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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdmaddpi 7101 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
 |- 
 dom  +N  =  ( N.  X.  N. )
 
Theoremdmmulpi 7102 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
 |- 
 dom  .N  =  ( N.  X.  N. )
 
Theoremaddclpi 7103 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  +N  B )  e.  N. )
 
Theoremmulclpi 7104 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  .N  B )  e.  N. )
 
Theoremaddcompig 7105 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 7106 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 7107 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 7108 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 7109 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 7110 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 7111 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 7112 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 7113* 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 7114 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 7115 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 7116 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
 |- 
 1o  <N  ( 1o  +N  1o )
 
Theoremnlt1pig 7117 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
 |-  ( A  e.  N.  ->  -.  A  <N  1o )
 
Theoremindpi 7118* 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 7119 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 7120* 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 7125) works with the equivalence classes of these ordered pairs determined by the equivalence relation  ~Q (df-enq 7123). (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 7121* 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 7122* 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 7123* 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 7124 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 7125* 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 7126* 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 7127 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 7128* 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 7129* 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 7130* 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 7131* 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 7132 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 7133 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 7134 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 7135 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 7136 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
 |- 
 ~Q  e.  _V
 
Theoremenqdc 7137 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 7138 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 7139 The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
 |- 
 Q.  e.  _V
 
Theorem0nnq 7140 The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
 |- 
 -.  (/)  e.  Q.
 
Theoremltrelnq 7141 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 7142 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
 |- 
 1Q  e.  Q.
 
Theoremaddcmpblnq 7143 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 7144 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 7145 Lemma for addpipqqs 7146. (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 7146 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 7147 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 7148 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 7149 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 7150 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 7151 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  e.  Q. )
 
Theoremmulclnq 7152 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  e.  Q. )
 
Theoremdmaddpqlem 7153* Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 7155. (Contributed by Jim Kingdon, 15-Sep-2019.)
 |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v >. ]  ~Q  )
 
Theoremnqpi 7154* Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 7153 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 7155 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  +Q  =  ( Q.  X.  Q. )
 
Theoremdmmulpq 7156 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  .Q  =  ( Q.  X.  Q. )
 
Theoremaddcomnqg 7157 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 7158 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 7159 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 7160 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 7161 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 7162 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 7163 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 7164 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
 |-  ( A  e.  N.  ->  1Q  =  [ <. A ,  A >. ]  ~Q  )
 
Theoremmulidnq 7165 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
 |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
 
Theoremrecexnq 7166* 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 7167 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 7168 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 7169 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 7170 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 7171 Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
 |-  ( *Q `  1Q )  =  1Q
 
Theoremnqtri3or 7172 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 7173 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 7174 '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 7175 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 7176 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 7177 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 7178 Ordering property of addition for positive fractions. One direction of ltanqg 7176. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  ( C  +Q  A )  <Q  ( C  +Q  B ) )
 
Theoremltmnqi 7179 Ordering property of multiplication for positive fractions. One direction of ltmnqg 7177. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  ( C  .Q  A )  <Q  ( C  .Q  B ) )
 
Theoremlt2addnq 7180 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 7181 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 7182 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 7183 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 7184* 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 7185* 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 7186* 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 7187* 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 7188* There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  x  <Q  A )
 
Theoremnsmallnq 7189* 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 7190* 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 7186). (Contributed by Jim Kingdon, 25-Nov-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  A )
 
Theoremltbtwnnqq 7191* 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 7192* 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 7193* 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 7194* A version of the Archimedean property. This variation is "stronger" than archnqq 7193 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 7195* Like prarloclemarch 7194 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 7279. (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 7196 Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7197. (Contributed by Jim Kingdon, 29-Dec-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  <Q  B  <-> 
 ( *Q `  B )  <Q  ( *Q `  A ) ) )
 
Theoremltrnqi 7197 Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7196. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  <Q  B  ->  ( *Q `  B ) 
 <Q  ( *Q `  A ) )
 
Theoremnnnq 7198 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 7199 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 7200* 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 ) ) ) }
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