Definition df-itg 22573

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 22571 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 22571 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 22568	. 2 class S. A B _d x
5		cc0 9541	. . . 4 class 0
6		c3 10662	. . . 4 class 3
7		cfz 11786	. . . 4 class ...
8	5, 6, 7	co 6303	. . 3 class ( 0 ... 3 )
9		ci 9543	. . . . 5 class _i
10		vk	. . . . . 6 setvar k
11	10	cv 1437	. . . . 5 class k
12		cexp 12273	. . . . 5 class ^
13	9, 11, 12	co 6303	. . . 4 class ( _i ^ k )
14		cr 9540	. . . . . 6 class RR
15		vy	. . . . . . 7 setvar y
16		cdiv 10271	. . . . . . . . 9 class /
17	3, 13, 16	co 6303	. . . . . . . 8 class ( B / ( _i ^ k ) )
18		cre 13154	. . . . . . . 8 class Re
19	17, 18	cfv 5599	. . . . . . 7 class ( Re ` ( B / ( _i ^ k ) ) )
20	1	cv 1437	. . . . . . . . . 10 class x
21	20, 2	wcel 1869	. . . . . . . . 9 wff x e. A
22	15	cv 1437	. . . . . . . . . 10 class y
23		cle 9678	. . . . . . . . . 10 class <_
24	5, 22, 23	wbr 4421	. . . . . . . . 9 wff 0 <_ y
25	21, 24	wa 371	. . . . . . . 8 wff ( x e. A /\ 0 <_ y )
26	25, 22, 5	cif 3910	. . . . . . 7 class if ( ( x e. A /\ 0 <_ y ) , y , 0 )
27	15, 19, 26	csb 3396	. . . . . 6 class [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 )
28	1, 14, 27	cmpt 4480	. . . . 5 class ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) )
29		citg2 22566	. . . . 5 class S.2
30	28, 29	cfv 5599	. . . 4 class ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) )
31		cmul 9546	. . . 4 class x.
32	13, 30, 31	co 6303	. . 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 13745	. 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 1438	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  22719  itgex  22720  nfitg1  22723