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Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-dv 22901* 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 22902* Define the  n-th derivative operator on functions on the complex numbers. This just iterates the derivative operation according to the last argument. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |- 
 Dn  =  ( s  e.  ~P CC ,  f  e.  ( CC  ^pm  s )  |->  seq 0 ( ( ( x  e.  _V  |->  ( s  _D  x ) )  o.  1st ) ,  ( NN0  X.  {
 f } ) ) )
 
Definitiondf-cpn 22903* 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  Dn f ) `  x )  e.  ( dom  f -cn->
 CC ) } )
 )
 
Theoremreldv 22904 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 22905* Lemma for ellimc 22907. (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 22906* 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 22907* 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 22908 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 22909 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( F lim CC  B )  C_  CC
 
Theoremlimcdif 22910 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 22911* Write the definition of a limit directly in terms of open sets of the topology on the complex numbers. (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 22912 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 22913* 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 22914 Lemma for limcflf 22915. (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 22915 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 complex numbers. (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 22916* 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 22917* 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 22918* 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 22919 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 22920 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 22921 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 22922*  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 22923 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 22924* 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 22925 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 22926* 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 22927* 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 22928* 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 22929 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 22930 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 22931* 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 22932* 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 22933 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 22934 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 22935 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 22936 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 22937 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 22938 Both  RR and  CC are subsets of  CC. (Contributed by Mario Carneiro, 10-Feb-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  S 
 C_  CC )
 
Theoremrecnperf 22939 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 22940 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 22941 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 22942 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  ( CC  _D  F ) : dom  ( CC 
 _D  F ) --> CC
 
Theoremdvreslem 22943* Lemma for dvres 22945. (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 22944* Lemma for dvres2 22946. (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 22945 Restriction of a derivative. Note that our definition of derivative df-dv 22901 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 22946 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 22945, 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 22947 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 22948 Restriction of a complex differentiable function to the reals. This version of dvres3 22947 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 22949* Lemma for dvid 22951 and dvconst 22950. (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 22950 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 22951 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 22952* 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 22953 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 22954 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 22955* 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  Dn F )  =  seq 0 ( ( G  o.  1st ) ,  ( NN0  X.  { F } ) ) )
 
Theoremdvnff 22956 The iterated derivative is a function. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC 
 ^pm  S ) )  ->  ( S  Dn F ) : NN0 --> ( CC  ^pm  dom  F ) )
 
Theoremdvn0 22957 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  Dn F ) `  0
 )  =  F )
 
Theoremdvnp1 22958 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  Dn F ) `  ( N  +  1 )
 )  =  ( S  _D  ( ( S  Dn F ) `
  N ) ) )
 
Theoremdvn1 22959 One times iterated derivative. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S ) ) 
 ->  ( ( S  Dn F ) `  1
 )  =  ( S  _D  F ) )
 
Theoremdvnf 22960 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  Dn F ) `  N ) : dom  ( ( S  Dn F ) `  N ) --> CC )
 
Theoremdvnbss 22961 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  Dn F ) `  N )  C_  dom  F )
 
Theoremdvnadd 22962 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  Dn
 ( ( S  Dn F ) `  M ) ) `  N )  =  ( ( S  Dn F ) `
  ( M  +  N ) ) )
 
Theoremdvn2bss 22963 An N-times differentiable point is an M-times differentiable 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  Dn F ) `  N )  C_  dom  ( ( S  Dn F ) `
  M ) )
 
Theoremdvnres 22964 Multiple derivative version of dvres3a 22948. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  N  e.  NN0 )  /\  dom  ( ( CC 
 Dn F ) `
  N )  = 
 dom  F )  ->  (
 ( S  Dn
 ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
 
Theoremcpnfval 22965* 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  Dn f ) `
  n )  e.  ( dom  f -cn-> CC ) } ) )
 
Theoremfncpn 22966 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 22967 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  Dn F ) `
  N )  e.  ( dom  F -cn-> CC ) ) ) )
 
Theoremcpnord 22968  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 22969 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 22970 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 22971 The sum rule for derivatives at a point. For the (simpler but more limited) function version, see dvadd 22973. (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  oF  +  G ) ) ( K  +  L ) )
 
Theoremdvmulbr 22972 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmul 22974. (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  oF  x.  G ) ) ( ( K  x.  ( G `
  C ) )  +  ( L  x.  ( F `  C ) ) ) )
 
Theoremdvadd 22973 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddbr 22971. (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  oF  +  G ) ) `  C )  =  ( (
 ( S  _D  F ) `  C )  +  ( ( S  _D  G ) `  C ) ) )
 
Theoremdvmul 22974 The product rule for derivatives at a point. For the (more general) relation version, see dvmulbr 22972. (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  oF  x.  G ) ) `  C )  =  ( (
 ( ( S  _D  F ) `  C )  x.  ( G `  C ) )  +  ( ( ( S  _D  G ) `  C )  x.  ( F `  C ) ) ) )
 
Theoremdvaddf 22975 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  oF  +  G )
 )  =  ( ( S  _D  F )  oF  +  ( S  _D  G ) ) )
 
Theoremdvmulf 22976 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  oF  x.  G )
 )  =  ( ( ( S  _D  F )  oF  x.  G )  oF  +  (
 ( S  _D  G )  oF  x.  F ) ) )
 
Theoremdvcmul 22977 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 } )  oF  x.  F ) ) `  C )  =  ( A  x.  ( ( S  _D  F ) `  C ) ) )
 
Theoremdvcmulf 22978 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 }
 )  oF  x.  F ) )  =  ( ( S  X.  { A } )  oF  x.  ( S  _D  F ) ) )
 
Theoremdvcobr 22979 The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 22980. (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 22980 The chain rule for derivatives at a point. For the (more general) relation version, see dvcobr 22979. (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 22981 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 )  oF  x.  ( T  _D  G ) ) )
 
Theoremdvcjbr 22982 The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 22983. (This doesn't follow from dvcobr 22979 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 22983 The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 22982. (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 22984 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 22985 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 
 Dn F ) `
  N ) : dom  ( ( RR 
 Dn F ) `
  N ) --> RR )
 
Theoremdvexp 22986* 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 22987* 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 22988* 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 22989* 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 22990* 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 ) )
 
Theoremdvmptc 22991* Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  0 ) )
 
Theoremdvmptcl 22992* Closure lemma for dvmptcmul 22997 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   =>    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  CC )
 
Theoremdvmptadd 22993* Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  D  e.  W )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  C ) )  =  ( x  e.  X  |->  D ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( A  +  C ) ) )  =  ( x  e.  X  |->  ( B  +  D ) ) )
 
Theoremdvmptmul 22994* Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  D  e.  W )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  C ) )  =  ( x  e.  X  |->  D ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( A  x.  C ) ) )  =  ( x  e.  X  |->  ( ( B  x.  C )  +  ( D  x.  A ) ) ) )
 
Theoremdvmptres2 22995* Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ph  ->  Z 
 C_  X )   &    |-  J  =  ( Kt  S )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  ( ph  ->  ( ( int `  J ) `  Z )  =  Y )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  Z  |->  A ) )  =  ( x  e.  Y  |->  B ) )
 
Theoremdvmptres 22996* Function-builder for derivative: restrict a derivative to an open subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ph  ->  Y 
 C_  X )   &    |-  J  =  ( Kt  S )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  ( ph  ->  Y  e.  J )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  Y  |->  A ) )  =  ( x  e.  Y  |->  B ) )
 
Theoremdvmptcmul 22997* Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( C  x.  A ) ) )  =  ( x  e.  X  |->  ( C  x.  B ) ) )
 
Theoremdvmptdivc 22998* Function-builder for derivative, division rule for constant divisor. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  C  =/=  0
 )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( A  /  C ) ) )  =  ( x  e.  X  |->  ( B  /  C ) ) )
 
Theoremdvmptneg 22999* Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  -u A ) )  =  ( x  e.  X  |->  -u B ) )
 
Theoremdvmptsub 23000* Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  D  e.  W )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  C ) )  =  ( x  e.  X  |->  D ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( A  -  C ) ) )  =  ( x  e.  X  |->  ( B  -  D ) ) )
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