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Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltsonq 6901 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.)
 |- 
 <Q  Or  Q.
 
Theoremnqtric 6902 Trichotomy for positive fractions. (Contributed by Jim Kingdon, 21-Sep-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  <Q  B  <->  -.  ( A  =  B  \/  B  <Q  A )
 ) )
 
Theoremltanqg 6903 Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
 
Theoremltmnqg 6904 Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by Jim Kingdon, 22-Sep-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )
 
Theoremltanqi 6905 Ordering property of addition for positive fractions. One direction of ltanqg 6903. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  ( C  +Q  A )  <Q  ( C  +Q  B ) )
 
Theoremltmnqi 6906 Ordering property of multiplication for positive fractions. One direction of ltmnqg 6904. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  ( C  .Q  A )  <Q  ( C  .Q  B ) )
 
Theoremlt2addnq 6907 Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.)
 |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. ) )  ->  ( ( A  <Q  B 
 /\  C  <Q  D ) 
 ->  ( A  +Q  C )  <Q  ( B  +Q  D ) ) )
 
Theoremlt2mulnq 6908 Ordering property of multiplication for positive fractions. (Contributed by Jim Kingdon, 18-Jul-2021.)
 |-  ( ( ( A  e.  Q.  /\  B  e.  Q. )  /\  ( C  e.  Q.  /\  D  e.  Q. ) )  ->  ( ( A  <Q  B 
 /\  C  <Q  D ) 
 ->  ( A  .Q  C )  <Q  ( B  .Q  D ) ) )
 
Theorem1lt2nq 6909 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 |- 
 1Q  <Q  ( 1Q  +Q  1Q )
 
Theoremltaddnq 6910 The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  A  <Q  ( A  +Q  B ) )
 
Theoremltexnqq 6911* Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by Jim Kingdon, 23-Sep-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  <Q  B  <->  E. x  e.  Q.  ( A  +Q  x )  =  B )
 )
 
Theoremltexnqi 6912* Ordering on positive fractions in terms of existence of sum. (Contributed by Jim Kingdon, 30-Apr-2020.)
 |-  ( A  <Q  B  ->  E. x  e.  Q.  ( A  +Q  x )  =  B )
 
Theoremhalfnqq 6913* One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  =  A )
 
Theoremhalfnq 6914* One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  ( A  e.  Q.  ->  E. x ( x  +Q  x )  =  A )
 
Theoremnsmallnqq 6915* There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  x  <Q  A )
 
Theoremnsmallnq 6916* There is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  ( A  e.  Q.  ->  E. x  x  <Q  A )
 
Theoremsubhalfnqq 6917* There is a number which is less than half of any positive fraction. The case where  A is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 6913). (Contributed by Jim Kingdon, 25-Nov-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  A )
 
Theoremltbtwnnqq 6918* There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  <Q  B  <->  E. x  e.  Q.  ( A  <Q  x  /\  x  <Q  B ) )
 
Theoremltbtwnnq 6919* There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  ( A  <Q  B  <->  E. x ( A 
 <Q  x  /\  x  <Q  B ) )
 
Theoremarchnqq 6920* For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Jim Kingdon, 1-Dec-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  N.  A  <Q  [ <. x ,  1o >. ]  ~Q  )
 
Theoremprarloclemarch 6921* A version of the Archimedean property. This variation is "stronger" than archnqq 6920 in the sense that we provide an integer which is larger than a given rational  A even after being multiplied by a second rational  B. (Contributed by Jim Kingdon, 30-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  E. x  e.  N.  A  <Q  ( [ <. x ,  1o >. ]  ~Q  .Q  B ) )
 
Theoremprarloclemarch2 6922* Like prarloclemarch 6921 but the integer must be at least two, and there is also  B added to the right hand side. These details follow straightforwardly but are chosen to be helpful in the proof of prarloc 7006. (Contributed by Jim Kingdon, 25-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q.  /\  C  e.  Q. )  ->  E. x  e.  N.  ( 1o  <N  x  /\  A  <Q  ( B  +Q  ( [ <. x ,  1o >. ]  ~Q  .Q  C ) ) ) )
 
Theoremltrnqg 6923 Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6924. (Contributed by Jim Kingdon, 29-Dec-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  <Q  B  <-> 
 ( *Q `  B )  <Q  ( *Q `  A ) ) )
 
Theoremltrnqi 6924 Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 6923. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  <Q  B  ->  ( *Q `  B ) 
 <Q  ( *Q `  A ) )
 
Theoremnnnq 6925 The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.)
 |-  ( A  e.  N.  ->  [ <. A ,  1o >. ]  ~Q  e.  Q. )
 
Theoremltnnnq 6926 Ordering of positive integers via 
<N or  <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  <N  B  <->  [ <. A ,  1o >. ]  ~Q  <Q  [ <. B ,  1o >. ]  ~Q  )
 )
 
Definitiondf-enq0 6927* Define equivalence relation for non-negative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
 |- ~Q0  =  { <. x ,  y >.  |  ( ( x  e.  ( om  X.  N. )  /\  y  e.  ( om  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .o  u )  =  ( w  .o  v ) ) ) }
 
Definitiondf-nq0 6928 Define class of non-negative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
 |- Q0  =  ( ( om  X.  N. ) /. ~Q0  )
 
Definitiondf-0nq0 6929 Define non-negative fraction constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 5-Nov-2019.)
 |- 0Q0  =  [ <. (/) ,  1o >. ] ~Q0
 
Definitiondf-plq0 6930* Define addition on non-negative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
 |- +Q0  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e. Q0  /\  y  e. Q0 )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( ( w  .o  f )  +o  (
 v  .o  u )
 ) ,  ( v  .o  f ) >. ] ~Q0  )
 ) }
 
Definitiondf-mq0 6931* Define multiplication on non-negative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
 |- ·Q0  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e. Q0  /\  y  e. Q0 )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u ) ,  ( v  .o  f ) >. ] ~Q0  ) ) }
 
Theoremdfmq0qs 6932* Multiplication on non-negative fractions. This definition is similar to df-mq0 6931 but expands Q0 (Contributed by Jim Kingdon, 22-Nov-2019.)
 |- ·Q0  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e.  (
 ( om  X.  N. ) /. ~Q0  ) 
 /\  y  e.  (
 ( om  X.  N. ) /. ~Q0  ) )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u ) ,  ( v  .o  f ) >. ] ~Q0  ) ) }
 
Theoremdfplq0qs 6933* Addition on non-negative fractions. This definition is similar to df-plq0 6930 but expands Q0 (Contributed by Jim Kingdon, 24-Nov-2019.)
 |- +Q0  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  (
 ( om  X.  N. ) /. ~Q0  ) 
 /\  y  e.  (
 ( om  X.  N. ) /. ~Q0  ) )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( ( w  .o  f )  +o  (
 v  .o  u )
 ) ,  ( v  .o  f ) >. ] ~Q0  )
 ) }
 
Theoremenq0enq 6934 Equivalence on positive fractions in terms of equivalence on non-negative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.)
 |- 
 ~Q  =  ( ~Q0  i^i  ( ( N. 
 X.  N. )  X.  ( N.  X.  N. ) ) )
 
Theoremenq0sym 6935 The equivalence relation for non-negative fractions is symmetric. Lemma for enq0er 6938. (Contributed by Jim Kingdon, 14-Nov-2019.)
 |-  ( f ~Q0  g  ->  g ~Q0  f )
 
Theoremenq0ref 6936 The equivalence relation for non-negative fractions is reflexive. Lemma for enq0er 6938. (Contributed by Jim Kingdon, 14-Nov-2019.)
 |-  ( f  e.  ( om  X.  N. )  <->  f ~Q0  f )
 
Theoremenq0tr 6937 The equivalence relation for non-negative fractions is transitive. Lemma for enq0er 6938. (Contributed by Jim Kingdon, 14-Nov-2019.)
 |-  ( ( f ~Q0  g  /\  g ~Q0  h )  ->  f ~Q0  h )
 
Theoremenq0er 6938 The equivalence relation for non-negative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
 |- ~Q0  Er  ( om  X.  N. )
 
Theoremenq0breq 6939 Equivalence relation for non-negative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.)
 |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. ) )  ->  ( <. A ,  B >. ~Q0  <. C ,  D >.  <->  ( A  .o  D )  =  ( B  .o  C ) ) )
 
Theoremenq0eceq 6940 Equivalence class equality of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.)
 |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ] ~Q0  =  [ <. C ,  D >. ] ~Q0  <->  ( A  .o  D )  =  ( B  .o  C ) ) )
 
Theoremnqnq0pi 6941 A non-negative fraction is a positive fraction if its numerator and denominator are positive integers. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  [ <. A ,  B >. ] ~Q0  =  [ <. A ,  B >. ]  ~Q  )
 
Theoremenq0ex 6942 The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
 |- ~Q0  e.  _V
 
Theoremnq0ex 6943 The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
 |- Q0  e.  _V
 
Theoremnqnq0 6944 A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
 |- 
 Q.  C_ Q0
 
Theoremnq0nn 6945* Decomposition of a non-negative fraction into numerator and denominator. (Contributed by Jim Kingdon, 24-Nov-2019.)
 |-  ( A  e. Q0  ->  E. w E. v
 ( ( w  e. 
 om  /\  v  e.  N. )  /\  A  =  [ <. w ,  v >. ] ~Q0  ) )
 
Theoremaddcmpblnq0 6946 Lemma showing compatibility of addition on non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
 |-  ( ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. )
 )  /\  ( ( F  e.  om  /\  G  e.  N. )  /\  ( R  e.  om  /\  S  e.  N. ) ) ) 
 ->  ( ( ( A  .o  D )  =  ( B  .o  C )  /\  ( F  .o  S )  =  ( G  .o  R ) ) 
 ->  <. ( ( A  .o  G )  +o  ( B  .o  F ) ) ,  ( B  .o  G ) >. ~Q0  <. ( ( C  .o  S )  +o  ( D  .o  R ) ) ,  ( D  .o  S ) >. ) )
 
Theoremmulcmpblnq0 6947 Lemma showing compatibility of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
 |-  ( ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. )
 )  /\  ( ( F  e.  om  /\  G  e.  N. )  /\  ( R  e.  om  /\  S  e.  N. ) ) ) 
 ->  ( ( ( A  .o  D )  =  ( B  .o  C )  /\  ( F  .o  S )  =  ( G  .o  R ) ) 
 ->  <. ( A  .o  F ) ,  ( B  .o  G ) >. ~Q0  <. ( C  .o  R ) ,  ( D  .o  S ) >. ) )
 
Theoremmulcanenq0ec 6948 Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  om  /\  C  e.  N. )  ->  [ <. ( A  .o  B ) ,  ( A  .o  C ) >. ] ~Q0  =  [ <. B ,  C >. ] ~Q0  )
 
Theoremnnnq0lem1 6949* Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6952 and mulnnnq0 6953. (Contributed by Jim Kingdon, 23-Nov-2019.)
 |-  ( ( ( A  e.  ( ( om  X. 
 N. ) /. ~Q0  )  /\  B  e.  ( ( om  X.  N. ) /. ~Q0  ) )  /\  (
 ( ( A  =  [ <. w ,  v >. ] ~Q0  /\  B  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ C ] ~Q0  )  /\  ( ( A  =  [ <. s ,  f >. ] ~Q0  /\  B  =  [ <. g ,  h >. ] ~Q0  )  /\  q  =  [ D ] ~Q0  ) ) )  ->  ( ( ( ( w  e.  om  /\  v  e.  N. )  /\  ( s  e.  om  /\  f  e.  N. )
 )  /\  ( ( u  e.  om  /\  t  e.  N. )  /\  (
 g  e.  om  /\  h  e.  N. )
 ) )  /\  (
 ( w  .o  f
 )  =  ( v  .o  s )  /\  ( u  .o  h )  =  ( t  .o  g ) ) ) )
 
Theoremaddnq0mo 6950* There is at most one result from adding non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
 |-  ( ( A  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  B  e.  ( ( om  X.  N. ) /. ~Q0  ) )  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ] ~Q0  /\  B  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ <. ( ( w  .o  t )  +o  (
 v  .o  u )
 ) ,  ( v  .o  t ) >. ] ~Q0  )
 )
 
Theoremmulnq0mo 6951* There is at most one result from multiplying non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
 |-  ( ( A  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  B  e.  ( ( om  X.  N. ) /. ~Q0  ) )  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ] ~Q0  /\  B  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u ) ,  ( v  .o  t ) >. ] ~Q0  ) )
 
Theoremaddnnnq0 6952 Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
 |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ] ~Q0 +Q0  [ <. C ,  D >. ] ~Q0  )  =  [ <. ( ( A  .o  D )  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )
 
Theoremmulnnnq0 6953 Multiplication of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)
 |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ] ~Q0 ·Q0  [ <. C ,  D >. ] ~Q0  )  =  [ <. ( A  .o  C ) ,  ( B  .o  D ) >. ] ~Q0  )
 
Theoremaddclnq0 6954 Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0 ) 
 ->  ( A +Q0  B )  e. Q0 )
 
Theoremmulclnq0 6955 Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0 ) 
 ->  ( A ·Q0  B )  e. Q0 )
 
Theoremnqpnq0nq 6956 A positive fraction plus a non-negative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e. Q0 )  ->  ( A +Q0  B )  e.  Q. )
 
Theoremnqnq0a 6957 Addition of positive fractions is equal with  +Q or +Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  =  ( A +Q0  B ) )
 
Theoremnqnq0m 6958 Multiplication of positive fractions is equal with  .Q or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  =  ( A ·Q0  B ) )
 
Theoremnq0m0r 6959 Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
 |-  ( A  e. Q0  ->  (0Q0 ·Q0  A )  = 0Q0 )
 
Theoremnq0a0 6960 Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
 |-  ( A  e. Q0  ->  ( A +Q0 0Q0 )  =  A )
 
Theoremnnanq0 6961 Addition of non-negative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.)
 |-  ( ( N  e.  om 
 /\  M  e.  om  /\  A  e.  N. )  ->  [ <. ( N  +o  M ) ,  A >. ] ~Q0  =  ( [ <. N ,  A >. ] ~Q0 +Q0  [ <. M ,  A >. ] ~Q0  ) )
 
Theoremdistrnq0 6962 Multiplication of non-negative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0  /\  C  e. Q0 )  ->  ( A ·Q0  ( B +Q0  C ) )  =  ( ( A ·Q0  B ) +Q0  ( A ·Q0  C ) ) )
 
Theoremmulcomnq0 6963 Multiplication of non-negative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0 ) 
 ->  ( A ·Q0  B )  =  ( B ·Q0  A ) )
 
Theoremaddassnq0lemcl 6964 A natural number closure law. Lemma for addassnq0 6965. (Contributed by Jim Kingdon, 3-Dec-2019.)
 |-  ( ( ( I  e.  om  /\  J  e.  N. )  /\  ( K  e.  om  /\  L  e.  N. ) )  ->  ( ( ( I  .o  L )  +o  ( J  .o  K ) )  e.  om  /\  ( J  .o  L )  e.  N. ) )
 
Theoremaddassnq0 6965 Addition of non-negaative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0  /\  C  e. Q0 )  ->  ( ( A +Q0  B ) +Q0  C )  =  ( A +Q0  ( B +Q0  C ) ) )
 
Theoremdistnq0r 6966 Multiplication of non-negative fractions is distributive. Version of distrnq0 6962 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0  /\  C  e. Q0 )  ->  ( ( B +Q0  C ) ·Q0  A )  =  ( ( B ·Q0  A ) +Q0  ( C ·Q0  A ) ) )
 
Theoremaddpinq1 6967 Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
 |-  ( A  e.  N.  ->  [ <. ( A  +N  1o ) ,  1o >. ] 
 ~Q  =  ( [ <. A ,  1o >. ] 
 ~Q  +Q  1Q )
 )
 
Theoremnq02m 6968 Multiply a non-negative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( A  e. Q0  ->  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  A )  =  ( A +Q0  A ) )
 
Definitiondf-inp 6969* Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other.

Here we follow the definition of a Dedekind cut from Definition 11.2.1 of [HoTT], p. (varies) with the one exception that we define it over positive rational numbers rather than all rational numbers.

A Dedekind cut is an ordered pair of a lower set  l and an upper set  u which is inhabited ( E. q  e. 
Q. q  e.  l  /\  E. r  e. 
Q. r  e.  u), rounded ( A. q  e.  Q. ( q  e.  l  <->  E. r  e.  Q. ( q  <Q  r  /\  r  e.  l
) ) and likewise for  u), disjoint ( A. q  e. 
Q. -.  ( q  e.  l  /\  q  e.  u )) and located ( A. q  e. 
Q. A. r  e.  Q. ( q  <Q  r  ->  ( q  e.  l  \/  r  e.  u
) )). See HoTT for more discussion of those terms and different ways of defining Dedekind cuts.

(Note: This is a "temporary" definition used in the construction of complex numbers, and is intended to be used only by the construction.) (Contributed by Jim Kingdon, 25-Sep-2019.)

 |- 
 P.  =  { <. l ,  u >.  |  ( ( ( l  C_  Q. 
 /\  u  C_  Q. )  /\  ( E. q  e. 
 Q.  q  e.  l  /\  E. r  e.  Q.  r  e.  u )
 )  /\  ( ( A. q  e.  Q.  ( q  e.  l  <->  E. r  e.  Q.  (
 q  <Q  r  /\  r  e.  l ) )  /\  A. r  e.  Q.  (
 r  e.  u  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  u )
 ) )  /\  A. q  e.  Q.  -.  (
 q  e.  l  /\  q  e.  u )  /\  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  l  \/  r  e.  u ) ) ) ) }
 
Definitiondf-i1p 6970* Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by Jim Kingdon, 25-Sep-2019.)
 |- 
 1P  =  <. { l  |  l  <Q  1Q } ,  { u  |  1Q  <Q  u } >.
 
Definitiondf-iplp 6971* Define addition on positive reals. From Section 11.2.1 of [HoTT], p. (varies). We write this definition to closely resemble the definition in HoTT although some of the conditions are redundant (for example,  r  e.  ( 1st `  x ) implies 
r  e.  Q.) and can be simplified as shown at genpdf 7011.

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 26-Sep-2019.)

 |- 
 +P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  <. { q  e. 
 Q.  |  E. r  e.  Q.  E. s  e. 
 Q.  ( r  e.  ( 1st `  x )  /\  s  e.  ( 1st `  y )  /\  q  =  ( r  +Q  s ) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
 r  e.  ( 2nd `  x )  /\  s  e.  ( 2nd `  y
 )  /\  q  =  ( r  +Q  s
 ) ) } >. )
 
Definitiondf-imp 6972* Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 6971 or the definition of multiplication on positive reals in Metamath Proof Explorer. This is as opposed to the more complicated definition of multiplication given in Section 11.2.1 of [HoTT], p. (varies), which appears to be motivated by handling negative numbers or handling modified Dedekind cuts in which locatedness is omitted.

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.)

 |- 
 .P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  <. { q  e. 
 Q.  |  E. r  e.  Q.  E. s  e. 
 Q.  ( r  e.  ( 1st `  x )  /\  s  e.  ( 1st `  y )  /\  q  =  ( r  .Q  s ) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
 r  e.  ( 2nd `  x )  /\  s  e.  ( 2nd `  y
 )  /\  q  =  ( r  .Q  s
 ) ) } >. )
 
Definitiondf-iltp 6973* Define ordering on positive reals. We define  x 
<P  y if there is a positive fraction  q which is an element of the upper cut of  x and the lower cut of  y. From the definition of < in Section 11.2.1 of [HoTT], p. (varies).

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.)

 |- 
 <P  =  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
 
Theoremnpsspw 6974 Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |- 
 P.  C_  ( ~P Q.  X. 
 ~P Q. )
 
Theorempreqlu 6975 Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  =  B 
 <->  ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B ) ) ) )
 
Theoremnpex 6976 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
 |- 
 P.  e.  _V
 
Theoremelinp 6977* Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  <->  ( ( ( L  C_  Q.  /\  U  C_ 
 Q. )  /\  ( E. q  e.  Q.  q  e.  L  /\  E. r  e.  Q.  r  e.  U ) )  /\  ( ( A. q  e.  Q.  ( q  e.  L  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  L )
 )  /\  A. r  e. 
 Q.  ( r  e.  U  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  U )
 ) )  /\  A. q  e.  Q.  -.  (
 q  e.  L  /\  q  e.  U )  /\  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  L  \/  r  e.  U )
 ) ) ) )
 
Theoremprop 6978 A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( A  e.  P.  -> 
 <. ( 1st `  A ) ,  ( 2nd `  A ) >.  e.  P. )
 
Theoremelnp1st2nd 6979* Membership in positive reals, using  1st and  2nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)
 |-  ( A  e.  P.  <->  (
 ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A ) ) )  /\  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  A ) 
 <-> 
 E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) ) 
 /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  ( q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) 
 /\  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  ( 1st `  A )  \/  r  e.  ( 2nd `  A ) ) ) ) ) )
 
Theoremprml 6980* A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  E. x  e.  Q.  x  e.  L )
 
Theoremprmu 6981* A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  E. x  e.  Q.  x  e.  U )
 
Theoremprssnql 6982 The lower cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  L  C_ 
 Q. )
 
Theoremprssnqu 6983 The upper cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  U  C_ 
 Q. )
 
Theoremelprnql 6984 An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L )  ->  B  e.  Q. )
 
Theoremelprnqu 6985 An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  U )  ->  B  e.  Q. )
 
Theorem0npr 6986 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
 |- 
 -.  (/)  e.  P.
 
Theoremprcdnql 6987 A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L )  ->  ( C  <Q  B  ->  C  e.  L ) )
 
Theoremprcunqu 6988 An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  C  e.  U )  ->  ( C  <Q  B  ->  B  e.  U ) )
 
Theoremprubl 6989 A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.)
 |-  ( ( ( <. L ,  U >.  e.  P.  /\  B  e.  L ) 
 /\  C  e.  Q. )  ->  ( -.  C  e.  L  ->  B  <Q  C ) )
 
Theoremprltlu 6990 An element of a lower cut is less than an element of the corresponding upper cut. (Contributed by Jim Kingdon, 15-Oct-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L  /\  C  e.  U )  ->  B  <Q  C )
 
Theoremprnmaxl 6991* A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L )  ->  E. x  e.  L  B  <Q  x )
 
Theoremprnminu 6992* An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  U )  ->  E. x  e.  U  x  <Q  B )
 
Theoremprnmaddl 6993* A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L )  ->  E. x  e.  Q.  ( B  +Q  x )  e.  L )
 
Theoremprloc 6994 A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  A  <Q  B )  ->  ( A  e.  L  \/  B  e.  U ) )
 
Theoremprdisj 6995 A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  A  e.  Q. )  ->  -.  ( A  e.  L  /\  A  e.  U ) )
 
Theoremprarloclemlt 6996 Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7006. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( ( X  e.  om  /\  ( <. L ,  U >.  e. 
 P.  /\  A  e.  L  /\  P  e.  Q. ) )  /\  y  e. 
 om )  ->  ( A  +Q  ( [ <. ( y  +o  1o ) ,  1o >. ]  ~Q  .Q  P ) )  <Q  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) ) )
 
Theoremprarloclemlo 6997* Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 7006. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( ( X  e.  om  /\  ( <. L ,  U >.  e. 
 P.  /\  A  e.  L  /\  P  e.  Q. ) )  /\  y  e. 
 om )  ->  (
 ( A  +Q  ( [ <. ( y  +o  1o ) ,  1o >. ] 
 ~Q  .Q  P )
 )  e.  L  ->  ( ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  suc 
 X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) 
 ->  E. y  e.  om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) ) ) )
 
Theoremprarloclemup 6998 Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 7006. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( ( X  e.  om  /\  ( <. L ,  U >.  e. 
 P.  /\  A  e.  L  /\  P  e.  Q. ) )  /\  y  e. 
 om )  ->  (
 ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U  ->  ( ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  suc 
 X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) 
 ->  E. y  e.  om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) ) ) )
 
Theoremprarloclem3step 6999* Induction step for prarloclem3 7000. (Contributed by Jim Kingdon, 9-Nov-2019.)
 |-  ( ( ( X  e.  om  /\  ( <. L ,  U >.  e. 
 P.  /\  A  e.  L  /\  P  e.  Q. ) )  /\  E. y  e.  om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  suc  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )  ->  E. y  e.  om  (
 ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )
 
Theoremprarloclem3 7000* Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7006. (Contributed by Jim Kingdon, 27-Oct-2019.)
 |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L ) 
 /\  ( X  e.  om 
 /\  P  e.  Q. )  /\  E. y  e. 
 om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )  ->  E. j  e.  om  (
 ( A +Q0  ( [ <. j ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( j  +o  2o ) ,  1o >. ] 
 ~Q  .Q  P )
 )  e.  U ) )
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