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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremitg2lr 19101* 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 19102 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 19103 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 19104 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 19105* 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 19106 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 19107 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 19108 The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  ( S.2 `  ( RR  X.  { 0 } ) )  =  0
 
Theoremitg2lecl 19109 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 19110 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 19111* 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 19112* 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 19113* 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 19126, 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 19114* Approximate version of itg2ub 19104. 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 19115* Approximate version of itg2le 19110. 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 19116* 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 19117 Lemma for itg2mulc 19118. (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 19118 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 19119* Lemma for itg2split 19120. (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 19120* The  S.2 integral splits under an almost disjoint union. (The proof avoids the use of itg2add 19130 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 19121* Lemma for itg2mono 19124. 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 19124. 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 19085. 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 19122* Lemma for itg2mono 19124. (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 19123* Lemma for itg2mono 19124. (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 19124* 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 19125* Subject to the conditions coming from mbfi1fseq 19092, 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 19126* Subject to the conditions coming from mbfi1fseq 19092, 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 19127* In an extension to the results of itg2i1fseq 19126, 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 19128* Special case of itg2i1fseq2 19127: 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 19129* Lemma for itg2add 19130. (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 19130 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 19131* 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 19132* Lemma for itgcn 19213. (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 19133* Lemma for itgcn 19213. (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 19134* A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 19400 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 19135 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 19136* 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 19137* 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 19138 An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
 |-  ( F  e.  L ^1  ->  F  e. MblFn )
 
Theoremiblitg 19139* 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 19140* Evaluate the class substitution in df-itg 18995. (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 19141 An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |- 
 S. A B  _d x  e.  _V
 
Theoremitgeq1f 19142 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 19143* 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 19144 Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
 |-  F/_ x S. A B  _d x
 
Theoremnfitg 19145* 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 19146* 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 19147* 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 19148 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 19149 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 19150 The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |- 
 S. (/) A  _d x  =  0
 
Theoremitgz 19151 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 19152* 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 19153* 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 19154* 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 19155* 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 19156* Lemma for itgposval 19166 and itgreval 19167. (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 19157 The zero function is integrable on any measurable set. (Unlike iblconst 19188, 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 19158* Lemma for iblcnlem 19159. (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 19159* Expand out the forall in isibl2 19137. (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 19160* Expand out the sum in dfitg 19140. (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 19161* 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 19162* Lemma for iblpos 19163. (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 19163* 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 19164* 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 19165* Lemma for itgposval 19166 and itgreval 19167. (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 19166* 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 19167* 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 19168* 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 )
 
Theoremiblcn 19169* Integrability of a complex function. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  L ^1 
 <->  ( ( x  e.  A  |->  ( Re `  B ) )  e.  L ^1  /\  ( x  e.  A  |->  ( Im
 `  B ) )  e.  L ^1 )
 ) )
 
Theoremitgcnval 19170* Decompose the integral of a complex function into real and imaginary parts. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  S. A B  _d x  =  ( S. A ( Re `  B )  _d x  +  ( _i  x.  S. A ( Im `  B )  _d x ) ) )
 
Theoremitgre 19171* Real part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  ( Re `  S. A B  _d x )  =  S. A ( Re `  B )  _d x )
 
Theoremitgim 19172* Imaginary part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  ( Im `  S. A B  _d x )  =  S. A ( Im `  B )  _d x )
 
Theoremiblneg 19173* The negative of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  -u B )  e.  L ^1 )
 
Theoremitgneg 19174* Negation of an integral. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  -u S. A B  _d x  =  S. A -u B  _d x )
 
Theoremiblss 19175* A subset of an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  A  e.  dom  vol )   &    |-  (
 ( ph  /\  x  e.  B )  ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  B  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )
 
Theoremiblss2 19176* Change the domain of an integrability predicate. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  e.  dom  vol )   &    |-  (
 ( ph  /\  x  e.  A )  ->  C  e.  V )   &    |-  ( ( ph  /\  x  e.  ( B 
 \  A ) ) 
 ->  C  =  0 )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  B  |->  C )  e.  L ^1 )
 
Theoremitgitg2 19177* Transfer an integral using  S.2 to an equivalent integral using  S.. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( ( ph  /\  x  e.  RR )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  RR )  ->  0  <_  A )   &    |-  ( ph  ->  ( x  e. 
 RR  |->  A )  e.  L ^1 )   =>    |-  ( ph  ->  S. RR A  _d x  =  ( S.2 `  ( x  e.  RR  |->  A ) ) )
 
Theoremi1fibl 19178 A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( F  e.  dom  S.1 
 ->  F  e.  L ^1 )
 
Theoremitgitg1 19179* Transfer an integral using  S.1 to an equivalent integral using  S.. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( F  e.  dom  S.1 
 ->  S. RR ( F `
  x )  _d x  =  ( S.1 `  F ) )
 
Theoremitgle 19180* Monotonicity of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  <_  C )   =>    |-  ( ph  ->  S. A B  _d x  <_  S. A C  _d x )
 
Theoremitgge0 19181* The integral of a positive function is positive. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  0  <_  S. A B  _d x )
 
Theoremitgss 19182* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  x  e.  ( B 
 \  A ) ) 
 ->  C  =  0 )   =>    |-  ( ph  ->  S. A C  _d x  =  S. B C  _d x )
 
Theoremitgss2 19183* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( A  C_  B  ->  S. A C  _d x  =  S. B if ( x  e.  A ,  C ,  0 )  _d x )
 
Theoremitgeqa 19184* Approximate equality of integrals. If  C ( x )  =  D ( x ) for almost all  x, then  S. B C ( x )  _d x  =  S. B D ( x )  _d x and one is integrable iff the other is. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
 |-  ( ( ph  /\  x  e.  B )  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  D  e.  CC )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( B  \  A ) )  ->  C  =  D )   =>    |-  ( ph  ->  (
 ( ( x  e.  B  |->  C )  e.  L ^1  <->  ( x  e.  B  |->  D )  e.  L ^1 )  /\  S. B C  _d x  =  S. B D  _d x ) )
 
Theoremitgss3 19185* Expand the set of an integral by a nullset. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  RR )   &    |-  ( ph  ->  ( vol * `  ( B  \  A ) )  =  0 )   &    |-  (
 ( ph  /\  x  e.  B )  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( ( x  e.  A  |->  C )  e.  L ^1  <->  ( x  e.  B  |->  C )  e.  L ^1 )  /\  S. A C  _d x  =  S. B C  _d x ) )
 
Theoremitgioo 19186* Equality of integrals on open and closed intervals. (Contributed by Mario Carneiro, 2-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  C  e.  CC )   =>    |-  ( ph  ->  S. ( A (,) B ) C  _d x  =  S. ( A [,] B ) C  _d x )
 
Theoremitgless 19187* Expand the integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Aug-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  A  e.  dom  vol )   &    |-  (
 ( ph  /\  x  e.  B )  ->  C  e.  RR )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  0  <_  C )   &    |-  ( ph  ->  ( x  e.  B  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  S. A C  _d x  <_  S. B C  _d x )
 
Theoremiblconst 19188 A constant function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  ( vol `  A )  e.  RR  /\  B  e.  CC )  ->  ( A  X.  { B }
 )  e.  L ^1 )
 
Theoremitgconst 19189* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  ( vol `  A )  e.  RR  /\  B  e.  CC )  ->  S. A B  _d x  =  ( B  x.  ( vol `  A ) ) )
 
Theoremibladdlem 19190* Lemma for ibladd 19191. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  D  =  ( B  +  C ) )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e. MblFn )   &    |-  ( ph  ->  (
 S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 ) ) )  e.  RR )   &    |-  ( ph  ->  ( S.2 `  ( x  e.  RR  |->  if (
 ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) ) )  e. 
 RR )   =>    |-  ( ph  ->  ( S.2 `  ( x  e. 
 RR  |->  if ( ( x  e.  A  /\  0  <_  D ) ,  D ,  0 ) ) )  e.  RR )
 
Theoremibladd 19191* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e.  L ^1 )
 
Theoremiblsub 19192* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  L ^1 )
 
Theoremitgaddlem1 19193* Lemma for itgadd 19195. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  0  <_  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  C )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgaddlem2 19194* Lemma for itgadd 19195. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgadd 19195* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgsub 19196* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  S. A ( B  -  C )  _d x  =  ( S. A B  _d x  -  S. A C  _d x ) )
 
Theoremitgfsum 19197* Take a finite sum of integrals over the same domain. (Contributed by Mario Carneiro, 24-Aug-2014.)
 |-  ( ph  ->  A  e.  dom  vol )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  k  e.  B ) )  ->  C  e.  V )   &    |-  (
 ( ph  /\  k  e.  B )  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  ( ( x  e.  A  |->  sum_ k  e.  B  C )  e.  L ^1  /\  S. A sum_ k  e.  B  C  _d x  =  sum_ k  e.  B  S. A C  _d x ) )
 
Theoremiblabslem 19198* Lemma for iblabs 19199. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( abs `  ( F `  B ) ) ,  0 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  ( F `  B ) )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  B )  e. 
 RR )   =>    |-  ( ph  ->  ( G  e. MblFn  /\  ( S.2 `  G )  e.  RR ) )
 
Theoremiblabs 19199* The absolute value of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B ) )  e.  L ^1 )
 
Theoremiblabsr 19200* A measurable function is integrable iff its absolute value is integrable. (See iblabs 19199 for the forward implication.) (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B )
 )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )
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