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Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvnfre 19301 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 19302* 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 19303* 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 19304* 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 19305* 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 19306* 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 19307* 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 19308* Closure lemma for dvmptcmul 19313 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 19309* 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 19310* 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 19311* 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 19312* 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 19313* 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 19314* 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 19315* 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 19316* 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 19317* 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 19318* 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 19319* 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 19320* 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 19321* 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 19322* 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 19323 Lemma for dvcnvre 19366. (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 19324* A weak version of dvcnvre 19366, 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 19325* 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 19326 Derivative of the exponential function at 0. The key step in the proof is eftlub 12389, 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 19327 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 19328 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 19329 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( CC  _D  sin )  =  cos
 
Theoremdvcos 19330 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) )
 
Theoremdvferm1lem 19331* Lemma for dvferm 19335. (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 19332* One-sided version of dvferm 19335. 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 19333* Lemma for dvferm 19335. (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 19334* One-sided version of dvferm 19335. 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 19335* 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 19336* Lemma for rolle 19337. (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 19337* 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 19338* 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 19339* 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 19340* 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 19341* 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 19342* Combine the results of dvlip 19340 and dvlipcn 19341 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 19343* Lemma for c1lip1 19344. (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 19344* 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 19345* 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 19346* 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 19347 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 19348 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 19349 Lemma for dvgt0 19351 and dvlt0 19352. (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 19350* Lemma for dvgt0 19351 and dvlt0 19352. (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 19351 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 19352 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 19353 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 19354* 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 19355* Lemma for dvivth 19357. (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 19356* Lemma for dvivth 19357. (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 19357 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 18818 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 19358 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 19359 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 19360* Lemma for lhop1 19361. (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 19361* 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 19362* 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 19363* 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 19364 Lemma for dvcnvre 19366. (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 19365 Lemma for dvcnvre 19366. (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 19366* 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 19367 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 19368* 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 ) )
 
Theoremdvfsumge 19369* 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 ) ) ) )  ->  B  <_  X )   =>    |-  ( ph  ->  ( D  -  C )  <_  sum_ k  e.  ( M..^ N ) X )
 
Theoremdvfsumabs 19370* 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-> CC ) )   &    |-  ( ( 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.  CC )   &    |-  (
 ( ph  /\  k  e.  ( M..^ N ) )  ->  Y  e.  RR )   &    |-  ( ( ph  /\  ( k  e.  ( M..^ N )  /\  x  e.  ( k (,) (
 k  +  1 ) ) ) )  ->  ( abs `  ( X  -  B ) )  <_  Y )   =>    |-  ( ph  ->  ( abs `  ( sum_ k  e.  ( M..^ N ) X  -  ( D  -  C ) ) )  <_  sum_ k  e.  ( M..^ N ) Y )
 
Theoremdvmptrecl 19371* Real closure of a derivative. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  ( ph  ->  S  C_ 
 RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  A  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  B  e.  V )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   =>    |-  ( ( ph  /\  x  e.  S )  ->  B  e.  RR )
 
Theoremdvfsumrlimf 19372* Lemma for dvfsumrlim 19378. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  S  =  ( T (,)  +oo )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M 
 <_  ( D  +  1 ) )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   &    |-  ( x  =  k  ->  B  =  C )   &    |-  G  =  ( x  e.  S  |->  (
 sum_ k  e.  ( M ... ( |_ `  x ) ) C  -  A ) )   =>    |-  ( ph  ->  G : S --> RR )
 
Theoremdvfsumlem1 19373* Lemma for dvfsumrlim 19378. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  S  =  ( T (,)  +oo )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M 
 <_  ( D  +  1 ) )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   &    |-  ( x  =  k  ->  B  =  C )   &    |-  ( ph  ->  U  e.  RR* )   &    |-  ( ( ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x 
 /\  x  <_  k  /\  k  <_  U ) )  ->  C  <_  B )   &    |-  H  =  ( x  e.  S  |->  ( ( ( x  -  ( |_ `  x ) )  x.  B )  +  ( sum_ k  e.  ( M ... ( |_ `  x ) ) C  -  A ) ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   &    |-  ( ph  ->  D  <_  X )   &    |-  ( ph  ->  X 
 <_  Y )   &    |-  ( ph  ->  Y 
 <_  U )   &    |-  ( ph  ->  Y 
 <_  ( ( |_ `  X )  +  1 )
 )   =>    |-  ( ph  ->  ( H `  Y )  =  ( ( ( ( Y  -  ( |_ `  X ) )  x.  [_ Y  /  x ]_ B )  -  [_ Y  /  x ]_ A )  +  sum_ k  e.  ( M ... ( |_ `  X ) ) C ) )
 
Theoremdvfsumlem2 19374* Lemma for dvfsumrlim 19378. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  S  =  ( T (,)  +oo )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M 
 <_  ( D  +  1 ) )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   &    |-  ( x  =  k  ->  B  =  C )   &    |-  ( ph  ->  U  e.  RR* )   &    |-  ( ( ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x 
 /\  x  <_  k  /\  k  <_  U ) )  ->  C  <_  B )   &    |-  H  =  ( x  e.  S  |->  ( ( ( x  -  ( |_ `  x ) )  x.  B )  +  ( sum_ k  e.  ( M ... ( |_ `  x ) ) C  -  A ) ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   &    |-  ( ph  ->  D  <_  X )   &    |-  ( ph  ->  X 
 <_  Y )   &    |-  ( ph  ->  Y 
 <_  U )   &    |-  ( ph  ->  Y 
 <_  ( ( |_ `  X )  +  1 )
 )   =>    |-  ( ph  ->  (
 ( H `  Y )  <_  ( H `  X )  /\  ( ( H `  X )  -  [_ X  /  x ]_ B )  <_  ( ( H `  Y )  -  [_ Y  /  x ]_ B ) ) )
 
Theoremdvfsumlem3 19375* Lemma for dvfsumrlim 19378. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  S  =  ( T (,)  +oo )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M 
 <_  ( D  +  1 ) )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   &    |-  ( x  =  k  ->  B  =  C )   &    |-  ( ph  ->  U  e.  RR* )   &    |-  ( ( ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x 
 /\  x  <_  k  /\  k  <_  U ) )  ->  C  <_  B )   &    |-  H  =  ( x  e.  S  |->  ( ( ( x  -  ( |_ `  x ) )  x.  B )  +  ( sum_ k  e.  ( M ... ( |_ `  x ) ) C  -  A ) ) )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   &    |-  ( ph  ->  D  <_  X )   &    |-  ( ph  ->  X 
 <_  Y )   &    |-  ( ph  ->  Y 
 <_  U )   =>    |-  ( ph  ->  (
 ( H `  Y )  <_  ( H `  X )  /\  ( ( H `  X )  -  [_ X  /  x ]_ B )  <_  ( ( H `  Y )  -  [_ Y  /  x ]_ B ) ) )
 
Theoremdvfsumlem4 19376* Lemma for dvfsumrlim 19378. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  S  =  ( T (,)  +oo )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M 
 <_  ( D  +  1 ) )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   &    |-  ( x  =  k  ->  B  =  C )   &    |-  ( ph  ->  U  e.  RR* )   &    |-  ( ( ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x 
 /\  x  <_  k  /\  k  <_  U ) )  ->  C  <_  B )   &    |-  G  =  ( x  e.  S  |->  (
 sum_ k  e.  ( M ... ( |_ `  x ) ) C  -  A ) )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  D  <_  x  /\  x  <_  U ) )  -> 
 0  <_  B )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   &    |-  ( ph  ->  D 
 <_  X )   &    |-  ( ph  ->  X 
 <_  Y )   &    |-  ( ph  ->  Y 
 <_  U )   =>    |-  ( ph  ->  ( abs `  ( ( G `
  Y )  -  ( G `  X ) ) )  <_  [_ X  /  x ]_ B )
 
Theoremdvfsumrlimge0 19377* Lemma for dvfsumrlim 19378. Satisfy the assumption of dvfsumlem4 19376. (Contributed by Mario Carneiro, 18-May-2016.)
 |-  S  =  ( T (,)  +oo )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M 
 <_  ( D  +  1 ) )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   &    |-  ( x  =  k  ->  B  =  C )   &    |-  ( ( ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x 
 /\  x  <_  k
 ) )  ->  C  <_  B )   &    |-  G  =  ( x  e.  S  |->  (
 sum_ k  e.  ( M ... ( |_ `  x ) ) C  -  A ) )   &    |-  ( ph  ->  ( x  e.  S  |->  B )  ~~> r  0 )   =>    |-  ( ( ph  /\  ( x  e.  S  /\  D  <_  x ) ) 
 ->  0  <_  B )
 
Theoremdvfsumrlim 19378* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). The statement here says that if  x  e.  S  |->  B is a decreasing function with antiderivative  A converging to zero, then the difference between  sum_ k  e.  ( M ... ( |_ `  x ) ) B ( k ) and  A ( x )  =  S. u  e.  ( M [,] x
) B ( u )  _d u converges to a constant limit value, with the remainder term bounded by  B
( x ). (Contributed by Mario Carneiro, 18-May-2016.)
 |-  S  =  ( T (,)  +oo )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M 
 <_  ( D  +  1 ) )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   &    |-  ( x  =  k  ->  B  =  C )   &    |-  ( ( ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x 
 /\  x  <_  k
 ) )  ->  C  <_  B )   &    |-  G  =  ( x  e.  S  |->  (
 sum_ k  e.  ( M ... ( |_ `  x ) ) C  -  A ) )   &    |-  ( ph  ->  ( x  e.  S  |->  B )  ~~> r  0 )   =>    |-  ( ph  ->  G  e.  dom  ~~> r  )
 
Theoremdvfsumrlim2 19379* Compare a finite sum to an integral (the integral here is given as a function with a known derivative). The statement here says that if  x  e.  S  |->  B is a decreasing function with antiderivative  A converging to zero, then the difference between  sum_ k  e.  ( M ... ( |_ `  x ) ) B ( k ) and  S. u  e.  ( M [,] x
) B ( u )  _d u  =  A ( x ) converges to a constant limit value, with the remainder term bounded by  B
( x ). (Contributed by Mario Carneiro, 18-May-2016.)
 |-  S  =  ( T (,)  +oo )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M 
 <_  ( D  +  1 ) )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   &    |-  ( x  =  k  ->  B  =  C )   &    |-  ( ( ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x 
 /\  x  <_  k
 ) )  ->  C  <_  B )   &    |-  G  =  ( x  e.  S  |->  (
 sum_ k  e.  ( M ... ( |_ `  x ) ) C  -  A ) )   &    |-  ( ph  ->  ( x  e.  S  |->  B )  ~~> r  0 )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  D  <_  X )   =>    |-  ( ( ph  /\  G  ~~> r  L )  ->  ( abs `  ( ( G `
  X )  -  L ) )  <_  [_ X  /  x ]_ B )
 
Theoremdvfsumrlim3 19380* Conjoin the statements of dvfsumrlim 19378 and dvfsumrlim2 19379. (This is useful as a target for lemmas, because the hypotheses to this theorem are complex and we don't want to repeat ourselves.) (Contributed by Mario Carneiro, 18-May-2016.)
 |-  S  =  ( T (,)  +oo )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  M 
 <_  ( D  +  1 ) )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   &    |-  ( x  =  k  ->  B  =  C )   &    |-  ( ( ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x 
 /\  x  <_  k
 ) )  ->  C  <_  B )   &    |-  G  =  ( x  e.  S  |->  (
 sum_ k  e.  ( M ... ( |_ `  x ) ) C  -  A ) )   &    |-  ( ph  ->  ( x  e.  S  |->  B )  ~~> r  0 )   &    |-  ( x  =  X  ->  B  =  E )   =>    |-  ( ph  ->  ( G : S --> RR  /\  G  e.  dom  ~~> r  /\  ( ( G  ~~> r  L  /\  X  e.  S  /\  D  <_  X )  ->  ( abs `  ( ( G `  X )  -  L ) )  <_  E ) ) )
 
Theoremdvfsum2 19381* The reverse of dvfsumrlim 19378, when comparing a finite sum of increasing terms to an integral. In this case there is no point in stating the limit properties, because the terms of the sum aren't approaching zero, but there is nevertheless still a natural asymptotic statement that can be made. (Contributed by Mario Carneiro, 20-May-2016.)
 |-  S  =  ( T (,)  +oo )   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  U  e.  RR* )   &    |-  ( ph  ->  M 
 <_  ( D  +  1 ) )   &    |-  ( ph  ->  T  e.  RR )   &    |-  (
 ( ph  /\  x  e.  S )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  Z )  ->  B  e.  RR )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )   &    |-  ( x  =  k  ->  B  =  C )   &    |-  ( ( ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x 
 /\  x  <_  k  /\  k  <_  U ) )  ->  B  <_  C )   &    |-  G  =  ( x  e.  S  |->  (
 sum_ k  e.  ( M ... ( |_ `  x ) ) C  -  A ) )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  D  <_  x ) )  -> 
 0  <_  B )   &    |-  ( ph  ->  X  e.  S )   &    |-  ( ph  ->  Y  e.  S )   &    |-  ( ph  ->  D 
 <_  X )   &    |-  ( ph  ->  X 
 <_  Y )   &    |-  ( ph  ->  Y 
 <_  U )   &    |-  ( x  =  Y  ->  B  =  E )   =>    |-  ( ph  ->  ( abs `  ( ( G `
  Y )  -  ( G `  X ) ) )  <_  E )
 
Theoremftc1lem1 19382* Lemma for ftc1a 19384 and ftc1 19389. (Contributed by Mario Carneiro, 31-Aug-2014.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  F : D --> CC )   &    |-  ( ph  ->  X  e.  ( A [,] B ) )   &    |-  ( ph  ->  Y  e.  ( A [,] B ) )   =>    |-  ( ( ph  /\  X  <_  Y )  ->  (
 ( G `  Y )  -  ( G `  X ) )  =  S. ( X (,) Y ) ( F `  t )  _d t
 )
 
Theoremftc1lem2 19383* Lemma for ftc1 19389. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  F : D --> CC )   =>    |-  ( ph  ->  G : ( A [,] B ) --> CC )
 
Theoremftc1a 19384* The Fundamental Theorem of Calculus, part one. The function  G formed by varying the right endpoint of an integral of  F is continuous if  F is integrable. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  F : D --> CC )   =>    |-  ( ph  ->  G  e.  ( ( A [,] B ) -cn-> CC ) )
 
Theoremftc1lem3 19385* Lemma for ftc1 19389. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `  C ) )   &    |-  J  =  ( Lt  RR )   &    |-  K  =  ( Lt  D )   &    |-  L  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  F : D --> CC )
 
Theoremftc1lem4 19386* Lemma for ftc1 19389. (Contributed by Mario Carneiro, 31-Aug-2014.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `  C ) )   &    |-  J  =  ( Lt  RR )   &    |-  K  =  ( Lt  D )   &    |-  L  =  (
 TopOpen ` fld )   &    |-  H  =  ( z  e.  ( ( A [,] B ) 
 \  { C }
 )  |->  ( ( ( G `  z )  -  ( G `  C ) )  /  ( z  -  C ) ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  y  e.  D ) 
 ->  ( ( abs `  (
 y  -  C ) )  <  R  ->  ( abs `  ( ( F `  y )  -  ( F `  C ) ) )  <  E ) )   &    |-  ( ph  ->  X  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( X  -  C ) )  <  R )   &    |-  ( ph  ->  Y  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( Y  -  C ) )  <  R )   =>    |-  ( ( ph  /\  X  <  Y )  ->  ( abs `  ( ( ( ( G `  Y )  -  ( G `  X ) )  /  ( Y  -  X ) )  -  ( F `  C ) ) )  <  E )
 
Theoremftc1lem5 19387* Lemma for ftc1 19389. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `  C ) )   &    |-  J  =  ( Lt  RR )   &    |-  K  =  ( Lt  D )   &    |-  L  =  (
 TopOpen ` fld )   &    |-  H  =  ( z  e.  ( ( A [,] B ) 
 \  { C }
 )  |->  ( ( ( G `  z )  -  ( G `  C ) )  /  ( z  -  C ) ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  y  e.  D ) 
 ->  ( ( abs `  (
 y  -  C ) )  <  R  ->  ( abs `  ( ( F `  y )  -  ( F `  C ) ) )  <  E ) )   &    |-  ( ph  ->  X  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( X  -  C ) )  <  R )   =>    |-  ( ( ph  /\  X  =/=  C )  ->  ( abs `  ( ( H `
  X )  -  ( F `  C ) ) )  <  E )
 
Theoremftc1lem6 19388* Lemma for ftc1 19389. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `  C ) )   &    |-  J  =  ( Lt  RR )   &    |-  K  =  ( Lt  D )   &    |-  L  =  (
 TopOpen ` fld )   &    |-  H  =  ( z  e.  ( ( A [,] B ) 
 \  { C }
 )  |->  ( ( ( G `  z )  -  ( G `  C ) )  /  ( z  -  C ) ) )   =>    |-  ( ph  ->  ( F `  C )  e.  ( H lim CC  C ) )
 
Theoremftc1 19389* The Fundamental Theorem of Calculus, part one. The function formed by varying the right endpoint of an integral is differentiable at  C with derivative  F ( C ) if the original function is continuous at  C. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B ) 
 C_  D )   &    |-  ( ph  ->  D  C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  C  e.  ( A (,) B ) )   &    |-  ( ph  ->  F  e.  ( ( K  CnP  L ) `  C ) )   &    |-  J  =  ( Lt  RR )   &    |-  K  =  ( Lt  D )   &    |-  L  =  (
 TopOpen ` fld )   =>    |-  ( ph  ->  C ( RR  _D  G ) ( F `  C ) )
 
Theoremftc1cn 19390* Strengthen the assumptions of ftc1 19389 to when the function  F is continuous on the entire interval  ( A ,  B ); in this case we can calculate  _D  G exactly. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `
  t )  _d t )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  F  e.  L ^1 )   =>    |-  ( ph  ->  ( RR  _D  G )  =  F )
 
Theoremftc2 19391* The Fundamental Theorem of Calculus, part two. If  F is a function continuous on  [ A ,  B ] and continuously differentiable on  ( A ,  B ), then the integral of the derivative of  F is equal to  F ( B )  -  F ( A ). (Contributed by Mario Carneiro, 2-Sep-2014.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( RR  _D  F )  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  L ^1 )   &    |-  ( ph  ->  F  e.  ( ( A [,] B ) -cn-> CC ) )   =>    |-  ( ph  ->  S. ( A (,) B ) ( ( RR  _D  F ) `  t
 )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
 
Theoremftc2ditglem 19392* Lemma for ftc2ditg 19393. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  A  e.  ( X [,] Y ) )   &    |-  ( ph  ->  B  e.  ( X [,] Y ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  ( ( X (,) Y ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  L ^1 )   &    |-  ( ph  ->  F  e.  (
 ( X [,] Y ) -cn-> CC ) )   =>    |-  ( ( ph  /\  A  <_  B )  ->  S__ [ A  ->  B ] ( ( RR 
 _D  F ) `  t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
 
Theoremftc2ditg 19393* Directed integral analog of ftc2 19391. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  A  e.  ( X [,] Y ) )   &    |-  ( ph  ->  B  e.  ( X [,] Y ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  ( ( X (,) Y ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  L ^1 )   &    |-  ( ph  ->  F  e.  (
 ( X [,] Y ) -cn-> CC ) )   =>    |-  ( ph  ->  S__
 [ A  ->  B ] ( ( RR 
 _D  F ) `  t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
 
Theoremitgparts 19394* Integration by parts. If  B ( x ) is the derivative of  A ( x ) and  D ( x ) is the derivative of  C ( x ), and  E  =  ( A  x.  B ) ( X ) and  F  =  ( A  x.  B ) ( Y ), then under suitable integrability and differentiability assumptions, the integral of  A  x.  D from  X to  Y is equal to  F  -  E minus the integral of  B  x.  C. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  X  <_  Y )   &    |-  ( ph  ->  ( x  e.  ( X [,] Y )  |->  A )  e.  ( ( X [,] Y ) -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  ( X [,] Y )  |->  C )  e.  ( ( X [,] Y )
 -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( X (,) Y ) -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  D )  e.  ( ( X (,) Y ) -cn-> CC ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  ( A  x.  D ) )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  ( B  x.  C ) )  e.  L ^1 )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( X [,] Y )  |->  C ) )  =  ( x  e.  ( X (,) Y )  |->  D ) )   &    |-  ( ( ph  /\  x  =  X )  ->  ( A  x.  C )  =  E )   &    |-  ( ( ph  /\  x  =  Y ) 
 ->  ( A  x.  C )  =  F )   =>    |-  ( ph  ->  S. ( X (,) Y ) ( A  x.  D )  _d x  =  ( ( F  -  E )  -  S. ( X (,) Y ) ( B  x.  C )  _d x ) )
 
Theoremitgsubstlem 19395* Lemma for itgsubst 19396. (Contributed by Mario Carneiro, 12-Sep-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  X  <_  Y )   &    |-  ( ph  ->  Z  e.  RR* )   &    |-  ( ph  ->  W  e.  RR* )   &    |-  ( ph  ->  ( x  e.  ( X [,] Y )  |->  A )  e.  ( ( X [,] Y )
 -cn-> ( Z (,) W ) ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( ( X (,) Y ) -cn-> CC )  i^i  L ^1 )
 )   &    |-  ( ph  ->  ( u  e.  ( Z (,) W )  |->  C )  e.  ( ( Z (,) W ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )   &    |-  ( u  =  A  ->  C  =  E )   &    |-  ( x  =  X  ->  A  =  K )   &    |-  ( x  =  Y  ->  A  =  L )   &    |-  ( ph  ->  M  e.  ( Z (,) W ) )   &    |-  ( ph  ->  N  e.  ( Z (,) W ) )   &    |-  ( ( ph  /\  x  e.  ( X [,] Y ) ) 
 ->  A  e.  ( M (,) N ) )   =>    |-  ( ph  ->  S__ [ K  ->  L ] C  _d u  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
 
Theoremitgsubst 19396* Integration by  u-substitution. If  A ( x ) is a continuous, differentiable function from  [ X ,  Y ] to  ( Z ,  W ), whose derivative is continuous and integrable, and  C ( u ) is a continuous function on  ( Z ,  W ), then the integral of  C ( u ) from  K  =  A ( X ) to  L  =  A ( Y ) is equal to the integral of  C ( A ( x ) )  _D  A ( x ) from  X to  Y. In this part of the proof we discharge the assumptions in itgsubstlem 19395, which use the fact that  ( Z ,  W ) is open to shrink the interval a little to  ( M ,  N ) where  Z  <  M  <  N  <  W- this is possible because  A ( x ) is a continuous function on a closed interval, so its range is in fact a closed interval and we have some wiggle room on the edges. (Contributed by Mario Carneiro, 7-Sep-2014.)
 |-  ( ph  ->  X  e.  RR )   &    |-  ( ph  ->  Y  e.  RR )   &    |-  ( ph  ->  X  <_  Y )   &    |-  ( ph  ->  Z  e.  RR* )   &    |-  ( ph  ->  W  e.  RR* )   &    |-  ( ph  ->  ( x  e.  ( X [,] Y )  |->  A )  e.  ( ( X [,] Y )
 -cn-> ( Z (,) W ) ) )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( ( X (,) Y ) -cn-> CC )  i^i  L ^1 )
 )   &    |-  ( ph  ->  ( u  e.  ( Z (,) W )  |->  C )  e.  ( ( Z (,) W ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  ( X [,] Y )  |->  A ) )  =  ( x  e.  ( X (,) Y )  |->  B ) )   &    |-  ( u  =  A  ->  C  =  E )   &    |-  ( x  =  X  ->  A  =  K )   &    |-  ( x  =  Y  ->  A  =  L )   =>    |-  ( ph  ->  S__ [ K  ->  L ] C  _d u  =  S__ [ X  ->  Y ] ( E  x.  B )  _d x )
 
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
 
13.1  Polynomials
 
13.1.1  Abstract polynomials, continued
 
Theoremevlslem6 19397* Lemma for evlseu 19400. Finiteness and consistency of the top level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  C  =  ( Base `  S )   &    |-  K  =  ( Base `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  T  =  (mulGrp `  S )   &    |-  .^  =  (.g `  T )   &    |- 
 .x.  =  ( .r `  S )   &    |-  V  =  ( I mVar  R )   &    |-  E  =  ( p  e.  B  |->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( p `  b ) )  .x.  ( T  gsumg  (
 b  o F  .^  G ) ) ) ) ) )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  F  e.  ( R RingHom  S ) )   &    |-  ( ph  ->  G : I --> C )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T  gsumg  (
 b  o F  .^  G ) ) ) ) : D --> C  /\  ( `' ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T  gsumg  (
 b  o F  .^  G ) ) ) ) " ( _V  \  { ( 0g `  S ) } )
 )  e.  Fin )
 )
 
Theoremevlslem3 19398* Lemma for evlseu 19400. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  C  =  ( Base `  S )   &    |-  K  =  ( Base `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  T  =  (mulGrp `  S )   &    |-  .^  =  (.g `  T )   &    |- 
 .x.  =  ( .r `  S )   &    |-  V  =  ( I mVar  R )   &    |-  E  =  ( p  e.  B  |->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( p `  b ) )  .x.  ( T  gsumg  (
 b  o F  .^  G ) ) ) ) ) )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  F  e.  ( R RingHom  S ) )   &    |-  ( ph  ->  G : I --> C )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  A  e.  D )   &    |-  ( ph  ->  H  e.  K )   =>    |-  ( ph  ->  ( E `  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) )  =  ( ( F `  H )  .x.  ( T 
 gsumg  ( A  o F  .^  G ) ) ) )
 
Theoremevlslem1 19399* Lemma for evlseu 19400, give a formula for (the unique) polynomial evaluation homomorphism. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  C  =  ( Base `  S )   &    |-  K  =  ( Base `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  T  =  (mulGrp `  S )   &    |-  .^  =  (.g `  T )   &    |- 
 .x.  =  ( .r `  S )   &    |-  V  =  ( I mVar  R )   &    |-  E  =  ( p  e.  B  |->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( p `  b ) )  .x.  ( T  gsumg  (
 b  o F  .^  G ) ) ) ) ) )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  F  e.  ( R RingHom  S ) )   &    |-  ( ph  ->  G : I --> C )   &    |-  A  =  (algSc `  P )   =>    |-  ( ph  ->  ( E  e.  ( P RingHom  S )  /\  ( E  o.  A )  =  F  /\  ( E  o.  V )  =  G ) )
 
Theoremevlseu 19400* For a given intepretation of the variables  G and of the scalars  F, this extends to a homomorphic interpretation of the polynomial ring in exactly one way. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  C  =  (
 Base `  S )   &    |-  A  =  (algSc `  P )   &    |-  V  =  ( I mVar  R )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  F  e.  ( R RingHom  S ) )   &    |-  ( ph  ->  G : I --> C )   =>    |-  ( ph  ->  E! m  e.  ( P RingHom  S )
 ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G )
 )
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