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Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulassprg 7201 Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( ( A  .P.  B )  .P.  C )  =  ( A  .P.  ( B  .P.  C ) ) )
 
Theoremdistrlem1prl 7202 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( A  .P.  ( B  +P.  C ) ) )  C_  ( 1st `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) )
 
Theoremdistrlem1pru 7203 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) )  C_  ( 2nd `  ( ( A  .P.  B )  +P.  ( A  .P.  C ) ) ) )
 
Theoremdistrlem4prl 7204* Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 1st `  A )  /\  y  e.  ( 1st `  B ) )  /\  ( f  e.  ( 1st `  A )  /\  z  e.  ( 1st `  C ) ) ) )  ->  (
 ( x  .Q  y
 )  +Q  ( f  .Q  z ) )  e.  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
 
Theoremdistrlem4pru 7205* Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  ( 2nd `  A )  /\  y  e.  ( 2nd `  B ) )  /\  ( f  e.  ( 2nd `  A )  /\  z  e.  ( 2nd `  C ) ) ) )  ->  (
 ( x  .Q  y
 )  +Q  ( f  .Q  z ) )  e.  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
 
Theoremdistrlem5prl 7206 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  (
 ( A  .P.  B )  +P.  ( A  .P.  C ) ) )  C_  ( 1st `  ( A  .P.  ( B  +P.  C ) ) ) )
 
Theoremdistrlem5pru 7207 Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  (
 ( A  .P.  B )  +P.  ( A  .P.  C ) ) )  C_  ( 2nd `  ( A  .P.  ( B  +P.  C ) ) ) )
 
Theoremdistrprg 7208 Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) )  =  ( ( A 
 .P.  B )  +P.  ( A  .P.  C ) ) )
 
Theoremltprordil 7209 If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.)
 |-  ( A  <P  B  ->  ( 1st `  A )  C_  ( 1st `  B ) )
 
Theorem1idprl 7210 Lemma for 1idpr 7212. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  1P ) )  =  ( 1st `  A ) )
 
Theorem1idpru 7211 Lemma for 1idpr 7212. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |-  ( A  e.  P.  ->  ( 2nd `  ( A  .P.  1P ) )  =  ( 2nd `  A ) )
 
Theorem1idpr 7212 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.)
 |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  A )
 
Theoremltnqpr 7213* We can order fractions via  <Q or  <P. (Contributed by Jim Kingdon, 19-Jun-2021.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  <Q  B  <->  <. { l  |  l 
 <Q  A } ,  { u  |  A  <Q  u } >.  <P  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) )
 
Theoremltnqpri 7214* We can order fractions via  <Q or  <P. (Contributed by Jim Kingdon, 8-Jan-2021.)
 |-  ( A  <Q  B  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. )
 
Theoremltpopr 7215 Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 7216. (Contributed by Jim Kingdon, 15-Dec-2019.)
 |- 
 <P  Po  P.
 
Theoremltsopr 7216 Positive real 'less than' is a weak linear order (in the sense of df-iso 4133). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
 |- 
 <P  Or  P.
 
Theoremltaddpr 7217 The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  A  <P  ( A 
 +P.  B ) )
 
Theoremltexprlemell 7218* Element in lower cut of the constructed difference. Lemma for ltexpri 7233. (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 ) ) } >.   =>    |-  ( q  e.  ( 1st `  C )  <->  ( q  e. 
 Q.  /\  E. y
 ( y  e.  ( 2nd `  A )  /\  ( y  +Q  q
 )  e.  ( 1st `  B ) ) ) )
 
Theoremltexprlemelu 7219* Element in upper cut of the constructed difference. Lemma for ltexpri 7233. (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 ) ) } >.   =>    |-  ( r  e.  ( 2nd `  C )  <->  ( r  e. 
 Q.  /\  E. y
 ( y  e.  ( 1st `  A )  /\  ( y  +Q  r
 )  e.  ( 2nd `  B ) ) ) )
 
Theoremltexprlemm 7220* Our constructed difference is inhabited. Lemma for ltexpri 7233. (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  ->  ( E. q  e.  Q.  q  e.  ( 1st `  C )  /\  E. r  e.  Q.  r  e.  ( 2nd `  C ) ) )
 
Theoremltexprlemopl 7221* The lower cut of our constructed difference is open. Lemma for ltexpri 7233. (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.  /\  q  e.  ( 1st `  C ) )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )
 
Theoremltexprlemlol 7222* The lower cut of our constructed difference is lower. Lemma for ltexpri 7233. (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 7223* The upper cut of our constructed difference is open. Lemma for ltexpri 7233. (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 7224* The upper cut of our constructed difference is upper. Lemma for ltexpri 7233. (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 7225* Our constructed difference is rounded. Lemma for ltexpri 7233. (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 7226* Our constructed difference is disjoint. Lemma for ltexpri 7233. (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 7227* Our constructed difference is located. Lemma for ltexpri 7233. (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 7228* Our constructed difference is a positive real. Lemma for ltexpri 7233. (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 7229* Lemma for ltexpri 7233. One directon 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 7230* Lemma for ltexpri 7233. Reverse directon 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 7231* Lemma for ltexpri 7233. 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 7232* Lemma for ltexpri 7233. 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 7233* 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 7234 Lemma for addcanprg 7236. (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 7235 Lemma for addcanprg 7236. (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 7236 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 7237* The difference from ltexpri 7233 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
 |-  ( A  <P  B  ->  E! x  e.  P.  ( A  +P.  x )  =  B )
 
Theoremltaprlem 7238 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 7239 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 7240* 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 7241 Strong extensionality of addition (ordering version). This is similar to addext 8148 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 7242* Membership in the lower cut of  B. Lemma for recexpr 7258. (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 7243* Membership in the upper cut of  B. Lemma for recexpr 7258. (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 7244*  B is inhabited. Lemma for recexpr 7258. (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 7245* The lower cut of  B is open. Lemma for recexpr 7258. (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 7246* The lower cut of  B is lower. Lemma for recexpr 7258. (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 7247* The upper cut of  B is open. Lemma for recexpr 7258. (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 7248* The upper cut of  B is upper. Lemma for recexpr 7258. (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 7249*  B is rounded. Lemma for recexpr 7258. (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 7250*  B is disjoint. Lemma for recexpr 7258. (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 7251*  B is located. Lemma for recexpr 7258. (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 7252*  B is a positive real. Lemma for recexpr 7258. (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 7253* The lower cut of one is a subset of the lower cut of  A  .P.  B. Lemma for recexpr 7258. (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 7254* The upper cut of one is a subset of the upper cut of  A  .P.  B. Lemma for recexpr 7258. (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 7255* The lower cut of  A  .P.  B is a subset of the lower cut of one. Lemma for recexpr 7258. (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 7256* The upper cut of  A  .P.  B is a subset of the upper cut of one. Lemma for recexpr 7258. (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 7257*  B is the reciprocal of  A. Lemma for recexpr 7258. (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 7258* 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 7259 Lemma for aptipr 7261. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  B  <P  A ) 
 ->  ( 1st `  A )  C_  ( 1st `  B ) )
 
Theoremaptiprlemu 7260 Lemma for aptipr 7261. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  B  <P  A ) 
 ->  ( 2nd `  B )  C_  ( 2nd `  A ) )
 
Theoremaptipr 7261 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 7262 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 7263* 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 7173. (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 7264* Lemma for cauappcvgprlemladdrl 7277. 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 7265* Lemma for cauappcvgpr 7282. 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 7266* Lemma for cauappcvgpr 7282. 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 7267* Lemma for cauappcvgpr 7282. 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 7268* Lemma for cauappcvgpr 7282. 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 7269* Lemma for cauappcvgpr 7282. 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 7270* Lemma for cauappcvgpr 7282. 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 7271* Lemma for cauappcvgpr 7282. 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 7272* Lemma for cauappcvgpr 7282. 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 7273* Lemma for cauappcvgpr 7282. 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 7274* Lemma for cauappcvgprlemladd 7278. 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 7275* Lemma for cauappcvgprlemladd 7278. 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 7276* Lemma for cauappcvgprlemladd 7278. 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 7277* Lemma for cauappcvgprlemladd 7278. 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 7278* Lemma for cauappcvgpr 7282. 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 7279* Lemma for cauappcvgpr 7282. 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 7280* Lemma for cauappcvgpr 7282. 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 7281* Lemma for cauappcvgpr 7282. 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 7282* 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 7302 and caucvgprpr 7332 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 7283* 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 7284* 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 7285 Lemma for caucvgpr 7302. 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 7286* Lemma for caucvgpr 7302. 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 7287* Lemma for caucvgpr 7302. 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 7288* Lemma for caucvgpr 7302. 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 7289* Lemma for caucvgpr 7302. 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 7290* Lemma for caucvgpr 7302. 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 7291* Lemma for caucvgpr 7302. 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 7292* Lemma for caucvgpr 7302. 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 7293* Lemma for caucvgpr 7302. 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 7294* Lemma for caucvgpr 7302. 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 7295* Lemma for caucvgpr 7302. 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 7296* Lemma for caucvgpr 7302. 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 7297* Lemma for caucvgpr 7302. 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 7298* Lemma for caucvgpr 7302. 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 7299* Lemma for caucvgpr 7302. 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 7300* Lemma for caucvgpr 7302. 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 } >. )
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