Definition df-itg 21118

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 21116 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 21116 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 21113	. 2 class S. A B _d x
5		cc0 9297	. . . 4 class 0
6		c3 10387	. . . 4 class 3
7		cfz 11452	. . . 4 class ...
8	5, 6, 7	co 6106	. . 3 class ( 0 ... 3 )
9		ci 9299	. . . . 5 class _i
10		vk	. . . . . 6 setvar k
11	10	cv 1368	. . . . 5 class k
12		cexp 11880	. . . . 5 class ^
13	9, 11, 12	co 6106	. . . 4 class ( _i ^ k )
14		cr 9296	. . . . . 6 class RR
15		vy	. . . . . . 7 setvar y
16		cdiv 10008	. . . . . . . . 9 class /
17	3, 13, 16	co 6106	. . . . . . . 8 class ( B / ( _i ^ k ) )
18		cre 12601	. . . . . . . 8 class Re
19	17, 18	cfv 5433	. . . . . . 7 class ( Re ` ( B / ( _i ^ k ) ) )
20	1	cv 1368	. . . . . . . . . 10 class x
21	20, 2	wcel 1756	. . . . . . . . 9 wff x e. A
22	15	cv 1368	. . . . . . . . . 10 class y
23		cle 9434	. . . . . . . . . 10 class <_
24	5, 22, 23	wbr 4307	. . . . . . . . 9 wff 0 <_ y
25	21, 24	wa 369	. . . . . . . 8 wff ( x e. A /\ 0 <_ y )
26	25, 22, 5	cif 3806	. . . . . . 7 class if ( ( x e. A /\ 0 <_ y ) , y , 0 )
27	15, 19, 26	csb 3303	. . . . . 6 class [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 )
28	1, 14, 27	cmpt 4365	. . . . 5 class ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) )
29		citg2 21111	. . . . 5 class S.2
30	28, 29	cfv 5433	. . . 4 class ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) )
31		cmul 9302	. . . 4 class x.
32	13, 30, 31	co 6106	. . 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 13178	. 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 1369	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  21262  itgex  21263  nfitg1  21266