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Theorem List for Intuitionistic Logic Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsuplociccex 13201* An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7965 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
 |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  <  C )   &    |-  ( ph  ->  A  C_  ( B [,] C ) )   &    |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  A. x  e.  ( B [,] C ) A. y  e.  ( B [,] C ) ( x  <  y  ->  ( E. z  e.  A  x  <  z  \/  A. z  e.  A  z  <  y ) ) )   =>    |-  ( ph  ->  E. x  e.  ( B [,] C ) ( A. y  e.  A  -.  x  < 
 y  /\  A. y  e.  ( B [,] C ) ( y  < 
 x  ->  E. z  e.  A  y  <  z
 ) ) )
 
Theoremdedekindicclemuub 13202* Lemma for dedekindicc 13209. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  A. z  e.  L  z  <  C )
 
Theoremdedekindicclemub 13203* Lemma for dedekindicc 13209. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  E. x  e.  ( A [,] B ) A. y  e.  L  y  <  x )
 
Theoremdedekindicclemloc 13204* Lemma for dedekindicc 13209. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   =>    |-  ( ph  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( E. z  e.  L  x  <  z  \/  A. z  e.  L  z  <  y ) ) )
 
Theoremdedekindicclemlub 13205* Lemma for dedekindicc 13209. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  E. x  e.  ( A [,] B ) ( A. y  e.  L  -.  x  < 
 y  /\  A. y  e.  ( A [,] B ) ( y  < 
 x  ->  E. z  e.  L  y  <  z
 ) ) )
 
Theoremdedekindicclemlu 13206* Lemma for dedekindicc 13209. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  E. x  e.  ( A [,] B ) ( A. q  e.  L  q  <  x  /\  A. r  e.  U  x  <  r ) )
 
Theoremdedekindicclemeu 13207* Lemma for dedekindicc 13209. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  C  e.  ( A [,] B ) )   &    |-  ( ph  ->  (
 A. q  e.  L  q  <  C  /\  A. r  e.  U  C  <  r ) )   &    |-  ( ph  ->  D  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( A. q  e.  L  q  <  D  /\  A. r  e.  U  D  <  r
 ) )   &    |-  ( ph  ->  C  <  D )   =>    |-  ( ph  -> F.  )
 
Theoremdedekindicclemicc 13208* Lemma for dedekindicc 13209. Same as dedekindicc 13209, except that we merely show  x to be an element of  ( A [,] B ). Later we will strengthen that to  ( A (,) B
). (Contributed by Jim Kingdon, 5-Jan-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  E! x  e.  ( A [,] B ) ( A. q  e.  L  q  <  x  /\  A. r  e.  U  x  <  r
 ) )
 
Theoremdedekindicc 13209* A Dedekind cut identifies a unique real number. Similar to df-inp 7401 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  L  C_  ( A [,] B ) )   &    |-  ( ph  ->  U  C_  ( A [,] B ) )   &    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )   &    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )   &    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )   &    |-  ( ph  ->  ( L  i^i  U )  =  (/) )   &    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U ) ) )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  E! x  e.  ( A (,) B ) ( A. q  e.  L  q  <  x  /\  A. r  e.  U  x  <  r
 ) )
 
8.0.2  Intermediate value theorem
 
Theoremivthinclemlm 13210* Lemma for ivthinc 13219. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )
 
Theoremivthinclemum 13211* Lemma for ivthinc 13219. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  R )
 
Theoremivthinclemlopn 13212* Lemma for ivthinc 13219. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   &    |-  ( ph  ->  Q  e.  L )   =>    |-  ( ph  ->  E. r  e.  L  Q  <  r
 )
 
Theoremivthinclemlr 13213* Lemma for ivthinc 13219. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
 
Theoremivthinclemuopn 13214* Lemma for ivthinc 13219. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   &    |-  ( ph  ->  S  e.  R )   =>    |-  ( ph  ->  E. q  e.  R  q  <  S )
 
Theoremivthinclemur 13215* Lemma for ivthinc 13219. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  R  <->  E. q  e.  R  q  <  r ) )
 
Theoremivthinclemdisj 13216* Lemma for ivthinc 13219. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  ( L  i^i  R )  =  (/) )
 
Theoremivthinclemloc 13217* Lemma for ivthinc 13219. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  (
 q  e.  L  \/  r  e.  R )
 ) )
 
Theoremivthinclemex 13218* Lemma for ivthinc 13219. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   &    |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }   &    |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `
  w ) }   =>    |-  ( ph  ->  E! z  e.  ( A (,) B ) ( A. q  e.  L  q  <  z  /\  A. r  e.  R  z  <  r ) )
 
Theoremivthinc 13219* The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `  B ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  x )  <  ( F `  y ) )   =>    |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
 
Theoremivthdec 13220* The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  U  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  ( A [,] B ) 
 C_  D )   &    |-  ( ph  ->  F  e.  ( D -cn-> CC ) )   &    |-  (
 ( ph  /\  x  e.  ( A [,] B ) )  ->  ( F `
  x )  e. 
 RR )   &    |-  ( ph  ->  ( ( F `  B )  <  U  /\  U  <  ( F `  A ) ) )   &    |-  (
 ( ( ph  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  < 
 y ) )  ->  ( F `  y )  <  ( F `  x ) )   =>    |-  ( ph  ->  E. c  e.  ( A (,) B ) ( F `  c )  =  U )
 
8.1  Derivatives
 
8.1.1  Real and complex differentiation
 
8.1.1.1  Derivatives of functions of one complex or real variable
 
Syntaxclimc 13221 The limit operator.
 class lim CC
 
Syntaxcdv 13222 The derivative operator.
 class  _D
 
Definitiondf-limced 13223* Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
 |- lim
 CC  =  ( f  e.  ( CC  ^pm  CC ) ,  x  e. 
 CC  |->  { y  e.  CC  |  ( ( f : dom  f --> CC  /\  dom  f  C_  CC )  /\  ( x  e.  CC  /\ 
 A. e  e.  RR+  E. d  e.  RR+  A. z  e.  dom  f ( ( z #  x  /\  ( abs `  ( z  -  x ) )  < 
 d )  ->  ( abs `  ( ( f `
  z )  -  y ) )  < 
 e ) ) ) } )
 
Definitiondf-dvap 13224* Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set  s here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of  CC and is well-behaved when  s contains no isolated points, we will restrict our attention to the cases  s  =  RR or  s  =  CC for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
 |- 
 _D  =  ( s  e.  ~P CC ,  f  e.  ( CC  ^pm  s )  |->  U_ x  e.  ( ( int `  (
 ( MetOpen `  ( abs  o. 
 -  ) )t  s ) ) `  dom  f
 ) ( { x }  X.  ( ( z  e.  { w  e. 
 dom  f  |  w #  x }  |->  ( ( ( f `  z
 )  -  ( f `
  x ) ) 
 /  ( z  -  x ) ) ) lim
 CC  x ) ) )
 
Theoremlimcrcl 13225 Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( C  e.  ( F lim CC  B )  ->  ( F : dom  F --> CC  /\  dom  F  C_  CC  /\  B  e.  CC )
 )
 
Theoremlimccl 13226 Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  ( F lim CC  B )  C_  CC
 
Theoremellimc3apf 13227* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  F/_ z F   =>    |-  ( ph  ->  ( C  e.  ( F lim
 CC  B )  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
 y )  ->  ( abs `  ( ( F `
  z )  -  C ) )  < 
 x ) ) ) )
 
Theoremellimc3ap 13228* Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( C  e.  ( F lim CC  B )  <->  ( C  e.  CC  /\  A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B ) )  < 
 y )  ->  ( abs `  ( ( F `
  z )  -  C ) )  < 
 x ) ) ) )
 
Theoremlimcdifap 13229* It suffices to consider functions which are not defined at  B to define the limit of a function. In particular, the value of the original function  F at  B does not affect the limit of  F. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   =>    |-  ( ph  ->  ( F lim CC  B )  =  ( ( F  |`  { x  e.  A  |  x #  B } ) lim CC  B ) )
 
Theoremlimcmpted 13230* Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
 |-  ( ph  ->  A  C_ 
 CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  (
 ( ph  /\  z  e.  A )  ->  D  e.  CC )   =>    |-  ( ph  ->  ( C  e.  ( (
 z  e.  A  |->  D ) lim CC  B )  <-> 
 ( C  e.  CC  /\ 
 A. x  e.  RR+  E. y  e.  RR+  A. z  e.  A  ( ( z #  B  /\  ( abs `  ( z  -  B ) )  <  y ) 
 ->  ( abs `  ( D  -  C ) )  <  x ) ) ) )
 
Theoremlimcimolemlt 13231* Lemma for limcimo 13232. (Contributed by Jim Kingdon, 3-Jul-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  ( Kt  S ) )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  { q  e.  C  |  q #  B }  C_  A )   &    |-  K  =  ( MetOpen `  ( abs  o. 
 -  ) )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  X  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  Y  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  A. z  e.  A  ( ( z #  B  /\  ( abs `  (
 z  -  B ) )  <  D ) 
 ->  ( abs `  (
 ( F `  z
 )  -  X ) )  <  ( ( abs `  ( X  -  Y ) )  / 
 2 ) ) )   &    |-  ( ph  ->  G  e.  RR+ )   &    |-  ( ph  ->  A. w  e.  A  ( ( w #  B  /\  ( abs `  ( w  -  B ) )  <  G )  ->  ( abs `  ( ( F `  w )  -  Y ) )  <  ( ( abs `  ( X  -  Y ) )  / 
 2 ) ) )   =>    |-  ( ph  ->  ( abs `  ( X  -  Y ) )  <  ( abs `  ( X  -  Y ) ) )
 
Theoremlimcimo 13232* Conditions which ensure there is at most one limit value of  F at  B. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 8-Jul-2023.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  B  e.  S )   &    |-  ( ph  ->  C  e.  ( Kt  S ) )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  { q  e.  C  |  q #  B }  C_  A )   &    |-  K  =  ( MetOpen `  ( abs  o. 
 -  ) )   =>    |-  ( ph  ->  E* x  x  e.  ( F lim CC  B ) )
 
Theoremlimcresi 13233 Any limit of  F is also a limit of the restriction of  F. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( F lim CC  B )  C_  ( ( F  |`  C ) lim CC  B )
 
Theoremcnplimcim 13234 If a function is continuous at  B, its limit at  B equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( Kt  A )   =>    |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B ) 
 ->  ( F : A --> CC  /\  ( F `  B )  e.  ( F lim CC  B ) ) ) )
 
Theoremcnplimclemle 13235 Lemma for cnplimccntop 13237. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( Kt  A )   &    |-  ( ph  ->  A 
 C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  ( F `  B )  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  Z  e.  A )   &    |-  (
 ( ph  /\  Z #  B  /\  ( abs `  ( Z  -  B ) )  <  D )  ->  ( abs `  ( ( F `  Z )  -  ( F `  B ) ) )  <  ( E  /  2 ) )   &    |-  ( ph  ->  ( abs `  ( Z  -  B ) )  <  D )   =>    |-  ( ph  ->  ( abs `  ( ( F `  Z )  -  ( F `  B ) ) )  <  E )
 
Theoremcnplimclemr 13236 Lemma for cnplimccntop 13237. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( Kt  A )   &    |-  ( ph  ->  A 
 C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  ( F `  B )  e.  ( F lim CC  B ) )   =>    |-  ( ph  ->  F  e.  ( ( J  CnP  K ) `  B ) )
 
Theoremcnplimccntop 13237 A function is continuous at  B iff its limit at  B equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( Kt  A )   =>    |-  ( ( A  C_  CC  /\  B  e.  A )  ->  ( F  e.  ( ( J  CnP  K ) `  B )  <-> 
 ( F : A --> CC  /\  ( F `  B )  e.  ( F lim CC  B ) ) ) )
 
Theoremcnlimcim 13238* If  F is a continuous function, the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.)
 |-  ( A  C_  CC  ->  ( F  e.  ( A -cn-> CC )  ->  ( F : A --> CC  /\  A. x  e.  A  ( F `  x )  e.  ( F lim CC  x ) ) ) )
 
Theoremcnlimc 13239*  F is a continuous function iff the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( A  C_  CC  ->  ( F  e.  ( A -cn-> CC )  <->  ( F : A
 --> CC  /\  A. x  e.  A  ( F `  x )  e.  ( F lim CC  x ) ) ) )
 
Theoremcnlimci 13240 If  F is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  F  e.  ( A -cn-> D ) )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  ( F `  B )  e.  ( F lim CC  B ) )
 
Theoremcnmptlimc 13241* If  F is a continuous function, then the limit of the function at any point equals its value. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  ( ph  ->  ( x  e.  A  |->  X )  e.  ( A -cn-> D ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( x  =  B  ->  X  =  Y )   =>    |-  ( ph  ->  Y  e.  ( ( x  e.  A  |->  X ) lim
 CC  B ) )
 
Theoremlimccnpcntop 13242 If the limit of  F at  B is  C and  G is continuous at  C, then the limit of  G  o.  F at  B is  G ( C ). (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 18-Jun-2023.)
 |-  ( ph  ->  F : A --> D )   &    |-  ( ph  ->  D  C_  CC )   &    |-  K  =  ( MetOpen `  ( abs  o.  -  )
 )   &    |-  J  =  ( Kt  D )   &    |-  ( ph  ->  C  e.  ( F lim CC  B ) )   &    |-  ( ph  ->  G  e.  (
 ( J  CnP  K ) `  C ) )   =>    |-  ( ph  ->  ( G `  C )  e.  (
 ( G  o.  F ) lim CC  B ) )
 
Theoremlimccnp2lem 13243* Lemma for limccnp2cntop 13244. This is most of the result, expressed in epsilon-delta form, with a large number of hypotheses so that lengthy expressions do not need to be repeated. (Contributed by Jim Kingdon, 9-Nov-2023.)
 |-  ( ( ph  /\  x  e.  A )  ->  R  e.  X )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  S  e.  Y )   &    |-  ( ph  ->  X  C_  CC )   &    |-  ( ph  ->  Y  C_ 
 CC )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   &    |-  J  =  ( ( K  tX  K )t  ( X  X.  Y ) )   &    |-  ( ph  ->  C  e.  ( ( x  e.  A  |->  R ) lim
 CC  B ) )   &    |-  ( ph  ->  D  e.  ( ( x  e.  A  |->  S ) lim CC  B ) )   &    |-  ( ph  ->  H  e.  (
 ( J  CnP  K ) `  <. C ,  D >. ) )   &    |-  F/ x ph   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  L  e.  RR+ )   &    |-  ( ph  ->  A. r  e.  X  A. s  e.  Y  (
 ( ( C ( ( abs  o.  -  )  |`  ( X  X.  X ) ) r )  <  L  /\  ( D ( ( abs 
 o.  -  )  |`  ( Y  X.  Y ) ) s )  <  L )  ->  ( ( C H D ) ( abs  o.  -  )
 ( r H s ) )  <  E ) )   &    |-  ( ph  ->  F  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  A  ( ( x #  B  /\  ( abs `  ( x  -  B ) )  <  F )  ->  ( abs `  ( R  -  C ) )  <  L ) )   &    |-  ( ph  ->  G  e.  RR+ )   &    |-  ( ph  ->  A. x  e.  A  ( ( x #  B  /\  ( abs `  ( x  -  B ) )  <  G )  ->  ( abs `  ( S  -  D ) )  <  L ) )   =>    |-  ( ph  ->  E. d  e.  RR+  A. x  e.  A  ( ( x #  B  /\  ( abs `  ( x  -  B ) )  <  d )  ->  ( abs `  ( ( R H S )  -  ( C H D ) ) )  <  E ) )
 
Theoremlimccnp2cntop 13244* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Nov-2023.)
 |-  ( ( ph  /\  x  e.  A )  ->  R  e.  X )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  S  e.  Y )   &    |-  ( ph  ->  X  C_  CC )   &    |-  ( ph  ->  Y  C_ 
 CC )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   &    |-  J  =  ( ( K  tX  K )t  ( X  X.  Y ) )   &    |-  ( ph  ->  C  e.  ( ( x  e.  A  |->  R ) lim
 CC  B ) )   &    |-  ( ph  ->  D  e.  ( ( x  e.  A  |->  S ) lim CC  B ) )   &    |-  ( ph  ->  H  e.  (
 ( J  CnP  K ) `  <. C ,  D >. ) )   =>    |-  ( ph  ->  ( C H D )  e.  ( ( x  e.  A  |->  ( R H S ) ) lim CC  B ) )
 
Theoremlimccoap 13245* Composition of two limits. This theorem is only usable in the case where  x #  X implies R(x) #  C so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
 |-  ( ( ph  /\  x  e.  { w  e.  A  |  w #  X }
 )  ->  R  e.  { w  e.  B  |  w #  C } )   &    |-  (
 ( ph  /\  y  e. 
 { w  e.  B  |  w #  C }
 )  ->  S  e.  CC )   &    |-  ( ph  ->  C  e.  ( ( x  e.  { w  e.  A  |  w #  X }  |->  R ) lim CC  X ) )   &    |-  ( ph  ->  D  e.  (
 ( y  e.  { w  e.  B  |  w #  C }  |->  S ) lim
 CC  C ) )   &    |-  ( y  =  R  ->  S  =  T )   =>    |-  ( ph  ->  D  e.  ( ( x  e. 
 { w  e.  A  |  w #  X }  |->  T ) lim CC  X ) )
 
Theoremreldvg 13246 The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
 |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S ) ) 
 ->  Rel  ( S  _D  F ) )
 
Theoremdvlemap 13247* Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
 |-  ( ph  ->  F : D --> CC )   &    |-  ( ph  ->  D  C_  CC )   &    |-  ( ph  ->  B  e.  D )   =>    |-  ( ( ph  /\  A  e.  { w  e.  D  |  w #  B }
 )  ->  ( (
 ( F `  A )  -  ( F `  B ) )  /  ( A  -  B ) )  e.  CC )
 
Theoremdvfvalap 13248* Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
 |-  T  =  ( Kt  S )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  ( ( S  _D  F )  = 
 U_ x  e.  (
 ( int `  T ) `  A ) ( { x }  X.  (
 ( z  e.  { w  e.  A  |  w #  x }  |->  ( ( ( F `  z
 )  -  ( F `
  x ) ) 
 /  ( z  -  x ) ) ) lim
 CC  x ) ) 
 /\  ( S  _D  F )  C_  ( ( ( int `  T ) `  A )  X.  CC ) ) )
 
Theoremeldvap 13249* The differentiable predicate. A function  F is differentiable at  B with derivative  C iff  F is defined in a neighborhood of  B and the difference quotient has limit  C at  B. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
 |-  T  =  ( Kt  S )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   &    |-  G  =  ( z  e.  { w  e.  A  |  w #  B }  |->  ( ( ( F `  z )  -  ( F `  B ) )  /  ( z  -  B ) ) )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ph  ->  ( B ( S  _D  F ) C  <->  ( B  e.  ( ( int `  T ) `  A )  /\  C  e.  ( G lim CC  B ) ) ) )
 
Theoremdvcl 13250 The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ( ph  /\  B ( S  _D  F ) C )  ->  C  e.  CC )
 
Theoremdvbssntrcntop 13251 The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   &    |-  J  =  ( Kt  S )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ph  ->  dom  ( S  _D  F )  C_  ( ( int `  J ) `  A ) )
 
Theoremdvbss 13252 The set of differentiable points is a subset of the domain of the function. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  A  C_  S )   =>    |-  ( ph  ->  dom  ( S  _D  F )  C_  A )
 
Theoremdvbsssg 13253 The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
 |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S ) ) 
 ->  dom  ( S  _D  F )  C_  S )
 
Theoremrecnprss 13254 Both  RR and  CC are subsets of  CC. (Contributed by Mario Carneiro, 10-Feb-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  S 
 C_  CC )
 
Theoremdvfgg 13255 Explicitly write out the functionality condition on derivative for  S  =  RR and 
CC. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
 |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC 
 ^pm  S ) )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
 
Theoremdvfpm 13256 The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.)
 |-  ( F  e.  ( CC  ^pm  RR )  ->  ( RR  _D  F ) : dom  ( RR 
 _D  F ) --> CC )
 
Theoremdvfcnpm 13257 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jul-2023.)
 |-  ( F  e.  ( CC  ^pm  CC )  ->  ( CC  _D  F ) : dom  ( CC 
 _D  F ) --> CC )
 
Theoremdvidlemap 13258* Lemma for dvid 13260 and dvconst 13259. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
 |-  ( ph  ->  F : CC --> CC )   &    |-  (
 ( ph  /\  ( x  e.  CC  /\  z  e.  CC  /\  z #  x ) )  ->  ( ( ( F `  z
 )  -  ( F `
  x ) ) 
 /  ( z  -  x ) )  =  B )   &    |-  B  e.  CC   =>    |-  ( ph  ->  ( CC  _D  F )  =  ( CC  X.  { B }
 ) )
 
Theoremdvconst 13259 Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
 |-  ( A  e.  CC  ->  ( CC  _D  ( CC  X.  { A }
 ) )  =  ( CC  X.  { 0 } ) )
 
Theoremdvid 13260 Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
 |-  ( CC  _D  (  _I  |`  CC ) )  =  ( CC  X.  { 1 } )
 
Theoremdvcnp2cntop 13261 A function is continuous at each point for which it is differentiable. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  J  =  ( Kt  A )   &    |-  K  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ( ( S 
 C_  CC  /\  F : A
 --> CC  /\  A  C_  S )  /\  B  e.  dom  ( S  _D  F ) )  ->  F  e.  ( ( J  CnP  K ) `  B ) )
 
Theoremdvcn 13262 A differentiable function is continuous. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-Sep-2015.)
 |-  ( ( ( S 
 C_  CC  /\  F : A
 --> CC  /\  A  C_  S )  /\  dom  ( S  _D  F )  =  A )  ->  F  e.  ( A -cn-> CC )
 )
 
Theoremdvaddxxbr 13263 The sum rule for derivatives at a point. That is, if the derivative of  F at  C is  K and the derivative of  G at  C is  L, then the derivative of the pointwise sum of those two functions at  C is  K  +  L. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  C ( S  _D  F ) K )   &    |-  ( ph  ->  C ( S  _D  G ) L )   &    |-  J  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ph  ->  C ( S  _D  ( F  oF  +  G ) ) ( K  +  L ) )
 
Theoremdvmulxxbr 13264 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 13266. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  C ( S  _D  F ) K )   &    |-  ( ph  ->  C ( S  _D  G ) L )   &    |-  J  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ph  ->  C ( S  _D  ( F  oF  x.  G ) ) ( ( K  x.  ( G `
  C ) )  +  ( L  x.  ( F `  C ) ) ) )
 
Theoremdvaddxx 13265 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 13263. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  C  e.  dom  ( S  _D  F ) )   &    |-  ( ph  ->  C  e.  dom  ( S  _D  G ) )   =>    |-  ( ph  ->  ( ( S  _D  ( F  oF  +  G ) ) `  C )  =  ( (
 ( S  _D  F ) `  C )  +  ( ( S  _D  G ) `  C ) ) )
 
Theoremdvmulxx 13266 The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 13264. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  C  e.  dom  ( S  _D  F ) )   &    |-  ( ph  ->  C  e.  dom  ( S  _D  G ) )   =>    |-  ( ph  ->  ( ( S  _D  ( F  oF  x.  G ) ) `  C )  =  ( (
 ( ( S  _D  F ) `  C )  x.  ( G `  C ) )  +  ( ( ( S  _D  G ) `  C )  x.  ( F `  C ) ) ) )
 
Theoremdviaddf 13267 The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X 
 C_  S )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  dom  ( S  _D  F )  =  X )   &    |-  ( ph  ->  dom  ( S  _D  G )  =  X )   =>    |-  ( ph  ->  ( S  _D  ( F  oF  +  G )
 )  =  ( ( S  _D  F )  oF  +  ( S  _D  G ) ) )
 
Theoremdvimulf 13268 The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  X 
 C_  S )   &    |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  G : X --> CC )   &    |-  ( ph  ->  dom  ( S  _D  F )  =  X )   &    |-  ( ph  ->  dom  ( S  _D  G )  =  X )   =>    |-  ( ph  ->  ( S  _D  ( F  oF  x.  G )
 )  =  ( ( ( S  _D  F )  oF  x.  G )  oF  +  (
 ( S  _D  G )  oF  x.  F ) ) )
 
Theoremdvcoapbr 13269* The chain rule for derivatives at a point. The  u #  C  -> 
( G `  u
) #  ( G `  C ) hypothesis constrains what functions work for  G. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  S )   &    |-  ( ph  ->  G : Y --> X )   &    |-  ( ph  ->  Y  C_  T )   &    |-  ( ph  ->  A. u  e.  Y  ( u #  C  ->  ( G `  u ) #  ( G `  C ) ) )   &    |-  ( ph  ->  S  C_  CC )   &    |-  ( ph  ->  T  C_ 
 CC )   &    |-  ( ph  ->  ( G `  C ) ( S  _D  F ) K )   &    |-  ( ph  ->  C ( T  _D  G ) L )   &    |-  J  =  (
 MetOpen `  ( abs  o.  -  ) )   =>    |-  ( ph  ->  C ( T  _D  ( F  o.  G ) ) ( K  x.  L ) )
 
Theoremdvcjbr 13270 The derivative of the conjugate of a function. For the (simpler but more limited) function version, see dvcj 13271. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ph  ->  F : X --> CC )   &    |-  ( ph  ->  X  C_  RR )   &    |-  ( ph  ->  C  e.  dom  ( RR  _D  F ) )   =>    |-  ( ph  ->  C ( RR  _D  ( *  o.  F ) ) ( * `  (
 ( RR  _D  F ) `  C ) ) )
 
Theoremdvcj 13271 The derivative of the conjugate of a function. For the (more general) relation version, see dvcjbr 13270. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( ( F : X
 --> CC  /\  X  C_  RR )  ->  ( RR 
 _D  ( *  o.  F ) )  =  ( *  o.  ( RR  _D  F ) ) )
 
Theoremdvfre 13272 The derivative of a real function is real. (Contributed by Mario Carneiro, 1-Sep-2014.)
 |-  ( ( F : A
 --> RR  /\  A  C_  RR )  ->  ( RR 
 _D  F ) : dom  ( RR  _D  F ) --> RR )
 
Theoremdvexp 13273* Derivative of a power function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( N  e.  NN  ->  ( CC  _D  ( x  e.  CC  |->  ( x ^ N ) ) )  =  ( x  e.  CC  |->  ( N  x.  ( x ^
 ( N  -  1
 ) ) ) ) )
 
Theoremdvexp2 13274* Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
 |-  ( N  e.  NN0  ->  ( CC  _D  ( x  e.  CC  |->  ( x ^ N ) ) )  =  ( x  e.  CC  |->  if ( N  =  0 , 
 0 ,  ( N  x.  ( x ^
 ( N  -  1
 ) ) ) ) ) )
 
Theoremdvrecap 13275* Derivative of the reciprocal function. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 28-Dec-2016.)
 |-  ( A  e.  CC  ->  ( CC  _D  ( x  e.  { w  e.  CC  |  w #  0 }  |->  ( A  /  x ) ) )  =  ( x  e. 
 { w  e.  CC  |  w #  0 }  |->  -u ( A  /  ( x ^ 2 ) ) ) )
 
Theoremdvmptidcn 13276 Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
 |-  ( CC  _D  ( x  e.  CC  |->  x ) )  =  ( x  e.  CC  |->  1 )
 
Theoremdvmptccn 13277* Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( CC  _D  ( x  e. 
 CC  |->  A ) )  =  ( x  e. 
 CC  |->  0 ) )
 
Theoremdvmptclx 13278* Closure lemma for dvmptmulx 13280 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ph  ->  X 
 C_  S )   =>    |-  ( ( ph  /\  x  e.  X ) 
 ->  B  e.  CC )
 
Theoremdvmptaddx 13279* Function-builder for derivative, addition rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ph  ->  X 
 C_  S )   &    |-  (
 ( ph  /\  x  e.  X )  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  D  e.  W )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  C ) )  =  ( x  e.  X  |->  D ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( A  +  C ) ) )  =  ( x  e.  X  |->  ( B  +  D ) ) )
 
Theoremdvmptmulx 13280* Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  A  e.  CC )   &    |-  (
 ( ph  /\  x  e.  X )  ->  B  e.  V )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ph  ->  X 
 C_  S )   &    |-  (
 ( ph  /\  x  e.  X )  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  D  e.  W )   &    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  C ) )  =  ( x  e.  X  |->  D ) )   =>    |-  ( ph  ->  ( S  _D  ( x  e.  X  |->  ( A  x.  C ) ) )  =  ( x  e.  X  |->  ( ( B  x.  C )  +  ( D  x.  A ) ) ) )
 
Theoremdvmptcmulcn 13281* Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
 |-  ( ( ph  /\  x  e.  CC )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  CC )  ->  B  e.  V )   &    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  A ) )  =  ( x  e.  CC  |->  B ) )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( CC  _D  ( x  e. 
 CC  |->  ( C  x.  A ) ) )  =  ( x  e. 
 CC  |->  ( C  x.  B ) ) )
 
Theoremdvmptnegcn 13282* Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
 |-  ( ( ph  /\  x  e.  CC )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  CC )  ->  B  e.  V )   &    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  A ) )  =  ( x  e.  CC  |->  B ) )   =>    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  -u A ) )  =  ( x  e.  CC  |->  -u B ) )
 
Theoremdvmptsubcn 13283* Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
 |-  ( ( ph  /\  x  e.  CC )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  CC )  ->  B  e.  V )   &    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  A ) )  =  ( x  e.  CC  |->  B ) )   &    |-  (
 ( ph  /\  x  e. 
 CC )  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  CC )  ->  D  e.  W )   &    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  C ) )  =  ( x  e.  CC  |->  D ) )   =>    |-  ( ph  ->  ( CC  _D  ( x  e.  CC  |->  ( A  -  C ) ) )  =  ( x  e.  CC  |->  ( B  -  D ) ) )
 
Theoremdvmptcjx 13284* Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.)
 |-  ( ( ph  /\  x  e.  X )  ->  A  e.  CC )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  B  e.  V )   &    |-  ( ph  ->  ( RR  _D  ( x  e.  X  |->  A ) )  =  ( x  e.  X  |->  B ) )   &    |-  ( ph  ->  X  C_  RR )   =>    |-  ( ph  ->  ( RR  _D  ( x  e.  X  |->  ( * `  A ) ) )  =  ( x  e.  X  |->  ( * `  B ) ) )
 
Theoremdveflem 13285 Derivative of the exponential function at 0. The key step in the proof is eftlub 11625, 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 13286 Derivative of the exponential function. (Contributed by Mario Carneiro, 9-Aug-2014.) (Proof shortened by Mario Carneiro, 10-Feb-2015.)
 |-  ( CC  _D  exp )  =  exp
 
PART 9  BASIC REAL AND COMPLEX FUNCTIONS
 
9.1  Basic trigonometry
 
9.1.1  The exponential, sine, and cosine functions (cont.)
 
Theoremefcn 13287 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)
 |- 
 exp  e.  ( CC -cn-> CC )
 
Theoremsincn 13288 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
 |- 
 sin  e.  ( CC -cn-> CC )
 
Theoremcoscn 13289 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)
 |- 
 cos  e.  ( CC -cn-> CC )
 
Theoremreeff1olem 13290* Lemma for reeff1o 13292. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  ( ( U  e.  RR  /\  1  <  U )  ->  E. x  e.  RR  ( exp `  x )  =  U )
 
Theoremreeff1oleme 13291* Lemma for reeff1o 13292. (Contributed by Jim Kingdon, 15-May-2024.)
 |-  ( U  e.  (
 0 (,) _e )  ->  E. x  e.  RR  ( exp `  x )  =  U )
 
Theoremreeff1o 13292 The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)
 |-  ( exp  |`  RR ) : RR
 -1-1-onto-> RR+
 
Theoremefltlemlt 13293 Lemma for eflt 13294. The converse of efltim 11633 plus the epsilon-delta setup. (Contributed by Jim Kingdon, 22-May-2024.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( exp `  A )  <  ( exp `  B ) )   &    |-  ( ph  ->  D  e.  RR+ )   &    |-  ( ph  ->  ( ( abs `  ( A  -  B ) )  <  D  ->  ( abs `  ( ( exp `  A )  -  ( exp `  B ) ) )  <  ( ( exp `  B )  -  ( exp `  A ) ) ) )   =>    |-  ( ph  ->  A  <  B )
 
Theoremeflt 13294 The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( exp `  A )  <  ( exp `  B ) ) )
 
Theoremefle 13295 The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <-> 
 ( exp `  A )  <_  ( exp `  B ) ) )
 
Theoremreefiso 13296 The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) (Revised by Mario Carneiro, 11-Mar-2014.)
 |-  ( exp  |`  RR )  Isom  <  ,  <  ( RR ,  RR+ )
 
Theoremreapef 13297 Apartness and the exponential function for reals. (Contributed by Jim Kingdon, 11-Jul-2024.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( exp `  A ) #  ( exp `  B )
 ) )
 
9.1.2  Properties of pi = 3.14159...
 
Theorempilem1 13298 Lemma for pire , pigt2lt4 and sinpi . (Contributed by Mario Carneiro, 9-May-2014.)
 |-  ( A  e.  ( RR+ 
 i^i  ( `' sin " { 0 } )
 ) 
 <->  ( A  e.  RR+  /\  ( sin `  A )  =  0 )
 )
 
Theoremcosz12 13299 Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.)
 |- 
 E. p  e.  (
 1 (,) 2 ) ( cos `  p )  =  0
 
Theoremsin0pilem1 13300* Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
 |- 
 E. p  e.  (
 1 (,) 2 ) ( ( cos `  p )  =  0  /\  A. x  e.  ( p (,) ( 2  x.  p ) ) 0  <  ( sin `  x ) )
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