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Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvge0 19301 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 19302* 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 19303* Lemma for dvivth 19305. (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 19304* Lemma for dvivth 19305. (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 19305 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 18766 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 19306 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 19307 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 19308* Lemma for lhop1 19309. (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 19309* 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 19310* 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 19311* 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 19312 Lemma for dvcnvre 19314. (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 19313 Lemma for dvcnvre 19314. (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 19314* 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 19315 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 19316* 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 19317* 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 19318* 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 19319* 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 19320* Lemma for dvfsumrlim 19326. (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 19321* Lemma for dvfsumrlim 19326. (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 19322* Lemma for dvfsumrlim 19326. (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 19323* Lemma for dvfsumrlim 19326. (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 19324* Lemma for dvfsumrlim 19326. (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 19325* Lemma for dvfsumrlim 19326. Satisfy the assumption of dvfsumlem4 19324. (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 19326* 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 19327* 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 19328* Conjoin the statements of dvfsumrlim 19326 and dvfsumrlim2 19327. (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 19329* The reverse of dvfsumrlim 19326, 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 19330* Lemma for ftc1a 19332 and ftc1 19337. (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 19331* Lemma for ftc1 19337. (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 19332* 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 19333* Lemma for ftc1 19337. (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 19334* Lemma for ftc1 19337. (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 19335* Lemma for ftc1 19337. (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 19336* Lemma for ftc1 19337. (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 19337* 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 19338* Strengthen the assumptions of ftc1 19337 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 19339* 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 19340* Lemma for ftc2ditg 19341. (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 19341* Directed integral analog of ftc2 19339. (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 19342* 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 19343* Lemma for itgsubst 19344. (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 19344* 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 19343, 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 19345* Lemma for evlseu 19348. 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 19346* Lemma for evlseu 19348. 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 19347* Lemma for evlseu 19348, 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 19348* 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 19349 Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |- 
 Rel  dom evalSub
 
Theoremmpfrcl 19350 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 19351* 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 19352* 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 19353 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 19354 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 19355* 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 19356 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 19357 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 19358* 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 )
 
Theoremevl1val 19359* 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 )   &    |-  M  =  ( 1o mPoly  R )   &    |-  K  =  ( Base `  M )   =>    |-  (
 ( R  e.  CRing  /\  A  e.  K ) 
 ->  ( O `  A )  =  ( ( Q `  A )  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )
 
Theoremevl1rhm 19360 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  T  =  ( R 
 ^s 
 B )   &    |-  B  =  (
 Base `  R )   =>    |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  T ) )
 
Theoremevl1sca 19361 Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  CRing  /\  X  e.  B ) 
 ->  ( O `  ( A `  X ) )  =  ( B  X.  { X } ) )
 
Theoremevl1scad 19362 Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( ( A `  X )  e.  U  /\  ( ( O `  ( A `  X ) ) `  Y )  =  X ) )
 
Theoremevl1var 19363 Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  CRing  ->  ( O `  X )  =  (  _I  |`  B ) )
 
Theoremevl1vard 19364 Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  X  =  (var1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  e.  U  /\  ( ( O `  X ) `  Y )  =  Y )
 )
 
Theoremevl1addd 19365 Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y )  =  V )
 )   &    |-  ( ph  ->  ( N  e.  U  /\  ( ( O `  N ) `  Y )  =  W )
 )   &    |-  .+b  =  ( +g  `  P )   &    |- 
 .+  =  ( +g  `  R )   =>    |-  ( ph  ->  (
 ( M  .+b  N )  e.  U  /\  (
 ( O `  ( M  .+b  N ) ) `
  Y )  =  ( V  .+  W ) ) )
 
Theoremevl1subd 19366 Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y )  =  V )
 )   &    |-  ( ph  ->  ( N  e.  U  /\  ( ( O `  N ) `  Y )  =  W )
 )   &    |-  .-  =  ( -g `  P )   &    |-  D  =  (
 -g `  R )   =>    |-  ( ph  ->  ( ( M 
 .-  N )  e.  U  /\  ( ( O `  ( M 
 .-  N ) ) `
  Y )  =  ( V D W ) ) )
 
Theoremevl1muld 19367 Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y )  =  V )
 )   &    |-  ( ph  ->  ( N  e.  U  /\  ( ( O `  N ) `  Y )  =  W )
 )   &    |-  .xb  =  ( .r `  P )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ph  ->  ( ( M  .xb  N )  e.  U  /\  (
 ( O `  ( M  .xb  N ) ) `
  Y )  =  ( V  .x.  W ) ) )
 
Theoremevl1vsd 19368 Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y )  =  V )
 )   &    |-  ( ph  ->  N  e.  B )   &    |-  .xb  =  ( .s `  P )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ph  ->  ( ( N  .xb  M )  e.  U  /\  ( ( O `  ( N 
 .xb  M ) ) `  Y )  =  ( N  .x.  V ) ) )
 
Theoremevl1expd 19369 Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  R )   &    |-  U  =  (
 Base `  P )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  ( M  e.  U  /\  ( ( O `  M ) `  Y )  =  V )
 )   &    |-  .xb  =  (.g `  (mulGrp `  P ) )   &    |-  .^  =  (.g `  (mulGrp `  R ) )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  (
 ( N  .xb  M )  e.  U  /\  (
 ( O `  ( N  .xb  M ) ) `
  Y )  =  ( N  .^  V ) ) )
 
Theoremmpfconst 19370 Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  B  =  ( Base `  S )   &    |-  Q  =  ran  ( ( I evalSub  S ) `  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  X  e.  R )   =>    |-  ( ph  ->  ( ( B  ^m  I
 )  X.  { X } )  e.  Q )
 
Theoremmpfproj 19371* Projections are multivariate polynomial functions. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  B  =  ( Base `  S )   &    |-  Q  =  ran  ( ( I evalSub  S ) `  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  S  e.  CRing )   &    |-  ( ph  ->  R  e.  (SubRing `  S ) )   &    |-  ( ph  ->  J  e.  I )   =>    |-  ( ph  ->  ( f  e.  ( B 
 ^m  I )  |->  ( f `  J ) )  e.  Q )
 
Theoremmpfsubrg 19372 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  Q  =  ran  (
 ( I evalSub  S ) `  R )   =>    |-  ( ( I  e. 
 _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  ->  Q  e.  (SubRing `  ( S  ^s  (
 ( Base `  S )  ^m  I ) ) ) )
 
Theoremmpff 19373 Polynomial functions are functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  Q  =  ran  (
 ( I evalSub  S ) `  R )   &    |-  B  =  (
 Base `  S )   =>    |-  ( F  e.  Q  ->  F : ( B  ^m  I ) --> B )
 
Theoremmpfaddcl 19374 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  Q  =  ran  (
 ( I evalSub  S ) `  R )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ( F  e.  Q  /\  G  e.  Q )  ->  ( F  o F  .+  G )  e.  Q )
 
Theoremmpfmulcl 19375 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  Q  =  ran  (
 ( I evalSub  S ) `  R )   &    |-  .x.  =  ( .r `  S )   =>    |-  ( ( F  e.  Q  /\  G  e.  Q )  ->  ( F  o F  .x.  G )  e.  Q )
 
Theoremmpfind 19376* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 19-Mar-2015.)
 |-  B  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  Q  =  ran  ( ( I evalSub  S ) `  R )   &    |-  ( ( ph  /\  (
 ( f  e.  Q  /\  ta )  /\  (
 g  e.  Q  /\  et ) ) )  ->  ze )   &    |-  ( ( ph  /\  ( ( f  e.  Q  /\  ta )  /\  ( g  e.  Q  /\  et ) ) ) 
 ->  si )   &    |-  ( x  =  ( ( B  ^m  I )  X.  { f } )  ->  ( ps  <->  ch ) )   &    |-  ( x  =  ( g  e.  ( B  ^m  I )  |->  ( g `  f ) )  ->  ( ps  <->  th ) )   &    |-  ( x  =  f  ->  ( ps  <->  ta ) )   &    |-  ( x  =  g  ->  ( ps  <->  et ) )   &    |-  ( x  =  ( f  o F  .+  g )  ->  ( ps 
 <->  ze ) )   &    |-  ( x  =  ( f  o F  .x.  g ) 
 ->  ( ps  <->  si ) )   &    |-  ( x  =  A  ->  ( ps  <->  rh ) )   &    |-  (
 ( ph  /\  f  e.  R )  ->  ch )   &    |-  (
 ( ph  /\  f  e.  I )  ->  th )   &    |-  ( ph  ->  A  e.  Q )   =>    |-  ( ph  ->  rh )
 
Theorempf1const 19377 Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  Q  =  ran  (eval1 `  R )   =>    |-  ( ( R  e.  CRing  /\  X  e.  B ) 
 ->  ( B  X.  { X } )  e.  Q )
 
Theorempf1id 19378 The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  Q  =  ran  (eval1 `  R )   =>    |-  ( R  e.  CRing  ->  (  _I  |`  B )  e.  Q )
 
Theorempf1subrg 19379 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  Q  =  ran  (eval1 `  R )   =>    |-  ( R  e.  CRing  ->  Q  e.  (SubRing `  ( R  ^s  B ) ) )
 
Theorempf1rcl 19380 Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   =>    |-  ( X  e.  Q  ->  R  e.  CRing )
 
Theorempf1f 19381 Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( F  e.  Q  ->  F : B --> B )
 
Theoremmpfpf1 19382* Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   &    |-  B  =  (
 Base `  R )   &    |-  E  =  ran  ( 1o eval  R )   =>    |-  ( F  e.  E  ->  ( F  o.  (
 y  e.  B  |->  ( 1o  X.  { y } ) ) )  e.  Q )
 
Theorempf1mpf 19383* Convert a univariate polynomial function to multivariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   &    |-  B  =  (
 Base `  R )   &    |-  E  =  ran  ( 1o eval  R )   =>    |-  ( F  e.  Q  ->  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
  (/) ) ) )  e.  E )
 
Theorempf1addcl 19384 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( F  e.  Q  /\  G  e.  Q )  ->  ( F  o F  .+  G )  e.  Q )
 
Theorempf1mulcl 19385 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Q  =  ran  (eval1 `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( F  e.  Q  /\  G  e.  Q )  ->  ( F  o F  .x.  G )  e.  Q )
 
Theorempf1ind 19386* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  Q  =  ran  (eval1 `  R )   &    |-  ( ( ph  /\  (
 ( f  e.  Q  /\  ta )  /\  (
 g  e.  Q  /\  et ) ) )  ->  ze )   &    |-  ( ( ph  /\  ( ( f  e.  Q  /\  ta )  /\  ( g  e.  Q  /\  et ) ) ) 
 ->  si )   &    |-  ( x  =  ( B  X.  {
 f } )  ->  ( ps  <->  ch ) )   &    |-  ( x  =  (  _I  |`  B )  ->  ( ps 
 <-> 
 th ) )   &    |-  ( x  =  f  ->  ( ps  <->  ta ) )   &    |-  ( x  =  g  ->  ( ps  <->  et ) )   &    |-  ( x  =  ( f  o F  .+  g ) 
 ->  ( ps  <->  ze ) )   &    |-  ( x  =  ( f  o F  .x.  g ) 
 ->  ( ps  <->  si ) )   &    |-  ( x  =  A  ->  ( ps  <->  rh ) )   &    |-  (
 ( ph  /\  f  e.  B )  ->  ch )   &    |-  ( ph  ->  th )   &    |-  ( ph  ->  A  e.  Q )   =>    |-  ( ph  ->  rh )
 
13.1.2  Polynomial degrees
 
Syntaxcmdg 19387 Multivariate polynomial degree.
 class mDeg
 
Syntaxcdg1 19388 Univariate polynomial degree.
 class deg1
 
Definitiondf-mdeg 19389* Define the degree of a polynomial. Note (SO): as an experiment I am using a definition which makes the degree of the zero polynomial  -oo, contrary to the convention used in df-dgr 19521. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |- mDeg  =  ( i  e.  _V ,  r  e.  _V  |->  ( f  e.  ( Base `  ( i mPoly  r
 ) )  |->  sup ( ran  (  h  e.  ( `' f " ( _V  \  { ( 0g `  r ) } )
 )  |->  (fld 
 gsumg  h ) ) , 
 RR* ,  <  ) ) )
 
Definitiondf-deg1 19390 Define the degree of a univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |- deg1  =  ( r  e.  _V  |->  ( 1o mDeg  r )
 )
 
Theoremreldmmdeg 19391 Multivariate degree is a binary operation. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |- 
 Rel  dom mDeg
 
Theoremtdeglem1 19392* Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( I  e.  V  ->  H : A --> NN0 )
 
Theoremtdeglem3 19393* Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( ( I  e.  V  /\  X  e.  A  /\  Y  e.  A )  ->  ( H `  ( X  o F  +  Y ) )  =  ( ( H `  X )  +  ( H `  Y ) ) )
 
Theoremtdeglem4 19394* There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( ( I  e.  V  /\  X  e.  A )  ->  ( ( H `  X )  =  0  <->  X  =  ( I  X.  { 0 } ) ) )
 
Theoremtdeglem2 19395 Simplification of total degree for the univariate case. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  ( h  e.  ( NN0  ^m  1o )  |->  ( h `  (/) ) )  =  ( h  e.  ( NN0  ^m  1o )  |->  (fld 
 gsumg  h ) )
 
Theoremmdegfval 19396* Value of the multivariate degree function. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  D  =  ( f  e.  B  |->  sup (
 ( H " ( `' f " ( _V  \  {  .0.  } )
 ) ) ,  RR* ,  <  ) )
 
Theoremmdegval 19397* Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H " ( `' F " ( _V  \  {  .0.  } )
 ) ) ,  RR* ,  <  ) )
 
Theoremmdegleb 19398* Property of being of limited degree. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   =>    |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ( ( D `
  F )  <_  G 
 <-> 
 A. x  e.  A  ( G  <  ( H `
  x )  ->  ( F `  x )  =  .0.  ) ) )
 
Theoremmdeglt 19399* If there is an upper limit on the degree of a polynomial that is lower than the degree of some exponent bag, then that exponent bag is unrepresented in the polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  ( D `  F )  < 
 ( H `  X ) )   =>    |-  ( ph  ->  ( F `  X )  =  .0.  )
 
Theoremmdegldg 19400* A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-Mar-2015.)
 |-  D  =  ( I mDeg 
 R )   &    |-  P  =  ( I mPoly  R )   &    |-  B  =  ( Base `  P )   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  { m  e.  ( NN0  ^m  I
 )  |  ( `' m " NN )  e.  Fin }   &    |-  H  =  ( h  e.  A  |->  (fld  gsumg  h ) )   &    |-  Y  =  ( 0g `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/=  Y )  ->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `  x )  =  ( D `  F ) ) )
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