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Theorem List for Intuitionistic Logic Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnqprloc 6701* A cut produced from a rational is located. Lemma for nqprlu 6703. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( A  e.  Q.  ->  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  { x  |  x  <Q  A }  \/  r  e.  { x  |  A  <Q  x }
 ) ) )
 
Theoremnqprxx 6702* The canonical embedding of the rationals into the reals, expressed with the same variable for the lower and upper cuts. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( A  e.  Q.  -> 
 <. { x  |  x  <Q  A } ,  { x  |  A  <Q  x } >.  e.  P. )
 
Theoremnqprlu 6703* The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
 |-  ( A  e.  Q.  -> 
 <. { l  |  l 
 <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
 
Theoremrecnnpr 6704* The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
 |-  ( A  e.  N.  -> 
 <. { l  |  l 
 <Q  ( *Q `  [ <. A ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. A ,  1o >. ]  ~Q  )  <Q  u } >.  e. 
 P. )
 
Theoremltnqex 6705 The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |- 
 { x  |  x  <Q  A }  e.  _V
 
Theoremgtnqex 6706 The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |- 
 { x  |  A  <Q  x }  e.  _V
 
Theoremnqprl 6707* Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by 
<P. (Contributed by Jim Kingdon, 8-Jul-2020.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  P. )  ->  ( A  e.  ( 1st `  B )  <->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B ) )
 
Theoremnqpru 6708* Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by 
<P. (Contributed by Jim Kingdon, 29-Nov-2020.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  P. )  ->  ( A  e.  ( 2nd `  B )  <->  B 
 <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) )
 
Theoremnnprlu 6709* The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
 |-  ( A  e.  N.  -> 
 <. { l  |  l 
 <Q  [ <. A ,  1o >. ]  ~Q  } ,  { u  |  [ <. A ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
 
Theorem1pr 6710 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
 |- 
 1P  e.  P.
 
Theorem1prl 6711 The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
 |-  ( 1st `  1P )  =  { x  |  x  <Q  1Q }
 
Theorem1pru 6712 The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
 |-  ( 2nd `  1P )  =  { x  |  1Q  <Q  x }
 
Theoremaddnqprlemrl 6713* Lemma for addnqpr 6717. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
 |-  ( ( 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 } >. ) )
 
Theoremaddnqprlemru 6714* Lemma for addnqpr 6717. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
 |-  ( ( 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 } >. ) )
 
Theoremaddnqprlemfl 6715* Lemma for addnqpr 6717. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
 |-  ( ( 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 } >. ) ) )
 
Theoremaddnqprlemfu 6716* Lemma for addnqpr 6717. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
 |-  ( ( 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 } >. ) ) )
 
Theoremaddnqpr 6717* Addition of fractions embedded into positive reals. One can either add the fractions as fractions, or embed them into positive reals and add them as positive reals, and get the same result. (Contributed by Jim Kingdon, 19-Aug-2020.)
 |-  ( ( 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 } >. ) )
 
Theoremaddnqpr1 6718* Addition of one to a fraction embedded into a positive real. One can either add the fraction one to the fraction, or the positive real one to the positive real, and get the same result. Special case of addnqpr 6717. (Contributed by Jim Kingdon, 26-Apr-2020.)
 |-  ( A  e.  Q.  -> 
 <. { l  |  l 
 <Q  ( A  +Q  1Q ) } ,  { u  |  ( A  +Q  1Q )  <Q  u } >.  =  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  1P ) )
 
Theoremappdivnq 6719* 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 6720* Approximate division for positive rationals. This can be thought of as a variation of appdivnq 6719 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 6721 Calculations for prmuloc 6722. (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 6722* 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 6723* Positive reals are multiplicatively located. This is a variation of prmuloc 6722 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 6724 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 6725 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 6726 Calculations for mullocpr 6727. (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 6727* 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 6728 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 6729* Lemma for mulnqpr 6733. 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 6730* Lemma for mulnqpr 6733. 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 6731* Lemma for mulnqpr 6733. 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 6732* Lemma for mulnqpr 6733. 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 6733* 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 6734 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 6735 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 6736 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 6737 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 6738 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 6739 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 6740* 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 6741* 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 6742 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 6743 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 6744 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 6745 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 6746 Lemma for 1idpr 6748. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  1P ) )  =  ( 1st `  A ) )
 
Theorem1idpru 6747 Lemma for 1idpr 6748. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |-  ( A  e.  P.  ->  ( 2nd `  ( A  .P.  1P ) )  =  ( 2nd `  A ) )
 
Theorem1idpr 6748 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 6749* 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 6750* 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 6751 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 6752. (Contributed by Jim Kingdon, 15-Dec-2019.)
 |- 
 <P  Po  P.
 
Theoremltsopr 6752 Positive real 'less than' is a weak linear order (in the sense of df-iso 4062). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
 |- 
 <P  Or  P.
 
Theoremltaddpr 6753 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 6754* Element in lower cut of the constructed difference. Lemma for ltexpri 6769. (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 6755* Element in upper cut of the constructed difference. Lemma for ltexpri 6769. (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 6756* Our constructed difference is inhabited. Lemma for ltexpri 6769. (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 6757* The lower cut of our constructed difference is open. Lemma for ltexpri 6769. (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 6758* The lower cut of our constructed difference is lower. Lemma for ltexpri 6769. (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 6759* The upper cut of our constructed difference is open. Lemma for ltexpri 6769. (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 6760* The upper cut of our constructed difference is upper. Lemma for ltexpri 6769. (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 6761* Our constructed difference is rounded. Lemma for ltexpri 6769. (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 6762* Our constructed difference is disjoint. Lemma for ltexpri 6769. (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 6763* Our constructed difference is located. Lemma for ltexpri 6769. (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 6764* Our constructed difference is a positive real. Lemma for ltexpri 6769. (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 6765* Lemma for ltexpri 6769. 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 6766* Lemma for ltexpri 6769. 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 6767* Lemma for ltexpri 6769. 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 6768* Lemma for ltexpri 6769. 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 6769* 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 6770 Lemma for addcanprg 6772. (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 6771 Lemma for addcanprg 6772. (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 6772 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 6773* The difference from ltexpri 6769 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
 |-  ( A  <P  B  ->  E! x  e.  P.  ( A  +P.  x )  =  B )
 
Theoremltaprlem 6774 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 6775 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 6776* 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 6777 Strong extensionality of addition (ordering version). This is similar to addext 7675 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 6778* Membership in the lower cut of  B. Lemma for recexpr 6794. (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 6779* Membership in the upper cut of  B. Lemma for recexpr 6794. (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 6780*  B is inhabited. Lemma for recexpr 6794. (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 6781* The lower cut of  B is open. Lemma for recexpr 6794. (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 6782* The lower cut of  B is lower. Lemma for recexpr 6794. (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 6783* The upper cut of  B is open. Lemma for recexpr 6794. (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 6784* The upper cut of  B is upper. Lemma for recexpr 6794. (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 6785*  B is rounded. Lemma for recexpr 6794. (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 6786*  B is disjoint. Lemma for recexpr 6794. (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 6787*  B is located. Lemma for recexpr 6794. (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 6788*  B is a positive real. Lemma for recexpr 6794. (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 6789* The lower cut of one is a subset of the lower cut of  A  .P.  B. Lemma for recexpr 6794. (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 6790* The upper cut of one is a subset of the upper cut of  A  .P.  B. Lemma for recexpr 6794. (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 6791* The lower cut of  A  .P.  B is a subset of the lower cut of one. Lemma for recexpr 6794. (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 6792* The upper cut of  A  .P.  B is a subset of the upper cut of one. Lemma for recexpr 6794. (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 6793*  B is the reciprocal of  A. Lemma for recexpr 6794. (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 6794* 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 6795 Lemma for aptipr 6797. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  B  <P  A ) 
 ->  ( 1st `  A )  C_  ( 1st `  B ) )
 
Theoremaptiprlemu 6796 Lemma for aptipr 6797. (Contributed by Jim Kingdon, 28-Jan-2020.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\ 
 -.  B  <P  A ) 
 ->  ( 2nd `  B )  C_  ( 2nd `  A ) )
 
Theoremaptipr 6797 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 6798 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 6799* 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 6709. (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 6800* Lemma for cauappcvgprlemladdrl 6813. 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 } >. ) ) ) )
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