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Definition df-itg 24139
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 24137 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 24137 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 24134 . 2 class 𝐴𝐵 d𝑥
5 cc0 10526 . . . 4 class 0
6 c3 11682 . . . 4 class 3
7 cfz 12882 . . . 4 class ...
85, 6, 7co 7148 . . 3 class (0...3)
9 ci 10528 . . . . 5 class i
10 vk . . . . . 6 setvar 𝑘
1110cv 1529 . . . . 5 class 𝑘
12 cexp 13419 . . . . 5 class
139, 11, 12co 7148 . . . 4 class (i↑𝑘)
14 cr 10525 . . . . . 6 class
15 vy . . . . . . 7 setvar 𝑦
16 cdiv 11286 . . . . . . . . 9 class /
173, 13, 16co 7148 . . . . . . . 8 class (𝐵 / (i↑𝑘))
18 cre 14446 . . . . . . . 8 class
1917, 18cfv 6352 . . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
201cv 1529 . . . . . . . . . 10 class 𝑥
2120, 2wcel 2107 . . . . . . . . 9 wff 𝑥𝐴
2215cv 1529 . . . . . . . . . 10 class 𝑦
23 cle 10665 . . . . . . . . . 10 class
245, 22, 23wbr 5063 . . . . . . . . 9 wff 0 ≤ 𝑦
2521, 24wa 396 . . . . . . . 8 wff (𝑥𝐴 ∧ 0 ≤ 𝑦)
2625, 22, 5cif 4470 . . . . . . 7 class if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
2715, 19, 26csb 3887 . . . . . 6 class (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
281, 14, 27cmpt 5143 . . . . 5 class (𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29 citg2 24132 . . . . 5 class 2
3028, 29cfv 6352 . . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31 cmul 10531 . . . 4 class ·
3213, 30, 31co 7148 . . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
338, 32, 10csu 15032 . 2 class Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
344, 33wceq 1530 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  24285  itgex  24286  nfitg1  24289
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