HomeHome Intuitionistic Logic Explorer
Theorem List (p. 77 of 149)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltexprlemlol 7601* The lower cut of our constructed difference is lower. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 21-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( ( A  <P  B 
 /\  q  e.  Q. )  ->  ( E. r  e.  Q.  ( q  <Q  r 
 /\  r  e.  ( 1st `  C ) ) 
 ->  q  e.  ( 1st `  C ) ) )
 
Theoremltexprlemopu 7602* The upper cut of our constructed difference is open. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 21-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( ( A  <P  B 
 /\  r  e.  Q.  /\  r  e.  ( 2nd `  C ) )  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) )
 
Theoremltexprlemupu 7603* The upper cut of our constructed difference is upper. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 21-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( ( A  <P  B 
 /\  r  e.  Q. )  ->  ( E. q  e.  Q.  ( q  <Q  r 
 /\  q  e.  ( 2nd `  C ) ) 
 ->  r  e.  ( 2nd `  C ) ) )
 
Theoremltexprlemrnd 7604* Our constructed difference is rounded. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 17-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( A  <P  B  ->  (
 A. q  e.  Q.  ( q  e.  ( 1st `  C )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) ) 
 /\  A. r  e.  Q.  ( r  e.  ( 2nd `  C )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) ) ) )
 
Theoremltexprlemdisj 7605* Our constructed difference is disjoint. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 17-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( A  <P  B  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C ) ) )
 
Theoremltexprlemloc 7606* Our constructed difference is located. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 17-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( A  <P  B  ->  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  ( 1st `  C )  \/  r  e.  ( 2nd `  C ) ) ) )
 
Theoremltexprlempr 7607* Our constructed difference is a positive real. Lemma for ltexpri 7612. (Contributed by Jim Kingdon, 17-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( A  <P  B  ->  C  e.  P. )
 
Theoremltexprlemfl 7608* Lemma for ltexpri 7612. One direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( A  <P  B  ->  ( 1st `  ( A  +P.  C ) )  C_  ( 1st `  B )
 )
 
Theoremltexprlemrl 7609* Lemma for ltexpri 7612. Reverse direction of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( A  <P  B  ->  ( 1st `  B )  C_  ( 1st `  ( A  +P.  C ) ) )
 
Theoremltexprlemfu 7610* Lemma for ltexpri 7612. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( A  <P  B  ->  ( 2nd `  ( A  +P.  C ) )  C_  ( 2nd `  B )
 )
 
Theoremltexprlemru 7611* Lemma for ltexpri 7612. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
 |-  C  =  <. { x  e.  Q.  |  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.   =>    |-  ( A  <P  B  ->  ( 2nd `  B )  C_  ( 2nd `  ( A  +P.  C ) ) )
 
Theoremltexpri 7612* Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
 |-  ( A  <P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
 
Theoremaddcanprleml 7613 Lemma for addcanprg 7615. (Contributed by Jim Kingdon, 25-Dec-2019.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A 
 +P.  B )  =  ( A  +P.  C ) )  ->  ( 1st `  B )  C_  ( 1st `  C ) )
 
Theoremaddcanprlemu 7614 Lemma for addcanprg 7615. (Contributed by Jim Kingdon, 25-Dec-2019.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( A 
 +P.  B )  =  ( A  +P.  C ) )  ->  ( 2nd `  B )  C_  ( 2nd `  C ) )
 
Theoremaddcanprg 7615 Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( ( A  +P.  B )  =  ( A 
 +P.  C )  ->  B  =  C ) )
 
Theoremlteupri 7616* The difference from ltexpri 7612 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
 |-  ( A  <P  B  ->  E! x  e.  P.  ( A  +P.  x )  =  B )
 
Theoremltaprlem 7617 Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.)
 |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A ) 
 <P  ( C  +P.  B ) ) )
 
Theoremltaprg 7618 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
 
Theoremprplnqu 7619* Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)
 |-  ( ph  ->  X  e.  P. )   &    |-  ( ph  ->  Q  e.  Q. )   &    |-  ( ph  ->  A  e.  ( 2nd `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) )   =>    |-  ( ph  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q )  =  A )
 
Theoremaddextpr 7620 Strong extensionality of addition (ordering version). This is similar to addext 8567 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  ->  ( ( A  +P.  B )  <P  ( C  +P.  D )  ->  ( A  <P  C  \/  B  <P  D ) ) )
 
Theoremrecexprlemell 7621* Membership in the lower cut of  B. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( C  e.  ( 1st `  B )  <->  E. y ( C 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) )
 
Theoremrecexprlemelu 7622* Membership in the upper cut of  B. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( C  e.  ( 2nd `  B )  <->  E. y ( y 
 <Q  C  /\  ( *Q `  y )  e.  ( 1st `  A ) ) )
 
Theoremrecexprlemm 7623*  B is inhabited. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( A  e.  P.  ->  ( E. q  e. 
 Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B ) ) )
 
Theoremrecexprlemopl 7624* The lower cut of  B is open. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 28-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( ( A  e.  P. 
 /\  q  e.  Q.  /\  q  e.  ( 1st `  B ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  B ) ) )
 
Theoremrecexprlemlol 7625* The lower cut of  B is lower. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 28-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( ( A  e.  P. 
 /\  q  e.  Q. )  ->  ( E. r  e.  Q.  ( q  <Q  r 
 /\  r  e.  ( 1st `  B ) ) 
 ->  q  e.  ( 1st `  B ) ) )
 
Theoremrecexprlemopu 7626* The upper cut of  B is open. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 28-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( ( A  e.  P. 
 /\  r  e.  Q.  /\  r  e.  ( 2nd `  B ) )  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) )
 
Theoremrecexprlemupu 7627* The upper cut of  B is upper. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 28-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( ( A  e.  P. 
 /\  r  e.  Q. )  ->  ( E. q  e.  Q.  ( q  <Q  r 
 /\  q  e.  ( 2nd `  B ) ) 
 ->  r  e.  ( 2nd `  B ) ) )
 
Theoremrecexprlemrnd 7628*  B is rounded. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( A  e.  P.  ->  ( A. q  e. 
 Q.  ( q  e.  ( 1st `  B ) 
 <-> 
 E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  B ) ) ) 
 /\  A. r  e.  Q.  ( r  e.  ( 2nd `  B )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) ) )
 
Theoremrecexprlemdisj 7629*  B is disjoint. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( A  e.  P.  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  B )  /\  q  e.  ( 2nd `  B ) ) )
 
Theoremrecexprlemloc 7630*  B is located. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( A  e.  P.  ->  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) ) )
 
Theoremrecexprlempr 7631*  B is a positive real. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( A  e.  P.  ->  B  e.  P. )
 
Theoremrecexprlem1ssl 7632* The lower cut of one is a subset of the lower cut of  A  .P.  B. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( A  e.  P.  ->  ( 1st `  1P )  C_  ( 1st `  ( A  .P.  B ) ) )
 
Theoremrecexprlem1ssu 7633* The upper cut of one is a subset of the upper cut of  A  .P.  B. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( A  e.  P.  ->  ( 2nd `  1P )  C_  ( 2nd `  ( A  .P.  B ) ) )
 
Theoremrecexprlemss1l 7634* The lower cut of  A  .P.  B is a subset of the lower cut of one. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  B ) ) 
 C_  ( 1st `  1P ) )
 
Theoremrecexprlemss1u 7635* The upper cut of  A  .P.  B is a subset of the upper cut of one. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( A  e.  P.  ->  ( 2nd `  ( A  .P.  B ) ) 
 C_  ( 2nd `  1P ) )
 
Theoremrecexprlemex 7636*  B is the reciprocal of  A. Lemma for recexpr 7637. (Contributed by Jim Kingdon, 27-Dec-2019.)
 |-  B  =  <. { x  |  E. y ( x 
 <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  { x  |  E. y ( y 
 <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.   =>    |-  ( A  e.  P.  ->  ( A  .P.  B )  =  1P )
 
Theoremrecexpr 7637* The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
 |-  ( A  e.  P.  ->  E. x  e.  P.  ( A  .P.  x )  =  1P )
 
Theoremaptiprleml 7638 Lemma for aptipr 7640. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  B  <P  A ) 
 ->  ( 1st `  A )  C_  ( 1st `  B ) )
 
Theoremaptiprlemu 7639 Lemma for aptipr 7640. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  B  <P  A ) 
 ->  ( 2nd `  B )  C_  ( 2nd `  A ) )
 
Theoremaptipr 7640 Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  ( A  <P  B  \/  B  <P  A ) )  ->  A  =  B )
 
Theoremltmprr 7641 Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( ( C  .P.  A )  <P  ( C  .P.  B )  ->  A  <P  B ) )
 
Theoremarchpr 7642* For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer  x is embedded into the reals as described at nnprlu 7552. (Contributed by Jim Kingdon, 22-Apr-2020.)
 |-  ( A  e.  P.  ->  E. x  e.  N.  A  <P  <. { l  |  l  <Q  [ <. x ,  1o >. ]  ~Q  } ,  { u  |  [ <. x ,  1o >. ] 
 ~Q  <Q  u } >. )
 
Theoremcaucvgprlemcanl 7643* Lemma for cauappcvgprlemladdrl 7656. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)
 |-  ( ph  ->  L  e.  P. )   &    |-  ( ph  ->  S  e.  Q. )   &    |-  ( ph  ->  R  e.  Q. )   &    |-  ( ph  ->  Q  e.  Q. )   =>    |-  ( ph  ->  (
 ( R  +Q  Q )  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  Q ) } ,  { u  |  ( S  +Q  Q ) 
 <Q  u } >. ) )  <->  R  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) ) )
 
Theoremcauappcvgprlemm 7644* Lemma for cauappcvgpr 7661. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   =>    |-  ( ph  ->  ( E. s  e.  Q.  s  e.  ( 1st `  L )  /\  E. r  e.  Q.  r  e.  ( 2nd `  L ) ) )
 
Theoremcauappcvgprlemopl 7645* Lemma for cauappcvgpr 7661. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   =>    |-  ( ( ph  /\  s  e.  ( 1st `  L ) )  ->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )
 
Theoremcauappcvgprlemlol 7646* Lemma for cauappcvgpr 7661. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   =>    |-  ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
 
Theoremcauappcvgprlemopu 7647* Lemma for cauappcvgpr 7661. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   =>    |-  ( ( ph  /\  r  e.  ( 2nd `  L ) )  ->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) )
 
Theoremcauappcvgprlemupu 7648* Lemma for cauappcvgpr 7661. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   =>    |-  ( ( ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  r  e.  ( 2nd `  L ) )
 
Theoremcauappcvgprlemrnd 7649* Lemma for cauappcvgpr 7661. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  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 ) ) ) ) )
 
Theoremcauappcvgprlemdisj 7650* Lemma for cauappcvgpr 7661. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   =>    |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
 
Theoremcauappcvgprlemloc 7651* Lemma for cauappcvgpr 7661. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   =>    |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
 s  <Q  r  ->  (
 s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L ) ) ) )
 
Theoremcauappcvgprlemcl 7652* Lemma for cauappcvgpr 7661. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   =>    |-  ( ph  ->  L  e.  P. )
 
Theoremcauappcvgprlemladdfu 7653* Lemma for cauappcvgprlemladd 7657. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ph  ->  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) )  C_  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  ( ( F `  q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
  q )  +Q  q )  +Q  S ) 
 <Q  u } >. ) )
 
Theoremcauappcvgprlemladdfl 7654* Lemma for cauappcvgprlemladd 7657. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ph  ->  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) )  C_  ( 1st `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  ( ( F `  q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
  q )  +Q  q )  +Q  S ) 
 <Q  u } >. ) )
 
Theoremcauappcvgprlemladdru 7655* Lemma for cauappcvgprlemladd 7657. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ph  ->  ( 2nd `  <. { l  e. 
 Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  ( ( F `  q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
  q )  +Q  q )  +Q  S ) 
 <Q  u } >. )  C_  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )
 
Theoremcauappcvgprlemladdrl 7656* Lemma for cauappcvgprlemladd 7657. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ph  ->  ( 1st `  <. { l  e. 
 Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  ( ( F `  q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
  q )  +Q  q )  +Q  S ) 
 <Q  u } >. )  C_  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )
 
Theoremcauappcvgprlemladd 7657* Lemma for cauappcvgpr 7661. This takes  L and offsets it by the positive fraction  S. (Contributed by Jim Kingdon, 23-Jun-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   &    |-  ( ph  ->  S  e.  Q. )   =>    |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  = 
 <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( ( F `
  q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
  q )  +Q  q )  +Q  S ) 
 <Q  u } >. )
 
Theoremcauappcvgprlem1 7658* Lemma for cauappcvgpr 7661. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   &    |-  ( ph  ->  Q  e.  Q. )   &    |-  ( ph  ->  R  e.  Q. )   =>    |-  ( ph  ->  <. { l  |  l  <Q  ( F `
  Q ) } ,  { u  |  ( F `  Q ) 
 <Q  u } >.  <P  ( L 
 +P.  <. { l  |  l  <Q  ( Q  +Q  R ) } ,  { u  |  ( Q  +Q  R )  <Q  u } >. ) )
 
Theoremcauappcvgprlem2 7659* Lemma for cauappcvgpr 7661. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   &    |-  ( ph  ->  Q  e.  Q. )   &    |-  ( ph  ->  R  e.  Q. )   =>    |-  ( ph  ->  L  <P 
 <. { l  |  l 
 <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) 
 <Q  u } >. )
 
Theoremcauappcvgprlemlim 7660* Lemma for cauappcvgpr 7661. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)
 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   &    |-  L  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
 )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
 )  <Q  u } >.   =>    |-  ( ph  ->  A. q  e.  Q.  A. r  e.  Q.  ( <. { l  |  l 
 <Q  ( F `  q
 ) } ,  { u  |  ( F `  q )  <Q  u } >. 
 <P  ( L  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  ( q  +Q  r ) 
 <Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  ( ( F `  q )  +Q  ( q  +Q  r
 ) ) } ,  { u  |  (
 ( F `  q
 )  +Q  ( q  +Q  r ) )  <Q  u } >. ) )
 
Theoremcauappcvgpr 7661* A Cauchy approximation has a limit. A Cauchy approximation, here  F, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of  F is  Q. rather than  P.. 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 caucvgpr 7681 and caucvgprpr 7711 but is somewhat simpler, so reading this one first may help understanding the other two.

(Contributed by Jim Kingdon, 19-Jun-2020.)

 |-  ( ph  ->  F : Q. --> Q. )   &    |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
 ( F `  p )  <Q  ( ( F `
  q )  +Q  ( p  +Q  q
 ) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  ( p  +Q  q ) ) ) )   &    |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )   =>    |-  ( ph  ->  E. y  e.  P.  A. q  e. 
 Q.  A. r  e.  Q.  ( <. { l  |  l  <Q  ( F `  q ) } ,  { u  |  ( F `  q )  <Q  u } >.  <P  ( y 
 +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
 q  +Q  r )  <Q  u } >. )  /\  y  <P  <. { l  |  l  <Q  ( ( F `  q )  +Q  ( q  +Q  r
 ) ) } ,  { u  |  (
 ( F `  q
 )  +Q  ( q  +Q  r ) )  <Q  u } >. ) )
 
Theoremarchrecnq 7662* 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 7663* 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 7664 Lemma for caucvgpr 7681. 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 7665* Lemma for caucvgpr 7681. 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 7666* Lemma for caucvgpr 7681. 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 7667* Lemma for caucvgpr 7681. 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 7668* Lemma for caucvgpr 7681. 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 7669* Lemma for caucvgpr 7681. 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 7670* Lemma for caucvgpr 7681. 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 7671* Lemma for caucvgpr 7681. 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 7672* Lemma for caucvgpr 7681. 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 7673* Lemma for caucvgpr 7681. 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 7674* Lemma for caucvgpr 7681. 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 7675* Lemma for caucvgpr 7681. 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 7676* Lemma for caucvgpr 7681. 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 7677* Lemma for caucvgpr 7681. 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 7678* Lemma for caucvgpr 7681. 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 7679* Lemma for caucvgpr 7681. 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 7680* Lemma for caucvgpr 7681. 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 7681* 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 7661 and caucvgprpr 7711. Reading cauappcvgpr 7661 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 7682* Lemma for caucvgprpr 7711. 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 7683* Lemma for caucvgprpr 7711. Rearranging some expressions for caucvgprprlemloc 7702. (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 7684* Lemma for caucvgprpr 7711. 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 7685* Lemma for caucvgprpr 7711. 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 7686* Lemma for caucvgprpr 7711. 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 7687* Lemma for caucvgprpr 7711. 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 7688* Lemma for caucvgprpr 7711. 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 7689* Lemma for caucvgprpr 7711. 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 7690* Lemma for caucvgprpr 7711. 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 7691* Lemma for caucvgprpr 7711. 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 7692* Lemma for caucvgprpr 7711. 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 7693* Lemma for caucvgprpr 7711. 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 7694* Lemma for caucvgprpr 7711. 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 7695* Lemma for caucvgprpr 7711. 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 7696* Lemma for caucvgprpr 7711. 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 7697* Lemma for caucvgprpr 7711. 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 7698* Lemma for caucvgprpr 7711. 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 7699* Lemma for caucvgprpr 7711. 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 7700* Lemma for caucvgprpr 7711. 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 ) ) ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14834
  Copyright terms: Public domain < Previous  Next >