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Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1idpr 7301 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 7302* 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 7303* 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 7304 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 7305. (Contributed by Jim Kingdon, 15-Dec-2019.)
 |- 
 <P  Po  P.
 
Theoremltsopr 7305 Positive real 'less than' is a weak linear order (in the sense of df-iso 4157). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
 |- 
 <P  Or  P.
 
Theoremltaddpr 7306 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 7307* Element in lower cut of the constructed difference. Lemma for ltexpri 7322. (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 7308* Element in upper cut of the constructed difference. Lemma for ltexpri 7322. (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 7309* Our constructed difference is inhabited. Lemma for ltexpri 7322. (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 7310* The lower cut of our constructed difference is open. Lemma for ltexpri 7322. (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 7311* The lower cut of our constructed difference is lower. Lemma for ltexpri 7322. (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 7312* The upper cut of our constructed difference is open. Lemma for ltexpri 7322. (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 7313* The upper cut of our constructed difference is upper. Lemma for ltexpri 7322. (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 7314* Our constructed difference is rounded. Lemma for ltexpri 7322. (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 7315* Our constructed difference is disjoint. Lemma for ltexpri 7322. (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 7316* Our constructed difference is located. Lemma for ltexpri 7322. (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 7317* Our constructed difference is a positive real. Lemma for ltexpri 7322. (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 7318* Lemma for ltexpri 7322. 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 7319* Lemma for ltexpri 7322. 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 7320* Lemma for ltexpri 7322. 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 7321* Lemma for ltexpri 7322. 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 7322* 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 7323 Lemma for addcanprg 7325. (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 7324 Lemma for addcanprg 7325. (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 7325 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 7326* The difference from ltexpri 7322 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
 |-  ( A  <P  B  ->  E! x  e.  P.  ( A  +P.  x )  =  B )
 
Theoremltaprlem 7327 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 7328 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 7329* 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 7330 Strong extensionality of addition (ordering version). This is similar to addext 8238 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 7331* Membership in the lower cut of  B. Lemma for recexpr 7347. (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 7332* Membership in the upper cut of  B. Lemma for recexpr 7347. (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 7333*  B is inhabited. Lemma for recexpr 7347. (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 7334* The lower cut of  B is open. Lemma for recexpr 7347. (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 7335* The lower cut of  B is lower. Lemma for recexpr 7347. (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 7336* The upper cut of  B is open. Lemma for recexpr 7347. (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 7337* The upper cut of  B is upper. Lemma for recexpr 7347. (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 7338*  B is rounded. Lemma for recexpr 7347. (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 7339*  B is disjoint. Lemma for recexpr 7347. (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 7340*  B is located. Lemma for recexpr 7347. (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 7341*  B is a positive real. Lemma for recexpr 7347. (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 7342* The lower cut of one is a subset of the lower cut of  A  .P.  B. Lemma for recexpr 7347. (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 7343* The upper cut of one is a subset of the upper cut of  A  .P.  B. Lemma for recexpr 7347. (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 7344* The lower cut of  A  .P.  B is a subset of the lower cut of one. Lemma for recexpr 7347. (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 7345* The upper cut of  A  .P.  B is a subset of the upper cut of one. Lemma for recexpr 7347. (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 7346*  B is the reciprocal of  A. Lemma for recexpr 7347. (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 7347* 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 7348 Lemma for aptipr 7350. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  B  <P  A ) 
 ->  ( 1st `  A )  C_  ( 1st `  B ) )
 
Theoremaptiprlemu 7349 Lemma for aptipr 7350. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  B  <P  A ) 
 ->  ( 2nd `  B )  C_  ( 2nd `  A ) )
 
Theoremaptipr 7350 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 7351 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 7352* 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 7262. (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 7353* Lemma for cauappcvgprlemladdrl 7366. 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 7354* Lemma for cauappcvgpr 7371. 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 7355* Lemma for cauappcvgpr 7371. 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 7356* Lemma for cauappcvgpr 7371. 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 7357* Lemma for cauappcvgpr 7371. 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 7358* Lemma for cauappcvgpr 7371. 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 7359* Lemma for cauappcvgpr 7371. 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 7360* Lemma for cauappcvgpr 7371. 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 7361* Lemma for cauappcvgpr 7371. 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 7362* Lemma for cauappcvgpr 7371. 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 7363* Lemma for cauappcvgprlemladd 7367. 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 7364* Lemma for cauappcvgprlemladd 7367. 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 7365* Lemma for cauappcvgprlemladd 7367. 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 7366* Lemma for cauappcvgprlemladd 7367. 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 7367* Lemma for cauappcvgpr 7371. 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 7368* Lemma for cauappcvgpr 7371. 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 7369* Lemma for cauappcvgpr 7371. 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 7370* Lemma for cauappcvgpr 7371. 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 7371* 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 7391 and caucvgprpr 7421 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 7372* 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 7373* 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 7374 Lemma for caucvgpr 7391. 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 7375* Lemma for caucvgpr 7391. 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 7376* Lemma for caucvgpr 7391. 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 7377* Lemma for caucvgpr 7391. 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 7378* Lemma for caucvgpr 7391. 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 7379* Lemma for caucvgpr 7391. 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 7380* Lemma for caucvgpr 7391. 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 7381* Lemma for caucvgpr 7391. 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 7382* Lemma for caucvgpr 7391. 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 7383* Lemma for caucvgpr 7391. 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 7384* Lemma for caucvgpr 7391. 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 7385* Lemma for caucvgpr 7391. 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 7386* Lemma for caucvgpr 7391. 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 7387* Lemma for caucvgpr 7391. 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 7388* Lemma for caucvgpr 7391. 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 7389* Lemma for caucvgpr 7391. 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 7390* Lemma for caucvgpr 7391. 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 7391* 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 7371 and caucvgprpr 7421. Reading cauappcvgpr 7371 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 7392* Lemma for caucvgprpr 7421. 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 7393* Lemma for caucvgprpr 7421. Rearranging some expressions for caucvgprprlemloc 7412. (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 7394* Lemma for caucvgprpr 7421. 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 7395* Lemma for caucvgprpr 7421. 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 7396* Lemma for caucvgprpr 7421. 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 7397* Lemma for caucvgprpr 7421. 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 7398* Lemma for caucvgprpr 7421. 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 7399* Lemma for caucvgprpr 7421. 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 7400* Lemma for caucvgprpr 7421. 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 } >. ) )
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