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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremappdivnq 7101* Approximate division for positive rationals. Proposition 12.7 of [BauerTaylor], p. 55 (a special case where  A and  B are positive, as well as  C). Our proof is simpler than the one in BauerTaylor because we have reciprocals. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  E. m  e.  Q.  ( A  <Q  ( m  .Q  C )  /\  ( m  .Q  C ) 
 <Q  B ) )
 
Theoremappdiv0nq 7102* Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7101 in which  A is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( B  e.  Q. 
 /\  C  e.  Q. )  ->  E. m  e.  Q.  ( m  .Q  C ) 
 <Q  B )
 
Theoremprmuloclemcalc 7103 Calculations for prmuloc 7104. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ph  ->  R  <Q  U )   &    |-  ( ph  ->  U 
 <Q  ( D  +Q  P ) )   &    |-  ( ph  ->  ( A  +Q  X )  =  B )   &    |-  ( ph  ->  ( P  .Q  B )  <Q  ( R  .Q  X ) )   &    |-  ( ph  ->  A  e.  Q. )   &    |-  ( ph  ->  B  e.  Q. )   &    |-  ( ph  ->  D  e.  Q. )   &    |-  ( ph  ->  P  e.  Q. )   &    |-  ( ph  ->  X  e.  Q. )   =>    |-  ( ph  ->  ( U  .Q  A ) 
 <Q  ( D  .Q  B ) )
 
Theoremprmuloc 7104* Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  A  <Q  B )  ->  E. d  e.  Q.  E. u  e.  Q.  (
 d  e.  L  /\  u  e.  U  /\  ( u  .Q  A ) 
 <Q  ( d  .Q  B ) ) )
 
Theoremprmuloc2 7105* Positive reals are multiplicatively located. This is a variation of prmuloc 7104 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio  B, there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  ->  E. x  e.  L  ( x  .Q  B )  e.  U )
 
Theoremmulnqprl 7106 Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
 |-  ( ( ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B ) ) ) 
 /\  X  e.  Q. )  ->  ( X  <Q  ( G  .Q  H ) 
 ->  X  e.  ( 1st `  ( A  .P.  B ) ) ) )
 
Theoremmulnqpru 7107 Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
 |-  ( ( ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B ) ) ) 
 /\  X  e.  Q. )  ->  ( ( G  .Q  H )  <Q  X 
 ->  X  e.  ( 2nd `  ( A  .P.  B ) ) ) )
 
Theoremmullocprlem 7108 Calculations for mullocpr 7109. (Contributed by Jim Kingdon, 10-Dec-2019.)
 |-  ( ph  ->  ( A  e.  P.  /\  B  e.  P. ) )   &    |-  ( ph  ->  ( U  .Q  Q )  <Q  ( E  .Q  ( D  .Q  U ) ) )   &    |-  ( ph  ->  ( E  .Q  ( D  .Q  U ) )  <Q  ( T  .Q  ( D  .Q  U ) ) )   &    |-  ( ph  ->  ( T  .Q  ( D  .Q  U ) )  <Q  ( D  .Q  R ) )   &    |-  ( ph  ->  ( Q  e.  Q.  /\  R  e.  Q. ) )   &    |-  ( ph  ->  ( D  e.  Q.  /\  U  e.  Q. )
 )   &    |-  ( ph  ->  ( D  e.  ( 1st `  A )  /\  U  e.  ( 2nd `  A ) ) )   &    |-  ( ph  ->  ( E  e.  Q. 
 /\  T  e.  Q. ) )   =>    |-  ( ph  ->  ( Q  e.  ( 1st `  ( A  .P.  B ) )  \/  R  e.  ( 2nd `  ( A  .P.  B ) ) ) )
 
Theoremmullocpr 7109* Locatedness of multiplication on positive reals. Lemma 12.9 in [BauerTaylor], p. 56 (but where both  A and  B are positive, not just  A). (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  ( 1st `  ( A  .P.  B ) )  \/  r  e.  ( 2nd `  ( A  .P.  B ) ) ) ) )
 
Theoremmulclpr 7110 Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  .P.  B )  e.  P. )
 
Theoremmulnqprlemrl 7111* Lemma for mulnqpr 7115. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( 1st `  ( <. { l  |  l 
 <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) 
 C_  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B ) 
 <Q  u } >. ) )
 
Theoremmulnqprlemru 7112* Lemma for mulnqpr 7115. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( 2nd `  ( <. { l  |  l 
 <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) 
 C_  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B ) 
 <Q  u } >. ) )
 
Theoremmulnqprlemfl 7113* Lemma for mulnqpr 7115. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( 1st `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B ) 
 <Q  u } >. )  C_  ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )
 
Theoremmulnqprlemfu 7114* Lemma for mulnqpr 7115. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( 2nd `  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B ) 
 <Q  u } >. )  C_  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )
 
Theoremmulnqpr 7115* Multiplication of fractions embedded into positive reals. One can either multiply the fractions as fractions, or embed them into positive reals and multiply them as positive reals, and get the same result. (Contributed by Jim Kingdon, 18-Jul-2021.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  <. { l  |  l  <Q  ( A  .Q  B ) } ,  { u  |  ( A  .Q  B )  <Q  u } >.  =  ( <. { l  |  l 
 <Q  A } ,  { u  |  A  <Q  u } >.  .P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) )
 
Theoremaddcomprg 7116 Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  +P.  B )  =  ( B 
 +P.  A ) )
 
Theoremaddassprg 7117 Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (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 ) ) )
 
Theoremmulcomprg 7118 Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  .P.  B )  =  ( B 
 .P.  A ) )
 
Theoremmulassprg 7119 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 7120 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 7121 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 7122* 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 7123* 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 7124 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 7125 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 7126 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 7127 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 7128 Lemma for 1idpr 7130. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  1P ) )  =  ( 1st `  A ) )
 
Theorem1idpru 7129 Lemma for 1idpr 7130. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |-  ( A  e.  P.  ->  ( 2nd `  ( A  .P.  1P ) )  =  ( 2nd `  A ) )
 
Theorem1idpr 7130 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 7131* 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 7132* 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 7133 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 7134. (Contributed by Jim Kingdon, 15-Dec-2019.)
 |- 
 <P  Po  P.
 
Theoremltsopr 7134 Positive real 'less than' is a weak linear order (in the sense of df-iso 4115). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
 |- 
 <P  Or  P.
 
Theoremltaddpr 7135 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 7136* Element in lower cut of the constructed difference. Lemma for ltexpri 7151. (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 7137* Element in upper cut of the constructed difference. Lemma for ltexpri 7151. (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 7138* Our constructed difference is inhabited. Lemma for ltexpri 7151. (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 7139* The lower cut of our constructed difference is open. Lemma for ltexpri 7151. (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 7140* The lower cut of our constructed difference is lower. Lemma for ltexpri 7151. (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 7141* The upper cut of our constructed difference is open. Lemma for ltexpri 7151. (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 7142* The upper cut of our constructed difference is upper. Lemma for ltexpri 7151. (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 7143* Our constructed difference is rounded. Lemma for ltexpri 7151. (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 7144* Our constructed difference is disjoint. Lemma for ltexpri 7151. (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 7145* Our constructed difference is located. Lemma for ltexpri 7151. (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 7146* Our constructed difference is a positive real. Lemma for ltexpri 7151. (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 7147* Lemma for ltexpri 7151. 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 7148* Lemma for ltexpri 7151. 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 7149* Lemma for ltexpri 7151. 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 7150* Lemma for ltexpri 7151. 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 7151* 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 7152 Lemma for addcanprg 7154. (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 7153 Lemma for addcanprg 7154. (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 7154 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 7155* The difference from ltexpri 7151 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
 |-  ( A  <P  B  ->  E! x  e.  P.  ( A  +P.  x )  =  B )
 
Theoremltaprlem 7156 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 7157 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 7158* 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 7159 Strong extensionality of addition (ordering version). This is similar to addext 8063 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 7160* Membership in the lower cut of  B. Lemma for recexpr 7176. (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 7161* Membership in the upper cut of  B. Lemma for recexpr 7176. (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 7162*  B is inhabited. Lemma for recexpr 7176. (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 7163* The lower cut of  B is open. Lemma for recexpr 7176. (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 7164* The lower cut of  B is lower. Lemma for recexpr 7176. (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 7165* The upper cut of  B is open. Lemma for recexpr 7176. (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 7166* The upper cut of  B is upper. Lemma for recexpr 7176. (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 7167*  B is rounded. Lemma for recexpr 7176. (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 7168*  B is disjoint. Lemma for recexpr 7176. (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 7169*  B is located. Lemma for recexpr 7176. (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 7170*  B is a positive real. Lemma for recexpr 7176. (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 7171* The lower cut of one is a subset of the lower cut of  A  .P.  B. Lemma for recexpr 7176. (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 7172* The upper cut of one is a subset of the upper cut of  A  .P.  B. Lemma for recexpr 7176. (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 7173* The lower cut of  A  .P.  B is a subset of the lower cut of one. Lemma for recexpr 7176. (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 7174* The upper cut of  A  .P.  B is a subset of the upper cut of one. Lemma for recexpr 7176. (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 7175*  B is the reciprocal of  A. Lemma for recexpr 7176. (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 7176* 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 7177 Lemma for aptipr 7179. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  B  <P  A ) 
 ->  ( 1st `  A )  C_  ( 1st `  B ) )
 
Theoremaptiprlemu 7178 Lemma for aptipr 7179. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  B  <P  A ) 
 ->  ( 2nd `  B )  C_  ( 2nd `  A ) )
 
Theoremaptipr 7179 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 7180 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 7181* 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 7091. (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 7182* Lemma for cauappcvgprlemladdrl 7195. 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 7183* Lemma for cauappcvgpr 7200. 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 7184* Lemma for cauappcvgpr 7200. 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 7185* Lemma for cauappcvgpr 7200. 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 7186* Lemma for cauappcvgpr 7200. 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 7187* Lemma for cauappcvgpr 7200. 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 7188* Lemma for cauappcvgpr 7200. 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 7189* Lemma for cauappcvgpr 7200. 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 7190* Lemma for cauappcvgpr 7200. 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 7191* Lemma for cauappcvgpr 7200. 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 7192* Lemma for cauappcvgprlemladd 7196. 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 7193* Lemma for cauappcvgprlemladd 7196. 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 7194* Lemma for cauappcvgprlemladd 7196. 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 7195* Lemma for cauappcvgprlemladd 7196. 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 7196* Lemma for cauappcvgpr 7200. 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 7197* Lemma for cauappcvgpr 7200. 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 7198* Lemma for cauappcvgpr 7200. 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 7199* Lemma for cauappcvgpr 7200. 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 7200* 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 7220 and caucvgprpr 7250 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 } >. ) )
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