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Theorem List for Metamath Proof Explorer - 19701-19800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremi1fibl 19701 A simple function is integrable. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( F  e.  dom  S.1 
 ->  F  e.  L ^1 )
 
Theoremitgitg1 19702* Transfer an integral using  S.1 to an equivalent integral using  S.. (Contributed by Mario Carneiro, 6-Aug-2014.)
 |-  ( F  e.  dom  S.1 
 ->  S. RR ( F `
  x )  _d x  =  ( S.1 `  F ) )
 
Theoremitgle 19703* Monotonicity of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  <_  C )   =>    |-  ( ph  ->  S. A B  _d x  <_  S. A C  _d x )
 
Theoremitgge0 19704* The integral of a positive function is positive. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  0  <_  S. A B  _d x )
 
Theoremitgss 19705* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ( ph  /\  x  e.  ( B 
 \  A ) ) 
 ->  C  =  0 )   =>    |-  ( ph  ->  S. A C  _d x  =  S. B C  _d x )
 
Theoremitgss2 19706* Expand the set of an integral by adding zeroes outside the domain. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( A  C_  B  ->  S. A C  _d x  =  S. B if ( x  e.  A ,  C ,  0 )  _d x )
 
Theoremitgeqa 19707* Approximate equality of integrals. If  C ( x )  =  D ( x ) for almost all  x, then  S. B C ( x )  _d x  =  S. B D ( x )  _d x and one is integrable iff the other is. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
 |-  ( ( ph  /\  x  e.  B )  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  D  e.  CC )   &    |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  ( vol * `  A )  =  0 )   &    |-  (
 ( ph  /\  x  e.  ( B  \  A ) )  ->  C  =  D )   =>    |-  ( ph  ->  (
 ( ( x  e.  B  |->  C )  e.  L ^1  <->  ( x  e.  B  |->  D )  e.  L ^1 )  /\  S. B C  _d x  =  S. B D  _d x ) )
 
Theoremitgss3 19708* Expand the set of an integral by a nullset. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  RR )   &    |-  ( ph  ->  ( vol * `  ( B  \  A ) )  =  0 )   &    |-  (
 ( ph  /\  x  e.  B )  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( ( x  e.  A  |->  C )  e.  L ^1  <->  ( x  e.  B  |->  C )  e.  L ^1 )  /\  S. A C  _d x  =  S. B C  _d x ) )
 
Theoremitgioo 19709* Equality of integrals on open and closed intervals. (Contributed by Mario Carneiro, 2-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  C  e.  CC )   =>    |-  ( ph  ->  S. ( A (,) B ) C  _d x  =  S. ( A [,] B ) C  _d x )
 
Theoremitgless 19710* Expand the integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Aug-2014.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  A  e.  dom  vol )   &    |-  (
 ( ph  /\  x  e.  B )  ->  C  e.  RR )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  0  <_  C )   &    |-  ( ph  ->  ( x  e.  B  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  S. A C  _d x  <_  S. B C  _d x )
 
Theoremiblconst 19711 A constant function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  ( vol `  A )  e.  RR  /\  B  e.  CC )  ->  ( A  X.  { B }
 )  e.  L ^1 )
 
Theoremitgconst 19712* Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ( A  e.  dom 
 vol  /\  ( vol `  A )  e.  RR  /\  B  e.  CC )  ->  S. A B  _d x  =  ( B  x.  ( vol `  A ) ) )
 
Theoremibladdlem 19713* Lemma for ibladd 19714. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  D  =  ( B  +  C ) )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e. MblFn )   &    |-  ( ph  ->  (
 S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 ) ) )  e.  RR )   &    |-  ( ph  ->  ( S.2 `  ( x  e.  RR  |->  if (
 ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) ) )  e. 
 RR )   =>    |-  ( ph  ->  ( S.2 `  ( x  e. 
 RR  |->  if ( ( x  e.  A  /\  0  <_  D ) ,  D ,  0 ) ) )  e.  RR )
 
Theoremibladd 19714* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e.  L ^1 )
 
Theoremiblsub 19715* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  L ^1 )
 
Theoremitgaddlem1 19716* Lemma for itgadd 19718. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  0  <_  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  C )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgaddlem2 19717* Lemma for itgadd 19718. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgadd 19718* Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgsub 19719* Subtract two integrals over the same domain. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  S. A ( B  -  C )  _d x  =  ( S. A B  _d x  -  S. A C  _d x ) )
 
Theoremitgfsum 19720* Take a finite sum of integrals over the same domain. (Contributed by Mario Carneiro, 24-Aug-2014.)
 |-  ( ph  ->  A  e.  dom  vol )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  k  e.  B ) )  ->  C  e.  V )   &    |-  (
 ( ph  /\  k  e.  B )  ->  ( x  e.  A  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  ( ( x  e.  A  |->  sum_ k  e.  B  C )  e.  L ^1  /\  S. A sum_ k  e.  B  C  _d x  =  sum_ k  e.  B  S. A C  _d x ) )
 
Theoremiblabslem 19721* Lemma for iblabs 19722. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( abs `  ( F `  B ) ) ,  0 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  ( F `  B ) )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  B )  e. 
 RR )   =>    |-  ( ph  ->  ( G  e. MblFn  /\  ( S.2 `  G )  e.  RR ) )
 
Theoremiblabs 19722* The absolute value of an integrable function is integrable. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B ) )  e.  L ^1 )
 
Theoremiblabsr 19723* A measurable function is integrable iff its absolute value is integrable. (See iblabs 19722 for the forward implication.) (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B )
 )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )
 
Theoremiblmulc2 19724* Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e.  L ^1 )
 
Theoremitgmulc2lem1 19725* Lemma for itgmulc2 19727: positive real case. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  0 
 <_  C )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgmulc2lem2 19726* Lemma for itgmulc2 19727: real case. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgmulc2 19727* Multiply an integral by a constant. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgabs 19728* The triangle inequality for integrals. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   =>    |-  ( ph  ->  ( abs `  S. A B  _d x )  <_  S. A ( abs `  B )  _d x )
 
Theoremitgsplit 19729* The  S. integral splits under an almost disjoint union. (Contributed by Mario Carneiro, 11-Aug-2014.)
 |-  ( ph  ->  ( vol * `  ( A  i^i  B ) )  =  0 )   &    |-  ( ph  ->  U  =  ( A  u.  B ) )   &    |-  ( ( ph  /\  x  e.  U ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  B  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  S. U C  _d x  =  ( S. A C  _d x  +  S. B C  _d x ) )
 
Theoremitgspliticc 19730* The  S. integral splits on closed intervals with matching endpoints. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  e.  ( A [,] C ) )   &    |-  ( ( ph  /\  x  e.  ( A [,] C ) )  ->  D  e.  V )   &    |-  ( ph  ->  ( x  e.  ( A [,] B )  |->  D )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  ( B [,] C )  |->  D )  e.  L ^1 )   =>    |-  ( ph  ->  S. ( A [,] C ) D  _d x  =  ( S. ( A [,] B ) D  _d x  +  S. ( B [,] C ) D  _d x ) )
 
Theoremitgsplitioo 19731* The  S. integral splits on open intervals with matching endpoints. (Contributed by Mario Carneiro, 2-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  e.  ( A [,] C ) )   &    |-  ( ( ph  /\  x  e.  ( A (,) C ) )  ->  D  e.  CC )   &    |-  ( ph  ->  ( x  e.  ( A (,) B )  |->  D )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  ( B (,) C )  |->  D )  e.  L ^1 )   =>    |-  ( ph  ->  S. ( A (,) C ) D  _d x  =  ( S. ( A (,) B ) D  _d x  +  S. ( B (,) C ) D  _d x ) )
 
Theorembddmulibl 19732* A bounded function times an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ( F  e. MblFn  /\  G  e.  L ^1 
 /\  E. x  e.  RR  A. y  e.  dom  F ( abs `  ( F `  y ) )  <_  x )  ->  ( F  o F  x.  G )  e.  L ^1 )
 
Theorembddibl 19733* A bounded function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ( F  e. MblFn  /\  ( vol `  dom  F )  e.  RR  /\  E. x  e.  RR  A. y  e.  dom  F ( abs `  ( F `  y ) )  <_  x )  ->  F  e.  L ^1 )
 
Theoremcniccibl 19734 A continuous function on a closed interval is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  F  e.  ( ( A [,] B )
 -cn-> CC ) )  ->  F  e.  L ^1 )
 
Theoremitggt0 19735* The integral of a strictly positive function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  ( ph  ->  0  <  ( vol `  A ) )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  RR+ )   =>    |-  ( ph  ->  0  <  S. A B  _d x )
 
Theoremitgcn 19736* Transfer itg2cn 19657 to the full Lebesgue integral. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  C  e.  RR+ )   =>    |-  ( ph  ->  E. d  e.  RR+  A. u  e.  dom  vol ( ( u  C_  A  /\  ( vol `  u )  <  d )  ->  S. u ( abs `  B )  _d x  <  C ) )
 
Theoremditgeq1 19737* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( A  =  B  ->  S__ [ A  ->  C ] D  _d x  =  S__ [ B  ->  C ] D  _d x )
 
Theoremditgeq2 19738* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( A  =  B  ->  S__ [ C  ->  A ] D  _d x  =  S__ [ C  ->  B ] D  _d x )
 
Theoremditgeq3 19739* Equality theorem for the directed integral. (The domain of the equality here is very rough; for more precise bounds one should decompose it with ditgpos 19745 first and use the equality theorems for df-itg 19518.) (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( A. x  e. 
 RR  D  =  E  ->  S__ [ A  ->  B ] D  _d x  =  S__ [ A  ->  B ] E  _d x )
 
Theoremditgeq3dv 19740* Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( ( ph  /\  x  e.  RR )  ->  D  =  E )   =>    |-  ( ph  ->  S__ [ A  ->  B ] D  _d x  =  S__ [ A  ->  B ] E  _d x )
 
Theoremditgex 19741 A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |- 
 S__ [ A  ->  B ] C  _d x  e.  _V
 
Theoremditg0 19742* Value of the directed integral from a point to itself. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |- 
 S__ [ A  ->  A ] B  _d x  =  0
 
Theoremcbvditg 19743* Change bound variable in a directed integral. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( x  =  y 
 ->  C  =  D )   &    |-  F/_ y C   &    |-  F/_ x D   =>    |-  S__ [ A  ->  B ] C  _d x  =  S__ [ A  ->  B ] D  _d y
 
Theoremcbvditgv 19744* Change bound variable in a directed integral. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( x  =  y 
 ->  C  =  D )   =>    |-  S__ [ A  ->  B ] C  _d x  =  S__ [ A  ->  B ] D  _d y
 
Theoremditgpos 19745* Value of the directed integral in the forward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  S__ [ A  ->  B ] C  _d x  =  S. ( A (,) B ) C  _d x )
 
Theoremditgneg 19746* Value of the directed integral in the backward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  S__ [ B  ->  A ] C  _d x  =  -u S. ( A (,) B ) C  _d x )
 
Theoremditgcl 19747* Closure of a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  A  e.  ( X [,] Y ) )   &    |-  ( ph  ->  B  e.  ( X [,] Y ) )   &    |-  ( ( ph  /\  x  e.  ( X (,) Y ) ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  S__
 [ A  ->  B ] C  _d x  e.  CC )
 
Theoremditgswap 19748* Reverse a directed integral. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  A  e.  ( X [,] Y ) )   &    |-  ( ph  ->  B  e.  ( X [,] Y ) )   &    |-  ( ( ph  /\  x  e.  ( X (,) Y ) ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  C )  e.  L ^1 )   =>    |-  ( ph  ->  S__
 [ B  ->  A ] C  _d x  =  -u S__ [ A  ->  B ] C  _d x )
 
Theoremditgsplitlem 19749* Lemma for ditgsplit 19750. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  A  e.  ( X [,] Y ) )   &    |-  ( ph  ->  B  e.  ( X [,] Y ) )   &    |-  ( ph  ->  C  e.  ( X [,] Y ) )   &    |-  ( ( ph  /\  x  e.  ( X (,) Y ) ) 
 ->  D  e.  V )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  D )  e.  L ^1 )   &    |-  (
 ( ps  /\  th ) 
 <->  ( A  <_  B  /\  B  <_  C )
 )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  th )  ->  S__ [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
 
Theoremditgsplit 19750* This theorem is the raison d'être for the directed integral, because unlike itgspliticc 19730, there is no constraint on the ordering of the points  A ,  B ,  C in the domain. (Contributed by Mario Carneiro, 13-Aug-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  A  e.  ( X [,] Y ) )   &    |-  ( ph  ->  B  e.  ( X [,] Y ) )   &    |-  ( ph  ->  C  e.  ( X [,] Y ) )   &    |-  ( ( ph  /\  x  e.  ( X (,) Y ) ) 
 ->  D  e.  V )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  D )  e.  L ^1 )   =>    |-  ( ph  ->  S__
 [ A  ->  C ] D  _d x  =  ( S__ [ A  ->  B ] D  _d x  +  S__ [ B  ->  C ] D  _d x ) )
 
12.3  Derivatives
 
12.3.1  Real and complex differentiation
 
Syntaxclimc 19751 The limit operator.
 class lim CC
 
Syntaxcdv 19752 The derivative operator.
 class  _D
 
Syntaxcdvn 19753 The  n-th derivative operator.
 class  D n
 
Syntaxccpn 19754 The set of  n-times continuously differentiable functions.
 class  C ^n
 
Definitiondf-limc 19755* Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |- lim
 CC  =  ( f  e.  ( CC  ^pm  CC ) ,  x  e. 
 CC  |->  { y  |  [. ( TopOpen ` fld )  /  j ]. ( z  e.  ( dom  f  u.  { x } )  |->  if (
 z  =  x ,  y ,  ( f `  z ) ) )  e.  ( ( ( jt  ( dom  f  u. 
 { x } )
 )  CnP  j ) `  x ) } )
 
Definitiondf-dv 19756* Define the derivative operator on functions on the reals. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set  s here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of  CC and is well-behaved when  s contains no isolated points, we will restrict our attention to the cases  s  =  RR or  s  =  CC for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.)
 |- 
 _D  =  ( s  e.  ~P CC ,  f  e.  ( CC  ^pm  s )  |->  U_ x  e.  ( ( int `  (
 ( TopOpen ` fld )t  s ) ) `  dom  f ) ( { x }  X.  (
 ( z  e.  ( dom  f  \  { x } )  |->  ( ( ( f `  z
 )  -  ( f `
  x ) ) 
 /  ( z  -  x ) ) ) lim
 CC  x ) ) )
 
Definitiondf-dvn 19757* Define the  n-th derivative operator on functions on the complexes. This just iterates the derivative operation according to the last argument. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |- 
 D n  =  ( s  e.  ~P CC ,  f  e.  ( CC  ^pm  s )  |->  seq  0 ( ( ( x  e.  _V  |->  ( s  _D  x ) )  o.  1st ) ,  ( NN0  X.  {
 f } ) ) )
 
Definitiondf-cpn 19758* Define the set of  n-times continuously differentiable functions. (Contributed by Stefan O'Rear, 15-Nov-2014.)
 |-  C ^n  =  ( s  e.  ~P CC  |->  ( x  e.  NN0  |->  { f  e.  ( CC  ^pm  s
 )  |  ( ( s  D n f ) `  x )  e.  ( dom  f -cn->
 CC ) } )
 )
 
Theoremreldv 19759 The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |- 
 Rel  ( S  _D  F )
 
Theoremlimcvallem 19760* Lemma for ellimc 19762. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  J  =  ( Kt  ( A  u.  { B } ) )   &    |-  K  =  ( TopOpen ` fld )   &    |-  G  =  ( z  e.  ( A  u.  { B }
 )  |->  if ( z  =  B ,  C ,  ( F `  z ) ) )   =>    |-  ( ( F : A
 --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  ( G  e.  ( ( J  CnP  K ) `  B ) 
 ->  C  e.  CC )
 )
 
Theoremlimcfval 19761* Value and set bounds on the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  J  =  ( Kt  ( A  u.  { B } ) )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  ( ( F : A
 --> CC  /\  A  C_  CC  /\  B  e.  CC )  ->  ( ( F lim
 CC  B )  =  { y  |  ( z  e.  ( A  u.  { B }
 )  |->  if ( z  =  B ,  y ,  ( F `  z
 ) ) )  e.  ( ( J  CnP  K ) `  B ) }  /\  ( F lim
 CC  B )  C_  CC ) )
 
Theoremellimc 19762* Value of the limit predicate.  C is the limit of the function  F at  B if the function  G, formed by adding  B to the domain of  F and setting it to  C, is continuous at  B. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  J  =  ( Kt  ( A  u.  { B } ) )   &    |-  K  =  ( TopOpen ` fld )   &    |-  G  =  ( z  e.  ( A  u.  { B }
 )  |->  if ( z  =  B ,  C ,  ( F `  z ) ) )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( C  e.  ( F lim CC  B )  <->  G  e.  (
 ( J  CnP  K ) `  B ) ) )
 
Theoremlimcrcl 19763 Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( C  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC )
 )
 
Theoremlimccl 19764 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( F lim CC  B )  C_  CC
 
Theoremlimcdif 19765 It suffices to consider functions which are not defined at  B to define the limit of a function. In particular, the value of the original function  F at  B does not affect the limit of  F. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( ph  ->  F : A --> CC )   =>    |-  ( ph  ->  ( F lim CC  B )  =  ( ( F  |`  ( A  \  { B } ) ) lim CC  B ) )
 
Theoremellimc2 19766* Write the definition of a limit directly in terms of open sets of the topology on the complexes. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  K  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  ( C  e.  ( F lim CC  B )  <->  ( C  e.  CC  /\  A. u  e.  K  ( C  e.  u  ->  E. w  e.  K  ( B  e.  w  /\  ( F " ( w  i^i  ( A  \  { B } ) ) )  C_  u )
 ) ) ) )
 
Theoremlimcnlp 19767 If  B is not a limit point of the domain of the function 
F, then every point is a limit of  F at  B. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  ( ph  ->  -.  B  e.  ( (
 limPt `  K ) `  A ) )   =>    |-  ( ph  ->  ( F lim CC  B )  =  CC )
 
Theoremellimc3 19768* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( C  e.  ( F lim CC  B )  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z  =/=  B  /\  ( abs `  (
 z  -  B ) )  <  y ) 
 ->  ( abs `  (
 ( F `  z
 )  -  C ) )  <  x ) ) ) )
 
Theoremlimcflflem 19769 Lemma for limcflf 19770. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  ( ( limPt `  K ) `  A ) )   &    |-  K  =  ( TopOpen ` fld )   &    |-  C  =  ( A  \  { B } )   &    |-  L  =  ( ( ( nei `  K ) `  { B }
 )t 
 C )   =>    |-  ( ph  ->  L  e.  ( Fil `  C ) )
 
Theoremlimcflf 19770 The limit operator can be expressed as a filter limit, from the filter of neighborhoods of  B restricted to  A  \  { B }, to the topology of the complexes. (If  B is not a limit point of  A, then it is still formally a filter limit, but the neighborhood filter is not a proper filter in this case.) (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  ( ( limPt `  K ) `  A ) )   &    |-  K  =  ( TopOpen ` fld )   &    |-  C  =  ( A  \  { B } )   &    |-  L  =  ( ( ( nei `  K ) `  { B }
 )t 
 C )   =>    |-  ( ph  ->  ( F lim CC  B )  =  ( ( K  fLimf  L ) `  ( F  |`  C ) ) )
 
Theoremlimcmo 19771* If  B is a limit point of the domain of the function  F, then there is at most one limit value of  F at  B. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  ( ( limPt `  K ) `  A ) )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  ( ph  ->  E* x  x  e.  ( F lim CC  B ) )
 
Theoremlimcmpt 19772* Express the limit operator for a function defined by a mapping. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( ph  ->  A  C_ 
 CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  (
 ( ph  /\  z  e.  A )  ->  D  e.  CC )   &    |-  J  =  ( Kt  ( A  u.  { B } ) )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  ( ph  ->  ( C  e.  ( (
 z  e.  A  |->  D ) lim CC  B )  <-> 
 ( z  e.  ( A  u.  { B }
 )  |->  if ( z  =  B ,  C ,  D ) )  e.  ( ( J  CnP  K ) `  B ) ) )
 
Theoremlimcmpt2 19773* Express the limit operator for a function defined by a mapping. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( ph  ->  A  C_ 
 CC )   &    |-  ( ph  ->  B  e.  A )   &    |-  (
 ( ph  /\  ( z  e.  A  /\  z  =/=  B ) )  ->  D  e.  CC )   &    |-  J  =  ( Kt  A )   &    |-  K  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  ( C  e.  ( (
 z  e.  ( A 
 \  { B }
 )  |->  D ) lim CC  B )  <->  ( z  e.  A  |->  if ( z  =  B ,  C ,  D ) )  e.  ( ( J  CnP  K ) `  B ) ) )
 
Theoremlimcresi 19774 Any limit of  F is also a limit of the restriction of  F. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( F lim CC  B )  C_  ( ( F  |`  C ) lim CC  B )
 
Theoremlimcres 19775 If  B is an interior point of  C  u.  { B } relative to the domain  A, then a limit point of  F  |`  C extends to a limit of  F. (Contributed by Mario Carneiro, 27-Dec-2016.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  C  C_  A )   &    |-  ( ph  ->  A  C_ 
 CC )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  J  =  ( Kt  ( A  u.  { B } ) )   &    |-  ( ph  ->  B  e.  (
 ( int `  J ) `  ( C  u.  { B } ) ) )   =>    |-  ( ph  ->  ( ( F  |`  C ) lim CC  B )  =  ( F lim CC  B ) )
 
Theoremcnplimc 19776 A function is continuous at  B iff its limit at  B equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  K  =  ( TopOpen ` fld )   &    |-  J  =  ( Kt  A )   =>    |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B )  <-> 
 ( F : A --> CC  /\  ( F `  B )  e.  ( F lim CC  B ) ) ) )
 
Theoremcnlimc 19777*  F is a continuous function iff the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( A  C_  CC  ->  ( F  e.  ( A -cn-> CC )  <->  ( F : A
 --> CC  /\  A. x  e.  A  ( F `  x )  e.  ( F lim CC  x ) ) ) )
 
Theoremcnlimci 19778 If  F is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  F  e.  ( A -cn-> D ) )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  ( F `  B )  e.  ( F lim CC  B ) )
 
Theoremcnmptlimc 19779* If  F is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  ( x  e.  A  |->  X )  e.  ( A -cn-> D ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( x  =  B  ->  X  =  Y )   =>    |-  ( ph  ->  Y  e.  ( ( x  e.  A  |->  X ) lim
 CC  B ) )
 
Theoremlimccnp 19780 If the limit of  F at  B is  C and  G is continuous at  C, then the limit of  G  o.  F at  B is  G ( C ). (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  F : A --> D )   &    |-  ( ph  ->  D  C_  CC )   &    |-  K  =  ( TopOpen ` fld )   &    |-  J  =  ( Kt  D )   &    |-  ( ph  ->  C  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  G  e.  (
 ( J  CnP  K ) `  C ) )   =>    |-  ( ph  ->  ( G `  C )  e.  (
 ( G  o.  F ) lim CC  B ) )
 
Theoremlimccnp2 19781* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( ( ph  /\  x  e.  A )  ->  R  e.  X )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  S  e.  Y )   &    |-  ( ph  ->  X  C_  CC )   &    |-  ( ph  ->  Y  C_ 
 CC )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  J  =  ( ( K  tX  K )t  ( X  X.  Y ) )   &    |-  ( ph  ->  C  e.  ( ( x  e.  A  |->  R ) lim
 CC  B ) )   &    |-  ( ph  ->  D  e.  ( ( x  e.  A  |->  S ) lim CC  B ) )   &    |-  ( ph  ->  H  e.  (
 ( J  CnP  K ) `  <. C ,  D >. ) )   =>    |-  ( ph  ->  ( C H D )  e.  ( ( x  e.  A  |->  ( R H S ) ) lim CC  B ) )
 
Theoremlimcco 19782* Composition of two limits. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ( ph  /\  ( x  e.  A  /\  R  =/=  C ) ) 
 ->  R  e.  B )   &    |-  ( ( ph  /\  y  e.  B )  ->  S  e.  CC )   &    |-  ( ph  ->  C  e.  ( ( x  e.  A  |->  R ) lim
 CC  X ) )   &    |-  ( ph  ->  D  e.  ( ( y  e.  B  |->  S ) lim CC  C ) )   &    |-  (
 y  =  R  ->  S  =  T )   &    |-  (
 ( ph  /\  ( x  e.  A  /\  R  =  C ) )  ->  T  =  D )   =>    |-  ( ph  ->  D  e.  (
 ( x  e.  A  |->  T ) lim CC  X ) )
 
Theoremlimciun 19783* A point is a limit of  F on the finite union  U_ x  e.  A B ( x ) iff it is the limit of the restriction of  F to each  B ( x ). (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A. x  e.  A  B  C_ 
 CC )   &    |-  ( ph  ->  F : U_ x  e.  A  B --> CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( F lim CC  C )  =  ( CC  i^i  |^|_ x  e.  A  ( ( F  |`  B ) lim CC  C ) ) )
 
Theoremlimcun 19784 A point is a limit of  F on  A  u.  B iff it is the limit of the restriction of  F to  A and to  B. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ph  ->  A  C_ 
 CC )   &    |-  ( ph  ->  B 
 C_  CC )   &    |-  ( ph  ->  F : ( A  u.  B ) --> CC )   =>    |-  ( ph  ->  ( F lim CC  C )  =  (
 ( ( F  |`  A ) lim
 CC  C )  i^i  ( ( F  |`  B ) lim
 CC  C ) ) )
 
Theoremdvlem 19785 Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( ph  ->  F : D --> CC )   &    |-  ( ph  ->  D  C_  CC )   &    |-  ( ph  ->  B  e.  D )   =>    |-  ( ( ph  /\  A  e.  ( D  \  { B } ) )  ->  ( ( ( F `
  A )  -  ( F `  B ) )  /  ( A  -  B ) )  e.  CC )
 
Theoremdvfval 19786* Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 25-Dec-2016.)
 |-  T  =  ( Kt  S )   &    |-  K  =  (
 TopOpen ` fld )   =>    |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  ( ( S  _D  F )  = 
 U_ x  e.  (
 ( int `  T ) `  A ) ( { x }  X.  (
 ( z  e.  ( A  \  { x }
 )  |->  ( ( ( F `  z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )  /\  ( S  _D  F ) 
 C_  ( ( ( int `  T ) `  A )  X.  CC ) ) )
 
Theoremeldv 19787* The differentiable predicate. A function  F is differentiable at  B with derivative  C iff  F is defined in a neighborhood of  B and the difference quotient has limit  C at  B. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 25-Dec-2016.)
 |-  T  =  ( Kt  S )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  G  =  ( z  e.  ( A 
 \  { B }
 )  |->  ( ( ( F `  z )  -  ( F `  B ) )  /  ( z  -  B ) ) )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ph  ->  ( B ( S  _D  F ) C  <->  ( B  e.  ( ( int `  T ) `  A )  /\  C  e.  ( G lim CC  B ) ) ) )
 
Theoremdvcl 19788 The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ( ph  /\  B ( S  _D  F ) C )  ->  C  e.  CC )
 
Theoremdvbssntr 19789 The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  J  =  ( Kt  S )   &    |-  K  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  dom  ( S  _D  F )  C_  ( ( int `  J ) `  A ) )
 
Theoremdvbss 19790 The set of differentiable points is a subset of the domain of the function. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ph  ->  dom  ( S  _D  F )  C_  A )
 
Theoremdvbsss 19791 The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.)
 |- 
 dom  ( S  _D  F )  C_  S
 
Theoremperfdvf 19792 The derivative is a function, whenever it is defined relative to a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  K  =  ( TopOpen ` fld )   =>    |-  (
 ( Kt  S )  e. Perf  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
 
Theoremrecnprss 19793 Both  RR and  CC are subsets of  CC. (Contributed by Mario Carneiro, 10-Feb-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  S 
 C_  CC )
 
Theoremrecnperf 19794 Both  RR and  CC are perfect subsets of  CC. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  K  =  ( TopOpen ` fld )   =>    |-  ( S  e.  { RR ,  CC }  ->  ( Kt  S )  e. Perf )
 
Theoremdvfg 19795 Explicitly write out the functionality condition on derivative for  S  =  RR and 
CC. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
 
Theoremdvf 19796 The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( RR  _D  F ) : dom  ( RR 
 _D  F ) --> CC
 
Theoremdvfcn 19797 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  ( CC  _D  F ) : dom  ( CC 
 _D  F ) --> CC
 
Theoremdvreslem 19798* Lemma for dvres 19800. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  K  =  ( TopOpen ` fld )   &    |-  T  =  ( Kt  S )   &    |-  G  =  ( z  e.  ( A 
 \  { x }
 )  |->  ( ( ( F `  z )  -  ( F `  x ) )  /  ( z  -  x ) ) )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  B  C_  S )   &    |-  ( ph  ->  y  e.  CC )   =>    |-  ( ph  ->  ( x ( S  _D  ( F  |`  B ) ) y  <->  ( x ( S  _D  F ) y  /\  x  e.  ( ( int `  T ) `  B ) ) ) )
 
Theoremdvres2lem 19799* Lemma for dvres2 19801. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  K  =  ( TopOpen ` fld )   &    |-  T  =  ( Kt  S )   &    |-  G  =  ( z  e.  ( A 
 \  { x }
 )  |->  ( ( ( F `  z )  -  ( F `  x ) )  /  ( z  -  x ) ) )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  B  C_  S )   &    |-  ( ph  ->  y  e.  CC )   &    |-  ( ph  ->  x ( S  _D  F ) y )   &    |-  ( ph  ->  x  e.  B )   =>    |-  ( ph  ->  x ( B  _D  ( F  |`  B ) ) y )
 
Theoremdvres 19800 Restriction of a derivative. Note that our definition of derivative df-dv 19756 would still make sense if we demanded that  x be an element of the domain instead of an interior point of the domain, but then it is possible for a non-differentiable function to have two different derivatives at a single point 
x when restricted to different subsets containing  x; a classic example is the absolute value function restricted to  [ 0 ,  +oo ) and  (  -oo ,  0 ]. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  K  =  ( TopOpen ` fld )   &    |-  T  =  ( Kt  S )   =>    |-  ( ( ( S 
 C_  CC  /\  F : A
 --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  ->  ( S  _D  ( F  |`  B ) )  =  ( ( S  _D  F )  |`  ( ( int `  T ) `  B ) ) )
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