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Definition df-itg 23115
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 23113 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 23113 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 23110 . 2 class 𝐴𝐵 d𝑥
5 cc0 9792 . . . 4 class 0
6 c3 10918 . . . 4 class 3
7 cfz 12152 . . . 4 class ...
85, 6, 7co 6527 . . 3 class (0...3)
9 ci 9794 . . . . 5 class i
10 vk . . . . . 6 setvar 𝑘
1110cv 1473 . . . . 5 class 𝑘
12 cexp 12677 . . . . 5 class
139, 11, 12co 6527 . . . 4 class (i↑𝑘)
14 cr 9791 . . . . . 6 class
15 vy . . . . . . 7 setvar 𝑦
16 cdiv 10533 . . . . . . . . 9 class /
173, 13, 16co 6527 . . . . . . . 8 class (𝐵 / (i↑𝑘))
18 cre 13631 . . . . . . . 8 class
1917, 18cfv 5790 . . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
201cv 1473 . . . . . . . . . 10 class 𝑥
2120, 2wcel 1976 . . . . . . . . 9 wff 𝑥𝐴
2215cv 1473 . . . . . . . . . 10 class 𝑦
23 cle 9931 . . . . . . . . . 10 class
245, 22, 23wbr 4577 . . . . . . . . 9 wff 0 ≤ 𝑦
2521, 24wa 382 . . . . . . . 8 wff (𝑥𝐴 ∧ 0 ≤ 𝑦)
2625, 22, 5cif 4035 . . . . . . 7 class if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
2715, 19, 26csb 3498 . . . . . 6 class (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
281, 14, 27cmpt 4637 . . . . 5 class (𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29 citg2 23108 . . . . 5 class 2
3028, 29cfv 5790 . . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31 cmul 9797 . . . 4 class ·
3213, 30, 31co 6527 . . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
338, 32, 10csu 14210 . 2 class Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
344, 33wceq 1474 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  23259  itgex  23260  nfitg1  23263
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