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Theorem List for Metamath Proof Explorer - 19001-19100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremitgss3 19001* 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 19002* 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 19003* 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 19004 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 19005* 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 19006* Lemma for ibladd 19007. (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 19007* 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 19008* 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 19009* Lemma for itgadd 19011. (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 19010* Lemma for itgadd 19011. (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 19011* 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 19012* 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 19013* 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 19014* Lemma for iblabs 19015. (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 19015* 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 19016* A measurable function is integrable iff its absolute value is integrable. (See iblabs 19015 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 19017* 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 19018* Lemma for itgmulc2 19020: 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 19019* Lemma for itgmulc2 19020: 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 19020* 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 19021* 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 19022* 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 19023* 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 19024* 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 19025* 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 19026* 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 19027 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 19028* 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 19029* Transfer itg2cn 18950 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 19030* 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 19031* 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 19032* 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 19038 first and use the equality theorems for df-itg 18811.) (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 19033* 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 19034 A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |- 
 S__ [ A  ->  B ] C  _d x  e.  _V
 
Theoremditg0 19035* 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 19036* 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 19037* 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 19038* 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 19039* 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 19040* 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 19041* 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 19042* Lemma for ditgsplit 19043. (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 19043* This theorem is the raison d'être for the directed integral, because unlike itgspliticc 19023, 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 19044 The limit operator.
 class lim CC
 
Syntaxcdv 19045 The derivative operator.
 class  _D
 
Syntaxcdvn 19046 The  n-th derivative operator.
 class  D n
 
Syntaxccpn 19047 The set of  n-times continuously differentiable functions.
 class  C ^n
 
Definitiondf-limc 19048* 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 19049* 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 19050* 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 19051* 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 19052 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 19053* Lemma for ellimc 19055. (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 19054* 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 19055* 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 19056 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 19057 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( F lim CC  B )  C_  CC
 
Theoremlimcdif 19058 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 19059* 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 19060 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 19061* 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 19062 Lemma for limcflf 19063. (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 19063 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 19064* 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 19065* 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 19066* 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 19067 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 19068 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 19069 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 19070*  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 19071 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 19072* 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 19073 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 19074* 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 19075* 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 19076* 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 19077 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 19078 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 19079* 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 19080* 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 19081 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 19082 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 19083 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 19084 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 19085 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 19086 Both  RR and  CC are subsets of  CC. (Contributed by Mario Carneiro, 10-Feb-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  S 
 C_  CC )
 
Theoremrecnperf 19087 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 19088 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 19089 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 19090 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  ( CC  _D  F ) : dom  ( CC 
 _D  F ) --> CC
 
Theoremdvreslem 19091* Lemma for dvres 19093. (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 19092* Lemma for dvres2 19094. (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 19093 Restriction of a derivative. Note that our definition of derivative df-dv 19049 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 ) ) )
 
Theoremdvres2 19094 Restriction of the base set of a derivative. The primary application of this theorem says that if a function is complex differentiable then it is also real differentiable. Unlike dvres 19093, there is no simple reverse relation relating real differentiable functions to complex differentiability, and indeed there are functions like  Re ( x ) which are everywhere real-differentiable but nowhere complex-differentiable.) (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  ( ( ( S 
 C_  CC  /\  F : A
 --> CC )  /\  ( A  C_  S  /\  B  C_  S ) )  ->  ( ( S  _D  F )  |`  B ) 
 C_  ( B  _D  ( F  |`  B ) ) )
 
Theoremdvres3 19095 Restriction of a complex differentiable function to the reals. (Contributed by Mario Carneiro, 10-Feb-2015.)
 |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A
 --> CC )  /\  ( A  C_  CC  /\  S  C_ 
 dom  ( CC  _D  F ) ) ) 
 ->  ( S  _D  ( F  |`  S ) )  =  ( ( CC 
 _D  F )  |`  S ) )
 
Theoremdvres3a 19096 Restriction of a complex differentiable function to the reals. This version of dvres3 19095 assumes that  F is differentiable on its domain, but does not require  F to be differentiable on the whole real line. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  (
 ( ( S  e.  { RR ,  CC }  /\  F : A --> CC )  /\  ( A  e.  J  /\  dom  ( CC  _D  F )  =  A ) )  ->  ( S  _D  ( F  |`  S ) )  =  ( ( CC  _D  F )  |`  S ) )
 
Theoremdvidlem 19097* Lemma for dvid 19099 and dvconst 19098. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( ph  ->  F : CC --> CC )   &    |-  (
 ( ph  /\  ( x  e.  CC  /\  z  e.  CC  /\  z  =/= 
 x ) )  ->  ( ( ( F `
  z )  -  ( F `  x ) )  /  ( z  -  x ) )  =  B )   &    |-  B  e.  CC   =>    |-  ( ph  ->  ( CC  _D  F )  =  ( CC  X.  { B } ) )
 
Theoremdvconst 19098 Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( A  e.  CC  ->  ( CC  _D  ( CC  X.  { A }
 ) )  =  ( CC  X.  { 0 } ) )
 
Theoremdvid 19099 Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( CC  _D  (  _I  |`  CC ) )  =  ( CC  X.  { 1 } )
 
Theoremdvcnp 19100* The difference quotient is continuous at  B when the original function is differentiable at  B. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  J  =  ( Kt  A )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  G  =  ( z  e.  A  |->  if ( z  =  B ,  ( ( S  _D  F ) `  B ) ,  ( (
 ( F `  z
 )  -  ( F `
  B ) ) 
 /  ( z  -  B ) ) ) )   =>    |-  ( ( ( S  e.  { RR ,  CC }  /\  F : A
 --> CC  /\  A  C_  S )  /\  B  e.  dom  (  S  _D  F ) )  ->  G  e.  ( ( J  CnP  K ) `  B ) )
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