MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-itg Structured version   Visualization version   GIF version

Definition df-itg 25658
Description: Define the full Lebesgue integral, for complex-valued functions to . The syntax is designed to be suggestive of the standard notation for integrals. For example, our notation for the integral of 𝑥↑2 from 0 to 1 is ∫(0[,]1)(𝑥↑2) d𝑥 = (1 / 3). The only real function of this definition is to break the integral up into nonnegative real parts and send it off to df-itg2 25656 for further processing. Note that this definition cannot handle integrals which evaluate to infinity, because addition and multiplication are not currently defined on extended reals. (You can use df-itg2 25656 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)
Assertion
Ref Expression
df-itg 𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
Distinct variable groups:   𝑦,𝑘,𝐴   𝐵,𝑘,𝑦   𝑥,𝑘,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Detailed syntax breakdown of Definition df-itg
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 cA . . 3 class 𝐴
3 cB . . 3 class 𝐵
41, 2, 3citg 25653 . 2 class 𝐴𝐵 d𝑥
5 cc0 11155 . . . 4 class 0
6 c3 12322 . . . 4 class 3
7 cfz 13547 . . . 4 class ...
85, 6, 7co 7431 . . 3 class (0...3)
9 ci 11157 . . . . 5 class i
10 vk . . . . . 6 setvar 𝑘
1110cv 1539 . . . . 5 class 𝑘
12 cexp 14102 . . . . 5 class
139, 11, 12co 7431 . . . 4 class (i↑𝑘)
14 cr 11154 . . . . . 6 class
15 vy . . . . . . 7 setvar 𝑦
16 cdiv 11920 . . . . . . . . 9 class /
173, 13, 16co 7431 . . . . . . . 8 class (𝐵 / (i↑𝑘))
18 cre 15136 . . . . . . . 8 class
1917, 18cfv 6561 . . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
201cv 1539 . . . . . . . . . 10 class 𝑥
2120, 2wcel 2108 . . . . . . . . 9 wff 𝑥𝐴
2215cv 1539 . . . . . . . . . 10 class 𝑦
23 cle 11296 . . . . . . . . . 10 class
245, 22, 23wbr 5143 . . . . . . . . 9 wff 0 ≤ 𝑦
2521, 24wa 395 . . . . . . . 8 wff (𝑥𝐴 ∧ 0 ≤ 𝑦)
2625, 22, 5cif 4525 . . . . . . 7 class if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
2715, 19, 26csb 3899 . . . . . 6 class (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
281, 14, 27cmpt 5225 . . . . 5 class (𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29 citg2 25651 . . . . 5 class 2
3028, 29cfv 6561 . . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31 cmul 11160 . . . 4 class ·
3213, 30, 31co 7431 . . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
338, 32, 10csu 15722 . 2 class Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
344, 33wceq 1540 1 wff 𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
Colors of variables: wff setvar class
This definition is referenced by:  dfitg  25804  itgex  25805  itgeq1f  25806  itgeq1  25808  nfitg1  25809  cbvitgv  25812  itgeq12i  36207  itgeq12sdv  36220  cbvitgvw2  36249  cbvitgdavw  36282  cbvitgdavw2  36298
  Copyright terms: Public domain W3C validator