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Theorem List for Metamath Proof Explorer - 19801-19900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvmptres2 19801* 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 19802* 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 19803* 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 19804* 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 19805* 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 19806* 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 19807* 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 19808* 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 19809* 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 19810* 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 19811* 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 19812* 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 19813 Lemma for dvcnvre 19856. (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 19814* A weak version of dvcnvre 19856, 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 19815* 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 19816 Derivative of the exponential function at 0. The key step in the proof is eftlub 12665, 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 19817 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 19818 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 19819 Derivative of the sine function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( CC  _D  sin )  =  cos
 
Theoremdvcos 19820 Derivative of the cosine function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( CC  _D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) )
 
Theoremdvferm1lem 19821* Lemma for dvferm 19825. (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 19822* One-sided version of dvferm 19825. 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 19823* Lemma for dvferm 19825. (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 19824* One-sided version of dvferm 19825. 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 19825* 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 19826* Lemma for rolle 19827. (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 19827* 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 19828* 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 19829* 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 19830* 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 19831* 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 19832* Combine the results of dvlip 19830 and dvlipcn 19831 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 19833* Lemma for c1lip1 19834. (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 19834* 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 19835* 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 19836* 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 19837 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 19838 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 19839 Lemma for dvgt0 19841 and dvlt0 19842. (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 19840* Lemma for dvgt0 19841 and dvlt0 19842. (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 19841 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 19842 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 19843 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 19844* 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 19845* Lemma for dvivth 19847. (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 19846* Lemma for dvivth 19847. (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 19847 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 19308 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 19848 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 19849 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 19850* Lemma for lhop1 19851. (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 19851* 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 19852* 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 19853* 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 19854 Lemma for dvcnvre 19856. (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 19855 Lemma for dvcnvre 19856. (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 19856* 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 19857 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 19858* 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 19859* 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 19860* 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 19861* 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 19862* Lemma for dvfsumrlim 19868. (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 19863* Lemma for dvfsumrlim 19868. (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 19864* Lemma for dvfsumrlim 19868. (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 19865* Lemma for dvfsumrlim 19868. (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 19866* Lemma for dvfsumrlim 19868. (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 19867* Lemma for dvfsumrlim 19868. Satisfy the assumption of dvfsumlem4 19866. (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 19868* 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 19869* 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 19870* Conjoin the statements of dvfsumrlim 19868 and dvfsumrlim2 19869. (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 19871* The reverse of dvfsumrlim 19868, 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 19872* Lemma for ftc1a 19874 and ftc1 19879. (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 19873* Lemma for ftc1 19879. (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 19874* 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 19875* Lemma for ftc1 19879. (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 19876* Lemma for ftc1 19879. (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 19877* Lemma for ftc1 19879. (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 19878* Lemma for ftc1 19879. (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 19879* 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 19880* Strengthen the assumptions of ftc1 19879 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 19881* 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 19882* Lemma for ftc2ditg 19883. (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 19883* Directed integral analog of ftc2 19881. (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 19884* 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 19885* Lemma for itgsubst 19886. (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 19886* 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 19885, 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 19887* Lemma for evlseu 19890. 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 19888* Lemma for evlseu 19890. 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 19889* Lemma for evlseu 19890, 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 19890* 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 )
 )
 
Theoremreldmevls 19891 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |- 
 Rel  dom evalSub
 
Theoremmpfrcl 19892 Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  Q  =  ran  (
 ( I evalSub  S ) `  R )   =>    |-  ( X  e.  Q  ->  ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
 ) )
 
Theoremevlsval 19893* Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 |-  Q  =  ( ( I evalSub  S ) `  R )   &    |-  W  =  ( I mPoly  U )   &    |-  V  =  ( I mVar  U )   &    |-  U  =  ( Ss  R )   &    |-  T  =  ( S  ^s  ( B  ^m  I
 ) )   &    |-  B  =  (
 Base `  S )   &    |-  A  =  (algSc `  W )   &    |-  X  =  ( x  e.  R  |->  ( ( B  ^m  I )  X.  { x } ) )   &    |-  Y  =  ( x  e.  I  |->  ( g  e.  ( B  ^m  I )  |->  ( g `  x ) ) )   =>    |-  ( ( I  e. 
 _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  ->  Q  =  ( iota_ f  e.  ( W RingHom  T ) ( ( f  o.  A )  =  X  /\  (
 f  o.  V )  =  Y ) ) )
 
Theoremevlsval2 19894* Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.)
 |-  Q  =  ( ( I evalSub  S ) `  R )   &    |-  W  =  ( I mPoly  U )   &    |-  V  =  ( I mVar  U )   &    |-  U  =  ( Ss  R )   &    |-  T  =  ( S  ^s  ( B  ^m  I
 ) )   &    |-  B  =  (
 Base `  S )   &    |-  A  =  (algSc `  W )   &    |-  X  =  ( x  e.  R  |->  ( ( B  ^m  I )  X.  { x } ) )   &    |-  Y  =  ( x  e.  I  |->  ( g  e.  ( B  ^m  I )  |->  ( g `  x ) ) )   =>    |-  ( ( I  e. 
 _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  ->  ( Q  e.  ( W RingHom  T ) 
 /\  ( ( Q  o.  A )  =  X  /\  ( Q  o.  V )  =  Y ) ) )
 
Theoremevlsrhm 19895 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Stefan O'Rear, 12-Mar-2015.)
 |-  Q  =  ( ( I evalSub  S ) `  R )   &    |-  W  =  ( I mPoly  U )   &    |-  U  =  ( Ss  R )   &    |-  T  =  ( S  ^s  ( B  ^m  I
 ) )   &    |-  B  =  (
 Base `  S )   =>    |-  ( ( I  e.  V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) ) 
 ->  Q  e.  ( W RingHom  T ) )
 
Theoremevlssca 19896 Polynomial evaluation maps scalars to constant functions. (Contributed by Stefan O'Rear, 13-Mar-2015.)
 |-  Q  =  ( ( I evalSub  S ) `  R )   &    |-  W  =  ( I mPoly  U )   &    |-  U  =  ( Ss  R )   &    |-  B  =  (
 Base `  S )   &    |-  A  =  (algSc `  W )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  X  e.  R )   =>    |-  ( ph  ->  ( Q `  ( A `
  X ) )  =  ( ( B 
 ^m  I )  X.  { X } ) )
 
Theoremevlsvar 19897* Polynomial evaluation maps variables to projections. (Contributed by Stefan O'Rear, 12-Mar-2015.)
 |-  Q  =  ( ( I evalSub  S ) `  R )   &    |-  V  =  ( I mVar 
 U )   &    |-  U  =  ( Ss  R )   &    |-  B  =  (
 Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( Q `  ( V `
  X ) )  =  ( g  e.  ( B  ^m  I
 )  |->  ( g `  X ) ) )
 
Theoremevlval 19898 Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ( I eval 
 R )   &    |-  B  =  (
 Base `  R )   =>    |-  Q  =  ( ( I evalSub  R ) `  B )
 
Theoremevlrhm 19899 The simple evaluation map is a ring homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ( I eval 
 R )   &    |-  B  =  (
 Base `  R )   &    |-  W  =  ( I mPoly  R )   &    |-  T  =  ( R  ^s  ( B  ^m  I ) )   =>    |-  ( ( I  e.  V  /\  R  e.  CRing
 )  ->  Q  e.  ( W RingHom  T ) )
 
Theoremevl1fval 19900* Value of the simple/same ring evalutation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  Q  =  ( 1o eval  R )   &    |-  B  =  (
 Base `  R )   =>    |-  O  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  Q )
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