Definition df-itg 21062

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 21060 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 21060 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 21057	. 2 class S. A B _d x
5		cc0 9278	. . . 4 class 0
6		c3 10368	. . . 4 class 3
7		cfz 11433	. . . 4 class ...
8	5, 6, 7	co 6090	. . 3 class ( 0 ... 3 )
9		ci 9280	. . . . 5 class _i
10		vk	. . . . . 6 setvar k
11	10	cv 1363	. . . . 5 class k
12		cexp 11861	. . . . 5 class ^
13	9, 11, 12	co 6090	. . . 4 class ( _i ^ k )
14		cr 9277	. . . . . 6 class RR
15		vy	. . . . . . 7 setvar y
16		cdiv 9989	. . . . . . . . 9 class /
17	3, 13, 16	co 6090	. . . . . . . 8 class ( B / ( _i ^ k ) )
18		cre 12582	. . . . . . . 8 class Re
19	17, 18	cfv 5415	. . . . . . 7 class ( Re ` ( B / ( _i ^ k ) ) )
20	1	cv 1363	. . . . . . . . . 10 class x
21	20, 2	wcel 1761	. . . . . . . . 9 wff x e. A
22	15	cv 1363	. . . . . . . . . 10 class y
23		cle 9415	. . . . . . . . . 10 class <_
24	5, 22, 23	wbr 4289	. . . . . . . . 9 wff 0 <_ y
25	21, 24	wa 369	. . . . . . . 8 wff ( x e. A /\ 0 <_ y )
26	25, 22, 5	cif 3788	. . . . . . 7 class if ( ( x e. A /\ 0 <_ y ) , y , 0 )
27	15, 19, 26	csb 3285	. . . . . 6 class [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 )
28	1, 14, 27	cmpt 4347	. . . . 5 class ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) )
29		citg2 21055	. . . . 5 class S.2
30	28, 29	cfv 5415	. . . 4 class ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) )
31		cmul 9283	. . . 4 class x.
32	13, 30, 31	co 6090	. . 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 13159	. 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 1364	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  21206  itgex  21207  nfitg1  21210