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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddcompig 7101 Addition of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  +N  B )  =  ( B  +N  A ) )
 
Theoremaddasspig 7102 Addition of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  ->  ( ( A  +N  B )  +N  C )  =  ( A  +N  ( B  +N  C ) ) )
 
Theoremmulcompig 7103 Multiplication of positive integers is commutative. (Contributed by Jim Kingdon, 26-Aug-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  .N  B )  =  ( B  .N  A ) )
 
Theoremmulasspig 7104 Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  ->  ( ( A  .N  B )  .N  C )  =  ( A  .N  ( B  .N  C ) ) )
 
Theoremdistrpig 7105 Multiplication of positive integers is distributive. (Contributed by Jim Kingdon, 26-Aug-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  ->  ( A  .N  ( B  +N  C ) )  =  ( ( A  .N  B )  +N  ( A  .N  C ) ) )
 
Theoremaddcanpig 7106 Addition cancellation law for positive integers. (Contributed by Jim Kingdon, 27-Aug-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  ->  ( ( A  +N  B )  =  ( A  +N  C )  <->  B  =  C ) )
 
Theoremmulcanpig 7107 Multiplication cancellation law for positive integers. (Contributed by Jim Kingdon, 29-Aug-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  ->  ( ( A  .N  B )  =  ( A  .N  C )  <->  B  =  C ) )
 
Theoremaddnidpig 7108 There is no identity element for addition on positive integers. (Contributed by NM, 28-Nov-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  -.  ( A  +N  B )  =  A )
 
Theoremltexpi 7109* Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  <N  B  <->  E. x  e.  N.  ( A  +N  x )  =  B )
 )
 
Theoremltapig 7110 Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  ->  ( A  <N  B  <->  ( C  +N  A )  <N  ( C  +N  B ) ) )
 
Theoremltmpig 7111 Ordering property of multiplication for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  ->  ( A  <N  B  <->  ( C  .N  A )  <N  ( C  .N  B ) ) )
 
Theorem1lt2pi 7112 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
 |- 
 1o  <N  ( 1o  +N  1o )
 
Theoremnlt1pig 7113 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
 |-  ( A  e.  N.  ->  -.  A  <N  1o )
 
Theoremindpi 7114* Principle of Finite Induction on positive integers. (Contributed by NM, 23-Mar-1996.)
 |-  ( x  =  1o  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +N  1o )  ->  ( ph 
 <-> 
 th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  N.  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  N.  ->  ta )
 
Theoremnnppipi 7115 A natural number plus a positive integer is a positive integer. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( A  e.  om 
 /\  B  e.  N. )  ->  ( A  +o  B )  e.  N. )
 
Definitiondf-plpq 7116* Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition  +Q (df-plqqs 7121) works with the equivalence classes of these ordered pairs determined by the equivalence relation  ~Q (df-enq 7119). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.)
 |- 
 +pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N.  X.  N. )  |-> 
 <. ( ( ( 1st `  x )  .N  ( 2nd `  y ) )  +N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) ) ,  ( ( 2nd `  x )  .N  ( 2nd `  y
 ) ) >. )
 
Definitiondf-mpq 7117* Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.)
 |- 
 .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N.  X.  N. )  |-> 
 <. ( ( 1st `  x )  .N  ( 1st `  y
 ) ) ,  (
 ( 2nd `  x )  .N  ( 2nd `  y
 ) ) >. )
 
Definitiondf-ltpq 7118* Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.)
 |- 
 <pQ  =  { <. x ,  y >.  |  ( ( x  e.  ( N. 
 X.  N. )  /\  y  e.  ( N.  X.  N. ) )  /\  ( ( 1st `  x )  .N  ( 2nd `  y
 ) )  <N  ( ( 1st `  y )  .N  ( 2nd `  x ) ) ) }
 
Definitiondf-enq 7119* Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.)
 |- 
 ~Q  =  { <. x ,  y >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X. 
 N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .N  u )  =  ( w  .N  v ) ) ) }
 
Definitiondf-nqqs 7120 Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 16-Aug-1995.)
 |- 
 Q.  =  ( ( N.  X.  N. ) /.  ~Q  )
 
Definitiondf-plqqs 7121* Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 24-Aug-1995.)
 |- 
 +Q  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] 
 ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [ ( <. w ,  v >.  +pQ  <. u ,  f >. ) ]  ~Q  ) ) }
 
Definitiondf-mqqs 7122* Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 24-Aug-1995.)
 |- 
 .Q  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  Q.  /\  y  e.  Q. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] 
 ~Q  /\  y  =  [ <. u ,  f >. ]  ~Q  )  /\  z  =  [ ( <. w ,  v >.  .pQ  <. u ,  f >. ) ]  ~Q  ) ) }
 
Definitiondf-1nqqs 7123 Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117. (Contributed by NM, 29-Oct-1995.)
 |- 
 1Q  =  [ <. 1o ,  1o >. ]  ~Q
 
Definitiondf-rq 7124* Define reciprocal on positive fractions. It means the same thing as one divided by the argument (although we don't define full division since we will never need it). This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to denote this unary operation. (Contributed by Jim Kingdon, 20-Sep-2019.)
 |- 
 *Q  =  { <. x ,  y >.  |  ( x  e.  Q.  /\  y  e.  Q.  /\  ( x  .Q  y )  =  1Q ) }
 
Definitiondf-ltnqqs 7125* Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.)
 |- 
 <Q  =  { <. x ,  y >.  |  ( ( x  e.  Q.  /\  y  e.  Q. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ] 
 ~Q  /\  y  =  [ <. v ,  u >. ]  ~Q  )  /\  ( z  .N  u )  <N  ( w  .N  v ) ) ) }
 
Theoremdfplpq2 7126* Alternate definition of pre-addition on positive fractions. (Contributed by Jim Kingdon, 12-Sep-2019.)
 |- 
 +pQ  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) ) 
 /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  = 
 <. ( ( w  .N  f )  +N  (
 v  .N  u )
 ) ,  ( v  .N  f ) >. ) ) }
 
Theoremdfmpq2 7127* Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.)
 |- 
 .pQ  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( N.  X.  N. )  /\  y  e.  ( N.  X.  N. ) ) 
 /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  = 
 <. ( w  .N  u ) ,  ( v  .N  f ) >. ) ) }
 
Theoremenqbreq 7128 Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( <. A ,  B >.  ~Q  <. C ,  D >.  <-> 
 ( A  .N  D )  =  ( B  .N  C ) ) )
 
Theoremenqbreq2 7129 Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. ) )  ->  ( A  ~Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B ) )  =  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) )
 
Theoremenqer 7130 The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |- 
 ~Q  Er  ( N.  X. 
 N. )
 
Theoremenqeceq 7131 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ]  ~Q  =  [ <. C ,  D >. ] 
 ~Q 
 <->  ( A  .N  D )  =  ( B  .N  C ) ) )
 
Theoremenqex 7132 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
 |- 
 ~Q  e.  _V
 
Theoremenqdc 7133 The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  -> DECID  <. A ,  B >.  ~Q  <. C ,  D >. )
 
Theoremenqdc1 7134 The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  C  e.  ( N.  X.  N. ) )  -> DECID  <. A ,  B >.  ~Q  C )
 
Theoremnqex 7135 The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
 |- 
 Q.  e.  _V
 
Theorem0nnq 7136 The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
 |- 
 -.  (/)  e.  Q.
 
Theoremltrelnq 7137 Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.)
 |- 
 <Q  C_  ( Q.  X.  Q. )
 
Theorem1nq 7138 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
 |- 
 1Q  e.  Q.
 
Theoremaddcmpblnq 7139 Lemma showing compatibility of addition. (Contributed by NM, 27-Aug-1995.)
 |-  ( ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
 )  /\  ( ( F  e.  N.  /\  G  e.  N. )  /\  ( R  e.  N.  /\  S  e.  N. ) ) ) 
 ->  ( ( ( A  .N  D )  =  ( B  .N  C )  /\  ( F  .N  S )  =  ( G  .N  R ) ) 
 ->  <. ( ( A  .N  G )  +N  ( B  .N  F ) ) ,  ( B  .N  G ) >.  ~Q 
 <. ( ( C  .N  S )  +N  ( D  .N  R ) ) ,  ( D  .N  S ) >. ) )
 
Theoremmulcmpblnq 7140 Lemma showing compatibility of multiplication. (Contributed by NM, 27-Aug-1995.)
 |-  ( ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. )
 )  /\  ( ( F  e.  N.  /\  G  e.  N. )  /\  ( R  e.  N.  /\  S  e.  N. ) ) ) 
 ->  ( ( ( A  .N  D )  =  ( B  .N  C )  /\  ( F  .N  S )  =  ( G  .N  R ) ) 
 ->  <. ( A  .N  F ) ,  ( B  .N  G ) >.  ~Q 
 <. ( C  .N  R ) ,  ( D  .N  S ) >. ) )
 
Theoremaddpipqqslem 7141 Lemma for addpipqqs 7142. (Contributed by Jim Kingdon, 11-Sep-2019.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  <. ( ( A  .N  D )  +N  ( B  .N  C ) ) ,  ( B  .N  D ) >.  e.  ( N.  X.  N. ) )
 
Theoremaddpipqqs 7142 Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ]  ~Q  +Q  [ <. C ,  D >. ] 
 ~Q  )  =  [ <. ( ( A  .N  D )  +N  ( B  .N  C ) ) ,  ( B  .N  D ) >. ]  ~Q  )
 
Theoremmulpipq2 7143 Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. ) )  ->  ( A  .pQ  B )  =  <. ( ( 1st `  A )  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
 >. )
 
Theoremmulpipq 7144 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( <. A ,  B >.  .pQ  <. C ,  D >. )  =  <. ( A  .N  C ) ,  ( B  .N  D ) >. )
 
Theoremmulpipqqs 7145 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ]  ~Q  .Q  [ <. C ,  D >. ] 
 ~Q  )  =  [ <. ( A  .N  C ) ,  ( B  .N  D ) >. ]  ~Q  )
 
Theoremordpipqqs 7146 Ordering of positive fractions in terms of positive integers. (Contributed by Jim Kingdon, 14-Sep-2019.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ]  ~Q  <Q  [ <. C ,  D >. ]  ~Q  <->  ( A  .N  D )  <N  ( B  .N  C ) ) )
 
Theoremaddclnq 7147 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  e.  Q. )
 
Theoremmulclnq 7148 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  e.  Q. )
 
Theoremdmaddpqlem 7149* Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7151. (Contributed by Jim Kingdon, 15-Sep-2019.)
 |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v >. ]  ~Q  )
 
Theoremnqpi 7150* Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7149 but also shows that the numerator and denominator are positive integers. (Contributed by Jim Kingdon, 20-Sep-2019.)
 |-  ( A  e.  Q.  ->  E. w E. v
 ( ( w  e. 
 N.  /\  v  e.  N. )  /\  A  =  [ <. w ,  v >. ]  ~Q  ) )
 
Theoremdmaddpq 7151 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  +Q  =  ( Q.  X.  Q. )
 
Theoremdmmulpq 7152 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  .Q  =  ( Q.  X.  Q. )
 
Theoremaddcomnqg 7153 Addition of positive fractions is commutative. (Contributed by Jim Kingdon, 15-Sep-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  =  ( B  +Q  A ) )
 
Theoremaddassnqg 7154 Addition of positive fractions is associative. (Contributed by Jim Kingdon, 16-Sep-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( ( A  +Q  B )  +Q  C )  =  ( A  +Q  ( B  +Q  C ) ) )
 
Theoremmulcomnqg 7155 Multiplication of positive fractions is commutative. (Contributed by Jim Kingdon, 17-Sep-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  =  ( B  .Q  A ) )
 
Theoremmulassnqg 7156 Multiplication of positive fractions is associative. (Contributed by Jim Kingdon, 17-Sep-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( ( A  .Q  B )  .Q  C )  =  ( A  .Q  ( B  .Q  C ) ) )
 
Theoremmulcanenq 7157 Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  -> 
 <. ( A  .N  B ) ,  ( A  .N  C ) >.  ~Q  <. B ,  C >. )
 
Theoremmulcanenqec 7158 Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 17-Sep-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  ->  [ <. ( A  .N  B ) ,  ( A  .N  C ) >. ] 
 ~Q  =  [ <. B ,  C >. ]  ~Q  )
 
Theoremdistrnqg 7159 Multiplication of positive fractions is distributive. (Contributed by Jim Kingdon, 17-Sep-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q.  /\  C  e.  Q. )  ->  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C ) ) )
 
Theorem1qec 7160 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
 |-  ( A  e.  N.  ->  1Q  =  [ <. A ,  A >. ]  ~Q  )
 
Theoremmulidnq 7161 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
 |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
 
Theoremrecexnq 7162* Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)
 |-  ( A  e.  Q.  ->  E. y ( y  e.  Q.  /\  ( A  .Q  y )  =  1Q ) )
 
Theoremrecmulnqg 7163 Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( ( *Q `  A )  =  B  <->  ( A  .Q  B )  =  1Q ) )
 
Theoremrecclnq 7164 Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
 |-  ( A  e.  Q.  ->  ( *Q `  A )  e.  Q. )
 
Theoremrecidnq 7165 A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.)
 |-  ( A  e.  Q.  ->  ( A  .Q  ( *Q `  A ) )  =  1Q )
 
Theoremrecrecnq 7166 Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.)
 |-  ( A  e.  Q.  ->  ( *Q `  ( *Q `  A ) )  =  A )
 
Theoremrec1nq 7167 Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
 |-  ( *Q `  1Q )  =  1Q
 
Theoremnqtri3or 7168 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 )
 )
 
Theoremltdcnq 7169 Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  -> DECID  A  <Q  B )
 
Theoremltsonq 7170 '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 7171 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 7172 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 7173 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 7174 Ordering property of addition for positive fractions. One direction of ltanqg 7172. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  ( C  +Q  A )  <Q  ( C  +Q  B ) )
 
Theoremltmnqi 7175 Ordering property of multiplication for positive fractions. One direction of ltmnqg 7173. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  ( C  .Q  A )  <Q  ( C  .Q  B ) )
 
Theoremlt2addnq 7176 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 7177 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 7178 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 7179 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 7180* 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 7181* 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 7182* 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 7183* 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 7184* There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  x  <Q  A )
 
Theoremnsmallnq 7185* 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 7186* 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 7182). (Contributed by Jim Kingdon, 25-Nov-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  A )
 
Theoremltbtwnnqq 7187* 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 7188* 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 7189* 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 7190* A version of the Archimedean property. This variation is "stronger" than archnqq 7189 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 7191* Like prarloclemarch 7190 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 7275. (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 7192 Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7193. (Contributed by Jim Kingdon, 29-Dec-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  <Q  B  <-> 
 ( *Q `  B )  <Q  ( *Q `  A ) ) )
 
Theoremltrnqi 7193 Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7192. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  <Q  B  ->  ( *Q `  B ) 
 <Q  ( *Q `  A ) )
 
Theoremnnnq 7194 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 7195 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 7196* Define equivalence relation for nonnegative 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 7197 Define class of nonnegative 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 7198 Define nonnegative 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 7199* Define addition on nonnegative 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 7200* Define multiplication on nonnegative 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  ) ) }
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