HomeHome Metamath Proof Explorer
Theorem List (p. 193 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21490)
  Hilbert Space Explorer  Hilbert Space Explorer
(21491-23013)
  Users' Mathboxes  Users' Mathboxes
(23014-31421)
 

Theorem List for Metamath Proof Explorer - 19201-19300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremditgneg 19201* 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 19202* 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 19203* 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 19204* Lemma for ditgsplit 19205. (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 19205* This theorem is the raison d'être for the directed integral, because unlike itgspliticc 19185, 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 19206 The limit operator.
 class lim CC
 
Syntaxcdv 19207 The derivative operator.
 class  _D
 
Syntaxcdvn 19208 The  n-th derivative operator.
 class  D n
 
Syntaxccpn 19209 The set of  n-times continuously differentiable functions.
 class  C ^n
 
Definitiondf-limc 19210* 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 19211* 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 19212* 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 19213* 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 19214 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 19215* Lemma for ellimc 19217. (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 19216* 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 19217* 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 19218 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 19219 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( F lim CC  B )  C_  CC
 
Theoremlimcdif 19220 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 19221* 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 19222 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 19223* 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 19224 Lemma for limcflf 19225. (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 19225 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 19226* 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 19227* 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 19228* 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 19229 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 19230 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 19231 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 19232*  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 19233 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 19234* 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 19235 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 19236* 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 19237* 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 19238* 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 19239 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 19240 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 19241* 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 19242* 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 19243 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 19244 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 19245 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 19246 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 19247 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 19248 Both  RR and  CC are subsets of  CC. (Contributed by Mario Carneiro, 10-Feb-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  S 
 C_  CC )
 
Theoremrecnperf 19249 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 19250 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 19251 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 19252 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  ( CC  _D  F ) : dom  ( CC 
 _D  F ) --> CC
 
Theoremdvreslem 19253* Lemma for dvres 19255. (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 19254* Lemma for dvres2 19256. (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 19255 Restriction of a derivative. Note that our definition of derivative df-dv 19211 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 19256 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 19255, 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 19257 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 19258 Restriction of a complex differentiable function to the reals. This version of dvres3 19257 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 19259* Lemma for dvid 19261 and dvconst 19260. (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 19260 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 19261 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 19262* 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 ) )
 
Theoremdvcnp2 19263 A function is continuous at each point for which it is differentiable. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  J  =  ( Kt  A )   &    |-  K  =  (
 TopOpen ` fld )   =>    |-  ( ( ( S 
 C_  CC  /\  F : A
 --> CC  /\  A  C_  S )  /\  B  e.  dom  (  S  _D  F ) )  ->  F  e.  ( ( J  CnP  K ) `  B ) )
 
Theoremdvcn 19264 A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)
 |-  ( ( ( S 
 C_  CC  /\  F : A
 --> CC  /\  A  C_  S )  /\  dom  (  S  _D  F )  =  A )  ->  F  e.  ( A -cn-> CC )
 )
 
Theoremdvnfval 19265* Value of the iterated derivative. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  G  =  ( x  e.  _V  |->  ( S  _D  x ) )   =>    |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S ) ) 
 ->  ( S  D n F )  =  seq  0 ( ( G  o.  1st ) ,  ( NN0  X.  { F } ) ) )
 
Theoremdvnff 19266 The iterated derivative is a function. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC 
 ^pm  S ) )  ->  ( S  D n F ) : NN0 --> ( CC  ^pm  dom  F ) )
 
Theoremdvn0 19267 Zero times iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S ) ) 
 ->  ( ( S  D n F ) `  0
 )  =  F )
 
Theoremdvnp1 19268 Successor iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S )  /\  N  e.  NN0 )  ->  ( ( S  D n F ) `  ( N  +  1 )
 )  =  ( S  _D  ( ( S  D n F ) `
  N ) ) )
 
Theoremdvn1 19269 One times iterated derivative. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S ) ) 
 ->  ( ( S  D n F ) `  1
 )  =  ( S  _D  F ) )
 
Theoremdvnf 19270 The N-times derivative is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC 
 ^pm  S )  /\  N  e.  NN0 )  ->  (
 ( S  D n F ) `  N ) : dom  ( ( S  D n F ) `  N ) --> CC )
 
Theoremdvnbss 19271 The set of N-times differentiable points is a subset of the domain of the function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC 
 ^pm  S )  /\  N  e.  NN0 )  ->  dom  (
 ( S  D n F ) `  N )  C_  dom  F )
 
Theoremdvnadd 19272 The  N-th derivative of the  M-th derivative of  F is the same as the  M  +  N-th derivative of  F. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S ) )  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  (
 ( S  D n
 ( ( S  D n F ) `  M ) ) `  N )  =  ( ( S  D n F ) `
  ( M  +  N ) ) )
 
Theoremdvn2bss 19273 An N-times differentiable point is an M-times differeentiable point, if  M  <_  N. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC 
 ^pm  S )  /\  M  e.  ( 0 ... N ) )  ->  dom  (
 ( S  D n F ) `  N )  C_  dom  ( ( S  D n F ) `
  M ) )
 
Theoremdvnres 19274 Multiple derivative version of dvres3a 19258. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  N  e.  NN0 )  /\  dom  ( ( CC 
 D n F ) `
  N )  = 
 dom  F )  ->  (
 ( S  D n
 ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N )  |`  S ) )
 
Theoremcpnfval 19275* Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( S  C_  CC  ->  ( C ^n `  S )  =  ( n  e.  NN0  |->  { f  e.  ( CC  ^pm  S )  |  ( ( S  D n f ) `
  n )  e.  ( dom  f -cn-> CC ) } ) )
 
Theoremfncpn 19276 The  C ^n object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( S  C_  CC  ->  ( C ^n `  S )  Fn  NN0 )
 
Theoremelcpn 19277 Condition for n-times continuous differentiability. (Contributed by Stefan O'Rear, 15-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( S  C_  CC  /\  N  e.  NN0 )  ->  ( F  e.  ( ( C ^n
 `  S ) `  N )  <->  ( F  e.  ( CC  ^pm  S ) 
 /\  ( ( S  D n F ) `
  N )  e.  ( dom  F -cn-> CC ) ) ) )
 
Theoremcpnord 19278  C ^n conditions are ordered by strength. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( S  e.  { RR ,  CC }  /\  M  e.  NN0  /\  N  e.  ( ZZ>= `  M )
 )  ->  ( ( C ^n `  S ) `
  N )  C_  ( ( C ^n
 `  S ) `  M ) )
 
Theoremcpncn 19279 A  C ^n function is continuous. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n `  S ) `  N ) ) 
 ->  F  e.  ( dom 
 F -cn-> CC ) )
 
Theoremcpnres 19280 The restriction of a  C ^n function is  C ^n. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( ( C ^n `  CC ) `  N ) ) 
 ->  ( F  |`  S )  e.  ( ( C ^n `  S ) `
  N ) )
 
Theoremdvaddbr 19281 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : Y --> CC )   &    |-  ( ph  ->  Y  C_  S )   &    |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  K  e.  V )   &    |-  ( ph  ->  L  e.  V )   &    |-  ( ph  ->  C ( S  _D  F ) K )   &    |-  ( ph  ->  C ( S  _D  G ) L )   &    |-  J  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  C ( S  _D  ( F  o F  +  G ) ) ( K  +  L ) )
 
Theoremdvmulbr 19282 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : Y --> CC )   &    |-  ( ph  ->  Y  C_  S )   &    |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  K  e.  V )   &    |-  ( ph  ->  L  e.  V )   &    |-  ( ph  ->  C ( S  _D  F ) K )   &    |-  ( ph  ->  C ( S  _D  G ) L )   &    |-  J  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  C ( S  _D  ( F  o F  x.  G ) ) ( ( K  x.  ( G `
  C ) )  +  ( L  x.  ( F `  C ) ) ) )
 
Theoremdvadd 19283 The sum rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : Y --> CC )   &    |-  ( ph  ->  Y  C_  S )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  C  e.  dom  (  S  _D  F ) )   &    |-  ( ph  ->  C  e.  dom  (  S  _D  G ) )   =>    |-  ( ph  ->  (
 ( S  _D  ( F  o F  +  G ) ) `  C )  =  ( (
 ( S  _D  F ) `  C )  +  ( ( S  _D  G ) `  C ) ) )
 
Theoremdvmul 19284 The product rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : Y --> CC )   &    |-  ( ph  ->  Y  C_  S )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  C  e.  dom  (  S  _D  F ) )   &    |-  ( ph  ->  C  e.  dom  (  S  _D  G ) )   =>    |-  ( ph  ->  (
 ( S  _D  ( F  o F  x.  G ) ) `  C )  =  ( (
 ( ( S  _D  F ) `  C )  x.  ( G `  C ) )  +  ( ( ( S  _D  G ) `  C )  x.  ( F `  C ) ) ) )
 
Theoremdvaddf 19285 The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( 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  o F  +  G )
 )  =  ( ( S  _D  F )  o F  +  ( S  _D  G ) ) )
 
Theoremdvmulf 19286 The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( 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  o F  x.  G ) )  =  ( ( ( S  _D  F )  o F  x.  G )  o F  +  (
 ( S  _D  G )  o F  x.  F ) ) )
 
Theoremdvcmul 19287 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  C  e.  dom  (  S  _D  F ) )   =>    |-  ( ph  ->  ( ( S  _D  (
 ( S  X.  { A } )  o F  x.  F ) ) `  C )  =  ( A  x.  ( ( S  _D  F ) `  C ) ) )
 
Theoremdvcmulf 19288 The product rule when one argument is a constant. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  dom  (  S  _D  F )  =  X )   =>    |-  ( ph  ->  ( S  _D  ( ( S  X.  { A }
 )  o F  x.  F ) )  =  ( ( S  X.  { A } )  o F  x.  ( S  _D  F ) ) )
 
Theoremdvcobr 19289 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : Y --> X )   &    |-  ( ph  ->  Y  C_  T )   &    |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  T 
 C_  CC )   &    |-  ( ph  ->  K  e.  V )   &    |-  ( ph  ->  L  e.  V )   &    |-  ( ph  ->  ( G `  C ) ( S  _D  F ) K )   &    |-  ( ph  ->  C ( T  _D  G ) L )   &    |-  J  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  C ( T  _D  ( F  o.  G ) ) ( K  x.  L ) )
 
Theoremdvco 19290 The chain rule for derivatives at a point. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : Y --> X )   &    |-  ( ph  ->  Y  C_  T )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  T  e.  { RR ,  CC } )   &    |-  ( ph  ->  ( G `  C )  e.  dom  (  S  _D  F ) )   &    |-  ( ph  ->  C  e.  dom  (  T  _D  G ) )   =>    |-  ( ph  ->  (
 ( T  _D  ( F  o.  G ) ) `
  C )  =  ( ( ( S  _D  F ) `  ( G `  C ) )  x.  ( ( T  _D  G ) `
  C ) ) )
 
Theoremdvcof 19291 The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  T  e.  { RR ,  CC } )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  G : Y --> X )   &    |-  ( ph  ->  dom  (  S  _D  F )  =  X )   &    |-  ( ph  ->  dom  (  T  _D  G )  =  Y )   =>    |-  ( ph  ->  ( T  _D  ( F  o.  G ) )  =  ( ( ( S  _D  F )  o.  G )  o F  x.  ( T  _D  G ) ) )
 
Theoremdvcjbr 19292 The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19289 because  * is not a function on the reals, and even if we used complex derivatives, 
* is not complex-differentiable.) (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  RR )   &    |-  ( ph  ->  C  e.  dom  ( RR  _D  F ) )   =>    |-  ( ph  ->  C ( RR  _D  ( *  o.  F ) ) ( * `  (
 ( RR  _D  F ) `  C ) ) )
 
Theoremdvcj 19293 The derivative of the conjugate of a function. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ( F : X
 --> CC  /\  X  C_  RR )  ->  ( RR 
 _D  ( *  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
 
Theoremdvfre 19294 The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ( F : A
 --> RR  /\  A  C_  RR )  ->  ( RR 
 _D  F ) : dom  ( RR  _D  F ) --> RR )
 
Theoremdvnfre 19295 The  N-th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ( F : A
 --> RR  /\  A  C_  RR  /\  N  e.  NN0 )  ->  ( ( RR 
 D n F ) `
  N ) : dom  ( ( RR 
 D n F ) `
  N ) --> RR )
 
Theoremdvexp 19296* Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( N  e.  NN  ->  ( CC  _D  ( x  e.  CC  |->  ( x ^ N ) ) )  =  ( x  e.  CC  |->  ( N  x.  ( x ^
 ( N  -  1
 ) ) ) ) )
 
Theoremdvexp2 19297* Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( N  e.  NN0  ->  ( CC  _D  ( x  e.  CC  |->  ( x ^ N ) ) )  =  ( x  e.  CC  |->  if ( N  =  0 , 
 0 ,  ( N  x.  ( x ^
 ( N  -  1
 ) ) ) ) ) )
 
Theoremdvrec 19298* Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  ( CC  \  { 0 } )  |->  ( A  /  x ) ) )  =  ( x  e.  ( CC  \  { 0 } )  |->  -u ( A  /  ( x ^ 2 ) ) ) )
 
Theoremdvmptres3 19299* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  J )   &    |-  ( ph  ->  ( S  i^i  X )  =  Y )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( CC  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  Y  |->  A ) )  =  ( x  e.  Y  |->  B ) )
 
Theoremdvmptid 19300* Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  S  |->  x ) )  =  ( x  e.  S  |->  1 ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31421
  Copyright terms: Public domain < Previous  Next >