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Theorem List for Metamath Proof Explorer - 19201-19300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremperfdvf 19201 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 19202 Both  RR and  CC are subsets of  CC. (Contributed by Mario Carneiro, 10-Feb-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  S 
 C_  CC )
 
Theoremrecnperf 19203 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 19204 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 19205 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 19206 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.)
 |-  ( CC  _D  F ) : dom  ( CC 
 _D  F ) --> CC
 
Theoremdvreslem 19207* Lemma for dvres 19209. (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 19208* Lemma for dvres2 19210. (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 19209 Restriction of a derivative. Note that our definition of derivative df-dv 19165 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 19210 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 19209, 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 19211 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 19212 Restriction of a complex differentiable function to the reals. This version of dvres3 19211 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 19213* Lemma for dvid 19215 and dvconst 19214. (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 19214 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 19215 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 19216* 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 19217 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 19218 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 19219* 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 19220 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 19221 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 19222 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 19223 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 19224 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 19225 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 19226 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 19227 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 19228 Multiple derivative version of dvres3a 19212. (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 19229* 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 19230 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 19231 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 19232  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 19233 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 19234 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 19235 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 19236 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 19237 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 19238 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 19239 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 19240 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 19241 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 19242 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 19243 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 19244 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 19245 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 19246 The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19243 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 19247 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 19248 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 19249 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 19250* 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 19251* 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 19252* 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 19253* 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 19254* 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 19255* 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 19256* Closure lemma for dvmptcmul 19261 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 19257* 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 19258* 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 19259* 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 19260* 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 19261* 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 19262* 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 19263* 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 19264* 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 ) ) )
 
Theoremdvmptcj 19265* Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ( ph  /\  x  e.  X )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   =>    |-  ( ph  ->  ( RR  _D  ( x  e.  X  |->  ( * `
  A ) ) )  =  ( x  e.  X  |->  ( * `
  B ) ) )
 
Theoremdvmptre 19266* Function-builder for derivative, real part. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ( ph  /\  x  e.  X )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   =>    |-  ( ph  ->  ( RR  _D  ( x  e.  X  |->  ( Re
 `  A ) ) )  =  ( x  e.  X  |->  ( Re
 `  B ) ) )
 
Theoremdvmptim 19267* Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ( ph  /\  x  e.  X )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   =>    |-  ( ph  ->  ( RR  _D  ( x  e.  X  |->  ( Im
 `  A ) ) )  =  ( x  e.  X  |->  ( Im
 `  B ) ) )
 
Theoremdvmptntr 19268* Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  X 
 C_  S )   &    |-  (
 ( ph  /\  x  e.  X )  ->  A  e.  CC )   &    |-  J  =  ( Kt  S )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  ( ph  ->  ( ( int `  J ) `  X )  =  Y )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( S  _D  ( x  e.  Y  |->  A ) ) )
 
Theoremdvmptco 19269* Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  T  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  Y )   &    |-  ( ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ( ph  /\  y  e.  Y ) 
 ->  C  e.  CC )   &    |-  (
 ( ph  /\  y  e.  Y )  ->  D  e.  W )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ph  ->  ( T  _D  ( y  e.  Y  |->  C ) )  =  ( y  e.  Y  |->  D ) )   &    |-  ( y  =  A  ->  C  =  E )   &    |-  ( y  =  A  ->  D  =  F )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  E ) )  =  ( x  e.  X  |->  ( F  x.  B ) ) )
 
Theoremdvmptfsum 19270* Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)
 |-  J  =  ( Kt  S )   &    |-  K  =  (
 TopOpen ` fld )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X  e.  J )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ( ph  /\  i  e.  I  /\  x  e.  X )  ->  A  e.  CC )   &    |-  ( ( ph  /\  i  e.  I  /\  x  e.  X )  ->  B  e.  CC )   &    |-  (
 ( ph  /\  i  e.  I )  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  sum_
 i  e.  I  A ) )  =  ( x  e.  X  |->  sum_ i  e.  I  B )
 )
 
Theoremdvcnvlem 19271 Lemma for dvcnvre 19314. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  S )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  F : X -1-1-onto-> Y )   &    |-  ( ph  ->  `' F  e.  ( Y -cn-> X ) )   &    |-  ( ph  ->  dom  (  S  _D  F )  =  X )   &    |-  ( ph  ->  -.  0  e.  ran  (  S  _D  F ) )   &    |-  ( ph  ->  C  e.  X )   =>    |-  ( ph  ->  ( F `  C ) ( S  _D  `' F ) ( 1 
 /  ( ( S  _D  F ) `  C ) ) )
 
Theoremdvcnv 19272* A weak version of dvcnvre 19314, valid for both real and complex domains but under the hypothesis that the inverse function is already known to be continuous, and the image set is known to be open. A more advanced proof can show that these conditions are unnecessary. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  S )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  F : X -1-1-onto-> Y )   &    |-  ( ph  ->  `' F  e.  ( Y -cn-> X ) )   &    |-  ( ph  ->  dom  (  S  _D  F )  =  X )   &    |-  ( ph  ->  -.  0  e.  ran  (  S  _D  F ) )   =>    |-  ( ph  ->  ( S  _D  `' F )  =  ( x  e.  Y  |->  ( 1  /  ( ( S  _D  F ) `  ( `' F `  x ) ) ) ) )
 
Theoremdvexp3 19273* Derivative of an exponential of integer exponent. (Contributed by Mario Carneiro, 26-Feb-2015.)
 |-  ( N  e.  ZZ  ->  ( CC  _D  ( x  e.  ( CC  \  { 0 } )  |->  ( x ^ N ) ) )  =  ( x  e.  ( CC  \  { 0 } )  |->  ( N  x.  ( x ^ ( N  -  1 ) ) ) ) )
 
Theoremdveflem 19274 Derivative of the exponential function at 0. The key step in the proof is eftlub 12337, to show that  abs ( exp ( x )  - 
1  -  x )  <_  abs ( x ) ^ 2  x.  (
3  /  4 ). (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  0 ( CC  _D  exp ) 1
 
Theoremdvef 19275 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)
 |-  ( CC  _D  exp )  =  exp
 
Theoremdvsincos 19276 Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( ( CC  _D  sin )  =  cos  /\  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )
 
Theoremdvsin 19277 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( CC  _D  sin )  =  cos
 
Theoremdvcos 19278 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) )
 
Theoremdvferm1lem 19279* Lemma for dvferm 19283. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  F : X --> RR )   &    |-  ( ph  ->  X  C_  RR )   &    |-  ( ph  ->  U  e.  ( A (,) B ) )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  X )   &    |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )   &    |-  ( ph  ->  A. y  e.  ( U (,) B ) ( F `  y ) 
 <_  ( F `  U ) )   &    |-  ( ph  ->  0  <  ( ( RR 
 _D  F ) `  U ) )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  ( ph  ->  A. z  e.  ( X  \  { U } ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  T )  ->  ( abs `  ( ( ( ( F `  z )  -  ( F `  U ) )  /  ( z  -  U ) )  -  (
 ( RR  _D  F ) `  U ) ) )  <  ( ( RR  _D  F ) `
  U ) ) )   &    |-  S  =  ( ( U  +  if ( B  <_  ( U  +  T ) ,  B ,  ( U  +  T ) ) )  /  2 )   =>    |-  -.  ph
 
Theoremdvferm1 19280* One-sided version of dvferm 19283. A point  U which is the local maximum of its right neighborhood has derivative at most zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  F : X --> RR )   &    |-  ( ph  ->  X  C_  RR )   &    |-  ( ph  ->  U  e.  ( A (,) B ) )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  X )   &    |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )   &    |-  ( ph  ->  A. y  e.  ( U (,) B ) ( F `  y ) 
 <_  ( F `  U ) )   =>    |-  ( ph  ->  (
 ( RR  _D  F ) `  U )  <_ 
 0 )
 
Theoremdvferm2lem 19281* Lemma for dvferm 19283. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  F : X --> RR )   &    |-  ( ph  ->  X  C_  RR )   &    |-  ( ph  ->  U  e.  ( A (,) B ) )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  X )   &    |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )   &    |-  ( ph  ->  A. y  e.  ( A (,) U ) ( F `  y ) 
 <_  ( F `  U ) )   &    |-  ( ph  ->  ( ( RR  _D  F ) `  U )  < 
 0 )   &    |-  ( ph  ->  T  e.  RR+ )   &    |-  ( ph  ->  A. z  e.  ( X 
 \  { U }
 ) ( ( z  =/=  U  /\  ( abs `  ( z  -  U ) )  <  T )  ->  ( abs `  ( ( ( ( F `  z )  -  ( F `  U ) )  /  ( z  -  U ) )  -  (
 ( RR  _D  F ) `  U ) ) )  <  -u (
 ( RR  _D  F ) `  U ) ) )   &    |-  S  =  ( ( if ( A 
 <_  ( U  -  T ) ,  ( U  -  T ) ,  A )  +  U )  /  2 )   =>    |-  -.  ph
 
Theoremdvferm2 19282* One-sided version of dvferm 19283. A point  U which is the local maximum of its left neighborhood has derivative at least zero. (Contributed by Mario Carneiro, 24-Feb-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  F : X --> RR )   &    |-  ( ph  ->  X  C_  RR )   &    |-  ( ph  ->  U  e.  ( A (,) B ) )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  X )   &    |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )   &    |-  ( ph  ->  A. y  e.  ( A (,) U ) ( F `  y ) 
 <_  ( F `  U ) )   =>    |-  ( ph  ->  0  <_  ( ( RR  _D  F ) `  U ) )
 
Theoremdvferm 19283* Fermat's theorem on stationary points. A point  U which is a local maximum has derivative equal to zero. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ph  ->  F : X --> RR )   &    |-  ( ph  ->  X  C_  RR )   &    |-  ( ph  ->  U  e.  ( A (,) B ) )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  X )   &    |-  ( ph  ->  U  e.  dom  ( RR  _D  F ) )   &    |-  ( ph  ->  A. y  e.  ( A (,) B ) ( F `  y ) 
 <_  ( F `  U ) )   =>    |-  ( ph  ->  (
 ( RR  _D  F ) `  U )  =  0 )
 
Theoremrollelem 19284* Lemma for rolle 19285. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  A. y  e.  ( A [,] B ) ( F `  y ) 
 <_  ( F `  U ) )   &    |-  ( ph  ->  U  e.  ( A [,] B ) )   &    |-  ( ph  ->  -.  U  e.  { A ,  B } )   =>    |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( RR  _D  F ) `  x )  =  0 )
 
Theoremrolle 19285* Rolle's theorem. If  F is a real continuous function on  [ A ,  B ] which is differentiable on  ( A ,  B
), and  F ( A )  =  F ( B ), then there is some  x  e.  ( A ,  B ) such that  ( RR  _D  F ) `  x  =  0. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  ( F `  A )  =  ( F `  B ) )   =>    |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( RR  _D  F ) `  x )  =  0 )
 
Theoremcmvth 19286* Cauchy's Mean Value Theorem. If  F ,  G are real continuous functions on  [ A ,  B ] differentiable on  ( A ,  B ), then there is some  x  e.  ( A ,  B ) such that  F'  ( x )  /  G'  ( x )  =  ( F ( A )  -  F
( B ) )  /  ( G ( A )  -  G
( B ) ). (We express the condition without division, so that we need no nonzero constraints.) (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  G  e.  (
 ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  dom  ( RR  _D  G )  =  ( A (,) B ) )   =>    |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR 
 _D  G ) `  x ) )  =  ( ( ( G `
  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `
  x ) ) )
 
Theoremmvth 19287* The Mean Value Theorem. If  F is a real continuous function on  [ A ,  B ] which is differentiable on  ( A ,  B
), then there is some  x  e.  ( A ,  B ) such that  ( RR  _D  F
) `  x is equal to the average slope over  [ A ,  B ]. (Contributed by Mario Carneiro, 1-Sep-2014.) (Proof shortened by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   =>    |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( RR 
 _D  F ) `  x )  =  (
 ( ( F `  B )  -  ( F `  A ) ) 
 /  ( B  -  A ) ) )
 
Theoremdvlip 19288* A function with derivative bounded by  M is Lipschitz continuous with Lipchitz constant equal to 
M. (Contributed by Mario Carneiro, 3-Mar-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> CC ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  (
 ( ph  /\  x  e.  ( A (,) B ) )  ->  ( abs `  ( ( RR  _D  F ) `  x ) )  <_  M )   =>    |-  ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B ) ) )  ->  ( abs `  ( ( F `  X )  -  ( F `  Y ) ) )  <_  ( M  x.  ( abs `  ( X  -  Y ) ) ) )
 
Theoremdvlipcn 19289* A complex function with derivative bounded by  M on an open ball is Lipschitz continuous with Lipchitz constant equal to  M. (Contributed by Mario Carneiro, 18-Mar-2015.)
 |-  ( ph  ->  X  C_ 
 CC )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  R  e.  RR* )   &    |-  B  =  ( A ( ball `  ( abs  o.  -  ) ) R )   &    |-  ( ph  ->  B 
 C_  dom  ( CC  _D  F ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ( ph  /\  x  e.  B )  ->  ( abs `  ( ( CC 
 _D  F ) `  x ) )  <_  M )   =>    |-  ( ( ph  /\  ( Y  e.  B  /\  Z  e.  B )
 )  ->  ( abs `  ( ( F `  Y )  -  ( F `  Z ) ) )  <_  ( M  x.  ( abs `  ( Y  -  Z ) ) ) )
 
Theoremdvlip2 19290* Combine the results of dvlip 19288 and dvlipcn 19289 into one. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  J  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  R  e.  RR* )   &    |-  B  =  ( A ( ball `  J ) R )   &    |-  ( ph  ->  B 
 C_  dom  (  S  _D  F ) )   &    |-  ( ph  ->  M  e.  RR )   &    |-  ( ( ph  /\  x  e.  B )  ->  ( abs `  ( ( S  _D  F ) `  x ) )  <_  M )   =>    |-  ( ( ph  /\  ( Y  e.  B  /\  Z  e.  B )
 )  ->  ( abs `  ( ( F `  Y )  -  ( F `  Z ) ) )  <_  ( M  x.  ( abs `  ( Y  -  Z ) ) ) )
 
Theoremc1liplem1 19291* Lemma for c1lip1 19292. (Contributed by Stefan O'Rear, 15-Nov-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( CC  ^pm  RR ) )   &    |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  (
 ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )   &    |-  K  =  sup ( ( abs " (
 ( RR  _D  F ) " ( A [,] B ) ) ) ,  RR ,  <  )   =>    |-  ( ph  ->  ( K  e.  RR  /\  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  <_  ( K  x.  ( abs `  (
 y  -  x ) ) ) ) ) )
 
Theoremc1lip1 19292* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  ( CC  ^pm  RR ) )   &    |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B )
 -cn-> RR ) )   &    |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )   =>    |-  ( ph  ->  E. k  e.  RR  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  <_  (
 k  x.  ( abs `  ( y  -  x ) ) ) )
 
Theoremc1lip2 19293* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( C ^n `  RR ) `  1 ) )   &    |-  ( ph  ->  ran 
 F  C_  RR )   &    |-  ( ph  ->  ( A [,] B )  C_  dom  F )   =>    |-  ( ph  ->  E. k  e.  RR  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  <_  (
 k  x.  ( abs `  ( y  -  x ) ) ) )
 
Theoremc1lip3 19294* C1 functions are Lipschitz continuous on closed intervals. (Contributed by Stefan O'Rear, 16-Nov-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( F  |`  RR )  e.  ( ( C ^n
 `  RR ) `  1 ) )   &    |-  ( ph  ->  ( F " RR )  C_  RR )   &    |-  ( ph  ->  ( A [,] B )  C_  dom  F )   =>    |-  ( ph  ->  E. k  e.  RR  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  <_  (
 k  x.  ( abs `  ( y  -  x ) ) ) )
 
Theoremdveq0 19295 If a continuous function has zero derivative at all points on the interior of a closed interval, then it must be a constant function. (Contributed by Mario Carneiro, 2-Sep-2014.) (Proof shortened by Mario Carneiro, 3-Mar-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  F )  =  (
 ( A (,) B )  X.  { 0 } ) )   =>    |-  ( ph  ->  F  =  ( ( A [,] B )  X.  { ( F `  A ) }
 ) )
 
Theoremdv11cn 19296 Two functions defined on a ball whose derivatives are the same and which are equal at any given point 
C in the ball must be equal everywhere. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  X  =  ( A ( ball `  ( abs  o. 
 -  ) ) R )   &    |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  R  e.  RR* )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  dom  ( CC  _D  F )  =  X )   &    |-  ( ph  ->  ( CC  _D  F )  =  ( CC  _D  G ) )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  ( F `  C )  =  ( G `  C ) )   =>    |-  ( ph  ->  F  =  G )
 
Theoremdvgt0lem1 19297 Lemma for dvgt0 19299 and dvlt0 19300. (Contributed by Mario Carneiro, 19-Feb-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  ( RR  _D  F ) : ( A (,) B ) --> S )   =>    |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B ) ) ) 
 /\  X  <  Y )  ->  ( ( ( F `  Y )  -  ( F `  X ) )  /  ( Y  -  X ) )  e.  S )
 
Theoremdvgt0lem2 19298* Lemma for dvgt0 19299 and dvlt0 19300. (Contributed by Mario Carneiro, 19-Feb-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  ( RR  _D  F ) : ( A (,) B ) --> S )   &    |-  O  Or  RR   &    |-  (
 ( ( ph  /\  ( x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y )  ->  ( F `  x ) O ( F `  y ) )   =>    |-  ( ph  ->  F 
 Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) )
 
Theoremdvgt0 19299 A function on a closed interval with positive derivative is increasing. (Contributed by Mario Carneiro, 19-Feb-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  ( RR  _D  F ) : ( A (,) B ) -->
 RR+ )   =>    |-  ( ph  ->  F  Isom  <  ,  <  (
 ( A [,] B ) ,  ran  F ) )
 
Theoremdvlt0 19300 A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  ( RR  _D  F ) : ( A (,) B ) --> (  -oo (,) 0
 ) )   =>    |-  ( ph  ->  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) )
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