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Theorem List for Metamath Proof Explorer - 37101-37200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvdivbd 37101* A sufficient condition for the derivative to be bounded, for the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  (
 ( ph  /\  x  e.  X )  ->  A  e.  CC )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  C ) )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  CC )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  R  e.  RR )   &    |-  ( ph  ->  T  e.  RR )   &    |-  ( ph  ->  Q  e.  RR )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( abs `  C )  <_  U )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  ( abs `  B )  <_  R )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( abs `  D )  <_  T )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  ( abs `  A )  <_  Q )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  B ) )  =  ( x  e.  X  |->  D ) )   &    |-  (
 ( ph  /\  x  e.  X )  ->  D  e.  CC )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  X  E  <_  ( abs `  B ) )   &    |-  F  =  ( S  _D  ( x  e.  X  |->  ( A 
 /  B ) ) )   =>    |-  ( ph  ->  E. b  e.  RR  A. x  e.  X  ( abs `  ( F `  x ) ) 
 <_  b )
 
Theoremdvsubcncf 37102 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  ( S  _D  F )  e.  ( X -cn-> CC ) )   &    |-  ( ph  ->  ( S  _D  G )  e.  ( X -cn-> CC ) )   =>    |-  ( ph  ->  ( S  _D  ( F  oF  -  G ) )  e.  ( X -cn-> CC ) )
 
Theoremdvmulcncf 37103 A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  ( S  _D  F )  e.  ( X -cn-> CC ) )   &    |-  ( ph  ->  ( S  _D  G )  e.  ( X -cn-> CC ) )   =>    |-  ( ph  ->  ( S  _D  ( F  oF  x.  G ) )  e.  ( X -cn-> CC ) )
 
Theoremdvcosax 37104* Derivative exercise: the derivative with respect to x of cos(Ax), given a constant  A. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  CC  |->  ( cos `  ( A  x.  x ) ) ) )  =  ( x  e. 
 CC  |->  ( A  x.  -u ( sin `  ( A  x.  x ) ) ) ) )
 
Theoremdvdivcncf 37105 A sufficient condition for the derivative of a quotient to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  G : X --> ( CC  \  { 0 } )
 )   &    |-  ( ph  ->  ( S  _D  F )  e.  ( X -cn-> CC )
 )   &    |-  ( ph  ->  ( S  _D  G )  e.  ( X -cn-> CC )
 )   =>    |-  ( ph  ->  ( S  _D  ( F  oF  /  G ) )  e.  ( X -cn-> CC ) )
 
Theoremdvbdfbdioolem1 37106* Given a function with bounded derivative, on an open interval, here is an absolute bound to the difference of the image of two points in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  K  e.  RR )   &    |-  ( ph  ->  A. x  e.  ( A (,) B ) ( abs `  ( ( RR  _D  F ) `  x ) )  <_  K )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  D  e.  ( C (,) B ) )   =>    |-  ( ph  ->  (
 ( abs `  ( ( F `  D )  -  ( F `  C ) ) )  <_  ( K  x.  ( D  -  C ) )  /\  ( abs `  ( ( F `  D )  -  ( F `  C ) ) )  <_  ( K  x.  ( B  -  A ) ) ) )
 
Theoremdvbdfbdioolem2 37107* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  K  e.  RR )   &    |-  ( ph  ->  A. x  e.  ( A (,) B ) ( abs `  (
 ( RR  _D  F ) `  x ) ) 
 <_  K )   &    |-  M  =  ( ( abs `  ( F `  ( ( A  +  B )  / 
 2 ) ) )  +  ( K  x.  ( B  -  A ) ) )   =>    |-  ( ph  ->  A. x  e.  ( A (,) B ) ( abs `  ( F `  x ) )  <_  M )
 
Theoremdvbdfbdioo 37108* A function on an open interval, with bounded derivative, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  E. a  e.  RR  A. x  e.  ( A (,) B ) ( abs `  ( ( RR  _D  F ) `  x ) )  <_  a )   =>    |-  ( ph  ->  E. b  e.  RR  A. x  e.  ( A (,) B ) ( abs `  ( F `  x ) ) 
 <_  b )
 
Theoremioodvbdlimc1lem1 37109* If  F has bounded derivative on  ( A (,) B ) then a sequence of points in its image converges to its  limsup. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F  e.  (
 ( A (,) B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  E. y  e.  RR  A. x  e.  ( A (,) B ) ( abs `  ( ( RR  _D  F ) `  x ) )  <_  y )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  R : ( ZZ>= `  M )
 --> ( A (,) B ) )   &    |-  S  =  ( j  e.  ( ZZ>= `  M )  |->  ( F `
  ( R `  j ) ) )   &    |-  ( ph  ->  R  e.  dom  ~~>  )   &    |-  K  =  sup ( { k  e.  ( ZZ>=
 `  M )  | 
 A. i  e.  ( ZZ>=
 `  k ) ( abs `  ( ( R `  i )  -  ( R `  k ) ) )  <  ( x  /  ( sup ( ran  ( z  e.  ( A (,) B )  |->  ( abs `  ( ( RR  _D  F ) `  z ) ) ) ,  RR ,  <  )  +  1 ) ) } ,  RR ,  `'  <  )   =>    |-  ( ph  ->  S  ~~>  ( limsup `  S )
 )
 
Theoremioodvbdlimc1lem2 37110* Limit at the lower bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  E. y  e.  RR  A. x  e.  ( A (,) B ) ( abs `  ( ( RR  _D  F ) `  x ) )  <_  y )   &    |-  Y  =  sup ( ran  ( x  e.  ( A (,) B )  |->  ( abs `  (
 ( RR  _D  F ) `  x ) ) ) ,  RR ,  <  )   &    |-  M  =  ( ( |_ `  (
 1  /  ( B  -  A ) ) )  +  1 )   &    |-  S  =  ( j  e.  ( ZZ>=
 `  M )  |->  ( F `  ( A  +  ( 1  /  j ) ) ) )   &    |-  R  =  ( j  e.  ( ZZ>= `  M )  |->  ( A  +  ( 1  /  j ) ) )   &    |-  N  =  if ( M  <_  ( ( |_ `  ( Y  /  ( x  /  2 ) ) )  +  1 ) ,  ( ( |_ `  ( Y  /  ( x  /  2 ) ) )  +  1 ) ,  M )   &    |-  ( ch 
 <->  ( ( ( ( ( ph  /\  x  e.  RR+ )  /\  j  e.  ( ZZ>= `  N )
 )  /\  ( abs `  ( ( S `  j )  -  ( limsup `
  S ) ) )  <  ( x 
 /  2 ) ) 
 /\  z  e.  ( A (,) B ) ) 
 /\  ( abs `  (
 z  -  A ) )  <  ( 1 
 /  j ) ) )   =>    |-  ( ph  ->  ( limsup `
  S )  e.  ( F lim CC  A ) )
 
Theoremioodvbdlimc1 37111* A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  E. y  e.  RR  A. x  e.  ( A (,) B ) ( abs `  ( ( RR  _D  F ) `  x ) )  <_  y )   =>    |-  ( ph  ->  ( F lim CC  A )  =/=  (/) )
 
Theoremioodvbdlimc2lem 37112* Limit at the upper bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  E. y  e.  RR  A. x  e.  ( A (,) B ) ( abs `  ( ( RR  _D  F ) `  x ) )  <_  y )   &    |-  Y  =  sup ( ran  ( x  e.  ( A (,) B )  |->  ( abs `  (
 ( RR  _D  F ) `  x ) ) ) ,  RR ,  <  )   &    |-  M  =  ( ( |_ `  (
 1  /  ( B  -  A ) ) )  +  1 )   &    |-  S  =  ( j  e.  ( ZZ>=
 `  M )  |->  ( F `  ( B  -  ( 1  /  j ) ) ) )   &    |-  R  =  ( j  e.  ( ZZ>= `  M )  |->  ( B  -  ( 1  /  j ) ) )   &    |-  N  =  if ( M  <_  ( ( |_ `  ( Y  /  ( x  /  2 ) ) )  +  1 ) ,  ( ( |_ `  ( Y  /  ( x  /  2 ) ) )  +  1 ) ,  M )   &    |-  ( ch 
 <->  ( ( ( ( ( ph  /\  x  e.  RR+ )  /\  j  e.  ( ZZ>= `  N )
 )  /\  ( abs `  ( ( S `  j )  -  ( limsup `
  S ) ) )  <  ( x 
 /  2 ) ) 
 /\  z  e.  ( A (,) B ) ) 
 /\  ( abs `  (
 z  -  B ) )  <  ( 1 
 /  j ) ) )   =>    |-  ( ph  ->  ( limsup `
  S )  e.  ( F lim CC  B ) )
 
Theoremioodvbdlimc2 37113* A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  E. y  e.  RR  A. x  e.  ( A (,) B ) ( abs `  ( ( RR  _D  F ) `  x ) )  <_  y )   =>    |-  ( ph  ->  ( F lim CC  B )  =/=  (/) )
 
Theoremdvdmsscn 37114  X is a subset of  CC. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   =>    |-  ( ph  ->  X 
 C_  CC )
 
Theoremdvmptmulf 37115* Function-builder for derivative, product rule. A version of dvmptmul 22658 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  D  e.  W )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  C ) )  =  ( x  e.  X  |->  D ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( A  x.  C ) ) )  =  ( x  e.  X  |->  ( ( B  x.  C )  +  ( D  x.  A ) ) ) )
 
Theoremdvnmptdivc 37116* Function-builder for iterated derivative, division rule for constant divisor. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  X  /\  n  e.  ( 0 ... M ) )  ->  B  e.  CC )   &    |-  (
 ( ph  /\  n  e.  ( 0 ... M ) )  ->  ( ( S  Dn ( x  e.  X  |->  A ) ) `  n )  =  ( x  e.  X  |->  B ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0 )   &    |-  ( ph  ->  M  e.  NN0 )   =>    |-  ( ( ph  /\  n  e.  ( 0 ... M ) )  ->  ( ( S  Dn ( x  e.  X  |->  ( A  /  C ) ) ) `  n )  =  ( x  e.  X  |->  ( B  /  C ) ) )
 
Theoremdvdsn1add 37117 If  K divides  N but  K does not divide  M, then  K does not divide  ( M  +  N ). (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  (
 ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( -.  K  ||  M  /\  K  ||  N )  ->  -.  K  ||  ( M  +  N ) ) )
 
Theoremdvxpaek 37118* Derivative of the polynomial  ( x  +  A
) ^ K. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  K  e.  NN )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( ( x  +  A ) ^ K ) ) )  =  ( x  e.  X  |->  ( K  x.  ( ( x  +  A ) ^ ( K  -  1 ) ) ) ) )
 
Theoremdvnmptconst 37119* The  N-th derivative of a constant function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  (
 ( S  Dn
 ( x  e.  X  |->  A ) ) `  N )  =  ( x  e.  X  |->  0 ) )
 
Theoremdvnxpaek 37120* The  n-th derivative of the polynomial (x+A)^K (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  F  =  ( x  e.  X  |->  ( ( x  +  A ) ^ K ) )   =>    |-  ( ( ph  /\  N  e.  NN0 )  ->  (
 ( S  Dn F ) `  N )  =  ( x  e.  X  |->  if ( K  <  N ,  0 ,  (
 ( ( ! `  K )  /  ( ! `  ( K  -  N ) ) )  x.  ( ( x  +  A ) ^
 ( K  -  N ) ) ) ) ) )
 
Theoremdvnmul 37121* Function-builder for the  N-th derivative, product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  (
 ( ph  /\  x  e.  X )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  B  e.  CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  F  =  ( x  e.  X  |->  A )   &    |-  G  =  ( x  e.  X  |->  B )   &    |-  (
 ( ph  /\  k  e.  ( 0 ... N ) )  ->  ( ( S  Dn F ) `  k ) : X --> CC )   &    |-  (
 ( ph  /\  k  e.  ( 0 ... N ) )  ->  ( ( S  Dn G ) `  k ) : X --> CC )   &    |-  C  =  ( k  e.  (
 0 ... N )  |->  ( ( S  Dn F ) `  k
 ) )   &    |-  D  =  ( k  e.  ( 0
 ... N )  |->  ( ( S  Dn G ) `  k
 ) )   =>    |-  ( ph  ->  (
 ( S  Dn
 ( x  e.  X  |->  ( A  x.  B ) ) ) `  N )  =  ( x  e.  X  |->  sum_ k  e.  ( 0 ... N ) ( ( N  _C  k )  x.  ( ( ( C `
  k ) `  x )  x.  (
 ( D `  ( N  -  k ) ) `
  x ) ) ) ) )
 
Theoremdvmptfprodlem 37122* Induction step for dvmptfprod 37123. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/ x ph   &    |-  F/ i ph   &    |-  F/ j ph   &    |-  F/_ i F   &    |-  F/_ j G   &    |-  ( ( ph  /\  i  e.  I  /\  x  e.  X )  ->  A  e.  CC )   &    |-  ( ph  ->  D  e.  Fin )   &    |-  ( ph  ->  E  e.  _V )   &    |-  ( ph  ->  -.  E  e.  D )   &    |-  ( ph  ->  ( D  u.  { E } )  C_  I )   &    |-  ( ph  ->  S  e.  { RR ,  CC }
 )   &    |-  ( ( ( ph  /\  x  e.  X ) 
 /\  j  e.  D )  ->  C  e.  CC )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  prod_ i  e.  D  A ) )  =  ( x  e.  X  |->  sum_
 j  e.  D  ( C  x.  prod_ i  e.  ( D  \  {
 j } ) A ) ) )   &    |-  (
 ( ph  /\  x  e.  X )  ->  G  e.  CC )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  F ) )  =  ( x  e.  X  |->  G ) )   &    |-  ( i  =  E  ->  A  =  F )   &    |-  ( j  =  E  ->  C  =  G )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  prod_ i  e.  ( D  u.  { E }
 ) A ) )  =  ( x  e.  X  |->  sum_ j  e.  ( D  u.  { E }
 ) ( C  x.  prod_ i  e.  ( ( D  u.  { E } )  \  { j } ) A ) ) )
 
Theoremdvmptfprod 37123* Function-builder for derivative, finite product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/ i ph   &    |-  F/ j ph   &    |-  J  =  ( Kt  S )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  J )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ( ph  /\  i  e.  I  /\  x  e.  X )  ->  A  e.  CC )   &    |-  ( ( ph  /\  i  e.  I  /\  x  e.  X )  ->  B  e.  CC )   &    |-  (
 ( ph  /\  i  e.  I )  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( i  =  j  ->  B  =  C )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  prod_ i  e.  I  A ) )  =  ( x  e.  X  |->  sum_ j  e.  I  ( C  x.  prod_ i  e.  ( I  \  { j }
 ) A ) ) )
 
Theoremdvnprodlem1 37124*  D is bijective. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  C  =  ( s  e.  ~P T  |->  ( n  e. 
 NN0  |->  { c  e.  (
 ( 0 ... n )  ^m  s )  | 
 sum_ t  e.  s  ( c `  t
 )  =  n }
 ) )   &    |-  ( ph  ->  J  e.  NN0 )   &    |-  D  =  ( c  e.  ( ( C `  ( R  u.  { Z }
 ) ) `  J )  |->  <. ( J  -  ( c `  Z ) ) ,  (
 c  |`  R ) >. )   &    |-  ( ph  ->  T  e.  Fin )   &    |-  ( ph  ->  Z  e.  T )   &    |-  ( ph  ->  -.  Z  e.  R )   &    |-  ( ph  ->  ( R  u.  { Z } )  C_  T )   =>    |-  ( ph  ->  D :
 ( ( C `  ( R  u.  { Z } ) ) `  J ) -1-1-onto-> U_ k  e.  (
 0 ... J ) ( { k }  X.  ( ( C `  R ) `  k
 ) ) )
 
Theoremdvnprodlem2 37125* Induction step for dvnprodlem2 37125. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  T  e.  Fin )   &    |-  ( ( ph  /\  t  e.  T )  ->  ( H `  t ) : X --> CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  t  e.  T  /\  j  e.  ( 0 ... N ) )  ->  ( ( S  Dn ( H `  t ) ) `  j ) : X --> CC )   &    |-  C  =  ( s  e.  ~P T  |->  ( n  e. 
 NN0  |->  { c  e.  (
 ( 0 ... n )  ^m  s )  | 
 sum_ t  e.  s  ( c `  t
 )  =  n }
 ) )   &    |-  ( ph  ->  R 
 C_  T )   &    |-  ( ph  ->  Z  e.  ( T  \  R ) )   &    |-  ( ph  ->  A. k  e.  ( 0 ... N ) ( ( S  Dn ( x  e.  X  |->  prod_ t  e.  R  ( ( H `
  t ) `  x ) ) ) `
  k )  =  ( x  e.  X  |->  sum_
 c  e.  ( ( C `  R ) `
  k ) ( ( ( ! `  k )  /  prod_ t  e.  R  ( ! `  ( c `  t
 ) ) )  x. 
 prod_ t  e.  R  ( ( ( S  Dn ( H `
  t ) ) `
  ( c `  t ) ) `  x ) ) ) )   &    |-  ( ph  ->  J  e.  ( 0 ...
 N ) )   &    |-  D  =  ( c  e.  (
 ( C `  ( R  u.  { Z }
 ) ) `  J )  |->  <. ( J  -  ( c `  Z ) ) ,  (
 c  |`  R ) >. )   =>    |-  ( ph  ->  ( ( S  Dn ( x  e.  X  |->  prod_ t  e.  ( R  u.  { Z } ) ( ( H `  t ) `
  x ) ) ) `  J )  =  ( x  e.  X  |->  sum_ c  e.  (
 ( C `  ( R  u.  { Z }
 ) ) `  J ) ( ( ( ! `  J ) 
 /  prod_ t  e.  ( R  u.  { Z }
 ) ( ! `  ( c `  t
 ) ) )  x. 
 prod_ t  e.  ( R  u.  { Z }
 ) ( ( ( S  Dn ( H `  t ) ) `  ( c `
  t ) ) `
  x ) ) ) )
 
Theoremdvnprodlem3 37126* The multinomial formula for the  k-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  T  e.  Fin )   &    |-  ( ( ph  /\  t  e.  T )  ->  ( H `  t ) : X --> CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  t  e.  T  /\  j  e.  ( 0 ... N ) )  ->  ( ( S  Dn ( H `  t ) ) `  j ) : X --> CC )   &    |-  F  =  ( x  e.  X  |->  prod_ t  e.  T  ( ( H `  t
 ) `  x )
 )   &    |-  D  =  ( s  e.  ~P T  |->  ( n  e.  NN0  |->  { c  e.  ( ( 0 ... n )  ^m  s
 )  |  sum_ t  e.  s  ( c `  t )  =  n } ) )   &    |-  C  =  ( n  e.  NN0  |->  { c  e.  ( ( 0 ... n ) 
 ^m  T )  | 
 sum_ t  e.  T  ( c `  t
 )  =  n }
 )   =>    |-  ( ph  ->  (
 ( S  Dn F ) `  N )  =  ( x  e.  X  |->  sum_ c  e.  ( C `  N ) ( ( ( ! `  N )  /  prod_ t  e.  T  ( ! `  ( c `  t
 ) ) )  x. 
 prod_ t  e.  T  ( ( ( S  Dn ( H `
  t ) ) `
  ( c `  t ) ) `  x ) ) ) )
 
Theoremdvnprod 37127* The multinomial formula for the  N-th derivative of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  T  e.  Fin )   &    |-  ( ( ph  /\  t  e.  T )  ->  ( H `  t ) : X --> CC )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ( ph  /\  t  e.  T  /\  k  e.  ( 0 ... N ) )  ->  ( ( S  Dn ( H `  t ) ) `  k ) : X --> CC )   &    |-  F  =  ( x  e.  X  |->  prod_ t  e.  T  ( ( H `  t
 ) `  x )
 )   &    |-  C  =  ( n  e.  NN0  |->  { c  e.  ( ( 0 ... n )  ^m  T )  |  sum_ t  e.  T  ( c `  t )  =  n } )   =>    |-  ( ph  ->  (
 ( S  Dn F ) `  N )  =  ( x  e.  X  |->  sum_ c  e.  ( C `  N ) ( ( ( ! `  N )  /  prod_ t  e.  T  ( ! `  ( c `  t
 ) ) )  x. 
 prod_ t  e.  T  ( ( ( S  Dn ( H `
  t ) ) `
  ( c `  t ) ) `  x ) ) ) )
 
21.30.11  Integrals
 
Theoremvolioo 37128 The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol `  ( A (,) B ) )  =  ( B  -  A ) )
 
Theoremitgsin0pilem1 37129* Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  C  =  ( t  e.  (
 0 [,] pi )  |->  -u ( cos `  t )
 )   =>    |- 
 S. ( 0 (,)
 pi ) ( sin `  x )  _d x  =  2
 
Theoremibliccsinexp 37130* sin^n on a closed interval is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  N  e.  NN0 )  ->  ( x  e.  ( A [,] B )  |->  ( ( sin `  x ) ^ N ) )  e.  L^1 )
 
Theoremitgsin0pi 37131 Calculation of the integral for sine on the (0,π) interval (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  S. ( 0 (,) pi ) ( sin `  x )  _d x  =  2
 
Theoremiblioosinexp 37132* sin^n on an open integral is integrable (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  N  e.  NN0 )  ->  ( x  e.  ( A (,) B )  |->  ( ( sin `  x ) ^ N ) )  e.  L^1 )
 
Theoremitgsinexplem1 37133* Integration by parts is applied to integrate sin^(N+1). (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  F  =  ( x  e.  CC  |->  ( ( sin `  x ) ^ N ) )   &    |-  G  =  ( x  e.  CC  |->  -u ( cos `  x ) )   &    |-  H  =  ( x  e.  CC  |->  ( ( N  x.  (
 ( sin `  x ) ^ ( N  -  1 ) ) )  x.  ( cos `  x ) ) )   &    |-  I  =  ( x  e.  CC  |->  ( ( ( sin `  x ) ^ N )  x.  ( sin `  x ) ) )   &    |-  L  =  ( x  e.  CC  |->  ( ( ( N  x.  ( ( sin `  x ) ^ ( N  -  1 ) ) )  x.  ( cos `  x ) )  x.  -u ( cos `  x ) ) )   &    |-  M  =  ( x  e.  CC  |->  ( ( ( cos `  x ) ^ 2
 )  x.  ( ( sin `  x ) ^ ( N  -  1 ) ) ) )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  S. ( 0 (,) pi ) ( ( ( sin `  x ) ^ N )  x.  ( sin `  x ) )  _d x  =  ( N  x.  S. (
 0 (,) pi ) ( ( ( cos `  x ) ^ 2 )  x.  ( ( sin `  x ) ^ ( N  -  1 ) ) )  _d x ) )
 
Theoremitgsinexp 37134* A recursive formula for the integral of sin^N on the interval (0,π) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  I  =  ( n  e.  NN0  |->  S. ( 0 (,) pi ) ( ( sin `  x ) ^ n )  _d x )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  2 ) )   =>    |-  ( ph  ->  ( I `  N )  =  ( ( ( N  -  1 )  /  N )  x.  ( I `  ( N  -  2
 ) ) ) )
 
Theoremiblconstmpt 37135* A constant function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  dom  vol  /\  ( vol `  A )  e.  RR  /\  B  e.  CC )  ->  ( x  e.  A  |->  B )  e.  L^1 )
 
Theoremitgeq1d 37136* Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  S. A C  _d x  =  S. B C  _d x )
 
Theoremmbf0 37137 The empty set is a measurable function (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (/)  e. MblFn
 
Theoremmbfres2cn 37138 Measurability of a piecewise function: if  F is measurable on subsets  B and  C of its domain, and these pieces make up all of  A, then  F is measurable on the whole domain. Similar to mbfres2 22346 but here the theorem is extended to complex valued functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  ( F  |`  B )  e. MblFn )   &    |-  ( ph  ->  ( F  |`  C )  e. MblFn )   &    |-  ( ph  ->  ( B  u.  C )  =  A )   =>    |-  ( ph  ->  F  e. MblFn )
 
Theoremvol0 37139 The measure of the empty set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( vol `  (/) )  =  0
 
Theoremditgeqiooicc 37140* A function  F on an open interval, has the same directed integral as its extension  G on the closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   =>    |-  ( ph  ->  S__
 [ A  ->  B ] ( F `  x )  _d x  =  S__ [ A  ->  B ] ( G `  x )  _d x )
 
Theoremvolge0 37141 The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  dom  vol  ->  0 
 <_  ( vol `  A ) )
 
Theoremcnbdibl 37142* A continuous bounded function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  dom  vol )   &    |-  ( ph  ->  ( vol `  A )  e.  RR )   &    |-  ( ph  ->  F  e.  ( A -cn-> CC ) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  dom  F ( abs `  ( F `  y ) )  <_  x )   =>    |-  ( ph  ->  F  e.  L^1 )
 
Theoremsnmbl 37143 A singleton is measurable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  RR  ->  { A }  e.  dom  vol )
 
Theoremditgeq3d 37144* Equality theorem for the directed integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  <_  B )   &    |-  ( ( ph  /\  x  e.  ( A (,) B ) )  ->  D  =  E )   =>    |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  =  S__ [ A  ->  B ] E  _d x )
 
Theoremiblempty 37145 The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (/)  e.  L^1
 
Theoremiblsplit 37146* The union of two integrable functions is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ( vol* `  ( A  i^i  B ) )  =  0
 )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ( ph  /\  x  e.  U ) 
 ->  C  e.  CC )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L^1 )   &    |-  ( ph  ->  ( x  e.  B  |->  C )  e.  L^1 )   =>    |-  ( ph  ->  ( x  e.  U  |->  C )  e.  L^1 )
 
Theoremvolsn 37147 A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  RR  ->  ( vol `  { A }
 )  =  0 )
 
Theoremitgvol0 37148* If the domani is negligible, the function is integrable and the integral is 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  ( vol* `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  B )  e.  L^1  /\  S. A B  _d x  =  0
 ) )
 
Theoremitgcoscmulx 37149* Exercise: the integral of  x  |->  cos a x on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <_  C )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  S. ( B (,) C ) ( cos `  ( A  x.  x ) )  _d x  =  ( ( ( sin `  ( A  x.  C ) )  -  ( sin `  ( A  x.  B ) ) )  /  A ) )
 
Theoremiblsplitf 37150* A version of iblsplit 37146 using bound-variable hypotheses instead of distinct variable conditions" (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F/ x ph   &    |-  ( ph  ->  ( vol* `  ( A  i^i  B ) )  =  0 )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ( ph  /\  x  e.  U ) 
 ->  C  e.  CC )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L^1 )   &    |-  ( ph  ->  ( x  e.  B  |->  C )  e.  L^1 )   =>    |-  ( ph  ->  ( x  e.  U  |->  C )  e.  L^1 )
 
Theoremibliooicc 37151* If a function is integrable on an open interval, it is integrable on the corresponding closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  ( A (,) B )  |->  C )  e.  L^1 )   &    |-  ( ( ph  /\  x  e.  ( A [,] B ) ) 
 ->  C  e.  CC )   =>    |-  ( ph  ->  ( x  e.  ( A [,] B )  |->  C )  e.  L^1 )
 
Theoremvolioc 37152 The measure of left open, right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol `  ( A (,] B ) )  =  ( B  -  A ) )
 
Theoremiblspltprt 37153* If a function is integrable on any interval of a partition, then it is integrable on the whole interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F/ t ph   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ( ZZ>=
 `  ( M  +  1 ) ) )   &    |-  ( ( ph  /\  i  e.  ( M ... N ) )  ->  ( P `
  i )  e. 
 RR )   &    |-  ( ( ph  /\  i  e.  ( M..^ N ) )  ->  ( P `  i )  <  ( P `  ( i  +  1
 ) ) )   &    |-  (
 ( ph  /\  t  e.  ( ( P `  M ) [,] ( P `  N ) ) )  ->  A  e.  CC )   &    |-  ( ( ph  /\  i  e.  ( M..^ N ) )  ->  ( t  e.  (
 ( P `  i
 ) [,] ( P `  ( i  +  1
 ) ) )  |->  A )  e.  L^1 )   =>    |-  ( ph  ->  (
 t  e.  ( ( P `  M ) [,] ( P `  N ) )  |->  A )  e.  L^1 )
 
Theoremitgsincmulx 37154* Exercise: the integral of  x  |->  sin a x on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <_  C )   =>    |-  ( ph  ->  S. ( B (,) C ) ( sin `  ( A  x.  x ) )  _d x  =  ( ( ( cos `  ( A  x.  B ) )  -  ( cos `  ( A  x.  C ) ) )  /  A ) )
 
Theoremitgsubsticclem 37155* lemma for itgsubsticc 37156 (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F  =  ( u  e.  ( K [,] L )  |->  C )   &    |-  G  =  ( u  e.  RR  |->  if ( u  e.  ( K [,] L ) ,  ( F `  u ) ,  if ( u  <  K ,  ( F `  K ) ,  ( F `  L ) ) ) )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  X  <_  Y )   &    |-  ( ph  ->  ( x  e.  ( X [,] Y )  |->  A )  e.  ( ( X [,] Y ) -cn-> ( K [,] L ) ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( ( X (,) Y ) -cn-> CC )  i^i  L^1 ) )   &    |-  ( ph  ->  F  e.  ( ( K [,] L ) -cn-> CC ) )   &    |-  ( ph  ->  K  e.  RR )   &    |-  ( ph  ->  L  e.  RR )   &    |-  ( ph  ->  K  <_  L )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )   &    |-  ( u  =  A  ->  C  =  E )   &    |-  ( x  =  X  ->  A  =  K )   &    |-  ( x  =  Y  ->  A  =  L )   =>    |-  ( ph  ->  S__ [ K  ->  L ] C  _d u  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
 
Theoremitgsubsticc 37156* Integration by u-substitution. The main difference with respect to itgsubst 22744 is that here we consider the range of  A ( x ) to be in the closed interval  ( K [,] L
). If  A ( x ) is a continuous, differentiable function from  [ X ,  Y ] to  ( Z ,  W ), whose derivative is continuous and integrable, and  C ( u ) is a continuous function on  ( Z ,  W ), then the integral of  C ( u ) from  K  =  A ( X ) to  L  =  A ( Y ) is equal to the integral of  C ( A ( x ) )  _D  A ( x ) from  X to  Y. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  X 
 <_  Y )   &    |-  ( ph  ->  ( x  e.  ( X [,] Y )  |->  A )  e.  ( ( X [,] Y )
 -cn-> ( K [,] L ) ) )   &    |-  ( ph  ->  ( u  e.  ( K [,] L )  |->  C )  e.  ( ( K [,] L ) -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( ( X (,) Y ) -cn-> CC )  i^i  L^1 ) )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )   &    |-  ( u  =  A  ->  C  =  E )   &    |-  ( x  =  X  ->  A  =  K )   &    |-  ( x  =  Y  ->  A  =  L )   &    |-  ( ph  ->  K  e.  RR )   &    |-  ( ph  ->  L  e.  RR )   =>    |-  ( ph  ->  S__
 [ K  ->  L ] C  _d u  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
 
Theoremitgioocnicc 37157* The integral of a piecewise continuous function  F on an open interval is equal to the integral of the continuous function  G, in the corresponding closed interval.  G is equal to  F on the open interval, but it is continuous at the two boundaries, also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F : dom  F --> CC )   &    |-  ( ph  ->  ( F  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  dom  F )   &    |-  ( ph  ->  R  e.  (
 ( F  |`  ( A (,) B ) ) lim
 CC  A ) )   &    |-  ( ph  ->  L  e.  ( ( F  |`  ( A (,) B ) ) lim
 CC  B ) )   &    |-  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `
  x ) ) ) )   =>    |-  ( ph  ->  ( G  e.  L^1  /\  S. ( A [,] B ) ( G `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x ) )
 
Theoremiblcncfioo 37158 A continuous function  F on an open interval  ( A (,) B ) with a finite right limit  R in  A and a finite left limit  L in  B is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  L  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  R  e.  ( F lim CC  A ) )   =>    |-  ( ph  ->  F  e.  L^1 )
 
Theoremitgspltprt 37159* The  S. integral splits on a given partition  P. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  ( M  +  1 ) ) )   &    |-  ( ph  ->  P : ( M ... N ) --> RR )   &    |-  (
 ( ph  /\  i  e.  ( M..^ N ) )  ->  ( P `  i )  <  ( P `  ( i  +  1 ) ) )   &    |-  ( ( ph  /\  t  e.  ( ( P `  M ) [,] ( P `  N ) ) )  ->  A  e.  CC )   &    |-  ( ( ph  /\  i  e.  ( M..^ N ) )  ->  ( t  e.  (
 ( P `  i
 ) [,] ( P `  ( i  +  1
 ) ) )  |->  A )  e.  L^1 )   =>    |-  ( ph  ->  S. ( ( P `  M ) [,] ( P `  N ) ) A  _d t  = 
 sum_ i  e.  ( M..^ N ) S. (
 ( P `  i
 ) [,] ( P `  ( i  +  1
 ) ) ) A  _d t )
 
Theoremitgiccshift 37160* The integral of a function,  F stays the same if its closed interval domain is shifted by  T. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> CC ) )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  G  =  ( x  e.  ( ( A  +  T ) [,] ( B  +  T ) )  |->  ( F `  ( x  -  T ) ) )   =>    |-  ( ph  ->  S. ( ( A  +  T ) [,] ( B  +  T )
 ) ( G `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
 
Theoremitgperiod 37161* The integral of a periodic function, with period  T stays the same if the domain of integration is shifted. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  ( ph  ->  F : RR --> CC )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  ( x  +  T ) )  =  ( F `  x ) )   &    |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )   =>    |-  ( ph  ->  S. ( ( A  +  T ) [,] ( B  +  T )
 ) ( F `  x )  _d x  =  S. ( A [,] B ) ( F `  x )  _d x )
 
Theoremitgsbtaddcnst 37162* Integral substitution, adding a constant to the function's argument. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> CC ) )   =>    |-  ( ph  ->  S__
 [ ( A  -  X )  ->  ( B  -  X ) ]
 ( F `  ( X  +  s )
 )  _d s  =  S__ [ A  ->  B ] ( F `  t )  _d t
 )
 
Theoremitgeq2d 37163* Equality theorem for an integral. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  =  C )   =>    |-  ( ph  ->  S. A B  _d x  =  S. A C  _d x )
 
21.30.12  Stone Weierstrass theorem - real version
 
Theoremstoweidlem1 37164 Lemma for stoweid 37226. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 12338. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A 
 <_  1 )   &    |-  ( ph  ->  D 
 <_  A )   =>    |-  ( ph  ->  (
 ( 1  -  ( A ^ N ) ) ^ ( K ^ N ) )  <_  ( 1  /  (
 ( K  x.  D ) ^ N ) ) )
 
Theoremstoweidlem2 37165* lemma for stoweid 37226: here we prove that the subalgebra of continuous functions, which contains constant functions, is closed under scaling. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   &    |-  ( ph  ->  E  e.  RR )   &    |-  ( ph  ->  F  e.  A )   =>    |-  ( ph  ->  (
 t  e.  T  |->  ( E  x.  ( F `
  t ) ) )  e.  A )
 
Theoremstoweidlem3 37166* Lemma for stoweid 37226: if  A is positive and all  M terms of a finite product are larger than  A, then the finite product is larger than A^M. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ i F   &    |- 
 F/ i ph   &    |-  X  =  seq 1 (  x.  ,  F )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  F : ( 1 ...
 M ) --> RR )   &    |-  (
 ( ph  /\  i  e.  ( 1 ... M ) )  ->  A  <  ( F `  i ) )   &    |-  ( ph  ->  A  e.  RR+ )   =>    |-  ( ph  ->  ( A ^ M )  < 
 ( X `  M ) )
 
Theoremstoweidlem4 37167* Lemma for stoweid 37226: a class variable replaces a setvar variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( ph  /\  x  e. 
 RR )  ->  (
 t  e.  T  |->  x )  e.  A )   =>    |-  ( ( ph  /\  B  e.  RR )  ->  (
 t  e.  T  |->  B )  e.  A )
 
Theoremstoweidlem5 37168* There exists a δ as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 0 < δ < 1 , p >= δ on  T  \  U. Here  D is used to represent δ in the paper and  Q to represent  T 
\  U in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  D  =  if ( C  <_  ( 1 
 /  2 ) ,  C ,  ( 1 
 /  2 ) )   &    |-  ( ph  ->  P : T
 --> RR )   &    |-  ( ph  ->  Q 
 C_  T )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A. t  e.  Q  C  <_  ( P `  t ) )   =>    |-  ( ph  ->  E. d
 ( d  e.  RR+  /\  d  <  1  /\  A. t  e.  Q  d 
 <_  ( P `  t
 ) ) )
 
Theoremstoweidlem6 37169* Lemma for stoweid 37226: two class variables replace two setvar variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t  f  =  F   &    |-  F/ t  g  =  G   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   =>    |-  ( ( ph  /\  F  e.  A  /\  G  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( G `  t ) ) )  e.  A )
 
Theoremstoweidlem7 37170* This lemma is used to prove that qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91, (at the top of page 91), is such that qn < ε on  T  \  U, and qn > 1 - ε on  V. Here it is proven that, for  n large enough, 1-(k*δ/2)^n > 1 - ε , and 1/(k*δ)^n < ε. The variable  A is used to represent (k*δ) in the paper, and  B is used to represent (k*δ/2). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F  =  ( i  e.  NN0  |->  ( ( 1  /  A ) ^ i
 ) )   &    |-  G  =  ( i  e.  NN0  |->  ( B ^ i ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  1  <  A )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  B  <  1 )   &    |-  ( ph  ->  E  e.  RR+ )   =>    |-  ( ph  ->  E. n  e.  NN  ( ( 1  -  E )  < 
 ( 1  -  ( B ^ n ) ) 
 /\  ( 1  /  ( A ^ n ) )  <  E ) )
 
Theoremstoweidlem8 37171* Lemma for stoweid 37226: two class variables replace two setvar variables, for the sum of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  F/_ t F   &    |-  F/_ t G   =>    |-  ( ( ph  /\  F  e.  A  /\  G  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  +  ( G `  t ) ) )  e.  A )
 
Theoremstoweidlem9 37172* Lemma for stoweid 37226: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  T  =  (/) )   &    |-  ( ph  ->  (
 t  e.  T  |->  1 )  e.  A )   =>    |-  ( ph  ->  E. g  e.  A  A. t  e.  T  ( abs `  (
 ( g `  t
 )  -  ( F `
  t ) ) )  <  E )
 
Theoremstoweidlem10 37173 Lemma for stoweid 37226. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90, this lemma is an application of Bernoulli's inequality. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
 1  -  ( N  x.  A ) ) 
 <_  ( ( 1  -  A ) ^ N ) )
 
Theoremstoweidlem11 37174* This lemma is used to prove that there is a function  g as in the proof of [BrosowskiDeutsh] p. 92 (at the top of page 92): this lemma proves that g(t) < ( j + 1 / 3 ) * ε. Here  E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  t  e.  T )   &    |-  ( ph  ->  j  e.  ( 1 ...
 N ) )   &    |-  (
 ( ph  /\  i  e.  ( 0 ... N ) )  ->  ( X `
  i ) : T --> RR )   &    |-  (
 ( ph  /\  i  e.  ( 0 ... N ) )  ->  ( ( X `  i ) `
  t )  <_ 
 1 )   &    |-  ( ( ph  /\  i  e.  ( j
 ... N ) ) 
 ->  ( ( X `  i ) `  t
 )  <  ( E  /  N ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  /  3
 ) )   =>    |-  ( ph  ->  (
 ( t  e.  T  |->  sum_
 i  e.  ( 0
 ... N ) ( E  x.  ( ( X `  i ) `
  t ) ) ) `  t )  <  ( ( j  +  ( 1  / 
 3 ) )  x.  E ) )
 
Theoremstoweidlem12 37175* Lemma for stoweid 37226. This Lemma is used by other three Lemmas. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) ) )   &    |-  ( ph  ->  P : T --> RR )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  K  e.  NN0 )   =>    |-  ( ( ph  /\  t  e.  T )  ->  ( Q `  t )  =  ( ( 1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) ) )
 
Theoremstoweidlem13 37176 Lemma for stoweid 37226. This lemma is used to prove the statement abs( f(t) - g(t) ) < 2 epsilon, in the last step of the proof in [BrosowskiDeutsh] p. 92. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  j  e.  RR )   &    |-  ( ph  ->  (
 ( j  -  (
 4  /  3 )
 )  x.  E )  <  X )   &    |-  ( ph  ->  X  <_  (
 ( j  -  (
 1  /  3 )
 )  x.  E ) )   &    |-  ( ph  ->  ( ( j  -  (
 4  /  3 )
 )  x.  E )  <  Y )   &    |-  ( ph  ->  Y  <  (
 ( j  +  (
 1  /  3 )
 )  x.  E ) )   =>    |-  ( ph  ->  ( abs `  ( Y  -  X ) )  < 
 ( 2  x.  E ) )
 
Theoremstoweidlem14 37177* There exists a  k as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90:  k is an integer and 1 < k * δ < 2.  D is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  A  =  { j  e.  NN  |  ( 1  /  D )  <  j }   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  D  <  1 )   =>    |-  ( ph  ->  E. k  e.  NN  ( 1  < 
 ( k  x.  D )  /\  ( ( k  x.  D )  / 
 2 )  <  1
 ) )
 
Theoremstoweidlem15 37178* This lemma is used to prove the existence of a function  p as in Lemma 1 from [BrosowskiDeutsh] p. 90:  p is in the subalgebra, such that 0 ≤ p ≤ 1, p(t_0) = 0, and p > 0 on T - U. Here  ( G `  I ) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  ( ph  ->  G : ( 1 ... M ) --> Q )   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   =>    |-  (
 ( ( ph  /\  I  e.  ( 1 ... M ) )  /\  S  e.  T )  ->  ( ( ( G `  I
 ) `  S )  e.  RR  /\  0  <_  ( ( G `  I ) `  S )  /\  ( ( G `
  I ) `  S )  <_  1 ) )
 
Theoremstoweidlem16 37179* Lemma for stoweid 37226. The subset  Y of functions in the algebra  A, with values in [ 0 , 1 ], is closed under multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  Y  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) }   &    |-  H  =  ( t  e.  T  |->  ( ( f `  t )  x.  (
 g `  t )
 ) )   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   =>    |-  ( ( ph  /\  f  e.  Y  /\  g  e.  Y )  ->  H  e.  Y )
 
Theoremstoweidlem17 37180* This lemma proves that the function 
g (as defined in [BrosowskiDeutsh] p. 91, at the end of page 91) belongs to the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  X : ( 0 ... N ) --> A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ph  ->  E  e.  RR )   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   =>    |-  ( ph  ->  ( t  e.  T  |->  sum_ i  e.  (
 0 ... N ) ( E  x.  ( ( X `  i ) `
  t ) ) )  e.  A )
 
Theoremstoweidlem18 37181* This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 92 when A is empty, the trivial case. Here D is used to denote the set A of Lemma 2, because the variable A is used for the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t D   &    |- 
 F/ t ph   &    |-  F  =  ( t  e.  T  |->  1 )   &    |-  T  =  U. J   &    |-  ( ( ph  /\  a  e.  RR )  ->  (
 t  e.  T  |->  a )  e.  A )   &    |-  ( ph  ->  B  e.  ( Clsd `  J )
 )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  D  =  (/) )   =>    |-  ( ph  ->  E. x  e.  A  ( A. t  e.  T  ( 0  <_  ( x `  t ) 
 /\  ( x `  t )  <_  1 ) 
 /\  A. t  e.  D  ( x `  t )  <  E  /\  A. t  e.  B  (
 1  -  E )  <  ( x `  t ) ) )
 
Theoremstoweidlem19 37182* If a set of real functions is closed under multiplication and it contains constants, then it is closed under finite exponentiation. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 t  e.  T  |->  ( ( F `  t
 ) ^ N ) )  e.  A )
 
Theoremstoweidlem20 37183* If a set A of real functions from a common domain T is closed under the sum of two functions, then it is closed under the sum of a finite number of functions, indexed by G. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  F  =  ( t  e.  T  |->  sum_ i  e.  ( 1 ...
 M ) ( ( G `  i ) `
  t ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  G : ( 1 ...
 M ) --> A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   =>    |-  ( ph  ->  F  e.  A )
 
Theoremstoweidlem21 37184* Once the Stone Weierstrass theorem has been proven for approximating nonnegative functions, then this lemma is used to extend the result to functions with (possibly) negative values. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t G   &    |-  F/_ t H   &    |-  F/_ t S   &    |-  F/ t ph   &    |-  G  =  ( t  e.  T  |->  ( ( H `  t
 )  +  S ) )   &    |-  ( ph  ->  F : T --> RR )   &    |-  ( ph  ->  S  e.  RR )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ph  ->  A. f  e.  A  f : T --> RR )   &    |-  ( ph  ->  H  e.  A )   &    |-  ( ph  ->  A. t  e.  T  ( abs `  ( ( H `  t )  -  ( ( F `  t )  -  S ) ) )  <  E )   =>    |-  ( ph  ->  E. f  e.  A  A. t  e.  T  ( abs `  (
 ( f `  t
 )  -  ( F `
  t ) ) )  <  E )
 
Theoremstoweidlem22 37185* If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  F/_ t F   &    |-  F/_ t G   &    |-  H  =  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )   &    |-  I  =  ( t  e.  T  |->  -u 1 )   &    |-  L  =  ( t  e.  T  |->  ( ( I `  t )  x.  ( G `  t ) ) )   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   =>    |-  ( ( ph  /\  F  e.  A  /\  G  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
 
Theoremstoweidlem23 37186* This lemma is used to prove the existence of a function pt as in the beginning of Lemma 1 [BrosowskiDeutsh] p. 90: for all t in T - U, there exists a function p in the subalgebra, such that pt ( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  F/_ t G   &    |-  H  =  ( t  e.  T  |->  ( ( G `  t )  -  ( G `  Z ) ) )   &    |-  ( ( ph  /\  f  e.  A ) 
 ->  f : T --> RR )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ph  ->  S  e.  T )   &    |-  ( ph  ->  Z  e.  T )   &    |-  ( ph  ->  G  e.  A )   &    |-  ( ph  ->  ( G `  S )  =/=  ( G `  Z ) )   =>    |-  ( ph  ->  ( H  e.  A  /\  ( H `  S )  =/=  ( H `  Z )  /\  ( H `
  Z )  =  0 ) )
 
Theoremstoweidlem24 37187* This lemma proves that for  n sufficiently large, qn( t ) > ( 1 - epsilon ), for all  t in  V: see Lemma 1 [BrosowskiDeutsh] p. 90, (at the bottom of page 90). 
Q is used to represent qn in the paper,  N to represent  n in the paper,  K to represent  k,  D to represent δ, and  E to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  V  =  { t  e.  T  |  ( P `  t
 )  <  ( D  /  2 ) }   &    |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) ) )   &    |-  ( ph  ->  P : T --> RR )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  K  e.  NN0 )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  ( 1  -  E )  <  ( 1  -  ( ( ( K  x.  D )  / 
 2 ) ^ N ) ) )   &    |-  ( ph  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t ) 
 <_  1 ) )   =>    |-  ( ( ph  /\  t  e.  V ) 
 ->  ( 1  -  E )  <  ( Q `  t ) )
 
Theoremstoweidlem25 37188* This lemma proves that for n sufficiently large, qn( t ) < ε, for all  t in  T  \  U: see Lemma 1 [BrosowskiDeutsh] p. 91 (at the top of page 91).  Q is used to represent qn in the paper,  N to represent n in the paper,  K to represent k,  D to represent δ,  P to represent p, and  E to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) ) )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  K  e.  NN )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  P : T --> RR )   &    |-  ( ph  ->  A. t  e.  T  ( 0  <_  ( P `  t )  /\  ( P `  t ) 
 <_  1 ) )   &    |-  ( ph  ->  A. t  e.  ( T  \  U ) D 
 <_  ( P `  t
 ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  ( 1  /  ( ( K  x.  D ) ^ N ) )  <  E )   =>    |-  ( ( ph  /\  t  e.  ( T 
 \  U ) ) 
 ->  ( Q `  t
 )  <  E )
 
Theoremstoweidlem26 37189* This lemma is used to prove that there is a function  g as in the proof of [BrosowskiDeutsh] p. 92: this lemma proves that g(t) > ( j - 4 / 3 ) * ε. Here  L is used to represnt j in the paper,  D is used to represent A in the paper,  S is used to represent t, and  E is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ j ph   &    |-  F/ t ph   &    |-  D  =  ( j  e.  (
 0 ... N )  |->  { t  e.  T  |  ( F `  t ) 
 <_  ( ( j  -  ( 1  /  3
 ) )  x.  E ) } )   &    |-  B  =  ( j  e.  ( 0
 ... N )  |->  { t  e.  T  |  ( ( j  +  ( 1  /  3
 ) )  x.  E )  <_  ( F `  t ) } )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ph  ->  L  e.  ( 1 ...
 N ) )   &    |-  ( ph  ->  S  e.  (
 ( D `  L )  \  ( D `  ( L  -  1
 ) ) ) )   &    |-  ( ph  ->  F : T
 --> RR )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   &    |-  (
 ( ph  /\  i  e.  ( 0 ... N ) )  ->  ( X `
  i ) : T --> RR )   &    |-  (
 ( ph  /\  i  e.  ( 0 ... N )  /\  t  e.  T )  ->  0  <_  (
 ( X `  i
 ) `  t )
 )   &    |-  ( ( ph  /\  i  e.  ( 0 ... N )  /\  t  e.  ( B `  i ) ) 
 ->  ( 1  -  ( E  /  N ) )  <  ( ( X `
  i ) `  t ) )   =>    |-  ( ph  ->  ( ( L  -  (
 4  /  3 )
 )  x.  E )  <  ( ( t  e.  T  |->  sum_ i  e.  ( 0 ... N ) ( E  x.  ( ( X `  i ) `  t
 ) ) ) `  S ) )
 
Theoremstoweidlem27 37190* This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Here  ( q `  i ) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  G  =  ( w  e.  X  |->  { h  e.  Q  |  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } } )   &    |-  ( ph  ->  Q  e.  _V )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Y  Fn  ran  G )   &    |-  ( ph  ->  ran  G  e.  _V )   &    |-  ( ( ph  /\  l  e.  ran  G )  ->  ( Y `  l )  e.  l
 )   &    |-  ( ph  ->  F : ( 1 ...
 M ) -1-1-onto-> ran  G )   &    |-  ( ph  ->  ( T  \  U )  C_  U. X )   &    |- 
 F/ t ph   &    |-  F/ w ph   &    |-  F/_ h Q   =>    |-  ( ph  ->  E. q
 ( M  e.  NN  /\  ( q : ( 1 ... M ) --> Q  /\  A. t  e.  ( T  \  U ) E. i  e.  (
 1 ... M ) 0  <  ( ( q `
  i ) `  t ) ) ) )
 
Theoremstoweidlem28 37191* There exists a δ as in Lemma 1 [BrosowskiDeutsh] p. 90: 0 < delta < 1 and p >= delta on 
T  \  U. Here  d is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t U   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  P  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  A. t  e.  ( T  \  U ) 0  <  ( P `  t ) )   &    |-  ( ph  ->  U  e.  J )   =>    |-  ( ph  ->  E. d
 ( d  e.  RR+  /\  d  <  1  /\  A. t  e.  ( T 
 \  U ) d 
 <_  ( P `  t
 ) ) )
 
Theoremstoweidlem29 37192* When the hypothesis for the extreme value theorem hold, then the inf of the range of the function belongs to the range, it is real and it a lower bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ t ph   &    |-  T  =  U. J   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  T  =/=  (/) )   =>    |-  ( ph  ->  ( sup ( ran  F ,  RR ,  `'  <  )  e.  ran  F  /\  sup ( ran  F ,  RR ,  `'  <  )  e.  RR  /\  A. t  e.  T  sup ( ran 
 F ,  RR ,  `'  <  )  <_  ( F `  t ) ) )
 
Theoremstoweidlem30 37193* This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p,  ( G `  i ) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `
  i ) `  t ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  G : ( 1 ...
 M ) --> Q )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   =>    |-  ( ( ph  /\  S  e.  T ) 
 ->  ( P `  S )  =  ( (
 1  /  M )  x.  sum_ i  e.  (
 1 ... M ) ( ( G `  i
 ) `  S )
 ) )
 
Theoremstoweidlem31 37194* This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that  R is a finite subset of  V,  x indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all  i ranging in the finite indexing set, 0 ≤ xi ≤ 1, xi < ε / m on V(ti), and xi > 1 - ε / m on  B. Here M is used to represent m in the paper,  E is used to represent ε in the paper, vi is used to represent V(ti). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ h ph   &    |-  F/ t ph   &    |-  F/ w ph   &    |-  Y  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) }   &    |-  V  =  { w  e.  J  |  A. e  e.  RR+  E. h  e.  A  (
 A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 )  /\  A. t  e.  w  ( h `  t )  < 
 e  /\  A. t  e.  ( T  \  U ) ( 1  -  e )  <  ( h `
  t ) ) }   &    |-  G  =  ( w  e.  R  |->  { h  e.  A  |  ( A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 )  /\  A. t  e.  w  ( h `  t )  < 
 ( E  /  M )  /\  A. t  e.  ( T  \  U ) ( 1  -  ( E  /  M ) )  <  ( h `
  t ) ) } )   &    |-  ( ph  ->  R 
 C_  V )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  v : ( 1 ...
 M ) -1-1-onto-> R )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  B 
 C_  ( T  \  U ) )   &    |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  ran 
 G  e.  Fin )   =>    |-  ( ph  ->  E. x ( x : ( 1 ...
 M ) --> Y  /\  A. i  e.  ( 1
 ... M ) (
 A. t  e.  (
 v `  i )
 ( ( x `  i ) `  t
 )  <  ( E  /  M )  /\  A. t  e.  B  (
 1  -  ( E 
 /  M ) )  <  ( ( x `
  i ) `  t ) ) ) )
 
Theoremstoweidlem32 37195* If a set A of real functions from a common domain T is a subalgebra and it contains constants, then it is closed under the sum of a finite number of functions, indexed by G and finally scaled by a real Y. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  P  =  ( t  e.  T  |->  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `
  i ) `  t ) ) )   &    |-  F  =  ( t  e.  T  |->  sum_ i  e.  (
 1 ... M ) ( ( G `  i
 ) `  t )
 )   &    |-  H  =  ( t  e.  T  |->  Y )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  G : ( 1 ... M ) --> A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  (
 g `  t )
 ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   =>    |-  ( ph  ->  P  e.  A )
 
Theoremstoweidlem33 37196* If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |-  F/_ t G   &    |-  F/ t ph   &    |-  (
 ( ph  /\  f  e.  A )  ->  f : T --> RR )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  +  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   =>    |-  ( ( ph  /\  F  e.  A  /\  G  e.  A )  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
 
Theoremstoweidlem34 37197* This lemma proves that for all  t in  T there is a  j as in the proof of [BrosowskiDeutsh] p. 91 (at the bottom of page 91 and at the top of page 92): (j-4/3) * ε < f(t) <= (j-1/3) * ε , g(t) < (j+1/3) * ε, and g(t) > (j-4/3) * ε. Here  E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ t F   &    |- 
 F/ j ph   &    |-  F/ t ph   &    |-  D  =  ( j  e.  (
 0 ... N )  |->  { t  e.  T  |  ( F `  t ) 
 <_  ( ( j  -  ( 1  /  3
 ) )  x.  E ) } )   &    |-  B  =  ( j  e.  ( 0
 ... N )  |->  { t  e.  T  |  ( ( j  +  ( 1  /  3
 ) )  x.  E )  <_  ( F `  t ) } )   &    |-  J  =  ( t  e.  T  |->  { j  e.  ( 1
 ... N )  |  t  e.  ( D `
  j ) }
 )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  T  e.  _V )   &    |-  ( ph  ->  F : T --> RR )   &    |-  ( ( ph  /\  t  e.  T ) 
 ->  0  <_  ( F `
  t ) )   &    |-  ( ( ph  /\  t  e.  T )  ->  ( F `  t )  < 
 ( ( N  -  1 )  x.  E ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  E  <  ( 1  / 
 3 ) )   &    |-  (
 ( ph  /\  j  e.  ( 0 ... N ) )  ->  ( X `
  j ) : T --> RR )   &    |-  (
 ( ph  /\  j  e.  ( 0 ... N )  /\  t  e.  T )  ->  0  <_  (
 ( X `  j
 ) `  t )
 )   &    |-  ( ( ph  /\  j  e.  ( 0 ... N )  /\  t  e.  T )  ->  ( ( X `
  j ) `  t )  <_  1 )   &    |-  ( ( ph  /\  j  e.  ( 0 ... N )  /\  t  e.  ( D `  j ) ) 
 ->  ( ( X `  j ) `  t
 )  <  ( E  /  N ) )   &    |-  (
 ( ph  /\  j  e.  ( 0 ... N )  /\  t  e.  ( B `  j ) ) 
 ->  ( 1  -  ( E  /  N ) )  <  ( ( X `
  j ) `  t ) )   =>    |-  ( ph  ->  A. t  e.  T  E. j  e.  RR  (
 ( ( ( j  -  ( 4  / 
 3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t ) 
 <_  ( ( j  -  ( 1  /  3
 ) )  x.  E ) )  /\  ( ( ( t  e.  T  |->  sum_
 i  e.  ( 0
 ... N ) ( E  x.  ( ( X `  i ) `
  t ) ) ) `  t )  <  ( ( j  +  ( 1  / 
 3 ) )  x.  E )  /\  (
 ( j  -  (
 4  /  3 )
 )  x.  E )  <  ( ( t  e.  T  |->  sum_ i  e.  ( 0 ... N ) ( E  x.  ( ( X `  i ) `  t
 ) ) ) `  t ) ) ) )
 
Theoremstoweidlem35 37198* This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Here  ( q `  i ) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ t ph   &    |-  F/ w ph   &    |-  F/ h ph   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  W  =  { w  e.  J  |  E. h  e.  Q  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } }   &    |-  G  =  ( w  e.  X  |->  { h  e.  Q  |  w  =  { t  e.  T  |  0  < 
 ( h `  t
 ) } } )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  X 
 C_  W )   &    |-  ( ph  ->  ( T  \  U )  C_  U. X )   &    |-  ( ph  ->  ( T  \  U )  =/=  (/) )   =>    |-  ( ph  ->  E. m E. q ( m  e. 
 NN  /\  ( q : ( 1 ... m ) --> Q  /\  A. t  e.  ( T 
 \  U ) E. i  e.  ( 1 ... m ) 0  < 
 ( ( q `  i ) `  t
 ) ) ) )
 
Theoremstoweidlem36 37199* This lemma is used to prove the existence of a function pt as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function p in the subalgebra, such that pt ( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. Z is used for t0 , S is used for t e. T - U , h is used for pt . G is used for (ht)^2 and the final h is a normalized version of G ( divided by its norm, see the variable N ). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ h Q   &    |-  F/_ t H   &    |-  F/_ t F   &    |-  F/_ t G   &    |- 
 F/ t ph   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  T  =  U. J   &    |-  G  =  ( t  e.  T  |->  ( ( F `  t
 )  x.  ( F `
  t ) ) )   &    |-  N  =  sup ( ran  G ,  RR ,  <  )   &    |-  H  =  ( t  e.  T  |->  ( ( G `  t
 )  /  N )
 )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  A 
 C_  ( J  Cn  K ) )   &    |-  (
 ( ph  /\  f  e.  A  /\  g  e.  A )  ->  (
 t  e.  T  |->  ( ( f `  t
 )  x.  ( g `
  t ) ) )  e.  A )   &    |-  ( ( ph  /\  x  e.  RR )  ->  (
 t  e.  T  |->  x )  e.  A )   &    |-  ( ph  ->  S  e.  T )   &    |-  ( ph  ->  Z  e.  T )   &    |-  ( ph  ->  F  e.  A )   &    |-  ( ph  ->  ( F `  S )  =/=  ( F `  Z ) )   &    |-  ( ph  ->  ( F `  Z )  =  0 )   =>    |-  ( ph  ->  E. h ( h  e.  Q  /\  0  < 
 ( h `  S ) ) )
 
Theoremstoweidlem37 37200* This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p,  ( G `  i ) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  Q  =  { h  e.  A  |  ( ( h `  Z )  =  0  /\  A. t  e.  T  ( 0  <_  ( h `  t )  /\  ( h `  t ) 
 <_  1 ) ) }   &    |-  P  =  ( t  e.  T  |->  ( ( 1  /  M )  x.  sum_ i  e.  ( 1 ... M ) ( ( G `
  i ) `  t ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  G : ( 1 ...
 M ) --> Q )   &    |-  ( ( ph  /\  f  e.  A )  ->  f : T --> RR )   &    |-  ( ph  ->  Z  e.  T )   =>    |-  ( ph  ->  ( P `  Z )  =  0 )
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