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Theorem List for Metamath Proof Explorer - 19001-19100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremitg1addlem3 19001* Lemma for itg1add 19004. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
 ( `' F " { i } )  i^i  ( `' G " { j } )
 ) ) ) )   =>    |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A I B )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
 
Theoremitg1addlem4 19002* Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
 ( `' F " { i } )  i^i  ( `' G " { j } )
 ) ) ) )   &    |-  P  =  (  +  |`  ( ran  F  X.  ran  G ) )   =>    |-  ( ph  ->  (
 S.1 `  ( F  o F  +  G ) )  =  sum_ y  e.  ran  F sum_ z  e.  ran  G (
 ( y  +  z
 )  x.  ( y I z ) ) )
 
Theoremitg1addlem5 19003* Lemma for itg1add . (Contributed by Mario Carneiro, 27-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  I  =  ( i  e.  RR ,  j  e.  RR  |->  if ( ( i  =  0  /\  j  =  0 ) ,  0 ,  ( vol `  (
 ( `' F " { i } )  i^i  ( `' G " { j } )
 ) ) ) )   &    |-  P  =  (  +  |`  ( ran  F  X.  ran  G ) )   =>    |-  ( ph  ->  (
 S.1 `  ( F  o F  +  G ) )  =  (
 ( S.1 `  F )  +  ( S.1 `  G ) ) )
 
Theoremitg1add 19004 The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   =>    |-  ( ph  ->  (
 S.1 `  ( F  o F  +  G ) )  =  (
 ( S.1 `  F )  +  ( S.1 `  G ) ) )
 
Theoremi1fmulclem 19005 Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ( (
 ph  /\  A  =/=  0 )  /\  B  e.  RR )  ->  ( `' ( ( RR  X.  { A } )  o F  x.  F )
 " { B }
 )  =  ( `' F " { ( B  /  A ) }
 ) )
 
Theoremi1fmulc 19006 A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( ( RR  X.  { A } )  o F  x.  F )  e.  dom  S.1 )
 
Theoremitg1mulc 19007 The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  (
 S.1 `  ( ( RR  X.  { A }
 )  o F  x.  F ) )  =  ( A  x.  ( S.1 `  F ) ) )
 
Theoremi1fres 19008* The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside  A.) (Contributed by Mario Carneiro, 29-Jun-2014.)
 |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( F `  x ) ,  0 ) )   =>    |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  e.  dom  S.1 )
 
Theoremi1fpos 19009* The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  G  =  ( x  e.  RR  |->  if (
 0  <_  ( F `  x ) ,  ( F `  x ) ,  0 ) )   =>    |-  ( F  e.  dom  S.1  ->  G  e.  dom  S.1 )
 
Theoremi1fposd 19010* Deduction form of i1fposd 19010. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( ph  ->  ( x  e.  RR  |->  A )  e.  dom  S.1 )   =>    |-  ( ph  ->  ( x  e.  RR  |->  if ( 0  <_  A ,  A ,  0 ) )  e.  dom  S.1 )
 
Theoremi1fsub 19011 The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( F  o F  -  G )  e.  dom  S.1 )
 
Theoremitg1sub 19012 The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( S.1 `  ( F  o F  -  G ) )  =  ( ( S.1 `  F )  -  ( S.1 `  G ) ) )
 
Theoremitg10a 19013* The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A 
 C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( RR  \  A ) )  ->  ( F `
  x )  =  0 )   =>    |-  ( ph  ->  ( S.1 `  F )  =  0 )
 
Theoremitg1ge0a 19014* The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A 
 C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( RR  \  A ) )  ->  0  <_  ( F `  x ) )   =>    |-  ( ph  ->  0  <_  ( S.1 `  F ) )
 
Theoremitg1lea 19015* Approximate version of itg1le 19016. If  F  <_  G for almost all  x, then  S.1 F  <_  S.1 G. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
 |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  ( ph  ->  A 
 C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  ( ( ph  /\  x  e.  ( RR  \  A ) )  ->  ( F `  x ) 
 <_  ( G `  x ) )   =>    |-  ( ph  ->  ( S.1 `  F )  <_  ( S.1 `  G )
 )
 
Theoremitg1le 19016 If one simple function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
 |-  ( ( F  e.  dom  S.1  /\  G  e.  dom  S.1  /\  F  o R  <_  G )  ->  ( S.1 `  F )  <_  ( S.1 `  G ) )
 
Theoremitg1climres 19017* Restricting the simple function  F to the increasing sequence  A ( n ) of measurable sets whose union is  RR yields a sequence of simple functions whose integrals approach the integral of  F. (Contributed by Mario Carneiro, 15-Aug-2014.)
 |-  ( ph  ->  A : NN --> dom  vol )   &    |-  (
 ( ph  /\  n  e. 
 NN )  ->  ( A `  n )  C_  ( A `  ( n  +  1 ) ) )   &    |-  ( ph  ->  U.
 ran  A  =  RR )   &    |-  ( ph  ->  F  e.  dom  S.1 )   &    |-  G  =  ( x  e.  RR  |->  if ( x  e.  ( A `  n ) ,  ( F `  x ) ,  0 )
 )   =>    |-  ( ph  ->  ( n  e.  NN  |->  ( S.1 `  G ) )  ~~>  ( S.1 `  F ) )
 
Theoremmbfi1fseqlem1 19018* Lemma for mbfi1fseq 19024. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   =>    |-  ( ph  ->  J : ( NN  X.  RR ) --> ( 0 [,)  +oo ) )
 
Theoremmbfi1fseqlem2 19019* Lemma for mbfi1fseq 19024. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x ) 
 <_  m ,  ( m J x ) ,  m ) ,  0 ) ) )   =>    |-  ( A  e.  NN  ->  ( G `  A )  =  ( x  e.  RR  |->  if ( x  e.  ( -u A [,] A ) ,  if ( ( A J x )  <_  A ,  ( A J x ) ,  A ) ,  0 ) ) )
 
Theoremmbfi1fseqlem3 19020* Lemma for mbfi1fseq 19024. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x ) 
 <_  m ,  ( m J x ) ,  m ) ,  0 ) ) )   =>    |-  ( ( ph  /\  A  e.  NN )  ->  ( G `  A ) : RR --> ran  (  m  e.  ( 0 ... ( A  x.  (
 2 ^ A ) ) )  |->  ( m 
 /  ( 2 ^ A ) ) ) )
 
Theoremmbfi1fseqlem4 19021* Lemma for mbfi1fseq 19024. This lemma is not as interesting as it is long - it is simply checking that  G is in fact a sequence of simple functions, by verifying that its range is in  ( 0 ... n 2 ^ n
)  /  ( 2 ^ n ) (which is to say, the numbers from  0 to  n in increments of  1  / 
( 2 ^ n
)), and also that the preimage of each point  k is measurable, because it is equal to  ( -u n [,] n )  i^i  ( `' F " ( k [,) k  +  1  /  ( 2 ^ n ) ) ) for  k  <  n and  ( -u n [,] n
)  i^i  ( `' F " ( k [,) 
+oo ) ) for  k  =  n. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x ) 
 <_  m ,  ( m J x ) ,  m ) ,  0 ) ) )   =>    |-  ( ph  ->  G : NN --> dom  S.1 )
 
Theoremmbfi1fseqlem5 19022* Lemma for mbfi1fseq 19024. Verify that  G describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x ) 
 <_  m ,  ( m J x ) ,  m ) ,  0 ) ) )   =>    |-  ( ( ph  /\  A  e.  NN )  ->  ( 0 p  o R  <_  ( G `  A )  /\  ( G `
  A )  o R  <_  ( G `  ( A  +  1 ) ) ) )
 
Theoremmbfi1fseqlem6 19023* Lemma for mbfi1fseq 19024. Verify that  G converges pointwise to  F, and wrap up the existence quantifier. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  J  =  ( m  e.  NN ,  y  e.  RR  |->  ( ( |_ `  (
 ( F `  y
 )  x.  ( 2 ^ m ) ) )  /  ( 2 ^ m ) ) )   &    |-  G  =  ( m  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m J x ) 
 <_  m ,  ( m J x ) ,  m ) ,  0 ) ) )   =>    |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\  A. n  e.  NN  (
 0 p  o R  <_  ( g `  n )  /\  ( g `  n )  o R  <_  ( g `  ( n  +  1 )
 ) )  /\  A. x  e.  RR  ( n  e.  NN  |->  ( ( g `  n ) `
  x ) )  ~~>  ( F `  x ) ) )
 
Theoremmbfi1fseq 19024* A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function  G and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\  A. n  e.  NN  (
 0 p  o R  <_  ( g `  n )  /\  ( g `  n )  o R  <_  ( g `  ( n  +  1 )
 ) )  /\  A. x  e.  RR  ( n  e.  NN  |->  ( ( g `  n ) `
  x ) )  ~~>  ( F `  x ) ) )
 
Theoremmbfi1flimlem 19025* Lemma for mbfi1flim 19026. (Contributed by Mario Carneiro, 5-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> RR )   =>    |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\ 
 A. x  e.  RR  ( n  e.  NN  |->  ( ( g `  n ) `  x ) )  ~~>  ( F `  x ) ) )
 
Theoremmbfi1flim 19026* Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : A --> RR )   =>    |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\ 
 A. x  e.  A  ( n  e.  NN  |->  ( ( g `  n ) `  x ) )  ~~>  ( F `  x ) ) )
 
Theoremmbfmullem2 19027* Lemma for mbfmul 19029. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  G  e. MblFn )   &    |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  G : A --> RR )   &    |-  ( ph  ->  P : NN --> dom  S.1 )   &    |-  ( ( ph  /\  x  e.  A )  ->  ( n  e.  NN  |->  ( ( P `  n ) `
  x ) )  ~~>  ( F `  x ) )   &    |-  ( ph  ->  Q : NN --> dom  S.1 )   &    |-  ( ( ph  /\  x  e.  A )  ->  ( n  e.  NN  |->  ( ( Q `  n ) `
  x ) )  ~~>  ( G `  x ) )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e. MblFn )
 
Theoremmbfmullem 19028 Lemma for mbfmul 19029. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  G  e. MblFn )   &    |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  G : A --> RR )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e. MblFn )
 
Theoremmbfmul 19029 The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  G  e. MblFn )   =>    |-  ( ph  ->  ( F  o F  x.  G )  e. MblFn )
 
Theoremitg2lcl 19030* The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g
 ) ) }   =>    |-  L  C_  RR*
 
Theoremitg2val 19031* Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g
 ) ) }   =>    |-  ( F : RR
 --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  =  sup ( L ,  RR* ,  <  )
 )
 
Theoremitg2l 19032* Elementhood in the set  L of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g
 ) ) }   =>    |-  ( A  e.  L 
 <-> 
 E. g  e.  dom  S.1 ( g  o R  <_  F  /\  A  =  ( S.1 `  g )
 ) )
 
Theoremitg2lr 19033* Sufficient condition for elementhood in the set  L. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( g  o R  <_  F  /\  x  =  ( S.1 `  g
 ) ) }   =>    |-  ( ( G  e.  dom  S.1  /\  G  o R  <_  F ) 
 ->  ( S.1 `  G )  e.  L )
 
Theoremxrge0f 19034 A real function is a nonnegative extended real function if all its values are greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 28-Jul-2014.)
 |-  ( ( F : RR
 --> RR  /\  0 p  o R  <_  F )  ->  F : RR --> ( 0 [,]  +oo ) )
 
Theoremitg2cl 19035 The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  e.  RR* )
 
Theoremitg2ub 19036 The integral of a nonnegative real function  F is an upper bound on the integrals of all simple functions  G dominated by  F. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ( F : RR
 --> ( 0 [,]  +oo )  /\  G  e.  dom  S.1  /\  G  o R  <_  F )  ->  ( S.1 `  G )  <_  ( S.2 `  F ) )
 
Theoremitg2leub 19037* Any upper bound on the integrals of all simple functions  G dominated by  F is greater than  ( S.2 `  F
), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ( F : RR
 --> ( 0 [,]  +oo )  /\  A  e.  RR* )  ->  ( ( S.2 `  F )  <_  A  <->  A. g  e.  dom  S.1 ( g  o R  <_  F  ->  ( S.1 `  g )  <_  A ) ) )
 
Theoremitg2ge0 19038 The integral of a nonnegative real function is greater or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( F : RR --> ( 0 [,]  +oo )  ->  0  <_  ( S.2 `  F ) )
 
Theoremitg2itg1 19039 The integral of a nonnegative simple function using  S.2 is the same as its value under  S.1. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ( F  e.  dom  S.1  /\  0 p  o R  <_  F )  ->  ( S.2 `  F )  =  ( S.1 `  F ) )
 
Theoremitg20 19040 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( S.2 `  ( RR  X.  { 0 } ) )  =  0
 
Theoremitg2lecl 19041 If an  S.2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ( F : RR
 --> ( 0 [,]  +oo )  /\  A  e.  RR  /\  ( S.2 `  F )  <_  A )  ->  ( S.2 `  F )  e.  RR )
 
Theoremitg2le 19042 If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( ( F : RR
 --> ( 0 [,]  +oo )  /\  G : RR --> ( 0 [,]  +oo )  /\  F  o R  <_  G )  ->  ( S.2 `  F )  <_  ( S.2 `  G )
 )
 
Theoremitg2const 19043* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  ( vol `  A )  e.  RR  /\  B  e.  ( 0 [,)  +oo ) )  ->  ( S.2 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 ) ) )  =  ( B  x.  ( vol `  A ) ) )
 
Theoremitg2const2 19044* When the base set of a constant function has infinite volume, the integral is also infinite and vice-versa. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  B  e.  RR+ )  ->  ( ( vol `  A )  e.  RR  <->  ( S.2 `  ( x  e. 
 RR  |->  if ( x  e.  A ,  B , 
 0 ) ) )  e.  RR ) )
 
Theoremitg2seq 19045* Definitional property of the  S.2 integral: for any function  F there is a countable sequence 
g of simple functions less than  F whose integrals converge to the integral of  F. (This theorem is for the most part unnecessary in lieu of itg2i1fseq 19058, but unlike that theorem this one doesn't require  F to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.)
 |-  ( F : RR --> ( 0 [,]  +oo )  ->  E. g ( g : NN --> dom  S.1  /\ 
 A. n  e.  NN  ( g `  n )  o R  <_  F  /\  ( S.2 `  F )  =  sup ( ran  (  n  e.  NN  |->  ( S.1 `  ( g `  n ) ) ) ,  RR* ,  <  )
 ) )
 
Theoremitg2uba 19046* Approximate version of itg2ub 19036. If  F approximately dominates  G, then  S.1 G  <_  S.2 F. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  G  e.  dom  S.1 )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( RR  \  A ) )  ->  ( G `
  x )  <_  ( F `  x ) )   =>    |-  ( ph  ->  ( S.1 `  G )  <_  ( S.2 `  F )
 )
 
Theoremitg2lea 19047* Approximate version of itg2le 19042. If  F  <_  G for almost all  x, then  S.2 F  <_  S.2 G. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  G : RR --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( RR  \  A ) )  ->  ( F `
  x )  <_  ( G `  x ) )   =>    |-  ( ph  ->  ( S.2 `  F )  <_  ( S.2 `  G )
 )
 
Theoremitg2eqa 19048* Approximate equality of integrals. If  F  =  G for almost all  x, then  S.2 F  = 
S.2 G. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  G : RR --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( RR  \  A ) )  ->  ( F `
  x )  =  ( G `  x ) )   =>    |-  ( ph  ->  ( S.2 `  F )  =  ( S.2 `  G ) )
 
Theoremitg2mulclem 19049 Lemma for itg2mulc 19050. (Contributed by Mario Carneiro, 8-Jul-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( S.2 `  ( ( RR 
 X.  { A } )  o F  x.  F ) )  <_  ( A  x.  ( S.2 `  F ) ) )
 
Theoremitg2mulc 19050 The integral of a nonnegative constant times a function is the constant times the integral of the original function. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  A  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  ( S.2 `  ( ( RR 
 X.  { A } )  o F  x.  F ) )  =  ( A  x.  ( S.2 `  F ) ) )
 
Theoremitg2splitlem 19051* Lemma for itg2split 19052. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  A  e.  dom  vol )   &    |-  ( ph  ->  B  e.  dom  vol )   &    |-  ( ph  ->  ( vol * `  ( A  i^i  B ) )  =  0
 )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ( ph  /\  x  e.  U ) 
 ->  C  e.  ( 0 [,]  +oo ) )   &    |-  F  =  ( x  e.  RR  |->  if ( x  e.  A ,  C ,  0 ) )   &    |-  G  =  ( x  e.  RR  |->  if ( x  e.  B ,  C ,  0 ) )   &    |-  H  =  ( x  e.  RR  |->  if ( x  e.  U ,  C ,  0 ) )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  (
 S.2 `  G )  e.  RR )   =>    |-  ( ph  ->  ( S.2 `  H )  <_  ( ( S.2 `  F )  +  ( S.2 `  G ) ) )
 
Theoremitg2split 19052* The  S.2 integral splits under an almost disjoint union. (The proof avoids the use of itg2add 19062 which requires CC.) (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  A  e.  dom  vol )   &    |-  ( ph  ->  B  e.  dom  vol )   &    |-  ( ph  ->  ( vol * `  ( A  i^i  B ) )  =  0
 )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ( ph  /\  x  e.  U ) 
 ->  C  e.  ( 0 [,]  +oo ) )   &    |-  F  =  ( x  e.  RR  |->  if ( x  e.  A ,  C ,  0 ) )   &    |-  G  =  ( x  e.  RR  |->  if ( x  e.  B ,  C ,  0 ) )   &    |-  H  =  ( x  e.  RR  |->  if ( x  e.  U ,  C ,  0 ) )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  (
 S.2 `  G )  e.  RR )   =>    |-  ( ph  ->  ( S.2 `  H )  =  ( ( S.2 `  F )  +  ( S.2 `  G ) ) )
 
Theoremitg2monolem1 19053* Lemma for itg2mono 19056. We show that for any constant  t less than one,  t  x.  S.1 H is less than  S, and so  S.1 H  <_  S, which is one half of the equality in itg2mono 19056. Consider the sequence  A ( n )  =  { x  |  t  x.  H  <_  F ( n ) }. This is an increasing sequence of measurable sets whose union is  RR, and so  H  |`  A ( n ) has an integral which equals  S.1 H in the limit, by itg1climres 19017. Then by taking the limit in  ( t  x.  H )  |`  A ( n )  <_  F
( n ), we get  t  x.  S.1 H  <_  S as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  G  =  ( x  e.  RR  |->  sup ( ran  (  n  e.  NN  |->  ( ( F `  n ) `  x ) ) ,  RR ,  <  ) )   &    |-  (
 ( ph  /\  n  e. 
 NN )  ->  ( F `  n )  e. MblFn
 )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n ) : RR --> ( 0 [,)  +oo ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n )  o R  <_  ( F `  ( n  +  1 ) ) )   &    |-  ( ( ph  /\  x  e.  RR )  ->  E. y  e.  RR  A. n  e. 
 NN  ( ( F `
  n ) `  x )  <_  y )   &    |-  S  =  sup ( ran  (  n  e.  NN  |->  ( S.2 `  ( F `  n ) ) ) ,  RR* ,  <  )   &    |-  ( ph  ->  T  e.  (
 0 (,) 1 ) )   &    |-  ( ph  ->  H  e.  dom  S.1 )   &    |-  ( ph  ->  H  o R  <_  G )   &    |-  ( ph  ->  S  e.  RR )   &    |-  A  =  ( n  e.  NN  |->  { x  e.  RR  |  ( T  x.  ( H `  x ) ) 
 <_  ( ( F `  n ) `  x ) } )   =>    |-  ( ph  ->  ( T  x.  ( S.1 `  H ) )  <_  S )
 
Theoremitg2monolem2 19054* Lemma for itg2mono 19056. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  G  =  ( x  e.  RR  |->  sup ( ran  (  n  e.  NN  |->  ( ( F `  n ) `  x ) ) ,  RR ,  <  ) )   &    |-  (
 ( ph  /\  n  e. 
 NN )  ->  ( F `  n )  e. MblFn
 )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n ) : RR --> ( 0 [,)  +oo ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n )  o R  <_  ( F `  ( n  +  1 ) ) )   &    |-  ( ( ph  /\  x  e.  RR )  ->  E. y  e.  RR  A. n  e. 
 NN  ( ( F `
  n ) `  x )  <_  y )   &    |-  S  =  sup ( ran  (  n  e.  NN  |->  ( S.2 `  ( F `  n ) ) ) ,  RR* ,  <  )   &    |-  ( ph  ->  P  e.  dom  S.1 )   &    |-  ( ph  ->  P  o R  <_  G )   &    |-  ( ph  ->  -.  ( S.1 `  P )  <_  S )   =>    |-  ( ph  ->  S  e.  RR )
 
Theoremitg2monolem3 19055* Lemma for itg2mono 19056. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  G  =  ( x  e.  RR  |->  sup ( ran  (  n  e.  NN  |->  ( ( F `  n ) `  x ) ) ,  RR ,  <  ) )   &    |-  (
 ( ph  /\  n  e. 
 NN )  ->  ( F `  n )  e. MblFn
 )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n ) : RR --> ( 0 [,)  +oo ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n )  o R  <_  ( F `  ( n  +  1 ) ) )   &    |-  ( ( ph  /\  x  e.  RR )  ->  E. y  e.  RR  A. n  e. 
 NN  ( ( F `
  n ) `  x )  <_  y )   &    |-  S  =  sup ( ran  (  n  e.  NN  |->  ( S.2 `  ( F `  n ) ) ) ,  RR* ,  <  )   &    |-  ( ph  ->  P  e.  dom  S.1 )   &    |-  ( ph  ->  P  o R  <_  G )   &    |-  ( ph  ->  -.  ( S.1 `  P )  <_  S )   =>    |-  ( ph  ->  ( S.1 `  P )  <_  S )
 
Theoremitg2mono 19056* The Monotone Convergence Theorem for nonnegative functions. If  { ( F `
 n ) : n  e.  NN } is a monotone increasing sequence of positive, measurable, real-valued functions, and  G is the pointwise limit of the sequence, then  ( S.2 `  G
) is the limit of the sequence  { ( S.2 `  ( F `  n
) ) : n  e.  NN }. (Contributed by Mario Carneiro, 16-Aug-2014.)
 |-  G  =  ( x  e.  RR  |->  sup ( ran  (  n  e.  NN  |->  ( ( F `  n ) `  x ) ) ,  RR ,  <  ) )   &    |-  (
 ( ph  /\  n  e. 
 NN )  ->  ( F `  n )  e. MblFn
 )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n ) : RR --> ( 0 [,)  +oo ) )   &    |-  ( ( ph  /\  n  e.  NN )  ->  ( F `  n )  o R  <_  ( F `  ( n  +  1 ) ) )   &    |-  ( ( ph  /\  x  e.  RR )  ->  E. y  e.  RR  A. n  e. 
 NN  ( ( F `
  n ) `  x )  <_  y )   &    |-  S  =  sup ( ran  (  n  e.  NN  |->  ( S.2 `  ( F `  n ) ) ) ,  RR* ,  <  )   =>    |-  ( ph  ->  ( S.2 `  G )  =  S )
 
Theoremitg2i1fseqle 19057* Subject to the conditions coming from mbfi1fseq 19024, the sequence of simple functions are all less than the target function  F. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  P : NN --> dom  S.1 )   &    |-  ( ph  ->  A. n  e.  NN  (
 0 p  o R  <_  ( P `  n )  /\  ( P `  n )  o R  <_  ( P `  ( n  +  1 )
 ) ) )   &    |-  ( ph  ->  A. x  e.  RR  ( n  e.  NN  |->  ( ( P `  n ) `  x ) )  ~~>  ( F `  x ) )   =>    |-  ( ( ph  /\  M  e.  NN )  ->  ( P `  M )  o R  <_  F )
 
Theoremitg2i1fseq 19058* Subject to the conditions coming from mbfi1fseq 19024, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  P : NN --> dom  S.1 )   &    |-  ( ph  ->  A. n  e.  NN  (
 0 p  o R  <_  ( P `  n )  /\  ( P `  n )  o R  <_  ( P `  ( n  +  1 )
 ) ) )   &    |-  ( ph  ->  A. x  e.  RR  ( n  e.  NN  |->  ( ( P `  n ) `  x ) )  ~~>  ( F `  x ) )   &    |-  S  =  ( m  e.  NN  |->  ( S.1 `  ( P `  m ) ) )   =>    |-  ( ph  ->  ( S.2 `  F )  =  sup ( ran  S ,  RR* ,  <  ) )
 
Theoremitg2i1fseq2 19059* In an extension to the results of itg2i1fseq 19058, if there is an upper bound on the integrals of the simple functions approaching  F, then  S.2 F is real and the standard limit relation applies. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  P : NN --> dom  S.1 )   &    |-  ( ph  ->  A. n  e.  NN  (
 0 p  o R  <_  ( P `  n )  /\  ( P `  n )  o R  <_  ( P `  ( n  +  1 )
 ) ) )   &    |-  ( ph  ->  A. x  e.  RR  ( n  e.  NN  |->  ( ( P `  n ) `  x ) )  ~~>  ( F `  x ) )   &    |-  S  =  ( m  e.  NN  |->  ( S.1 `  ( P `  m ) ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ( ph  /\  k  e.  NN )  ->  ( S.1 `  ( P `  k ) ) 
 <_  M )   =>    |-  ( ph  ->  S  ~~>  ( S.2 `  F )
 )
 
Theoremitg2i1fseq3 19060* Special case of itg2i1fseq2 19059: if the integral of  F is a real number, then the standard limit relation holds on the integrals of simple functions approaching 
F. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  P : NN --> dom  S.1 )   &    |-  ( ph  ->  A. n  e.  NN  (
 0 p  o R  <_  ( P `  n )  /\  ( P `  n )  o R  <_  ( P `  ( n  +  1 )
 ) ) )   &    |-  ( ph  ->  A. x  e.  RR  ( n  e.  NN  |->  ( ( P `  n ) `  x ) )  ~~>  ( F `  x ) )   &    |-  S  =  ( m  e.  NN  |->  ( S.1 `  ( P `  m ) ) )   &    |-  ( ph  ->  ( S.2 `  F )  e.  RR )   =>    |-  ( ph  ->  S  ~~>  ( S.2 `  F )
 )
 
Theoremitg2addlem 19061* Lemma for itg2add 19062. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  ( S.2 `  F )  e.  RR )   &    |-  ( ph  ->  G  e. MblFn )   &    |-  ( ph  ->  G : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  (
 S.2 `  G )  e.  RR )   &    |-  ( ph  ->  P : NN --> dom  S.1 )   &    |-  ( ph  ->  A. n  e.  NN  ( 0 p  o R  <_  ( P `  n )  /\  ( P `  n )  o R  <_  ( P `  ( n  +  1 ) ) ) )   &    |-  ( ph  ->  A. x  e.  RR  ( n  e.  NN  |->  ( ( P `  n ) `
  x ) )  ~~>  ( F `  x ) )   &    |-  ( ph  ->  Q : NN --> dom  S.1 )   &    |-  ( ph  ->  A. n  e.  NN  ( 0 p  o R  <_  ( Q `  n )  /\  ( Q `  n )  o R  <_  ( Q `  ( n  +  1 ) ) ) )   &    |-  ( ph  ->  A. x  e.  RR  ( n  e.  NN  |->  ( ( Q `  n ) `
  x ) )  ~~>  ( G `  x ) )   =>    |-  ( ph  ->  ( S.2 `  ( F  o F  +  G )
 )  =  ( (
 S.2 `  F )  +  ( S.2 `  G ) ) )
 
Theoremitg2add 19062 The  S.2 integral is linear. (Measurability is an essential component of this theorem; otherwise consider the characteristic function of a nonmeasurable set and its complement.) (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  ( S.2 `  F )  e.  RR )   &    |-  ( ph  ->  G  e. MblFn )   &    |-  ( ph  ->  G : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  (
 S.2 `  G )  e.  RR )   =>    |-  ( ph  ->  ( S.2 `  ( F  o F  +  G )
 )  =  ( (
 S.2 `  F )  +  ( S.2 `  G ) ) )
 
Theoremitg2gt0 19063* If the function  F is strictly positive on a set of positive measure, then the integral of the function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  ( ph  ->  A  e.  dom  vol )   &    |-  ( ph  ->  0  <  ( vol `  A ) )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  F  e. MblFn )   &    |-  (
 ( ph  /\  x  e.  A )  ->  0  <  ( F `  x ) )   =>    |-  ( ph  ->  0  <  ( S.2 `  F ) )
 
Theoremitg2cnlem1 19064* Lemma for itgcn 19145. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   =>    |-  ( ph  ->  sup ( ran  (  n  e.  NN  |->  ( S.2 `  ( x  e.  RR  |->  if ( ( F `
  x )  <_  n ,  ( F `  x ) ,  0 ) ) ) ) ,  RR* ,  <  )  =  ( S.2 `  F ) )
 
Theoremitg2cnlem2 19065* Lemma for itgcn 19145. (Contributed by Mario Carneiro, 31-Aug-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  -.  ( S.2 `  ( x  e.  RR  |->  if ( ( F `  x )  <_  M ,  ( F `  x ) ,  0 ) ) )  <_  ( ( S.2 `  F )  -  ( C  /  2
 ) ) )   =>    |-  ( ph  ->  E. d  e.  RR+  A. u  e.  dom  vol ( ( vol `  u )  <  d  ->  ( S.2 `  ( x  e.  RR  |->  if ( x  e.  u ,  ( F `  x ) ,  0 ) ) )  <  C ) )
 
Theoremitg2cn 19066* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 19332 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  E. d  e.  RR+  A. u  e.  dom  vol ( ( vol `  u )  <  d  ->  ( S.2 `  ( x  e. 
 RR  |->  if ( x  e.  u ,  ( F `
  x ) ,  0 ) ) )  <  C ) )
 
Theoremibllem 19067 Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) )
 
Theoremisibl 19068* The predicate " F is integrable". The "integrable" predicate corresponds roughly to the range of validity of  S. A B  _d x, which is to say that the expression  S. A B  _d x doesn't make sense unless  ( x  e.  A  |->  B )  e.  L ^1. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  G  =  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  T ) ,  T ,  0 ) ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  T  =  ( Re
 `  ( B  /  ( _i ^ k ) ) ) )   &    |-  ( ph  ->  dom  F  =  A )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( F `  x )  =  B )   =>    |-  ( ph  ->  ( F  e.  L ^1  <->  ( F  e. MblFn  /\ 
 A. k  e.  (
 0 ... 3 ) (
 S.2 `  G )  e.  RR ) ) )
 
Theoremisibl2 19069* The predicate " F is integrable" when  F is a mapping operation. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  G  =  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  T ) ,  T ,  0 ) ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  T  =  ( Re
 `  ( B  /  ( _i ^ k ) ) ) )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  L ^1 
 <->  ( ( x  e.  A  |->  B )  e. MblFn  /\  A. k  e.  (
 0 ... 3 ) (
 S.2 `  G )  e.  RR ) ) )
 
Theoremiblmbf 19070 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
 |-  ( F  e.  L ^1  ->  F  e. MblFn )
 
Theoremiblitg 19071* If a function is integrable, then the  S.2 integrals of the function's decompositions all exist. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  G  =  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  T ) ,  T ,  0 ) ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  T  =  ( Re
 `  ( B  /  ( _i ^ K ) ) ) )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ( ph  /\  K  e.  ZZ )  ->  ( S.2 `  G )  e. 
 RR )
 
Theoremdfitg 19072* Evaluate the class substitution in df-itg 18927. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  T  =  ( Re
 `  ( B  /  ( _i ^ k ) ) )   =>    |- 
 S. A B  _d x  =  sum_ k  e.  ( 0 ... 3
 ) ( ( _i
 ^ k )  x.  ( S.2 `  ( x  e.  RR  |->  if (
 ( x  e.  A  /\  0  <_  T ) ,  T ,  0 ) ) ) )
 
Theoremitgex 19073 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |- 
 S. A B  _d x  e.  _V
 
Theoremitgeq1f 19074 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  =  B  ->  S. A C  _d x  =  S. B C  _d x )
 
Theoremitgeq1 19075* Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( A  =  B  ->  S. A C  _d x  =  S. B C  _d x )
 
Theoremnfitg1 19076 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  F/_ x S. A B  _d x
 
Theoremnfitg 19077* Bound-variable hypothesis builder for an integral: if  y is (effectively) not free in  A and  B, it is not free in  S. A B  _d x. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  F/_ y A   &    |-  F/_ y B   =>    |-  F/_ y S. A B  _d x
 
Theoremcbvitg 19078* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( x  =  y 
 ->  B  =  C )   &    |-  F/_ y B   &    |-  F/_ x C   =>    |-  S. A B  _d x  =  S. A C  _d y
 
Theoremcbvitgv 19079* Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( x  =  y 
 ->  B  =  C )   =>    |-  S. A B  _d x  =  S. A C  _d y
 
Theoremitgeq2 19080 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( A. x  e.  A  B  =  C  ->  S. A B  _d x  =  S. A C  _d x )
 
Theoremitgresr 19081 The domain of an integral only matters in its intersection with  RR. (Contributed by Mario Carneiro, 29-Jun-2014.)
 |- 
 S. A B  _d x  =  S. ( A  i^i  RR ) B  _d x
 
Theoremitg0 19082 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |- 
 S. (/) A  _d x  =  0
 
Theoremitgz 19083 The integral of zero on any set is zero. (Contributed by Mario Carneiro, 29-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |- 
 S. A 0  _d x  =  0
 
Theoremitgeq2dv 19084* Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  S. A B  _d x  =  S. A C  _d x )
 
Theoremitgmpt 19085* Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ph  ->  S. A B  _d x  =  S. A ( ( x  e.  A  |->  B ) `  y )  _d y )
 
Theoremitgcl 19086* The integral of an integrable function is a complex number. (Contributed by Mario Carneiro, 29-Jun-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  S. A B  _d x  e.  CC )
 
Theoremitgvallem 19087* Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( _i ^ K )  =  T   =>    |-  ( k  =  K  ->  ( S.2 `  ( x  e.  RR  |->  if (
 ( x  e.  A  /\  0  <_  ( Re
 `  ( B  /  ( _i ^ k ) ) ) ) ,  ( Re `  ( B  /  ( _i ^
 k ) ) ) ,  0 ) ) )  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Re `  ( B 
 /  T ) ) ) ,  ( Re
 `  ( B  /  T ) ) ,  0 ) ) ) )
 
Theoremitgvallem3 19088* Lemma for itgposval 19098 and itgreval 19099. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  =  0 )   =>    |-  ( ph  ->  (
 S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 ) ) )  =  0 )
 
Theoremibl0 19089 The zero function is integrable on any measurable set. (Unlike iblconst 19120, this does not require  A to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)
 |-  ( A  e.  dom  vol 
 ->  ( A  X.  {
 0 } )  e.  L ^1 )
 
Theoremiblcnlem1 19090* Lemma for iblcnlem 19091. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  R  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Re `  B ) ) ,  ( Re
 `  B ) ,  0 ) ) )   &    |-  S  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  -u ( Re `  B ) ) ,  -u ( Re `  B ) ,  0 ) ) )   &    |-  T  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Im `  B ) ) ,  ( Im
 `  B ) ,  0 ) ) )   &    |-  U  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  -u ( Im `  B ) ) ,  -u ( Im `  B ) ,  0 ) ) )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  L ^1 
 <->  ( ( x  e.  A  |->  B )  e. MblFn  /\  ( R  e.  RR  /\  S  e.  RR )  /\  ( T  e.  RR  /\  U  e.  RR )
 ) ) )
 
Theoremiblcnlem 19091* Expand out the forall in isibl2 19069. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  R  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Re `  B ) ) ,  ( Re
 `  B ) ,  0 ) ) )   &    |-  S  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  -u ( Re `  B ) ) ,  -u ( Re `  B ) ,  0 ) ) )   &    |-  T  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Im `  B ) ) ,  ( Im
 `  B ) ,  0 ) ) )   &    |-  U  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  -u ( Im `  B ) ) ,  -u ( Im `  B ) ,  0 ) ) )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  L ^1 
 <->  ( ( x  e.  A  |->  B )  e. MblFn  /\  ( R  e.  RR  /\  S  e.  RR )  /\  ( T  e.  RR  /\  U  e.  RR )
 ) ) )
 
Theoremitgcnlem 19092* Expand out the sum in dfitg 19072. (Contributed by Mario Carneiro, 1-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  R  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Re `  B ) ) ,  ( Re
 `  B ) ,  0 ) ) )   &    |-  S  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  -u ( Re `  B ) ) ,  -u ( Re `  B ) ,  0 ) ) )   &    |-  T  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  ( Im `  B ) ) ,  ( Im
 `  B ) ,  0 ) ) )   &    |-  U  =  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  -u ( Im `  B ) ) ,  -u ( Im `  B ) ,  0 ) ) )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  S. A B  _d x  =  ( ( R  -  S )  +  ( _i  x.  ( T  -  U ) ) ) )
 
Theoremiblrelem 19093* Integrability of a real function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  L ^1 
 <->  ( ( x  e.  A  |->  B )  e. MblFn  /\  ( S.2 `  ( x  e.  RR  |->  if (
 ( x  e.  A  /\  0  <_  B ) ,  B ,  0 ) ) )  e. 
 RR  /\  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  -u B ) ,  -u B ,  0 ) ) )  e.  RR )
 ) )
 
Theoremiblposlem 19094* Lemma for iblpos 19095. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  -u B ) ,  -u B ,  0 ) ) )  =  0 )
 
Theoremiblpos 19095* Integrability of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  ( ( x  e.  A  |->  B )  e.  L ^1  <->  ( ( x  e.  A  |->  B )  e. MblFn  /\  ( S.2 `  ( x  e.  RR  |->  if ( x  e.  A ,  B ,  0 )
 ) )  e.  RR ) ) )
 
Theoremiblre 19096* Integrability of a real function. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  L ^1 
 <->  ( ( x  e.  A  |->  if ( 0  <_  B ,  B , 
 0 ) )  e.  L ^1  /\  ( x  e.  A  |->  if (
 0  <_  -u B ,  -u B ,  0 ) )  e.  L ^1 ) ) )
 
Theoremitgrevallem1 19097* Lemma for itgposval 19098 and itgreval 19099. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  S. A B  _d x  =  ( ( S.2 `  ( x  e.  RR  |->  if (
 ( x  e.  A  /\  0  <_  B ) ,  B ,  0 ) ) )  -  ( S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_ 
 -u B ) ,  -u B ,  0 ) ) ) ) )
 
Theoremitgposval 19098* The integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  S. A B  _d x  =  (
 S.2 `  ( x  e.  RR  |->  if ( x  e.  A ,  B , 
 0 ) ) ) )
 
Theoremitgreval 19099* Decompose the integral of a real function into positive and negative parts. (Contributed by Mario Carneiro, 31-Jul-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  S. A B  _d x  =  ( S. A if ( 0  <_  B ,  B ,  0 )  _d x  -  S. A if ( 0  <_  -u B ,  -u B ,  0 )  _d x ) )
 
Theoremitgrecl 19100* Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  S. A B  _d x  e.  RR )
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