Definition df-itg 21767

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 21765 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 21765 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 21762	. 2 class S. A B _d x
5		cc0 9488	. . . 4 class 0
6		c3 10582	. . . 4 class 3
7		cfz 11668	. . . 4 class ...
8	5, 6, 7	co 6282	. . 3 class ( 0 ... 3 )
9		ci 9490	. . . . 5 class _i
10		vk	. . . . . 6 setvar k
11	10	cv 1378	. . . . 5 class k
12		cexp 12130	. . . . 5 class ^
13	9, 11, 12	co 6282	. . . 4 class ( _i ^ k )
14		cr 9487	. . . . . 6 class RR
15		vy	. . . . . . 7 setvar y
16		cdiv 10202	. . . . . . . . 9 class /
17	3, 13, 16	co 6282	. . . . . . . 8 class ( B / ( _i ^ k ) )
18		cre 12889	. . . . . . . 8 class Re
19	17, 18	cfv 5586	. . . . . . 7 class ( Re ` ( B / ( _i ^ k ) ) )
20	1	cv 1378	. . . . . . . . . 10 class x
21	20, 2	wcel 1767	. . . . . . . . 9 wff x e. A
22	15	cv 1378	. . . . . . . . . 10 class y
23		cle 9625	. . . . . . . . . 10 class <_
24	5, 22, 23	wbr 4447	. . . . . . . . 9 wff 0 <_ y
25	21, 24	wa 369	. . . . . . . 8 wff ( x e. A /\ 0 <_ y )
26	25, 22, 5	cif 3939	. . . . . . 7 class if ( ( x e. A /\ 0 <_ y ) , y , 0 )
27	15, 19, 26	csb 3435	. . . . . 6 class [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 )
28	1, 14, 27	cmpt 4505	. . . . 5 class ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) )
29		citg2 21760	. . . . 5 class S.2
30	28, 29	cfv 5586	. . . 4 class ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) )
31		cmul 9493	. . . 4 class x.
32	13, 30, 31	co 6282	. . 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 13467	. 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 1379	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  21911  itgex  21912  nfitg1  21915