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Definition df-itg 23611
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 23609 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 23609 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 23606 . 2 class 𝐴𝐵 d𝑥
5 cc0 10148 . . . 4 class 0
6 c3 11283 . . . 4 class 3
7 cfz 12539 . . . 4 class ...
85, 6, 7co 6814 . . 3 class (0...3)
9 ci 10150 . . . . 5 class i
10 vk . . . . . 6 setvar 𝑘
1110cv 1631 . . . . 5 class 𝑘
12 cexp 13074 . . . . 5 class
139, 11, 12co 6814 . . . 4 class (i↑𝑘)
14 cr 10147 . . . . . 6 class
15 vy . . . . . . 7 setvar 𝑦
16 cdiv 10896 . . . . . . . . 9 class /
173, 13, 16co 6814 . . . . . . . 8 class (𝐵 / (i↑𝑘))
18 cre 14056 . . . . . . . 8 class
1917, 18cfv 6049 . . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
201cv 1631 . . . . . . . . . 10 class 𝑥
2120, 2wcel 2139 . . . . . . . . 9 wff 𝑥𝐴
2215cv 1631 . . . . . . . . . 10 class 𝑦
23 cle 10287 . . . . . . . . . 10 class
245, 22, 23wbr 4804 . . . . . . . . 9 wff 0 ≤ 𝑦
2521, 24wa 383 . . . . . . . 8 wff (𝑥𝐴 ∧ 0 ≤ 𝑦)
2625, 22, 5cif 4230 . . . . . . 7 class if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
2715, 19, 26csb 3674 . . . . . 6 class (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
281, 14, 27cmpt 4881 . . . . 5 class (𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29 citg2 23604 . . . . 5 class 2
3028, 29cfv 6049 . . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31 cmul 10153 . . . 4 class ·
3213, 30, 31co 6814 . . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
338, 32, 10csu 14635 . 2 class Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
344, 33wceq 1632 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  23755  itgex  23756  nfitg1  23759
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