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Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-mpq 6501* 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 6502* 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 6503* 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 6504 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 6505* 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 6506* 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 6507 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 6508* 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 6509* 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 6510* Alternative 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 6511* Alternative 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 6512 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 6513 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 6514 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 6515 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 6516 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
 |- 
 ~Q  e.  _V
 
Theoremenqdc 6517 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 6518 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 6519 The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
 |- 
 Q.  e.  _V
 
Theorem0nnq 6520 The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
 |- 
 -.  (/)  e.  Q.
 
Theoremltrelnq 6521 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 6522 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
 |- 
 1Q  e.  Q.
 
Theoremaddcmpblnq 6523 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 6524 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 6525 Lemma for addpipqqs 6526. (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 6526 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 6527 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 6528 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 6529 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 6530 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 6531 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  e.  Q. )
 
Theoremmulclnq 6532 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  e.  Q. )
 
Theoremdmaddpqlem 6533* Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6535. (Contributed by Jim Kingdon, 15-Sep-2019.)
 |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v >. ]  ~Q  )
 
Theoremnqpi 6534* Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 6533 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 6535 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  +Q  =  ( Q.  X.  Q. )
 
Theoremdmmulpq 6536 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  .Q  =  ( Q.  X.  Q. )
 
Theoremaddcomnqg 6537 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 6538 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 6539 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 6540 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 6541 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 6542 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 6543 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 6544 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
 |-  ( A  e.  N.  ->  1Q  =  [ <. A ,  A >. ]  ~Q  )
 
Theoremmulidnq 6545 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
 |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
 
Theoremrecexnq 6546* 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 6547 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 6548 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 6549 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 6550 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 6551 Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
 |-  ( *Q `  1Q )  =  1Q
 
Theoremnqtri3or 6552 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 6553 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 6554 '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 6555 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 6556 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 6557 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 6558 Ordering property of addition for positive fractions. One direction of ltanqg 6556. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  ( C  +Q  A )  <Q  ( C  +Q  B ) )
 
Theoremltmnqi 6559 Ordering property of multiplication for positive fractions. One direction of ltmnqg 6557. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  ( C  .Q  A )  <Q  ( C  .Q  B ) )
 
Theoremlt2addnq 6560 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 6561 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 6562 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 6563 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 6564* 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 6565* 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 6566* 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 6567* 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 6568* There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  x  <Q  A )
 
Theoremnsmallnq 6569* 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 6570* 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 6566). (Contributed by Jim Kingdon, 25-Nov-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  A )
 
Theoremltbtwnnqq 6571* 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 6572* 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 6573* 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 6574* A version of the Archimedean property. This variation is "stronger" than archnqq 6573 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 6575* Like prarloclemarch 6574 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 6659. (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 6576 Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6577. (Contributed by Jim Kingdon, 29-Dec-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  <Q  B  <-> 
 ( *Q `  B )  <Q  ( *Q `  A ) ) )
 
Theoremltrnqi 6577 Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 6576. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  <Q  B  ->  ( *Q `  B ) 
 <Q  ( *Q `  A ) )
 
Theoremnnnq 6578 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 6579 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 6580* 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 6581 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 6582 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 6583* 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 6584* 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 6585* Multiplication on non-negative fractions. This definition is similar to df-mq0 6584 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 6586* Addition on non-negative fractions. This definition is similar to df-plq0 6583 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 6587 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 6588 The equivalence relation for non-negative fractions is symmetric. Lemma for enq0er 6591. (Contributed by Jim Kingdon, 14-Nov-2019.)
 |-  ( f ~Q0  g  ->  g ~Q0  f )
 
Theoremenq0ref 6589 The equivalence relation for non-negative fractions is reflexive. Lemma for enq0er 6591. (Contributed by Jim Kingdon, 14-Nov-2019.)
 |-  ( f  e.  ( om  X.  N. )  <->  f ~Q0  f )
 
Theoremenq0tr 6590 The equivalence relation for non-negative fractions is transitive. Lemma for enq0er 6591. (Contributed by Jim Kingdon, 14-Nov-2019.)
 |-  ( ( f ~Q0  g  /\  g ~Q0  h )  ->  f ~Q0  h )
 
Theoremenq0er 6591 The equivalence relation for non-negative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
 |- ~Q0  Er  ( om  X.  N. )
 
Theoremenq0breq 6592 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 6593 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 6594 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 6595 The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
 |- ~Q0  e.  _V
 
Theoremnq0ex 6596 The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
 |- Q0  e.  _V
 
Theoremnqnq0 6597 A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
 |- 
 Q.  C_ Q0
 
Theoremnq0nn 6598* 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 6599 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 6600 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 ) >. ) )
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