Definition df-itg 22581

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 22579 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 22579 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 22576	. 2 class S. A B _d x
5		cc0 9539	. . . 4 class 0
6		c3 10660	. . . 4 class 3
7		cfz 11784	. . . 4 class ...
8	5, 6, 7	co 6290	. . 3 class ( 0 ... 3 )
9		ci 9541	. . . . 5 class _i
10		vk	. . . . . 6 setvar k
11	10	cv 1443	. . . . 5 class k
12		cexp 12272	. . . . 5 class ^
13	9, 11, 12	co 6290	. . . 4 class ( _i ^ k )
14		cr 9538	. . . . . 6 class RR
15		vy	. . . . . . 7 setvar y
16		cdiv 10269	. . . . . . . . 9 class /
17	3, 13, 16	co 6290	. . . . . . . 8 class ( B / ( _i ^ k ) )
18		cre 13160	. . . . . . . 8 class Re
19	17, 18	cfv 5582	. . . . . . 7 class ( Re ` ( B / ( _i ^ k ) ) )
20	1	cv 1443	. . . . . . . . . 10 class x
21	20, 2	wcel 1887	. . . . . . . . 9 wff x e. A
22	15	cv 1443	. . . . . . . . . 10 class y
23		cle 9676	. . . . . . . . . 10 class <_
24	5, 22, 23	wbr 4402	. . . . . . . . 9 wff 0 <_ y
25	21, 24	wa 371	. . . . . . . 8 wff ( x e. A /\ 0 <_ y )
26	25, 22, 5	cif 3881	. . . . . . 7 class if ( ( x e. A /\ 0 <_ y ) , y , 0 )
27	15, 19, 26	csb 3363	. . . . . 6 class [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 )
28	1, 14, 27	cmpt 4461	. . . . 5 class ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) )
29		citg2 22574	. . . . 5 class S.2
30	28, 29	cfv 5582	. . . 4 class ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) )
31		cmul 9544	. . . 4 class x.
32	13, 30, 31	co 6290	. . 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 13752	. 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 1444	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  22727  itgex  22728  nfitg1  22731