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Theorem List for Metamath Proof Explorer - 37801-37900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnfdmsn 37801* A function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( x  e.  { A }  |->  B )  e.  ( ~P { A }  Cn  ~P { B } ) )
 
Theoremcncfcompt 37802* Composition of continuous functions. A generalization of cncfmpt1f 21956 to arbitrary domains. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ( x  e.  A  |->  B )  e.  ( A -cn-> C ) )   &    |-  ( ph  ->  F  e.  ( C -cn-> D ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( F `
  B ) )  e.  ( A -cn-> D ) )
 
Theoremdivcncf 37803* The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X -cn-> ( CC  \  { 0 } ) ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A  /  B ) )  e.  ( X -cn-> CC )
 )
 
Theoremaddcncff 37804 The addition of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F  e.  ( X -cn-> CC ) )   &    |-  ( ph  ->  G  e.  ( X -cn-> CC ) )   =>    |-  ( ph  ->  ( F  oF  +  G )  e.  ( X -cn-> CC ) )
 
Theoremioccncflimc 37805 Limit at the upper bound, of a continuous function defined on a left open right closed interval. (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  ->  ( F `  B )  e.  ( ( F  |`  ( A (,) B ) ) lim CC  B ) )
 
Theoremcncfuni 37806* A function is continuous if it's domain is the union of sets over which the function is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  U. B )   &    |-  ( ( ph  /\  b  e.  B )  ->  ( A  i^i  b )  e.  ( ( TopOpen ` fld )t  A ) )   &    |-  (
 ( ph  /\  b  e.  B )  ->  ( F  |`  b )  e.  ( ( A  i^i  b ) -cn-> CC )
 )   =>    |-  ( ph  ->  F  e.  ( A -cn-> CC )
 )
 
Theoremicccncfext 37807* A continuous function on a closed interval can be extended to a continuous function on the whole real line. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F/_ x F   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  Y  =  U. K   &    |-  G  =  ( x  e.  RR  |->  if ( x  e.  ( A [,] B ) ,  ( F `  x ) ,  if ( x  <  A ,  ( F `  A ) ,  ( F `  B ) ) ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  K  e.  Top )   &    |-  ( ph  ->  F  e.  (
 ( Jt  ( A [,] B ) )  Cn  K ) )   =>    |-  ( ph  ->  ( G  e.  ( J  Cn  ( Kt  ran  F ) ) 
 /\  ( G  |`  ( A [,] B ) )  =  F ) )
 
Theoremcncficcgt0 37808* A the absolute value of a continuous function on a closed interval, that is never 0, has a strictly positive lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F  =  ( x  e.  ( A [,] B )  |->  C )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> ( RR  \  { 0 } ) ) )   =>    |-  ( ph  ->  E. y  e.  RR+  A. x  e.  ( A [,] B ) y 
 <_  ( abs `  C ) )
 
Theoremicocncflimc 37809 Limit at the lower bound, of a continuous function defined on a left closed right open interval. (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  ->  ( F `  A )  e.  ( ( F  |`  ( A (,) B ) ) lim CC  A ) )
 
Theoremcncfdmsn 37810* A complex function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( x  e.  { A }  |->  B )  e.  ( { A } -cn-> { B } )
 )
 
Theoremdivcncff 37811 The quotient of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F  e.  ( X -cn-> CC ) )   &    |-  ( ph  ->  G  e.  ( X -cn-> ( CC  \  { 0 } )
 ) )   =>    |-  ( ph  ->  ( F  oF  /  G )  e.  ( X -cn->
 CC ) )
 
Theoremcncfshiftioo 37812* A periodic continuous function stays continuous if the domain is an open interval that is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  C  =  ( A (,) B )   &    |-  ( ph  ->  T  e.  RR )   &    |-  D  =  ( ( A  +  T ) (,) ( B  +  T ) )   &    |-  ( ph  ->  F  e.  ( C -cn-> CC ) )   &    |-  G  =  ( x  e.  D  |->  ( F `  ( x  -  T ) ) )   =>    |-  ( ph  ->  G  e.  ( D -cn-> CC )
 )
 
Theoremcncfiooicclem1 37813* A continuous function  F on an open interval  ( A (,) B ) can be extended to a continuous function  G on the corresponding closed interval, if it has a finite right limit  R in  A and a finite left limit  L in  B.  F can be complex valued. This lemma assumes  A  <  B, the invoking theorem drops this assumption. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F/ x ph   &    |-  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  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  L  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  R  e.  ( F lim CC  A ) )   =>    |-  ( ph  ->  G  e.  ( ( A [,] B ) -cn-> CC ) )
 
Theoremcncfiooicc 37814* A continuous function  F on an open interval  ( A (,) B ) can be extended to a continuous function  G on the corresponding close interval, if it has a finite right limit  R in  A and a finite left limit  L in  B.  F can be complex valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F/ x ph   &    |-  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )   &    |-  ( 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  ->  G  e.  ( ( A [,] B ) -cn-> CC ) )
 
Theoremcncfiooiccre 37815* A continuous function  F on an open interval  ( A (,) B ) can be extended to a continuous function  G on the corresponding close interval, if it has a finite right limit  R in  A and a finite left limit  L in  B.  F is assumed to be real valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F/ x ph   &    |-  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  e.  ( ( A (,) B ) -cn-> RR ) )   &    |-  ( ph  ->  L  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  R  e.  ( F lim CC  A ) )   =>    |-  ( ph  ->  G  e.  ( ( A [,] B ) -cn-> RR ) )
 
Theoremcncfioobdlem 37816*  G actually extends  F. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F : ( A (,) B ) --> V )   &    |-  G  =  ( x  e.  ( A [,] B )  |->  if ( x  =  A ,  R ,  if ( x  =  B ,  L ,  ( F `  x ) ) ) )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   =>    |-  ( ph  ->  ( G `  C )  =  ( F `  C ) )
 
Theoremcncfioobd 37817* 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 bounded. (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  ->  E. x  e.  RR  A. y  e.  ( A (,) B ) ( abs `  ( F `  y ) ) 
 <_  x )
 
Theoremjumpncnp 37818 Jump discontinuity or discontinuity of the first kind: if the left and the right limit don't match, the function is discontinuous at the point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  K  =  ( TopOpen ` fld )   &    |-  ( ph  ->  A 
 C_  RR )   &    |-  J  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  B  e.  (
 ( limPt `  J ) `  ( A  i^i  ( -oo (,) B ) ) ) )   &    |-  ( ph  ->  B  e.  ( ( limPt `  J ) `  ( A  i^i  ( B (,) +oo ) ) ) )   &    |-  ( ph  ->  L  e.  ( ( F  |`  ( -oo (,)
 B ) ) lim CC  B ) )   &    |-  ( ph  ->  R  e.  (
 ( F  |`  ( B (,) +oo ) ) lim CC  B ) )   &    |-  ( ph  ->  L  =/=  R )   =>    |-  ( ph  ->  -.  F  e.  ( ( J  CnP  ( TopOpen ` fld ) ) `  B ) )
 
Theoremcncfcompt2 37819* Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  F/ x ph   &    |-  ( ph  ->  ( x  e.  A  |->  R )  e.  ( A
 -cn-> B ) )   &    |-  ( ph  ->  ( y  e.  C  |->  S )  e.  ( C -cn-> E ) )   &    |-  ( ph  ->  B 
 C_  C )   &    |-  (
 y  =  R  ->  S  =  T )   =>    |-  ( ph  ->  ( x  e.  A  |->  T )  e.  ( A
 -cn-> E ) )
 
Theoremcxpcncf2 37820* The complex power function is continuous with respect to its second argument. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( A  e.  ( CC  \  ( -oo (,] 0
 ) )  ->  ( x  e.  CC  |->  ( A 
 ^c  x ) )  e.  ( CC
 -cn-> CC ) )
 
Theoremfprodcncf 37821* The finite product of continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A  /\  k  e.  B )  ->  C  e.  CC )   &    |-  (
 ( ph  /\  k  e.  B )  ->  ( x  e.  A  |->  C )  e.  ( A -cn-> CC ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  prod_ k  e.  B  C )  e.  ( A -cn-> CC )
 )
 
21.30.10  Derivatives
 
Theoremdvsinexp 37822* The derivative of sin^N . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  ( CC  _D  ( x  e. 
 CC  |->  ( ( sin `  x ) ^ N ) ) )  =  ( x  e.  CC  |->  ( ( N  x.  ( ( sin `  x ) ^ ( N  -  1 ) ) )  x.  ( cos `  x ) ) ) )
 
Theoremdvcosre 37823 The real derivative of the cosine. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
 |-  ( RR  _D  ( x  e. 
 RR  |->  ( cos `  x ) ) )  =  ( x  e.  RR  |->  -u ( sin `  x ) )
 
Theoremdvrecg 37824* Derivative of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  ( CC  \  {
 0 } ) )   &    |-  ( ( ph  /\  x  e.  X )  ->  C  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  B ) )  =  ( x  e.  X  |->  C ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( A  /  B ) ) )  =  ( x  e.  X  |->  -u ( ( A  x.  C )  /  ( B ^ 2 ) ) ) )
 
Theoremdvsinax 37825* Derivative exercise: the derivative with respect to y of sin(Ay), given a constant  A. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  CC  ->  ( CC  _D  ( y  e.  CC  |->  ( sin `  ( A  x.  y
 ) ) ) )  =  ( y  e. 
 CC  |->  ( A  x.  ( cos `  ( A  x.  y ) ) ) ) )
 
Theoremdvsubf 37826 The subtraction rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  dom  ( S  _D  F )  =  X )   &    |-  ( ph  ->  dom  ( S  _D  G )  =  X )   =>    |-  ( ph  ->  ( S  _D  ( F  oF  -  G )
 )  =  ( ( S  _D  F )  oF  -  ( S  _D  G ) ) )
 
Theoremdvmptconst 37827* Function-builder for derivative: derivative of a constant. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  A  e.  (
 ( TopOpen ` fld )t  S ) )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  A  |->  B ) )  =  ( x  e.  A  |->  0 ) )
 
Theoremdvcnre 37828 From compex differentiation to real differentiation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( F : CC --> CC  /\  RR  C_  dom  ( CC  _D  F ) ) 
 ->  ( RR  _D  ( F  |`  RR ) )  =  ( ( CC 
 _D  F )  |`  RR ) )
 
Theoremdvmptidg 37829* Function-builder for derivative: derivative of the identity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  A  e.  (
 ( TopOpen ` fld )t  S ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  A  |->  x ) )  =  ( x  e.  A  |->  1 ) )
 
Theoremdvresntr 37830 Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  F : X --> CC )   &    |-  J  =  ( Kt  S )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  ( ph  ->  ( ( int `  J ) `  X )  =  Y )   =>    |-  ( ph  ->  ( S  _D  F )  =  ( S  _D  ( F  |`  Y ) ) )
 
Theoremdvmptdiv 37831* Function-builder for derivative, quotient rule. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( 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  \  {
 0 } ) )   &    |-  ( ( ph  /\  x  e.  X )  ->  D  e.  CC )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  C ) )  =  ( x  e.  X  |->  D ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( A  /  C ) ) )  =  ( x  e.  X  |->  ( ( ( B  x.  C )  -  ( D  x.  A ) )  /  ( C ^ 2 ) ) ) )
 
Theoremfperdvper 37832* The derivative of a periodic function is periodic. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : RR --> CC )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e. 
 RR )  ->  ( F `  ( x  +  T ) )  =  ( F `  x ) )   &    |-  G  =  ( RR  _D  F )   =>    |-  ( ( ph  /\  x  e.  dom  G )  ->  ( ( x  +  T )  e.  dom  G 
 /\  ( G `  ( x  +  T ) )  =  ( G `  x ) ) )
 
Theoremdvmptresicc 37833* Derivative of a function restricted to a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F  =  ( x  e.  CC  |->  A )   &    |-  ( ( ph  /\  x  e.  CC )  ->  A  e.  CC )   &    |-  ( ph  ->  ( CC  _D  F )  =  ( x  e.  CC  |->  B ) )   &    |-  ( ( ph  /\  x  e.  CC )  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   =>    |-  ( ph  ->  ( RR  _D  ( x  e.  ( C [,] D )  |->  A ) )  =  ( x  e.  ( C (,) D )  |->  B ) )
 
Theoremdvasinbx 37834* Derivative exercise: the derivative with respect to y of A x sin(By), given two constants  A and  B. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( CC  _D  (
 y  e.  CC  |->  ( A  x.  ( sin `  ( B  x.  y
 ) ) ) ) )  =  ( y  e.  CC  |->  ( ( A  x.  B )  x.  ( cos `  ( B  x.  y ) ) ) ) )
 
Theoremdvresioo 37835 Restriction of a derivative to an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  C_  RR  /\  F : A --> CC )  ->  ( RR  _D  ( F  |`  ( B (,) C ) ) )  =  ( ( RR  _D  F )  |`  ( B (,) C ) ) )
 
Theoremdvdivf 37836 The quotient rule for everywhere-differentiable functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  G : X --> ( CC  \  { 0 } )
 )   &    |-  ( ph  ->  dom  ( S  _D  F )  =  X )   &    |-  ( ph  ->  dom  ( S  _D  G )  =  X )   =>    |-  ( ph  ->  ( S  _D  ( F  oF  /  G ) )  =  ( ( ( ( S  _D  F )  oF  x.  G )  oF  -  (
 ( S  _D  G )  oF  x.  F ) )  oF  /  ( G  oF  x.  G ) ) )
 
Theoremdvdivbd 37837* 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 37838 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 37839 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 37840* 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 37841 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 37842* 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 37843* 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 37844* 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 37845* 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.) (Revised by AV, 3-Oct-2020.)
 |-  ( 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  = inf ( {
 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 37846* Limit at the lower bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)
 |-  ( 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 ) )
 
Theoremioodvbdlimc1lem1OLD 37847* 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.) Obsolete version of ioodvbdlimc1lem1 37845 as of 3-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( 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 )
 )
 
Theoremioodvbdlimc1lem2OLD 37848* Limit at the lower bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete version of ioodvbdlimc1lem2 37846 as of 3-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( 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 37849* A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by AV, 3-Oct-2020.)
 |-  ( 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 37850* Limit at the upper bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)
 |-  ( 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 ) )
 
Theoremioodvbdlimc2lemOLD 37851* Limit at the upper bound of an open interval, for a function with bounded derivative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete version of ioodvbdlimc2lem 37850 as of 3-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( 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 37852* A real function with bounded derivative, has a limit at the upper bound of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by AV, 3-Oct-2020.)
 |-  ( 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 37853  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 37854* Function-builder for derivative, product rule. A version of dvmptmul 22927 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 37855* 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 37856 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 37857* 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 37858* 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 37859* 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 37860* 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 37861* Induction step for dvmptfprod 37862. (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 37862* 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 37863*  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 37864* Induction step for dvnprodlem2 37864. (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 37865* 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 37866* 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 37867 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 37868* 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 37869* 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 37870 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 37871* 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 37872* 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 37873* 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 37874* 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 37875* 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 37876 The empty set is a measurable function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (/)  e. MblFn
 
Theoremmbfres2cn 37877 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 22613 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 37878 The measure of the empty set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( vol `  (/) )  =  0
 
Theoremditgeqiooicc 37879* 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 37880 The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  dom  vol  ->  0 
 <_  ( vol `  A ) )
 
Theoremcnbdibl 37881* 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 37882 A singleton is measurable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  RR  ->  { A }  e.  dom  vol )
 
Theoremditgeq3d 37883* 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 37884 The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (/)  e.  L^1
 
Theoremiblsplit 37885* 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 37886 A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  RR  ->  ( vol `  { A }
 )  =  0 )
 
Theoremitgvol0 37887* 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 37888* 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 37889* A version of iblsplit 37885 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 37890* 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 37891 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 37892* 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 37893* 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 37894* lemma for itgsubsticc 37895. (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 37895* Integration by u-substitution. The main difference with respect to itgsubst 23013 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 37896* 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 37897 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 37898* 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 37899* 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 37900* 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 )
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