HomeHome Metamath Proof Explorer
Theorem List (p. 192 of 309)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30843)
 

Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvcnp2 19101 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 19102 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 19103* 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 19104 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 19105 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 19106 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 19107 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 19108 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 19109 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 19110 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 19111 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 19112 Multiple derivative version of dvres3a 19096. (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 19113* 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 19114 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 19115 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 19116  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 19117 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 19118 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 19119 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 19120 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 19121 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 19122 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 19123 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 19124 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 19125 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 19126 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 19127 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 19128 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 19129 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 19130 The derivative of the conjugate of a function. (This doesn't follow from dvcobr 19127 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 19131 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 19132 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 19133 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 19134* 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 19135* 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 19136* 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 19137* 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 19138* 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 19139* 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 19140* Closure lemma for dvmptcmul 19145 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 19141* 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 19142* 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 19143* 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 19144* 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 19145* 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 19146* 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 19147* 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 19148* 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 19149* 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 19150* 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 19151* 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 19152* 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 19153* 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 19154* 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 19155 Lemma for dvcnvre 19198. (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 19156* A weak version of dvcnvre 19198, 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 19157* 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 19158 Derivative of the exponential function at 0. The key step in the proof is eftlub 12263, 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 19159 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 19160 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 19161 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( CC  _D  sin )  =  cos
 
Theoremdvcos 19162 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) )
 
Theoremdvferm1lem 19163* Lemma for dvferm 19167. (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 19164* One-sided version of dvferm 19167. 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 19165* Lemma for dvferm 19167. (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 19166* One-sided version of dvferm 19167. 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 19167* 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 19168* Lemma for rolle 19169. (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 19169* 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 19170* 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 19171* 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 19172* 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 19173* 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 19174* Combine the results of dvlip 19172 and dvlipcn 19173 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 19175* Lemma for c1lip1 19176. (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 19176* 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 19177* 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 19178* 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 19179 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 19180 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 19181 Lemma for dvgt0 19183 and dvlt0 19184. (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 19182* Lemma for dvgt0 19183 and dvlt0 19184. (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 19183 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 19184 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 ) )
 
Theoremdvge0 19185 A function on a closed interval with nonnegative derivative is weakly increasing. (Contributed by Mario Carneiro, 30-Apr-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  ( RR  _D  F ) : ( A (,) B ) --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  X  e.  ( A [,] B ) )   &    |-  ( ph  ->  Y  e.  ( A [,] B ) )   &    |-  ( ph  ->  X 
 <_  Y )   =>    |-  ( ph  ->  ( F `  X )  <_  ( F `  Y ) )
 
Theoremdvle 19186* If  A (
x ) ,  C
( x ) are differentiable functions and  A `  <_  C `
, then for  x  <_  y,  A ( y )  -  A ( x )  <_  C
( y )  -  C ( x ). (Contributed by Mario Carneiro, 16-May-2016.)
 |-  ( ph  ->  M  e.  RR )   &    |-  ( ph  ->  N  e.  RR )   &    |-  ( ph  ->  ( x  e.  ( M [,] N )  |->  A )  e.  ( ( M [,] N ) -cn-> RR ) )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  ( x  e.  ( M (,) N )  |->  B ) )   &    |-  ( ph  ->  ( x  e.  ( M [,] N )  |->  C )  e.  ( ( M [,] N )
 -cn-> RR ) )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( M (,) N )  |->  C ) )  =  ( x  e.  ( M (,) N )  |->  D ) )   &    |-  ( ( ph  /\  x  e.  ( M (,) N ) ) 
 ->  B  <_  D )   &    |-  ( ph  ->  X  e.  ( M [,] N ) )   &    |-  ( ph  ->  Y  e.  ( M [,] N ) )   &    |-  ( ph  ->  X 
 <_  Y )   &    |-  ( x  =  X  ->  A  =  P )   &    |-  ( x  =  X  ->  C  =  Q )   &    |-  ( x  =  Y  ->  A  =  R )   &    |-  ( x  =  Y  ->  C  =  S )   =>    |-  ( ph  ->  ( R  -  P )  <_  ( S  -  Q ) )
 
Theoremdvivthlem1 19187* Lemma for dvivth 19189. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  M  e.  ( A (,) B ) )   &    |-  ( ph  ->  N  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  M  <  N )   &    |-  ( ph  ->  C  e.  ( ( ( RR 
 _D  F ) `  N ) [,] (
 ( RR  _D  F ) `  M ) ) )   &    |-  G  =  ( y  e.  ( A (,) B )  |->  ( ( F `  y
 )  -  ( C  x.  y ) ) )   =>    |-  ( ph  ->  E. x  e.  ( M [,] N ) ( ( RR 
 _D  F ) `  x )  =  C )
 
Theoremdvivthlem2 19188* Lemma for dvivth 19189. (Contributed by Mario Carneiro, 20-Feb-2015.)
 |-  ( ph  ->  M  e.  ( A (,) B ) )   &    |-  ( ph  ->  N  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  M  <  N )   &    |-  ( ph  ->  C  e.  ( ( ( RR 
 _D  F ) `  N ) [,] (
 ( RR  _D  F ) `  M ) ) )   &    |-  G  =  ( y  e.  ( A (,) B )  |->  ( ( F `  y
 )  -  ( C  x.  y ) ) )   =>    |-  ( ph  ->  C  e.  ran  ( RR  _D  F ) )
 
Theoremdvivth 19189 Darboux' theorem, or the intermediate value theorem for derivatives. A differentiable function's derivative satisfies the intermediate value property, even though it may not be continuous (so that ivthicc 18650 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  M  e.  ( A (,) B ) )   &    |-  ( ph  ->  N  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   =>    |-  ( ph  ->  ( ( ( RR  _D  F ) `  M ) [,] ( ( RR 
 _D  F ) `  N ) )  C_  ran  ( RR  _D  F ) )
 
Theoremdvne0 19190 A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F ) )   =>    |-  ( ph  ->  ( F  Isom  <  ,  <  (
 ( A [,] B ) ,  ran  F )  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) , 
 ran  F ) ) )
 
Theoremdvne0f1 19191 A function on a closed interval with nonzero derivative is one-to-one. (Contributed by Mario Carneiro, 19-Feb-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F ) )   =>    |-  ( ph  ->  F :
 ( A [,] B ) -1-1-> RR )
 
Theoremlhop1lem 19192* Lemma for lhop1 19193. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  G : ( A (,) B ) --> RR )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  dom  ( RR  _D  G )  =  ( A (,) B ) )   &    |-  ( ph  ->  0  e.  ( F lim CC  A ) )   &    |-  ( ph  ->  0  e.  ( G lim CC  A ) )   &    |-  ( ph  ->  -.  0  e.  ran  G )   &    |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  G ) )   &    |-  ( ph  ->  C  e.  (
 ( z  e.  ( A (,) B )  |->  ( ( ( RR  _D  F ) `  z
 )  /  ( ( RR  _D  G ) `  z ) ) ) lim
 CC  A ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  D  <_  B )   &    |-  ( ph  ->  X  e.  ( A (,) D ) )   &    |-  ( ph  ->  A. t  e.  ( A (,) D ) ( abs `  ( (
 ( ( RR  _D  F ) `  t
 )  /  ( ( RR  _D  G ) `  t ) )  -  C ) )  <  E )   &    |-  R  =  ( A  +  ( r 
 /  2 ) )   =>    |-  ( ph  ->  ( abs `  ( ( ( F `
  X )  /  ( G `  X ) )  -  C ) )  <  ( 2  x.  E ) )
 
Theoremlhop1 19193* L'Hôpital's Rule for limits from the right. If  F and  G are differentiable real functions on  ( A ,  B ), and 
F and  G both approach 0 at  A, and  G ( x ) and  G'  ( x ) are not zero on  ( A ,  B ), and the limit of  F'  ( x )  /  G'  ( x ) at  A is  C, then the limit  F ( x )  /  G ( x ) at  A also exists and equals  C. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  G : ( A (,) B ) --> RR )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  dom  ( RR  _D  G )  =  ( A (,) B ) )   &    |-  ( ph  ->  0  e.  ( F lim CC  A ) )   &    |-  ( ph  ->  0  e.  ( G lim CC  A ) )   &    |-  ( ph  ->  -.  0  e.  ran  G )   &    |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  G ) )   &    |-  ( ph  ->  C  e.  (
 ( z  e.  ( A (,) B )  |->  ( ( ( RR  _D  F ) `  z
 )  /  ( ( RR  _D  G ) `  z ) ) ) lim
 CC  A ) )   =>    |-  ( ph  ->  C  e.  ( ( z  e.  ( A (,) B )  |->  ( ( F `
  z )  /  ( G `  z ) ) ) lim CC  A ) )
 
Theoremlhop2 19194* L'Hôpital's Rule for limits from the right. If  F and  G are differentiable real functions on  ( A ,  B ), and 
F and  G both approach 0 at  B, and  G ( x ) and  G'  ( x ) are not zero on  ( A ,  B ), and the limit of  F'  ( x )  /  G'  ( x ) at  B is  C, then the limit  F ( x )  /  G ( x ) at  B also exists and equals  C. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F : ( A (,) B ) --> RR )   &    |-  ( ph  ->  G : ( A (,) B ) --> RR )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )   &    |-  ( ph  ->  dom  ( RR  _D  G )  =  ( A (,) B ) )   &    |-  ( ph  ->  0  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  0  e.  ( G lim CC  B ) )   &    |-  ( ph  ->  -.  0  e.  ran  G )   &    |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  G ) )   &    |-  ( ph  ->  C  e.  ( ( z  e.  ( A (,) B )  |->  ( ( ( RR  _D  F ) `
  z )  /  ( ( RR  _D  G ) `  z
 ) ) ) lim CC  B ) )   =>    |-  ( ph  ->  C  e.  ( ( z  e.  ( A (,) B )  |->  ( ( F `
  z )  /  ( G `  z ) ) ) lim CC  B ) )
 
Theoremlhop 19195* L'Hôpital's Rule. If  I is an open set of the reals,  F and  G are real functions on  A containing all of  I except possibly  B, which are differentiable everywhere on  I  \  { B },  F and  G both approach 0, and the limit of  F'  ( x )  /  G'  ( x ) at  B is  C, then the limit  F ( x )  /  G ( x ) at  B also exists and equals  C. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  F : A --> RR )   &    |-  ( ph  ->  G : A --> RR )   &    |-  ( ph  ->  I  e.  ( topGen `  ran  (,) ) )   &    |-  ( ph  ->  B  e.  I )   &    |-  D  =  ( I  \  { B } )   &    |-  ( ph  ->  D 
 C_  dom  ( RR  _D  F ) )   &    |-  ( ph  ->  D  C_  dom  ( RR  _D  G ) )   &    |-  ( ph  ->  0  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  0  e.  ( G lim CC  B ) )   &    |-  ( ph  ->  -.  0  e.  ( G " D ) )   &    |-  ( ph  ->  -.  0  e.  ( ( RR  _D  G )
 " D ) )   &    |-  ( ph  ->  C  e.  ( ( z  e.  D  |->  ( ( ( RR  _D  F ) `
  z )  /  ( ( RR  _D  G ) `  z
 ) ) ) lim CC  B ) )   =>    |-  ( ph  ->  C  e.  ( ( z  e.  D  |->  ( ( F `  z ) 
 /  ( G `  z ) ) ) lim
 CC  B ) )
 
Theoremdvcnvrelem1 19196 Lemma for dvcnvre 19198. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  F  e.  ( X -cn-> RR )
 )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  X )   &    |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F ) )   &    |-  ( ph  ->  F : X
 -1-1-onto-> Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  (
 ( C  -  R ) [,] ( C  +  R ) )  C_  X )   =>    |-  ( ph  ->  ( F `  C )  e.  ( ( int `  ( topGen `
  ran  (,) ) ) `
  ( F "
 ( ( C  -  R ) [,] ( C  +  R )
 ) ) ) )
 
Theoremdvcnvrelem2 19197 Lemma for dvcnvre 19198. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  ( ph  ->  F  e.  ( X -cn-> RR )
 )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  X )   &    |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F ) )   &    |-  ( ph  ->  F : X
 -1-1-onto-> Y )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  (
 ( C  -  R ) [,] ( C  +  R ) )  C_  X )   &    |-  T  =  (
 topGen `  ran  (,) )   &    |-  J  =  ( TopOpen ` fld )   &    |-  M  =  ( Jt  X )   &    |-  N  =  ( Jt  Y )   =>    |-  ( ph  ->  (
 ( F `  C )  e.  ( ( int `  T ) `  Y )  /\  `' F  e.  ( ( N  CnP  M ) `  ( F `
  C ) ) ) )
 
Theoremdvcnvre 19198* The derivative rule for inverse functions. If  F is a continuous and differentiable bijective function from  X to  Y which never has derivative  0, then  `' F is also differentiable, and its derivative is the reciprocal of the derivative of  F. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  F  e.  ( X -cn-> RR )
 )   &    |-  ( ph  ->  dom  ( RR  _D  F )  =  X )   &    |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F ) )   &    |-  ( ph  ->  F : X
 -1-1-onto-> Y )   =>    |-  ( ph  ->  ( RR  _D  `' F )  =  ( x  e.  Y  |->  ( 1  /  ( ( RR  _D  F ) `  ( `' F `  x ) ) ) ) )
 
Theoremdvcvx 19199 A real function with strictly increasing derivative is strictly convex. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )   &    |-  ( ph  ->  ( RR  _D  F )  Isom  <  ,  <  ( ( A (,) B ) ,  W ) )   &    |-  ( ph  ->  T  e.  ( 0 (,) 1 ) )   &    |-  C  =  ( ( T  x.  A )  +  (
 ( 1  -  T )  x.  B ) )   =>    |-  ( ph  ->  ( F `  C )  <  (
 ( T  x.  ( F `  A ) )  +  ( ( 1  -  T )  x.  ( F `  B ) ) ) )
 
Theoremdvfsumle 19200* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.)
 |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  ( x  e.  ( M [,] N )  |->  A )  e.  ( ( M [,] N ) -cn-> RR ) )   &    |-  ( ( ph  /\  x  e.  ( M (,) N ) ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( M (,) N )  |->  A ) )  =  ( x  e.  ( M (,) N )  |->  B ) )   &    |-  ( x  =  M  ->  A  =  C )   &    |-  ( x  =  N  ->  A  =  D )   &    |-  ( ( ph  /\  k  e.  ( M..^ N ) )  ->  X  e.  RR )   &    |-  (
 ( ph  /\  ( k  e.  ( M..^ N )  /\  x  e.  (
 k (,) ( k  +  1 ) ) ) )  ->  X  <_  B )   =>    |-  ( ph  ->  sum_ k  e.  ( M..^ N ) X  <_  ( D  -  C ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30843
  Copyright terms: Public domain < Previous  Next >