Definition df-itg 19469

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 19467 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 19467 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 set x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	citg 19463	. 2 class S. A B _d x
5		cc0 8946	. . . 4 class 0
6		c3 10006	. . . 4 class 3
7		cfz 10999	. . . 4 class ...
8	5, 6, 7	co 6040	. . 3 class ( 0 ... 3 )
9		ci 8948	. . . . 5 class _i
10		vk	. . . . . 6 set k
11	10	cv 1648	. . . . 5 class k
12		cexp 11337	. . . . 5 class ^
13	9, 11, 12	co 6040	. . . 4 class ( _i ^ k )
14		cr 8945	. . . . . 6 class RR
15		vy	. . . . . . 7 set y
16		cdiv 9633	. . . . . . . . 9 class /
17	3, 13, 16	co 6040	. . . . . . . 8 class ( B / ( _i ^ k ) )
18		cre 11857	. . . . . . . 8 class Re
19	17, 18	cfv 5413	. . . . . . 7 class ( Re ` ( B / ( _i ^ k ) ) )
20	1	cv 1648	. . . . . . . . . 10 class x
21	20, 2	wcel 1721	. . . . . . . . 9 wff x e. A
22	15	cv 1648	. . . . . . . . . 10 class y
23		cle 9077	. . . . . . . . . 10 class <_
24	5, 22, 23	wbr 4172	. . . . . . . . 9 wff 0 <_ y
25	21, 24	wa 359	. . . . . . . 8 wff ( x e. A /\ 0 <_ y )
26	25, 22, 5	cif 3699	. . . . . . 7 class if ( ( x e. A /\ 0 <_ y ) , y , 0 )
27	15, 19, 26	csb 3211	. . . . . 6 class [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 )
28	1, 14, 27	cmpt 4226	. . . . 5 class ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) )
29		citg2 19461	. . . . 5 class S.2
30	28, 29	cfv 5413	. . . 4 class ( S.2 ` ( x e. RR |-> [_ ( Re ` ( B / ( _i ^ k ) ) ) / y ]_ if ( ( x e. A /\ 0 <_ y ) , y , 0 ) ) )
31		cmul 8951	. . . 4 class x.
32	13, 30, 31	co 6040	. . 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 12434	. 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 1649	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  19614  itgex  19615  nfitg1  19618