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Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcnr 6601 Set of signed reals.
 class  R.
 
Syntaxc0r 6602 The signed real constant 0.
 class  0R
 
Syntaxc1r 6603 The signed real constant 1.
 class  1R
 
Syntaxcm1r 6604 The signed real constant -1.
 class  -1R
 
Syntaxcplr 6605 Signed real addition.
 class  +R
 
Syntaxcmr 6606 Signed real multiplication.
 class  .R
 
Syntaxcltr 6607 Signed real ordering relation.
 class  <R
 
Definitiondf-ni 6608 Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.)
 |- 
 N.  =  ( om  \  { (/) } )
 
Definitiondf-pli 6609 Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)
 |- 
 +N  =  (  +o  |`  ( N.  X.  N. ) )
 
Definitiondf-mi 6610 Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.)
 |- 
 .N  =  (  .o  |`  ( N.  X.  N. ) )
 
Definitiondf-lti 6611 Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.)
 |- 
 <N  =  (  _E  i^i  ( N.  X.  N. ) )
 
Theoremelni 6612 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
 |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/= 
 (/) ) )
 
Theorempinn 6613 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
 |-  ( A  e.  N.  ->  A  e.  om )
 
Theorempion 6614 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
 |-  ( A  e.  N.  ->  A  e.  On )
 
Theorempiord 6615 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
 |-  ( A  e.  N.  ->  Ord  A )
 
Theoremniex 6616 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
 |- 
 N.  e.  _V
 
Theorem0npi 6617 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
 |- 
 -.  (/)  e.  N.
 
Theoremelni2 6618 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
 |-  ( A  e.  N.  <->  ( A  e.  om  /\  (/)  e.  A ) )
 
Theorem1pi 6619 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  N.
 
Theoremaddpiord 6620 Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  +N  B )  =  ( A  +o  B ) )
 
Theoremmulpiord 6621 Positive integer multiplication in terms of ordinal multiplication. (Contributed by NM, 27-Aug-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  .N  B )  =  ( A  .o  B ) )
 
Theoremmulidpi 6622 1 is an identity element for multiplication on positive integers. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.)
 |-  ( A  e.  N.  ->  ( A  .N  1o )  =  A )
 
Theoremltpiord 6623 Positive integer 'less than' in terms of ordinal membership. (Contributed by NM, 6-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B )
 )
 
Theoremltsopi 6624 Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.)
 |- 
 <N  Or  N.
 
Theorempitric 6625 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  <N  B  <->  -.  ( A  =  B  \/  B  <N  A )
 ) )
 
Theorempitri3or 6626 Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  <N  B  \/  A  =  B  \/  B  <N  A )
 )
 
Theoremltdcpi 6627 Less-than for positive integers is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  -> DECID  A  <N  B )
 
Theoremltrelpi 6628 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
 |- 
 <N  C_  ( N.  X.  N. )
 
Theoremdmaddpi 6629 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
 |- 
 dom  +N  =  ( N.  X.  N. )
 
Theoremdmmulpi 6630 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
 |- 
 dom  .N  =  ( N.  X.  N. )
 
Theoremaddclpi 6631 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  +N  B )  e.  N. )
 
Theoremmulclpi 6632 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  .N  B )  e.  N. )
 
Theoremaddcompig 6633 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 6634 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 6635 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 6636 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 6637 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 6638 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 6639 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 6640 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 6641* 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 6642 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 6643 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 6644 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
 |- 
 1o  <N  ( 1o  +N  1o )
 
Theoremnlt1pig 6645 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
 |-  ( A  e.  N.  ->  -.  A  <N  1o )
 
Theoremindpi 6646* 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 6647 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 6648* 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 6653) works with the equivalence classes of these ordered pairs determined by the equivalence relation  ~Q (df-enq 6651). (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 6649* 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 6650* 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 6651* 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 6652 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 6653* 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 6654* 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 6655 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 6656* 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 6657* 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 6658* 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 6659* 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 6660 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 6661 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 6662 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 6663 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 6664 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
 |- 
 ~Q  e.  _V
 
Theoremenqdc 6665 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 6666 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 6667 The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
 |- 
 Q.  e.  _V
 
Theorem0nnq 6668 The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.)
 |- 
 -.  (/)  e.  Q.
 
Theoremltrelnq 6669 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 6670 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.)
 |- 
 1Q  e.  Q.
 
Theoremaddcmpblnq 6671 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 6672 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 6673 Lemma for addpipqqs 6674. (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 6674 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 6675 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 6676 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 6677 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 6678 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 6679 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  e.  Q. )
 
Theoremmulclnq 6680 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  e.  Q. )
 
Theoremdmaddpqlem 6681* Decomposition of a positive fraction into numerator and denominator. Lemma for dmaddpq 6683. (Contributed by Jim Kingdon, 15-Sep-2019.)
 |-  ( x  e.  Q.  ->  E. w E. v  x  =  [ <. w ,  v >. ]  ~Q  )
 
Theoremnqpi 6682* Decomposition of a positive fraction into numerator and denominator. Similar to dmaddpqlem 6681 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 6683 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  +Q  =  ( Q.  X.  Q. )
 
Theoremdmmulpq 6684 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.)
 |- 
 dom  .Q  =  ( Q.  X.  Q. )
 
Theoremaddcomnqg 6685 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 6686 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 6687 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 6688 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 6689 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 6690 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 6691 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 6692 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
 |-  ( A  e.  N.  ->  1Q  =  [ <. A ,  A >. ]  ~Q  )
 
Theoremmulidnq 6693 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
 |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
 
Theoremrecexnq 6694* 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 6695 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 6696 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 6697 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 6698 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 6699 Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
 |-  ( *Q `  1Q )  =  1Q
 
Theoremnqtri3or 6700 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 )
 )
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