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Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremarchrecnq 7201* Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.)
 |-  ( A  e.  Q.  ->  E. j  e.  N.  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  A )
 
Theoremarchrecpr 7202* Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)
 |-  ( A  e.  P.  ->  E. j  e.  N.  <. { l  |  l  <Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  u } >.  <P  A )
 
Theoremcaucvgprlemk 7203 Lemma for caucvgpr 7220. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.)
 |-  ( ph  ->  J  <N  K )   &    |-  ( ph  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  Q )   =>    |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q )
 
Theoremcaucvgprlemnkj 7204* Lemma for caucvgpr 7220. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  K  e.  N. )   &    |-  ( ph  ->  J  e.  N. )   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ph  ->  -.  (
 ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
 )  <Q  ( F `  K )  /\  ( ( F `  J )  +Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) )  <Q  S ) )
 
Theoremcaucvgprlemnbj 7205* Lemma for caucvgpr 7220. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  B  e.  N. )   &    |-  ( ph  ->  J  e.  N. )   =>    |-  ( ph  ->  -.  (
 ( ( F `  B )  +Q  ( *Q `  [ <. B ,  1o >. ]  ~Q  )
 )  +Q  ( *Q ` 
 [ <. J ,  1o >. ]  ~Q  ) )  <Q  ( F `  J ) )
 
Theoremcaucvgprlemm 7206* Lemma for caucvgpr 7220. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   =>    |-  ( ph  ->  ( E. s  e.  Q.  s  e.  ( 1st `  L )  /\  E. r  e.  Q.  r  e.  ( 2nd `  L ) ) )
 
Theoremcaucvgprlemopl 7207* Lemma for caucvgpr 7220. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   =>    |-  ( ( ph  /\  s  e.  ( 1st `  L ) )  ->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )
 
Theoremcaucvgprlemlol 7208* Lemma for caucvgpr 7220. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   =>    |-  ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
 
Theoremcaucvgprlemopu 7209* Lemma for caucvgpr 7220. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   =>    |-  ( ( ph  /\  r  e.  ( 2nd `  L ) )  ->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) )
 
Theoremcaucvgprlemupu 7210* Lemma for caucvgpr 7220. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   =>    |-  ( ( ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  r  e.  ( 2nd `  L ) )
 
Theoremcaucvgprlemrnd 7211* Lemma for caucvgpr 7220. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   =>    |-  ( ph  ->  (
 A. s  e.  Q.  ( s  e.  ( 1st `  L )  <->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) ) 
 /\  A. r  e.  Q.  ( r  e.  ( 2nd `  L )  <->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) ) ) )
 
Theoremcaucvgprlemdisj 7212* Lemma for caucvgpr 7220. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   =>    |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
 
Theoremcaucvgprlemloc 7213* Lemma for caucvgpr 7220. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   =>    |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
 s  <Q  r  ->  (
 s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L ) ) ) )
 
Theoremcaucvgprlemcl 7214* Lemma for caucvgpr 7220. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   =>    |-  ( ph  ->  L  e.  P. )
 
Theoremcaucvgprlemladdfu 7215* Lemma for caucvgpr 7220. Adding  S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ph  ->  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) )  C_  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `
  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S ) 
 <Q  u } )
 
Theoremcaucvgprlemladdrl 7216* Lemma for caucvgpr 7220. Adding  S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ph  ->  { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  ( ( F `  j )  +Q  S ) }  C_  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )
 
Theoremcaucvgprlem1 7217* Lemma for caucvgpr 7220. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   &    |-  ( ph  ->  Q  e.  Q. )   &    |-  ( ph  ->  J  <N  K )   &    |-  ( ph  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  Q )   =>    |-  ( ph  ->  <. { l  |  l  <Q  ( F `  K ) } ,  { u  |  ( F `  K )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) )
 
Theoremcaucvgprlem2 7218* Lemma for caucvgpr 7220. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   &    |-  ( ph  ->  Q  e.  Q. )   &    |-  ( ph  ->  J  <N  K )   &    |-  ( ph  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  Q )   =>    |-  ( ph  ->  L 
 <P  <. { l  |  l  <Q  ( ( F `  K )  +Q  Q ) } ,  { u  |  (
 ( F `  K )  +Q  Q )  <Q  u } >. )
 
Theoremcaucvgprlemlim 7219* Lemma for caucvgpr 7220. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   &    |-  L  =  <. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
 )  <Q  u } >.   =>    |-  ( ph  ->  A. x  e.  Q.  E. j  e.  N.  A. k  e.  N.  ( j  <N  k 
 ->  ( <. { l  |  l  <Q  ( F `  k ) } ,  { u  |  ( F `  k )  <Q  u } >.  <P  ( L 
 +P.  <. { l  |  l  <Q  x } ,  { u  |  x  <Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  ( ( F `  k )  +Q  x ) } ,  { u  |  (
 ( F `  k
 )  +Q  x )  <Q  u } >. ) ) )
 
Theoremcaucvgpr 7220* A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within  1  /  n of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a fraction  A, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of cauappcvgpr 7200 and caucvgprpr 7250. Reading cauappcvgpr 7200 first (the simplest of the three) might help understanding the other two.

(Contributed by Jim Kingdon, 18-Jun-2020.)

 |-  ( ph  ->  F : N. --> Q. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <Q  ( ( F `
  k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `
  k )  <Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) ) ) ) )   &    |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j
 ) )   =>    |-  ( ph  ->  E. y  e.  P.  A. x  e. 
 Q.  E. j  e.  N.  A. k  e.  N.  (
 j  <N  k  ->  ( <. { l  |  l 
 <Q  ( F `  k
 ) } ,  { u  |  ( F `  k )  <Q  u } >. 
 <P  ( y  +P.  <. { l  |  l  <Q  x } ,  { u  |  x  <Q  u } >. )  /\  y  <P  <. { l  |  l  <Q  ( ( F `  k )  +Q  x ) } ,  { u  |  (
 ( F `  k
 )  +Q  x )  <Q  u } >. ) ) )
 
Theoremcaucvgprprlemk 7221* Lemma for caucvgprpr 7250. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)
 |-  ( ph  ->  J  <N  K )   &    |-  ( ph  ->  <. { l  |  l  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )   =>    |-  ( ph  ->  <. { l  |  l  <Q  ( *Q ` 
 [ <. K ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )
 
Theoremcaucvgprprlemloccalc 7222* Lemma for caucvgprpr 7250. Rearranging some expressions for caucvgprprlemloc 7241. (Contributed by Jim Kingdon, 8-Feb-2021.)
 |-  ( ph  ->  S  <Q  T )   &    |-  ( ph  ->  Y  e.  Q. )   &    |-  ( ph  ->  ( S  +Q  Y )  =  T )   &    |-  ( ph  ->  X  e.  Q. )   &    |-  ( ph  ->  ( X  +Q  X ) 
 <Q  Y )   &    |-  ( ph  ->  M  e.  N. )   &    |-  ( ph  ->  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  <Q  X )   =>    |-  ( ph  ->  (
 <. { l  |  l 
 <Q  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  )
 ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. M ,  1o >. ]  ~Q  ) )  <Q  u } >.  +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. M ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. M ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  <. { l  |  l  <Q  T } ,  { u  |  T  <Q  u } >. )
 
Theoremcaucvgprprlemell 7223* Lemma for caucvgprpr 7250. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)
 |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  (
 l  +Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >.  <P  ( F `
  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( X  e.  ( 1st `  L )  <->  ( X  e.  Q. 
 /\  E. b  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  b
 ) ) )
 
Theoremcaucvgprprlemelu 7224* Lemma for caucvgprpr 7250. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-Jan-2021.)
 |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  (
 l  +Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >.  <P  ( F `
  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( X  e.  ( 2nd `  L )  <->  ( X  e.  Q. 
 /\  E. b  e.  N.  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  X } ,  {
 q  |  X  <Q  q } >. ) )
 
Theoremcaucvgprprlemcbv 7225* Lemma for caucvgprpr 7250. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   =>    |-  ( ph  ->  A. a  e.  N.  A. b  e. 
 N.  ( a  <N  b 
 ->  ( ( F `  a )  <P  ( ( F `  b ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  b ) 
 <P  ( ( F `  a )  +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. a ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )
 
Theoremcaucvgprprlemval 7226* Lemma for caucvgprpr 7250. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   =>    |-  ( ( ph  /\  A  <N  B )  ->  (
 ( F `  A )  <P  ( ( F `
  B )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. A ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. A ,  1o >. ]  ~Q  )  <Q  q } >. ) 
 /\  ( F `  B )  <P  ( ( F `  A ) 
 +P.  <. { p  |  p  <Q  ( *Q `  [ <. A ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. A ,  1o >. ]  ~Q  )  <Q  q } >. ) ) )
 
Theoremcaucvgprprlemnkltj 7227* Lemma for caucvgprpr 7250. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  K  e.  N. )   &    |-  ( ph  ->  J  e.  N. )   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ( ph  /\  K  <N  J )  ->  -.  ( <. { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  K )  /\  ( ( F `
  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) 
 <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. ) )
 
Theoremcaucvgprprlemnkeqj 7228* Lemma for caucvgprpr 7250. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  K  e.  N. )   &    |-  ( ph  ->  J  e.  N. )   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ( ph  /\  K  =  J )  ->  -.  ( <. { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  K )  /\  ( ( F `
  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) 
 <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. ) )
 
Theoremcaucvgprprlemnjltk 7229* Lemma for caucvgprpr 7250. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  K  e.  N. )   &    |-  ( ph  ->  J  e.  N. )   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ( ph  /\  J  <N  K )  ->  -.  ( <. { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  K )  /\  ( ( F `
  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) 
 <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. ) )
 
Theoremcaucvgprprlemnkj 7230* Lemma for caucvgprpr 7250. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  K  e.  N. )   &    |-  ( ph  ->  J  e.  N. )   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ph  ->  -.  ( <. { p  |  p  <Q  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( S  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  K )  /\  ( ( F `
  J )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  q } >. ) 
 <P  <. { p  |  p  <Q  S } ,  { q  |  S  <Q  q } >. ) )
 
Theoremcaucvgprprlemnbj 7231* Lemma for caucvgprpr 7250. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  B  e.  N. )   &    |-  ( ph  ->  J  e.  N. )   =>    |-  ( ph  ->  -.  (
 ( ( F `  B )  +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. B ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. B ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 <P  ( F `  J ) )
 
Theoremcaucvgprprlemml 7232* Lemma for caucvgprpr 7250. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 1st `  L ) )
 
Theoremcaucvgprprlemmu 7233* Lemma for caucvgprpr 7250. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ph  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
 
Theoremcaucvgprprlemm 7234* Lemma for caucvgprpr 7250. The putative limit is inhabited. (Contributed by Jim Kingdon, 21-Dec-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ph  ->  ( E. s  e.  Q.  s  e.  ( 1st `  L )  /\  E. t  e.  Q.  t  e.  ( 2nd `  L ) ) )
 
Theoremcaucvgprprlemopl 7235* Lemma for caucvgprpr 7250. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ( ph  /\  s  e.  ( 1st `  L ) )  ->  E. t  e.  Q.  ( s  <Q  t 
 /\  t  e.  ( 1st `  L ) ) )
 
Theoremcaucvgprprlemlol 7236* Lemma for caucvgprpr 7250. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
 
Theoremcaucvgprprlemopu 7237* Lemma for caucvgprpr 7250. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ( ph  /\  t  e.  ( 2nd `  L ) )  ->  E. s  e.  Q.  ( s  <Q  t 
 /\  s  e.  ( 2nd `  L ) ) )
 
Theoremcaucvgprprlemupu 7238* Lemma for caucvgprpr 7250. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 21-Dec-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ( ph  /\  s  <Q  t  /\  s  e.  ( 2nd `  L ) )  ->  t  e.  ( 2nd `  L ) )
 
Theoremcaucvgprprlemrnd 7239* Lemma for caucvgprpr 7250. The putative limit is rounded. (Contributed by Jim Kingdon, 21-Dec-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ph  ->  ( A. s  e.  Q.  ( s  e.  ( 1st `  L )  <->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) ) 
 /\  A. t  e.  Q.  ( t  e.  ( 2nd `  L )  <->  E. s  e.  Q.  ( s  <Q  t  /\  s  e.  ( 2nd `  L ) ) ) ) )
 
Theoremcaucvgprprlemdisj 7240* Lemma for caucvgprpr 7250. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
 
Theoremcaucvgprprlemloc 7241* Lemma for caucvgprpr 7250. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ph  ->  A. s  e.  Q.  A. t  e. 
 Q.  ( s  <Q  t 
 ->  ( s  e.  ( 1st `  L )  \/  t  e.  ( 2nd `  L ) ) ) )
 
Theoremcaucvgprprlemcl 7242* Lemma for caucvgprpr 7250. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ph  ->  L  e.  P. )
 
Theoremcaucvgprprlemclphr 7243* Lemma for caucvgprpr 7250. The putative limit is a positive real. Like caucvgprprlemcl 7242 but without a distinct variable constraint between  ph and  r. (Contributed by Jim Kingdon, 19-Jun-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ph  ->  L  e.  P. )
 
Theoremcaucvgprprlemexbt 7244* Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   &    |-  ( ph  ->  Q  e.  Q. )   &    |-  ( ph  ->  T  e.  P. )   &    |-  ( ph  ->  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
 q  |  Q  <Q  q } >. )  <P  T )   =>    |-  ( ph  ->  E. b  e.  N.  ( ( ( F `  b ) 
 +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
 +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P  T )
 
Theoremcaucvgprprlemexb 7245* Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   &    |-  ( ph  ->  Q  e.  P. )   &    |-  ( ph  ->  R  e.  N. )   =>    |-  ( ph  ->  ( ( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( ( F `  R )  +P.  Q )  ->  E. b  e.  N.  ( ( ( F `
  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
 +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) )  <P  ( ( F `  R )  +P.  Q ) ) )
 
Theoremcaucvgprprlemaddq 7246* Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   &    |-  ( ph  ->  X  e.  P. )   &    |-  ( ph  ->  Q  e.  P. )   &    |-  ( ph  ->  E. r  e.  N.  ( X  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( ( F `  r
 )  +P.  Q )
 )   =>    |-  ( ph  ->  X  <P  ( L  +P.  Q ) )
 
Theoremcaucvgprprlem1 7247* Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   &    |-  ( ph  ->  Q  e.  P. )   &    |-  ( ph  ->  J 
 <N  K )   &    |-  ( ph  ->  <. { l  |  l  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )   =>    |-  ( ph  ->  ( F `  K )  <P  ( L  +P.  Q ) )
 
Theoremcaucvgprprlem2 7248* Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   &    |-  ( ph  ->  Q  e.  P. )   &    |-  ( ph  ->  J 
 <N  K )   &    |-  ( ph  ->  <. { l  |  l  <Q  ( *Q `  [ <. J ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  u } >.  <P  Q )   =>    |-  ( ph  ->  L  <P  ( ( F `  K )  +P.  Q ) )
 
Theoremcaucvgprprlemlim 7249* Lemma for caucvgprpr 7250. The putative limit is a limit. (Contributed by Jim Kingdon, 21-Nov-2020.)
 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   &    |-  L  =  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
 ) } ,  {
 q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >. 
 <P  ( F `  r
 ) } ,  { u  e.  Q.  |  E. r  e.  N.  (
 ( F `  r
 )  +P.  <. { p  |  p  <Q  ( *Q ` 
 [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  u } ,  {
 q  |  u  <Q  q } >. } >.   =>    |-  ( ph  ->  A. x  e.  P.  E. j  e. 
 N.  A. k  e.  N.  ( j  <N  k  ->  ( ( F `  k )  <P  ( L 
 +P.  x )  /\  L  <P  ( ( F `
  k )  +P.  x ) ) ) )
 
Theoremcaucvgprpr 7250* A Cauchy sequence of positive reals with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within  1  /  n of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a given value  A, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This is similar to caucvgpr 7220 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7200) might help in understanding this one, as they are slightly simpler but similarly structured. (Contributed by Jim Kingdon, 14-Nov-2020.)

 |-  ( ph  ->  F : N. --> P. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <P  ( ( F `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( F `  k )  <P  ( ( F `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )   =>    |-  ( ph  ->  E. y  e.  P.  A. x  e. 
 P.  E. j  e.  N.  A. k  e.  N.  (
 j  <N  k  ->  (
 ( F `  k
 )  <P  ( y  +P.  x )  /\  y  <P  ( ( F `  k
 )  +P.  x )
 ) ) )
 
Definitiondf-enr 7251* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)
 |- 
 ~R  =  { <. x ,  y >.  |  ( ( x  e.  ( P.  X.  P. )  /\  y  e.  ( P.  X. 
 P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  +P.  u )  =  ( w 
 +P.  v ) ) ) }
 
Definitiondf-nr 7252 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)
 |- 
 R.  =  ( ( P.  X.  P. ) /.  ~R  )
 
Definitiondf-plr 7253* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)
 |- 
 +R  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] 
 ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( w 
 +P.  u ) ,  ( v  +P.  f
 ) >. ]  ~R  )
 ) }
 
Definitiondf-mr 7254* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)
 |- 
 .R  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] 
 ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u ) 
 +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  ( v 
 .P.  u ) )
 >. ]  ~R  ) ) }
 
Definitiondf-ltr 7255* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.)
 |- 
 <R  =  { <. x ,  y >.  |  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ] 
 ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u ) 
 <P  ( w  +P.  v
 ) ) ) }
 
Definitiondf-0r 7256 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.)
 |- 
 0R  =  [ <. 1P ,  1P >. ]  ~R
 
Definitiondf-1r 7257 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.)
 |- 
 1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
 
Definitiondf-m1r 7258 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.)
 |- 
 -1R  =  [ <. 1P ,  ( 1P  +P.  1P ) >. ]  ~R
 
Theoremenrbreq 7259 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  ->  ( <. A ,  B >.  ~R  <. C ,  D >.  <-> 
 ( A  +P.  D )  =  ( B  +P.  C ) ) )
 
Theoremenrer 7260 The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |- 
 ~R  Er  ( P.  X. 
 P. )
 
Theoremenreceq 7261 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  ->  ( [ <. A ,  B >. ]  ~R  =  [ <. C ,  D >. ] 
 ~R 
 <->  ( A  +P.  D )  =  ( B  +P.  C ) ) )
 
Theoremenrex 7262 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)
 |- 
 ~R  e.  _V
 
Theoremltrelsr 7263 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)
 |- 
 <R  C_  ( R.  X.  R. )
 
Theoremaddcmpblnr 7264 Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.)
 |-  ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
 )  /\  ( ( F  e.  P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) ) 
 ->  ( ( ( A 
 +P.  D )  =  ( B  +P.  C ) 
 /\  ( F  +P.  S )  =  ( G 
 +P.  R ) )  ->  <. ( A  +P.  F ) ,  ( B  +P.  G ) >.  ~R  <. ( C  +P.  R ) ,  ( D  +P.  S ) >. ) )
 
Theoremmulcmpblnrlemg 7265 Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)
 |-  ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
 )  /\  ( ( F  e.  P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) ) 
 ->  ( ( ( A 
 +P.  D )  =  ( B  +P.  C ) 
 /\  ( F  +P.  S )  =  ( G 
 +P.  R ) )  ->  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  +P.  (
 ( C  .P.  S )  +P.  ( D  .P.  R ) ) ) )  =  ( ( D 
 .P.  F )  +P.  (
 ( ( A  .P.  G )  +P.  ( B 
 .P.  F ) )  +P.  ( ( C  .P.  R )  +P.  ( D 
 .P.  S ) ) ) ) ) )
 
Theoremmulcmpblnr 7266 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.)
 |-  ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
 )  /\  ( ( F  e.  P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) ) 
 ->  ( ( ( A 
 +P.  D )  =  ( B  +P.  C ) 
 /\  ( F  +P.  S )  =  ( G 
 +P.  R ) )  ->  <. ( ( A  .P.  F )  +P.  ( B 
 .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B 
 .P.  F ) ) >.  ~R 
 <. ( ( C  .P.  R )  +P.  ( D 
 .P.  S ) ) ,  ( ( C  .P.  S )  +P.  ( D 
 .P.  R ) ) >. ) )
 
Theoremprsrlem1 7267* Decomposing signed reals into positive reals. Lemma for addsrpr 7270 and mulsrpr 7271. (Contributed by Jim Kingdon, 30-Dec-2019.)
 |-  ( ( ( A  e.  ( ( P. 
 X.  P. ) /.  ~R  )  /\  B  e.  (
 ( P.  X.  P. ) /.  ~R  ) ) 
 /\  ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  ( A  =  [ <. s ,  f >. ] 
 ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )  ->  ( (
 ( ( w  e. 
 P.  /\  v  e.  P. )  /\  ( s  e.  P.  /\  f  e.  P. ) )  /\  ( ( u  e. 
 P.  /\  t  e.  P. )  /\  ( g  e.  P.  /\  h  e.  P. ) ) ) 
 /\  ( ( w 
 +P.  f )  =  ( v  +P.  s
 )  /\  ( u  +P.  h )  =  ( t  +P.  g ) ) ) )
 
Theoremaddsrmo 7268* There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
 |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  ->  E* z E. w E. v E. u E. t
 ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( w 
 +P.  u ) ,  ( v  +P.  t
 ) >. ]  ~R  )
 )
 
Theoremmulsrmo 7269* There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
 |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  ->  E* z E. w E. v E. u E. t
 ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u ) 
 +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  ( v 
 .P.  u ) )
 >. ]  ~R  ) )
 
Theoremaddsrpr 7270 Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  ->  ( [ <. A ,  B >. ]  ~R  +R  [ <. C ,  D >. ] 
 ~R  )  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )
 
Theoremmulsrpr 7271 Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ] 
 ~R  )  =  [ <. ( ( A  .P.  C )  +P.  ( B 
 .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B 
 .P.  C ) ) >. ] 
 ~R  )
 
Theoremltsrprg 7272 Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  ->  ( [ <. A ,  B >. ]  ~R  <R  [ <. C ,  D >. ]  ~R  <->  ( A  +P.  D )  <P  ( B  +P.  C ) ) )
 
Theoremgt0srpr 7273 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)
 |-  ( 0R  <R  [ <. A ,  B >. ]  ~R  <->  B  <P  A )
 
Theorem0nsr 7274 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.)
 |- 
 -.  (/)  e.  R.
 
Theorem0r 7275 The constant  0R is a signed real. (Contributed by NM, 9-Aug-1995.)
 |- 
 0R  e.  R.
 
Theorem1sr 7276 The constant  1R is a signed real. (Contributed by NM, 9-Aug-1995.)
 |- 
 1R  e.  R.
 
Theoremm1r 7277 The constant  -1R is a signed real. (Contributed by NM, 9-Aug-1995.)
 |- 
 -1R  e.  R.
 
Theoremaddclsr 7278 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  +R  B )  e.  R. )
 
Theoremmulclsr 7279 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  .R  B )  e.  R. )
 
Theoremaddcomsrg 7280 Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  +R  B )  =  ( B  +R  A ) )
 
Theoremaddasssrg 7281 Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\  C  e.  R. )  ->  ( ( A  +R  B )  +R  C )  =  ( A  +R  ( B  +R  C ) ) )
 
Theoremmulcomsrg 7282 Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  .R  B )  =  ( B  .R  A ) )
 
Theoremmulasssrg 7283 Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\  C  e.  R. )  ->  ( ( A  .R  B )  .R  C )  =  ( A  .R  ( B  .R  C ) ) )
 
Theoremdistrsrg 7284 Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  .R  ( B  +R  C ) )  =  ( ( A 
 .R  B )  +R  ( A  .R  C ) ) )
 
Theoremm1p1sr 7285 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
 |-  ( -1R  +R  1R )  =  0R
 
Theoremm1m1sr 7286 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
 |-  ( -1R  .R  -1R )  =  1R
 
Theoremlttrsr 7287* Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)
 |-  ( ( f  e. 
 R.  /\  g  e.  R. 
 /\  h  e.  R. )  ->  ( ( f 
 <R  g  /\  g  <R  h )  ->  f  <R  h ) )
 
Theoremltposr 7288 Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)
 |- 
 <R  Po  R.
 
Theoremltsosr 7289 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
 |- 
 <R  Or  R.
 
Theorem0lt1sr 7290 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)
 |- 
 0R  <R  1R
 
Theorem1ne0sr 7291 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)
 |- 
 -.  1R  =  0R
 
Theorem0idsr 7292 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)
 |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
 
Theorem1idsr 7293 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
 |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
 
Theorem00sr 7294 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
 |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )
 
Theoremltasrg 7295 Ordering property of addition. (Contributed by NM, 10-May-1996.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
 
Theorempn0sr 7296 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)
 |-  ( A  e.  R.  ->  ( A  +R  ( A  .R  -1R ) )  =  0R )
 
Theoremnegexsr 7297* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.)
 |-  ( A  e.  R.  ->  E. x  e.  R.  ( A  +R  x )  =  0R )
 
Theoremrecexgt0sr 7298* The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)
 |-  ( 0R  <R  A  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( A  .R  x )  =  1R ) )
 
Theoremrecexsrlem 7299* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.)
 |-  ( 0R  <R  A  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
 
Theoremaddgt0sr 7300 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.)
 |-  ( ( 0R  <R  A 
 /\  0R  <R  B ) 
 ->  0R  <R  ( A  +R  B ) )
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