Definition df-itg 22140

Description: Define the full Lebesgue integral, for complex-valued functions to RR. 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 S. ( 0 [,] 1 ) ( x ^ 2 ) _d x = ( 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 22138 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 22138 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)

Assertion

Ref	Expression
df-itg	|- S. A B _d x = sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 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

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	citg 22135	. 2 class S. A B _d x
5		cc0 9425	. . . 4 class 0
6		c3 10525	. . . 4 class 3
7		cfz 11615	. . . 4 class ...
8	5, 6, 7	co 6218	. . 3 class ( 0 ... 3 )
9		ci 9427	. . . . 5 class _i
10		vk	. . . . . 6 setvar k
11	10	cv 1398	. . . . 5 class k
12		cexp 12092	. . . . 5 class ^
13	9, 11, 12	co 6218	. . . 4 class ( _i ^ k )
14		cr 9424	. . . . . 6 class RR
15		vy	. . . . . . 7 setvar y
16		cdiv 10145	. . . . . . . . 9 class /
17	3, 13, 16	co 6218	. . . . . . . 8 class ( B / ( _i ^ k ) )
18		cre 12955	. . . . . . . 8 class Re
19	17, 18	cfv 5513	. . . . . . 7 class ( Re ` ( B / ( _i ^ k ) ) )
20	1	cv 1398	. . . . . . . . . 10 class x
21	20, 2	wcel 1836	. . . . . . . . 9 wff x e. A
22	15	cv 1398	. . . . . . . . . 10 class y
23		cle 9562	. . . . . . . . . 10 class <_
24	5, 22, 23	wbr 4384	. . . . . . . . 9 wff 0 <_ y
25	21, 24	wa 367	. . . . . . . 8 wff ( x e. A /\ 0 <_ y )
26	25, 22, 5	cif 3874	. . . . . . 7 class if ( ( x e. A /\ 0 <_ y ) , y , 0 )
27	15, 19, 26	csb 3365	. . . . . 6 class [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 )
28	1, 14, 27	cmpt 4442	. . . . 5 class ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) )
29		citg2 22133	. . . . 5 class S.2
30	28, 29	cfv 5513	. . . 4 class ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) )
31		cmul 9430	. . . 4 class x.
32	13, 30, 31	co 6218	. . 3 class ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) ) )
33	8, 32, 10	csu 13533	. 2 class sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) ) )
34	4, 33	wceq 1399	1 wff S. A B _d x = sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) ) )

This definition is referenced by:  dfitg  22284  itgex  22285  nfitg1  22288