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Theorem List for Intuitionistic Logic Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremivthinclemuopn 14201* Lemma for ivthinc 14206. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   &    |-  ( ph  ->  S  e.  R )   =>    |-  ( ph  ->  E. q  e.  R  q  <  S )
 
Theoremivthinclemur 14202* Lemma for ivthinc 14206. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  R  <->  E. q  e.  R  q  <  r ) )
 
Theoremivthinclemdisj 14203* Lemma for ivthinc 14206. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  ( L  i^i  R )  =  (/) )
 
Theoremivthinclemloc 14204* Lemma for ivthinc 14206. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  (
 q  e.  L  \/  r  e.  R )
 ) )
 
Theoremivthinclemex 14205* Lemma for ivthinc 14206. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  E! z  e.  ( A (,) B ) ( A. q  e.  L  q  <  z  /\  A. r  e.  R  z  <  r ) )
 
Theoremivthinc 14206* The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   =>    |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
 
Theoremivthdec 14207* The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  B )  <  U  /\  U  <  ( F `  A ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  y )  <  ( F `  x ) )   =>    |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
 
9.1  Derivatives
 
9.1.1  Real and complex differentiation
 
9.1.1.1  Derivatives of functions of one complex or real variable
 
Syntaxclimc 14208 The limit operator.
 class lim CC
 
Syntaxcdv 14209 The derivative operator.
 class  _D
 
Definitiondf-limced 14210* 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.) (Revised by Jim Kingdon, 3-Jun-2023.)
 |- lim
 CC  =  ( f  e.  ( CC  ^pm  CC ) ,  x  e. 
 CC  |->  { y  e.  CC  |  ( ( f : dom  f --> CC  /\  dom  f  C_  CC )  /\  ( x  e.  CC  /\ 
 A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  f ( ( z #  x  /\  ( abs `  ( z  -  x ) )  < 
 d )  ->  ( abs `  ( ( f `
  z )  -  y ) )  < 
 e ) ) ) } )
 
Definitiondf-dvap 14211* Define the derivative operator. 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.) (Revised by Jim Kingdon, 25-Jun-2023.)
 |- 
 _D  =  ( s  e.  ~P CC ,  f  e.  ( CC  ^pm  s )  |->  U_ x  e.  ( ( int `  (
 ( MetOpen `  ( abs  o. 
 -  ) ) ↾t  s ) ) `  dom  f
 ) ( { x }  X.  ( ( z  e.  { w  e. 
 dom  f  |  w #  x }  |->  ( ( ( f `  z
 )  -  ( f `
  x ) ) 
 /  ( z  -  x ) ) ) lim
 CC  x ) ) )
 
Theoremlimcrcl 14212 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 14213 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( F lim CC  B )  C_  CC
 
Theoremellimc3apf 14214* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  F/_ z F   =>    |-  ( 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 ) ) ) )
 
Theoremellimc3ap 14215* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.)
 |-  ( 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 ) ) ) )
 
Theoremlimcdifap 14216* 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.) (Revised by Jim Kingdon, 3-Jun-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   =>    |-  ( ph  ->  ( F lim CC  B )  =  ( ( F  |`  { x  e.  A  |  x #  B } ) lim CC  B ) )
 
Theoremlimcmpted 14217* Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
 |-  ( ph  ->  A  C_ 
 CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  (
 ( ph  /\  z  e.  A )  ->  D  e.  CC )   =>    |-  ( ph  ->  ( C  e.  ( (
 z  e.  A  |->  D ) 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 `  ( D  -  C ) )  <  x ) ) ) )
 
Theoremlimcimolemlt 14218* Lemma for limcimo 14219. (Contributed by Jim Kingdon, 3-Jul-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  ( K ↾t  S ) )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  { q  e.  C  |  q #  B }  C_  A )   &    |-  K  =  ( MetOpen `  ( abs  o. 
 -  ) )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  X  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  Y  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
 z  -  B ) )  <  D ) 
 ->  ( abs `  (
 ( F `  z
 )  -  X ) )  <  ( ( abs `  ( X  -  Y ) )  / 
 2 ) ) )   &    |-  ( ph  ->  G  e.  RR+ )   &    |-  ( ph  ->  A. w  e.  A  ( ( w #  B  /\  ( abs `  ( w  -  B ) )  <  G )  ->  ( abs `  ( ( F `  w )  -  Y ) )  <  ( ( abs `  ( X  -  Y ) )  / 
 2 ) ) )   =>    |-  ( ph  ->  ( abs `  ( X  -  Y ) )  <  ( abs `  ( X  -  Y ) ) )
 
Theoremlimcimo 14219* Conditions which ensure there is at most one limit value of  F at  B. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 8-Jul-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  ( K ↾t  S ) )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  { q  e.  C  |  q #  B }  C_  A )   &    |-  K  =  ( MetOpen `  ( abs  o. 
 -  ) )   =>    |-  ( ph  ->  E* x  x  e.  ( F lim CC  B ) )
 
Theoremlimcresi 14220 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 )
 
Theoremcnplimcim 14221 If a function is continuous at  B, its limit at  B equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( K ↾t  A )   =>    |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B ) 
 ->  ( F : A --> CC  /\  ( F `  B )  e.  ( F lim CC  B ) ) ) )
 
Theoremcnplimclemle 14222 Lemma for cnplimccntop 14224. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( K ↾t  A )   &    |-  ( ph  ->  A 
 C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  ( F `  B )  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  Z  e.  A )   &    |-  (
 ( ph  /\  Z #  B  /\  ( abs `  ( Z  -  B ) )  <  D )  ->  ( abs `  ( ( F `  Z )  -  ( F `  B ) ) )  <  ( E  /  2 ) )   &    |-  ( ph  ->  ( abs `  ( Z  -  B ) )  <  D )   =>    |-  ( ph  ->  ( abs `  ( ( F `  Z )  -  ( F `  B ) ) )  <  E )
 
Theoremcnplimclemr 14223 Lemma for cnplimccntop 14224. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( K ↾t  A )   &    |-  ( ph  ->  A 
 C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  ( F `  B )  e.  ( F lim CC  B ) )   =>    |-  ( ph  ->  F  e.  ( ( J  CnP  K ) `  B ) )
 
Theoremcnplimccntop 14224 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  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( K ↾t  A )   =>    |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B )  <-> 
 ( F : A --> CC  /\  ( F `  B )  e.  ( F lim CC  B ) ) ) )
 
Theoremcnlimcim 14225* If  F is a continuous function, the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.)
 |-  ( A  C_  CC  ->  ( F  e.  ( A -cn-> CC )  ->  ( F : A --> CC  /\  A. x  e.  A  ( F `  x )  e.  ( F lim CC  x ) ) ) )
 
Theoremcnlimc 14226*  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 14227 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 14228* 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 ) )
 
Theoremlimccnpcntop 14229 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.) (Revised by Jim Kingdon, 18-Jun-2023.)
 |-  ( ph  ->  F : A --> D )   &    |-  ( ph  ->  D  C_  CC )   &    |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( K ↾t  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 ) )
 
Theoremlimccnp2lem 14230* Lemma for limccnp2cntop 14231. This is most of the result, expressed in epsilon-delta form, with a large number of hypotheses so that lengthy expressions do not need to be repeated. (Contributed by Jim Kingdon, 9-Nov-2023.)
 |-  ( ( ph  /\  x  e.  A )  ->  R  e.  X )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  S  e.  Y )   &    |-  ( ph  ->  X  C_  CC )   &    |-  ( ph  ->  Y  C_ 
 CC )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   &    |-  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 >. ) )   &    |-  F/ x ph   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  L  e.  RR+ )   &    |-  ( ph  ->  A. r  e.  X  A. s  e.  Y  (
 ( ( C ( ( abs  o.  -  )  |`  ( X  X.  X ) ) r )  <  L  /\  ( D ( ( abs 
 o.  -  )  |`  ( Y  X.  Y ) ) s )  <  L )  ->  ( ( C H D ) ( abs  o.  -  )
 ( r H s ) )  <  E ) )   &    |-  ( ph  ->  F  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  A  ( ( x #  B  /\  ( abs `  ( x  -  B ) )  <  F )  ->  ( abs `  ( R  -  C ) )  <  L ) )   &    |-  ( ph  ->  G  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  A  ( ( x #  B  /\  ( abs `  ( x  -  B ) )  <  G )  ->  ( abs `  ( S  -  D ) )  <  L ) )   =>    |-  ( ph  ->  E. d  e.  RR+  A. x  e.  A  ( ( x #  B  /\  ( abs `  ( x  -  B ) )  <  d )  ->  ( abs `  ( ( R H S )  -  ( C H D ) ) )  <  E ) )
 
Theoremlimccnp2cntop 14231* 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.) (Revised by Jim Kingdon, 14-Nov-2023.)
 |-  ( ( ph  /\  x  e.  A )  ->  R  e.  X )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  S  e.  Y )   &    |-  ( ph  ->  X  C_  CC )   &    |-  ( ph  ->  Y  C_ 
 CC )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   &    |-  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 ) )
 
Theoremlimccoap 14232* Composition of two limits. This theorem is only usable in the case where  x #  X implies R(x) #  C so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
 |-  ( ( ph  /\  x  e.  { w  e.  A  |  w #  X }
 )  ->  R  e.  { w  e.  B  |  w #  C } )   &    |-  (
 ( ph  /\  y  e. 
 { w  e.  B  |  w #  C }
 )  ->  S  e.  CC )   &    |-  ( ph  ->  C  e.  ( ( x  e.  { w  e.  A  |  w #  X }  |->  R ) lim CC  X ) )   &    |-  ( ph  ->  D  e.  (
 ( y  e.  { w  e.  B  |  w #  C }  |->  S ) lim
 CC  C ) )   &    |-  ( y  =  R  ->  S  =  T )   =>    |-  ( ph  ->  D  e.  ( ( x  e. 
 { w  e.  A  |  w #  X }  |->  T ) lim CC  X ) )
 
Theoremreldvg 14233 The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
 |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S ) ) 
 ->  Rel  ( S  _D  F ) )
 
Theoremdvlemap 14234* Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
 |-  ( ph  ->  F : D --> CC )   &    |-  ( ph  ->  D  C_  CC )   &    |-  ( ph  ->  B  e.  D )   =>    |-  ( ( ph  /\  A  e.  { w  e.  D  |  w #  B }
 )  ->  ( (
 ( F `  A )  -  ( F `  B ) )  /  ( A  -  B ) )  e.  CC )
 
Theoremdvfvalap 14235* Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
 |-  T  =  ( K ↾t  S )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  ( ( S  _D  F )  = 
 U_ x  e.  (
 ( int `  T ) `  A ) ( { x }  X.  (
 ( z  e.  { w  e.  A  |  w #  x }  |->  ( ( ( F `  z
 )  -  ( F `
  x ) ) 
 /  ( z  -  x ) ) ) lim
 CC  x ) ) 
 /\  ( S  _D  F )  C_  ( ( ( int `  T ) `  A )  X.  CC ) ) )
 
Theoremeldvap 14236* 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 Jim Kingdon, 27-Jun-2023.)
 |-  T  =  ( K ↾t  S )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   &    |-  G  =  ( z  e.  { w  e.  A  |  w #  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 14237 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 )
 
Theoremdvbssntrcntop 14238 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 Jim Kingdon, 27-Jun-2023.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  J  =  ( K ↾t  S )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ph  ->  dom  ( S  _D  F )  C_  ( ( int `  J ) `  A ) )
 
Theoremdvbss 14239 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 )
 
Theoremdvbsssg 14240 The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
 |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S ) ) 
 ->  dom  ( S  _D  F )  C_  S )
 
Theoremrecnprss 14241 Both  RR and  CC are subsets of  CC. (Contributed by Mario Carneiro, 10-Feb-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  S 
 C_  CC )
 
Theoremdvfgg 14242 Explicitly write out the functionality condition on derivative for  S  =  RR and 
CC. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
 |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC 
 ^pm  S ) )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
 
Theoremdvfpm 14243 The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.)
 |-  ( F  e.  ( CC  ^pm  RR )  ->  ( RR  _D  F ) : dom  ( RR 
 _D  F ) --> CC )
 
Theoremdvfcnpm 14244 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jul-2023.)
 |-  ( F  e.  ( CC  ^pm  CC )  ->  ( CC  _D  F ) : dom  ( CC 
 _D  F ) --> CC )
 
Theoremdvidlemap 14245* Lemma for dvid 14247 and dvconst 14246. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
 |-  ( 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 14246 Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
 |-  ( A  e.  CC  ->  ( CC  _D  ( CC  X.  { A }
 ) )  =  ( CC  X.  { 0 } ) )
 
Theoremdvid 14247 Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
 |-  ( CC  _D  (  _I  |`  CC ) )  =  ( CC  X.  { 1 } )
 
Theoremdvcnp2cntop 14248 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  =  ( K ↾t  A )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ( ( S 
 C_  CC  /\  F : A
 --> CC  /\  A  C_  S )  /\  B  e.  dom  ( S  _D  F ) )  ->  F  e.  ( ( J  CnP  K ) `  B ) )
 
Theoremdvcn 14249 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 )
 )
 
Theoremdvaddxxbr 14250 The sum rule for derivatives at a point. That is, if the derivative of  F at  C is  K and the derivative of  G at  C is  L, then the derivative of the pointwise sum of those two functions at  C is  K  +  L. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  C ( S  _D  F ) K )   &    |-  ( ph  ->  C ( S  _D  G ) L )   &    |-  J  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ph  ->  C ( S  _D  ( F  oF  +  G ) ) ( K  +  L ) )
 
Theoremdvmulxxbr 14251 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 14253. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  C ( S  _D  F ) K )   &    |-  ( ph  ->  C ( S  _D  G ) L )   &    |-  J  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ph  ->  C ( S  _D  ( F  oF  x.  G ) ) ( ( K  x.  ( G `
  C ) )  +  ( L  x.  ( F `  C ) ) ) )
 
Theoremdvaddxx 14252 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 14250. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( 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 ) ) )
 
Theoremdvmulxx 14253 The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 14251. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( 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 ) ) ) )
 
Theoremdviaddf 14254 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  ->  X 
 C_  S )   &    |-  ( 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 ) ) )
 
Theoremdvimulf 14255 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  ->  X 
 C_  S )   &    |-  ( 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 ) ) )
 
Theoremdvcoapbr 14256* The chain rule for derivatives at a point. The  u #  C  -> 
( G `  u
) #  ( G `  C ) hypothesis constrains what functions work for  G. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : Y --> X )   &    |-  ( ph  ->  Y  C_  T )   &    |-  ( ph  ->  A. u  e.  Y  ( u #  C  ->  ( G `  u ) #  ( G `  C ) ) )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  T  C_ 
 CC )   &    |-  ( ph  ->  ( G `  C ) ( S  _D  F ) K )   &    |-  ( ph  ->  C ( T  _D  G ) L )   &    |-  J  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ph  ->  C ( T  _D  ( F  o.  G ) ) ( K  x.  L ) )
 
Theoremdvcjbr 14257 The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 14258. (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 14258 The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 14257. (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 14259 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 )
 
Theoremdvexp 14260* 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 14261* 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
 ) ) ) ) ) )
 
Theoremdvrecap 14262* 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.  { w  e.  CC  |  w #  0 }  |->  ( A  /  x ) ) )  =  ( x  e. 
 { w  e.  CC  |  w #  0 }  |->  -u ( A  /  ( x ^ 2 ) ) ) )
 
Theoremdvmptidcn 14263 Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
 |-  ( CC  _D  ( x  e.  CC  |->  x ) )  =  ( x  e.  CC  |->  1 )
 
Theoremdvmptccn 14264* Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( CC  _D  ( x  e. 
 CC  |->  A ) )  =  ( x  e. 
 CC  |->  0 ) )
 
Theoremdvmptclx 14265* Closure lemma for dvmptmulx 14267 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 
 C_  S )   =>    |-  ( ( ph  /\  x  e.  X ) 
 ->  B  e.  CC )
 
Theoremdvmptaddx 14266* 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 
 C_  S )   &    |-  (
 ( 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 ) ) )
 
Theoremdvmptmulx 14267* 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 
 C_  S )   &    |-  (
 ( 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 ) ) ) )
 
Theoremdvmptcmulcn 14268* Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
 |-  ( ( ph  /\  x  e.  CC )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  CC )  ->  B  e.  V )   &    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  A ) )  =  ( x  e.  CC  |->  B ) )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( CC  _D  ( x  e. 
 CC  |->  ( C  x.  A ) ) )  =  ( x  e. 
 CC  |->  ( C  x.  B ) ) )
 
Theoremdvmptnegcn 14269* Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
 |-  ( ( ph  /\  x  e.  CC )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  CC )  ->  B  e.  V )   &    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  A ) )  =  ( x  e.  CC  |->  B ) )   =>    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  -u A ) )  =  ( x  e.  CC  |->  -u B ) )
 
Theoremdvmptsubcn 14270* Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
 |-  ( ( ph  /\  x  e.  CC )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  CC )  ->  B  e.  V )   &    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  A ) )  =  ( x  e.  CC  |->  B ) )   &    |-  (
 ( ph  /\  x  e. 
 CC )  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  CC )  ->  D  e.  W )   &    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  C ) )  =  ( x  e.  CC  |->  D ) )   =>    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  ( A  -  C ) ) )  =  ( x  e.  CC  |->  ( B  -  D ) ) )
 
Theoremdvmptcjx 14271* Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.)
 |-  ( ( 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  ->  X  C_  RR )   =>    |-  ( ph  ->  ( RR  _D  ( x  e.  X  |->  ( * `  A ) ) )  =  ( x  e.  X  |->  ( * `  B ) ) )
 
Theoremdveflem 14272 Derivative of the exponential function at 0. The key step in the proof is eftlub 11700, 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 14273 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)
 |-  ( CC  _D  exp )  =  exp
 
PART 10  BASIC REAL AND COMPLEX FUNCTIONS
 
10.1  Basic trigonometry
 
10.1.1  The exponential, sine, and cosine functions (cont.)
 
Theoremefcn 14274 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |- 
 exp  e.  ( CC -cn-> CC )
 
Theoremsincn 14275 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
 |- 
 sin  e.  ( CC -cn-> CC )
 
Theoremcoscn 14276 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
 |- 
 cos  e.  ( CC -cn-> CC )
 
Theoremreeff1olem 14277* Lemma for reeff1o 14279. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( ( U  e.  RR  /\  1  <  U )  ->  E. x  e.  RR  ( exp `  x )  =  U )
 
Theoremreeff1oleme 14278* Lemma for reeff1o 14279. (Contributed by Jim Kingdon, 15-May-2024.)
 |-  ( U  e.  (
 0 (,) _e )  ->  E. x  e.  RR  ( exp `  x )  =  U )
 
Theoremreeff1o 14279 The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( exp  |`  RR ) : RR
 -1-1-onto-> RR+
 
Theoremefltlemlt 14280 Lemma for eflt 14281. The converse of efltim 11708 plus the epsilon-delta setup. (Contributed by Jim Kingdon, 22-May-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( exp `  A )  <  ( exp `  B ) )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  ( ( abs `  ( A  -  B ) )  <  D  ->  ( abs `  ( ( exp `  A )  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) )   =>    |-  ( ph  ->  A  <  B )
 
Theoremeflt 14281 The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( exp `  A )  <  ( exp `  B ) ) )
 
Theoremefle 14282 The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <-> 
 ( exp `  A )  <_  ( exp `  B ) ) )
 
Theoremreefiso 14283 The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) (Revised by Mario Carneiro, 11-Mar-2014.)
 |-  ( exp  |`  RR )  Isom  <  ,  <  ( RR ,  RR+ )
 
Theoremreapef 14284 Apartness and the exponential function for reals. (Contributed by Jim Kingdon, 11-Jul-2024.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( exp `  A ) #  ( exp `  B )
 ) )
 
10.1.2  Properties of pi = 3.14159...
 
Theorempilem1 14285 Lemma for pire , pigt2lt4 and sinpi . (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( A  e.  ( RR+ 
 i^i  ( `' sin " { 0 } )
 ) 
 <->  ( A  e.  RR+  /\  ( sin `  A )  =  0 )
 )
 
Theoremcosz12 14286 Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.)
 |- 
 E. p  e.  (
 1 (,) 2 ) ( cos `  p )  =  0
 
Theoremsin0pilem1 14287* Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
 |- 
 E. p  e.  (
 1 (,) 2 ) ( ( cos `  p )  =  0  /\  A. x  e.  ( p (,) ( 2  x.  p ) ) 0  <  ( sin `  x ) )
 
Theoremsin0pilem2 14288* Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
 |- 
 E. q  e.  (
 2 (,) 4 ) ( ( sin `  q
 )  =  0  /\  A. x  e.  ( 0 (,) q ) 0  <  ( sin `  x ) )
 
Theorempilem3 14289 Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
 |-  ( pi  e.  (
 2 (,) 4 )  /\  ( sin `  pi )  =  0 )
 
Theorempigt2lt4 14290  pi is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  ( 2  <  pi  /\  pi  <  4 )
 
Theoremsinpi 14291 The sine of  pi is 0. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( sin `  pi )  =  0
 
Theorempire 14292  pi is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  pi  e.  RR
 
Theorempicn 14293  pi is a complex number. (Contributed by David A. Wheeler, 6-Dec-2018.)
 |-  pi  e.  CC
 
Theorempipos 14294  pi is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)
 |-  0  <  pi
 
Theorempirp 14295  pi is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  pi  e.  RR+
 
Theoremnegpicn 14296  -u pi is a real number. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u pi  e.  CC
 
Theoremsinhalfpilem 14297 Lemma for sinhalfpi 14302 and coshalfpi 14303. (Contributed by Paul Chapman, 23-Jan-2008.)
 |-  ( ( sin `  ( pi  /  2 ) )  =  1  /\  ( cos `  ( pi  / 
 2 ) )  =  0 )
 
Theoremhalfpire 14298  pi  /  2 is real. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( pi  /  2
 )  e.  RR
 
Theoremneghalfpire 14299  -u pi  / 
2 is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u ( pi  /  2
 )  e.  RR
 
Theoremneghalfpirx 14300  -u pi  / 
2 is an extended real. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  -u ( pi  /  2
 )  e.  RR*
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