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Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaucvgprprlemdisj 7701* Lemma for caucvgprpr 7711. 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 7702* Lemma for caucvgprpr 7711. 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 7703* Lemma for caucvgprpr 7711. 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 7704* Lemma for caucvgprpr 7711. The putative limit is a positive real. Like caucvgprprlemcl 7703 but without a disjoint variable condition 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 7705* Lemma for caucvgprpr 7711. 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 7706* Lemma for caucvgprpr 7711. 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 7707* Lemma for caucvgprpr 7711. 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 7708* Lemma for caucvgprpr 7711. 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 7709* Lemma for caucvgprpr 7711. 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 7710* Lemma for caucvgprpr 7711. 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 7711* 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 7681 except that values of the sequence are positive reals rather than positive fractions. Reading that proof first (or cauappcvgpr 7661) 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 )
 ) ) )
 
Theoremsuplocexprlemell 7712* Lemma for suplocexpr 7724. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
 |-  ( B  e.  U. ( 1st " A )  <->  E. x  e.  A  B  e.  ( 1st `  x ) )
 
Theoremsuplocexprlem2b 7713 Lemma for suplocexpr 7724. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
 |-  B  =  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u } >.   =>    |-  ( A  C_  P.  ->  ( 2nd `  B )  =  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u }
 )
 
Theoremsuplocexprlemss 7714* Lemma for suplocexpr 7724. 
A is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   =>    |-  ( ph  ->  A  C_  P. )
 
Theoremsuplocexprlemml 7715* Lemma for suplocexpr 7724. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   =>    |-  ( ph  ->  E. s  e.  Q.  s  e.  U. ( 1st " A ) )
 
Theoremsuplocexprlemrl 7716* Lemma for suplocexpr 7724. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   =>    |-  ( ph  ->  A. q  e. 
 Q.  ( q  e. 
 U. ( 1st " A ) 
 <-> 
 E. r  e.  Q.  ( q  <Q  r  /\  r  e.  U. ( 1st " A ) ) ) )
 
Theoremsuplocexprlemmu 7717* Lemma for suplocexpr 7724. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   &    |-  B  =  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u } >.   =>    |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 2nd `  B ) )
 
Theoremsuplocexprlemru 7718* Lemma for suplocexpr 7724. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   &    |-  B  =  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u } >.   =>    |-  ( ph  ->  A. r  e. 
 Q.  ( r  e.  ( 2nd `  B ) 
 <-> 
 E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) )
 
Theoremsuplocexprlemdisj 7719* Lemma for suplocexpr 7724. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   &    |-  B  =  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u } >.   =>    |-  ( ph  ->  A. q  e. 
 Q.  -.  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )
 
Theoremsuplocexprlemloc 7720* Lemma for suplocexpr 7724. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   &    |-  B  =  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u } >.   =>    |-  ( ph  ->  A. q  e. 
 Q.  A. r  e.  Q.  ( q  <Q  r  ->  ( q  e.  U. ( 1st " A )  \/  r  e.  ( 2nd `  B ) ) ) )
 
Theoremsuplocexprlemex 7721* Lemma for suplocexpr 7724. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   &    |-  B  =  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u } >.   =>    |-  ( ph  ->  B  e.  P. )
 
Theoremsuplocexprlemub 7722* Lemma for suplocexpr 7724. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   &    |-  B  =  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u } >.   =>    |-  ( ph  ->  A. y  e.  A  -.  B  <P  y )
 
Theoremsuplocexprlemlub 7723* Lemma for suplocexpr 7724. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   &    |-  B  =  <. U. ( 1st " A ) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u } >.   =>    |-  ( ph  ->  ( y  <P  B  ->  E. z  e.  A  y  <P  z ) )
 
Theoremsuplocexpr 7724* An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
 |-  ( ph  ->  E. x  x  e.  A )   &    |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
 <P  x )   &    |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  ( x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )   =>    |-  ( ph  ->  E. x  e.  P.  ( A. y  e.  A  -.  x  <P  y 
 /\  A. y  e.  P.  ( y  <P  x  ->  E. z  e.  A  y  <P  z ) ) )
 
Definitiondf-enr 7725* 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 7726 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 7727* 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 7728* 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 7729* 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 7730 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 7731 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 7732 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 7733 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 7734 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 7735 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 7736 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)
 |- 
 ~R  e.  _V
 
Theoremltrelsr 7737 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)
 |- 
 <R  C_  ( R.  X.  R. )
 
Theoremaddcmpblnr 7738 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 7739 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 7740 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 7741* Decomposing signed reals into positive reals. Lemma for addsrpr 7744 and mulsrpr 7745. (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 7742* 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 7743* 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 7744 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 7745 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 7746 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 7747 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)
 |-  ( 0R  <R  [ <. A ,  B >. ]  ~R  <->  B  <P  A )
 
Theorem0nsr 7748 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.)
 |- 
 -.  (/)  e.  R.
 
Theorem0r 7749 The constant  0R is a signed real. (Contributed by NM, 9-Aug-1995.)
 |- 
 0R  e.  R.
 
Theorem1sr 7750 The constant  1R is a signed real. (Contributed by NM, 9-Aug-1995.)
 |- 
 1R  e.  R.
 
Theoremm1r 7751 The constant  -1R is a signed real. (Contributed by NM, 9-Aug-1995.)
 |- 
 -1R  e.  R.
 
Theoremaddclsr 7752 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  +R  B )  e.  R. )
 
Theoremmulclsr 7753 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  .R  B )  e.  R. )
 
Theoremaddcomsrg 7754 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 7755 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 7756 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 7757 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 7758 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 7759 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
 |-  ( -1R  +R  1R )  =  0R
 
Theoremm1m1sr 7760 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
 |-  ( -1R  .R  -1R )  =  1R
 
Theoremlttrsr 7761* 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 7762 Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)
 |- 
 <R  Po  R.
 
Theoremltsosr 7763 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
 |- 
 <R  Or  R.
 
Theorem0lt1sr 7764 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)
 |- 
 0R  <R  1R
 
Theorem1ne0sr 7765 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)
 |- 
 -.  1R  =  0R
 
Theorem0idsr 7766 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 7767 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
 |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
 
Theorem00sr 7768 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
 |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )
 
Theoremltasrg 7769 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 7770 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)
 |-  ( A  e.  R.  ->  ( A  +R  ( A  .R  -1R ) )  =  0R )
 
Theoremnegexsr 7771* 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 7772* 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 7773* 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 7774 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 ) )
 
Theoremltadd1sr 7775 Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
 |-  ( A  e.  R.  ->  A  <R  ( A  +R  1R ) )
 
Theoremltm1sr 7776 Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)
 |-  ( A  e.  R.  ->  ( A  +R  -1R )  <R  A )
 
Theoremmulgt0sr 7777 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)
 |-  ( ( 0R  <R  A 
 /\  0R  <R  B ) 
 ->  0R  <R  ( A  .R  B ) )
 
Theoremaptisr 7778 Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\ 
 -.  ( A  <R  B  \/  B  <R  A ) )  ->  A  =  B )
 
Theoremmulextsr1lem 7779 Lemma for mulextsr1 7780. (Contributed by Jim Kingdon, 17-Feb-2020.)
 |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. ) )  ->  ( ( ( ( X  .P.  U ) 
 +P.  ( Y  .P.  V ) )  +P.  (
 ( Z  .P.  V )  +P.  ( W  .P.  U ) ) )  <P  ( ( ( X  .P.  V )  +P.  ( Y 
 .P.  U ) )  +P.  ( ( Z  .P.  U )  +P.  ( W 
 .P.  V ) ) ) 
 ->  ( ( X  +P.  W )  <P  ( Y  +P.  Z )  \/  ( Z  +P.  Y )  <P  ( W  +P.  X ) ) ) )
 
Theoremmulextsr1 7780 Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\  C  e.  R. )  ->  ( ( A  .R  C )  <R  ( B 
 .R  C )  ->  ( A  <R  B  \/  B  <R  A ) ) )
 
Theoremarchsr 7781* For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression  [ <. ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  },  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R is the embedding of the positive integer  x into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
 |-  ( A  e.  R.  ->  E. x  e.  N.  A  <R  [ <. ( <. { l  |  l  <Q  [
 <. x ,  1o >. ] 
 ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
 
Theoremsrpospr 7782* Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)
 |-  ( ( A  e.  R. 
 /\  0R  <R  A ) 
 ->  E! x  e.  P.  [
 <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )
 
Theoremprsrcl 7783 Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)
 |-  ( A  e.  P.  ->  [ <. ( A  +P.  1P ) ,  1P >. ] 
 ~R  e.  R. )
 
Theoremprsrpos 7784 Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.)
 |-  ( A  e.  P.  ->  0R  <R  [ <. ( A 
 +P.  1P ) ,  1P >. ]  ~R  )
 
Theoremprsradd 7785 Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  [ <. ( ( A  +P.  B ) 
 +P.  1P ) ,  1P >. ]  ~R  =  ( [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  +R 
 [ <. ( B  +P.  1P ) ,  1P >. ] 
 ~R  ) )
 
Theoremprsrlt 7786 Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  <P  B  <->  [ <. ( A  +P.  1P ) ,  1P >. ] 
 ~R  <R  [ <. ( B 
 +P.  1P ) ,  1P >. ]  ~R  ) )
 
Theoremprsrriota 7787* Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
 |-  ( ( A  e.  R. 
 /\  0R  <R  A ) 
 ->  [ <. ( ( iota_ x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  +P.  1P ) ,  1P >. ]  ~R  =  A )
 
Theoremcaucvgsrlemcl 7788* Lemma for caucvgsr 7801. Terms of the sequence from caucvgsrlemgt1 7794 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   =>    |-  ( ( ph  /\  A  e.  N. )  ->  ( iota_
 y  e.  P.  ( F `  A )  =  [ <. ( y  +P.  1P ) ,  1P >. ] 
 ~R  )  e.  P. )
 
Theoremcaucvgsrlemasr 7789* Lemma for caucvgsr 7801. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
 |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `
  m ) )   =>    |-  ( ph  ->  A  e.  R. )
 
Theoremcaucvgsrlemfv 7790* Lemma for caucvgsr 7801. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   &    |-  G  =  ( x  e.  N.  |->  (
 iota_ y  e.  P.  ( F `  x )  =  [ <. ( y 
 +P.  1P ) ,  1P >. ]  ~R  ) )   =>    |-  ( ( ph  /\  A  e.  N. )  ->  [ <. ( ( G `
  A )  +P.  1P ) ,  1P >. ] 
 ~R  =  ( F `
  A ) )
 
Theoremcaucvgsrlemf 7791* Lemma for caucvgsr 7801. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   &    |-  G  =  ( x  e.  N.  |->  (
 iota_ y  e.  P.  ( F `  x )  =  [ <. ( y 
 +P.  1P ) ,  1P >. ]  ~R  ) )   =>    |-  ( ph  ->  G : N. --> P. )
 
Theoremcaucvgsrlemcau 7792* Lemma for caucvgsr 7801. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   &    |-  G  =  ( x  e.  N.  |->  (
 iota_ y  e.  P.  ( F `  x )  =  [ <. ( y 
 +P.  1P ) ,  1P >. ]  ~R  ) )   =>    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( G `  n )  <P  ( ( G `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( G `  k )  <P  ( ( G `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )
 
Theoremcaucvgsrlembound 7793* Lemma for caucvgsr 7801. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   &    |-  G  =  ( x  e.  N.  |->  (
 iota_ y  e.  P.  ( F `  x )  =  [ <. ( y 
 +P.  1P ) ,  1P >. ]  ~R  ) )   =>    |-  ( ph  ->  A. m  e.  N.  1P  <P  ( G `  m ) )
 
Theoremcaucvgsrlemgt1 7794* Lemma for caucvgsr 7801. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   =>    |-  ( ph  ->  E. y  e.  R.  A. x  e. 
 R.  ( 0R  <R  x 
 ->  E. j  e.  N.  A. i  e.  N.  (
 j  <N  i  ->  (
 ( F `  i
 )  <R  ( y  +R  x )  /\  y  <R  ( ( F `  i
 )  +R  x )
 ) ) ) )
 
Theoremcaucvgsrlemoffval 7795* Lemma for caucvgsr 7801. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   &    |-  G  =  ( a  e.  N.  |->  ( ( ( F `  a )  +R  1R )  +R  ( A  .R  -1R ) ) )   =>    |-  ( ( ph  /\  J  e.  N. )  ->  ( ( G `  J )  +R  A )  =  ( ( F `
  J )  +R  1R ) )
 
Theoremcaucvgsrlemofff 7796* Lemma for caucvgsr 7801. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   &    |-  G  =  ( a  e.  N.  |->  ( ( ( F `  a )  +R  1R )  +R  ( A  .R  -1R ) ) )   =>    |-  ( ph  ->  G : N. --> R. )
 
Theoremcaucvgsrlemoffcau 7797* Lemma for caucvgsr 7801. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   &    |-  G  =  ( a  e.  N.  |->  ( ( ( F `  a )  +R  1R )  +R  ( A  .R  -1R ) ) )   =>    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( G `  n )  <R  ( ( G `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( G `  k )  <R  ( ( G `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )
 
Theoremcaucvgsrlemoffgt1 7798* Lemma for caucvgsr 7801. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   &    |-  G  =  ( a  e.  N.  |->  ( ( ( F `  a )  +R  1R )  +R  ( A  .R  -1R ) ) )   =>    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( G `  m ) )
 
Theoremcaucvgsrlemoffres 7799* Lemma for caucvgsr 7801. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   &    |-  G  =  ( a  e.  N.  |->  ( ( ( F `  a )  +R  1R )  +R  ( A  .R  -1R ) ) )   =>    |-  ( ph  ->  E. y  e.  R.  A. x  e.  R.  ( 0R  <R  x  ->  E. j  e.  N.  A. k  e. 
 N.  ( j  <N  k 
 ->  ( ( F `  k )  <R  ( y  +R  x )  /\  y  <R  ( ( F `
  k )  +R  x ) ) ) ) )
 
Theoremcaucvgsrlembnd 7800* Lemma for caucvgsr 7801. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   =>    |-  ( ph  ->  E. y  e.  R.  A. x  e. 
 R.  ( 0R  <R  x 
 ->  E. j  e.  N.  A. k  e.  N.  (
 j  <N  k  ->  (
 ( F `  k
 )  <R  ( y  +R  x )  /\  y  <R  ( ( F `  k
 )  +R  x )
 ) ) ) )
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