ∈ (0(,)1)) & ⊢ (𝜑 → 𝐻 ∈ dom ∫1) & ⊢ (𝜑 → 𝐻 ∘r ≤ 𝐺) & ⊢ (𝜑 → 𝑆 ∈ ℝ) & ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻‘𝑥)) ≤ ((𝐹‘𝑛)‘𝑥)}) ⇒ ⊢ (𝜑 → (𝑇 · (∫1‘𝐻)) ≤ 𝑆) | ||
Theorem | itg2monolem2 24355* | Lemma for itg2mono 24357. (Contributed by Mario Carneiro, 16-Aug-2014.) |
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) & ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) & ⊢ (𝜑 → 𝑃 ∈ dom ∫1) & ⊢ (𝜑 → 𝑃 ∘r ≤ 𝐺) & ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆) ⇒ ⊢ (𝜑 → 𝑆 ∈ ℝ) | ||
Theorem | itg2monolem3 24356* | Lemma for itg2mono 24357. (Contributed by Mario Carneiro, 16-Aug-2014.) |
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) & ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) & ⊢ (𝜑 → 𝑃 ∈ dom ∫1) & ⊢ (𝜑 → 𝑃 ∘r ≤ 𝐺) & ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆) ⇒ ⊢ (𝜑 → (∫1‘𝑃) ≤ 𝑆) | ||
Theorem | itg2mono 24357* | The Monotone Convergence Theorem for nonnegative functions. If {(𝐹‘𝑛):𝑛 ∈ ℕ} is a monotone increasing sequence of positive, measurable, real-valued functions, and 𝐺 is the pointwise limit of the sequence, then (∫2‘𝐺) is the limit of the sequence {(∫2‘(𝐹‘𝑛)):𝑛 ∈ ℕ}. (Contributed by Mario Carneiro, 16-Aug-2014.) |
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1))) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) & ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ⇒ ⊢ (𝜑 → (∫2‘𝐺) = 𝑆) | ||
Theorem | itg2i1fseqle 24358* | Subject to the conditions coming from mbfi1fseq 24325, the sequence of simple functions are all less than the target function 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑃‘𝑀) ∘r ≤ 𝐹) | ||
Theorem | itg2i1fseq 24359* | Subject to the conditions coming from mbfi1fseq 24325, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))) ⇒ ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ*, < )) | ||
Theorem | itg2i1fseq2 24360* | In an extension to the results of itg2i1fseq 24359, 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.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫1‘(𝑃‘𝑘)) ≤ 𝑀) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹)) | ||
Theorem | itg2i1fseq3 24361* | Special case of itg2i1fseq2 24360: 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.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚))) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) ⇒ ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹)) | ||
Theorem | itg2addlem 24362* | Lemma for itg2add 24363. (Contributed by Mario Carneiro, 17-Aug-2014.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐺 ∈ MblFn) & ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ) & ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) & ⊢ (𝜑 → 𝑄:ℕ⟶dom ∫1) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑄‘𝑛) ∧ (𝑄‘𝑛) ∘r ≤ (𝑄‘(𝑛 + 1)))) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑄‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺))) | ||
Theorem | itg2add 24363 | 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.) |
⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐺 ∈ MblFn) & ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ) ⇒ ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺))) | ||
Theorem | itg2gt0 24364* | 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.) |
⊢ (𝜑 → 𝐴 ∈ dom vol) & ⊢ (𝜑 → 0 < (vol‘𝐴)) & ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → 0 < (∫2‘𝐹)) | ||
Theorem | itg2cnlem1 24365* | Lemma for itgcn 24448. (Contributed by Mario Carneiro, 30-Aug-2014.) |
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) ⇒ ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))), ℝ*, < ) = (∫2‘𝐹)) | ||
Theorem | itg2cnlem2 24366* | Lemma for itgcn 24448. (Contributed by Mario Carneiro, 31-Aug-2014.) |
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → ¬ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑀, (𝐹‘𝑥), 0))) ≤ ((∫2‘𝐹) − (𝐶 / 2))) ⇒ ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom vol((vol‘𝑢) < 𝑑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑢, (𝐹‘𝑥), 0))) < 𝐶)) | ||
Theorem | itg2cn 24367* | A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 24640 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.) |
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐹 ∈ MblFn) & ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom vol((vol‘𝑢) < 𝑑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑢, (𝐹‘𝑥), 0))) < 𝐶)) | ||
Theorem | ibllem 24368 | Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) | ||
Theorem | isibl 24369* | 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 (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘)))) & ⊢ (𝜑 → dom 𝐹 = 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ))) | ||
Theorem | isibl2 24370* | 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.) |
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘)))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ))) | ||
Theorem | iblmbf 24371 | An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.) |
⊢ (𝐹 ∈ 𝐿1 → 𝐹 ∈ MblFn) | ||
Theorem | iblitg 24372* | 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.) |
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝐾)))) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ ((𝜑 ∧ 𝐾 ∈ ℤ) → (∫2‘𝐺) ∈ ℝ) | ||
Theorem | dfitg 24373* | Evaluate the class substitution in df-itg 24227. (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((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))) | ||
Theorem | itgex 24374 | An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ ∫𝐴𝐵 d𝑥 ∈ V | ||
Theorem | itgeq1f 24375 | Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥) | ||
Theorem | itgeq1 24376* | Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥) | ||
Theorem | nfitg1 24377 | Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ Ⅎ𝑥∫𝐴𝐵 d𝑥 | ||
Theorem | nfitg 24378* | 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.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦∫𝐴𝐵 d𝑥 | ||
Theorem | cbvitg 24379* | Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) & ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 ⇒ ⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦 | ||
Theorem | cbvitgv 24380* | Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦 | ||
Theorem | itgeq2 24381 | Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥) | ||
Theorem | itgresr 24382 | The domain of an integral only matters in its intersection with ℝ. (Contributed by Mario Carneiro, 29-Jun-2014.) |
⊢ ∫𝐴𝐵 d𝑥 = ∫(𝐴 ∩ ℝ)𝐵 d𝑥 | ||
Theorem | itg0 24383 | The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.) |
⊢ ∫∅𝐴 d𝑥 = 0 | ||
Theorem | itgz 24384 | 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 | ||
Theorem | itgeq2dv 24385* | Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥) | ||
Theorem | itgmpt 24386* | Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) d𝑦) | ||
Theorem | itgcl 24387* | The integral of an integrable function is a complex number. This is Metamath 100 proof #86. (Contributed by Mario Carneiro, 29-Jun-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) ⇒ ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ) | ||
Theorem | itgvallem 24388* | 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((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0)))) | ||
Theorem | itgvallem3 24389* | Lemma for itgposval 24399 and itgreval 24400. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 0) ⇒ ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) = 0) | ||
Theorem | ibl0 24390 | The zero function is integrable on any measurable set. (Unlike iblconst 24421, this does not require 𝐴 to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.) |
⊢ (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1) | ||
Theorem | iblcnlem1 24391* | Lemma for iblcnlem 24392. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ 𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))) & ⊢ 𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))) & ⊢ 𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))) & ⊢ 𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))) | ||
Theorem | iblcnlem 24392* | Expand out the universal quantifier in isibl2 24370. (Contributed by Mario Carneiro, 6-Aug-2014.) |
⊢ 𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))) & ⊢ 𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))) & ⊢ 𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))) & ⊢ 𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))) | ||
Theorem | itgcnlem 24393* | Expand out the sum in dfitg 24373. (Contributed by Mario Carneiro, 1-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ 𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))) & ⊢ 𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))) & ⊢ 𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))) & ⊢ 𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) ⇒ ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ((𝑅 − 𝑆) + (i · (𝑇 − 𝑈)))) | ||
Theorem | iblrelem 24394* | Integrability of a real function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ))) | ||
Theorem | iblposlem 24395* | Lemma for iblpos 24396. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) = 0) | ||
Theorem | iblpos 24396* | Integrability of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ))) | ||
Theorem | iblre 24397* | Integrability of a real function. (Contributed by Mario Carneiro, 11-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1))) | ||
Theorem | itgrevallem1 24398* | Lemma for itgposval 24399 and itgreval 24400. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) ⇒ ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))))) | ||
Theorem | itgposval 24399* | The integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0)))) | ||
Theorem | itgreval 24400* | Decompose the integral of a real function into positive and negative parts. (Contributed by Mario Carneiro, 31-Jul-2014.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1) ⇒ ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)) |
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