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Theorem List for Metamath Proof Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-mpq 8501* Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8711, 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.) (New usage is discouraged.)
 |- 
 .pQ  =  ( x  e.  ( N.  X.  N. ) ,  y  e.  ( N.  X.  N. )  |-> 
 <. ( ( 1st `  x )  .N  ( 1st `  y
 ) ) ,  (
 ( 2nd `  x )  .N  ( 2nd `  y
 ) ) >. )
 
Definitiondf-ltpq 8502* Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)
 |- 
 <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 8503* Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8711, 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.) (New usage is discouraged.)
 |- 
 ~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-nq 8504* Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8711, 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.) (New usage is discouraged.)
 |- 
 Q.  =  { x  e.  ( N.  X.  N. )  |  A. y  e.  ( N.  X.  N. ) ( x  ~Q  y  ->  -.  ( 2nd `  y )  <N  ( 2nd `  x ) ) }
 
Definitiondf-erq 8505 Define a convenience function that "reduces" a fraction to lowest terms. Note that in this form, it is not obviously a function; we prove this in nqerf 8522. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
 |- 
 /Q  =  (  ~Q  i^i  ( ( N.  X.  N. )  X.  Q. )
 )
 
Definitiondf-plq 8506 Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8711, 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.) (New usage is discouraged.)
 |- 
 +Q  =  ( ( /Q  o.  +pQ  )  |`  ( Q.  X.  Q. ) )
 
Definitiondf-mq 8507 Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8711, 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.) (New usage is discouraged.)
 |- 
 .Q  =  ( ( /Q  o.  .pQ  )  |`  ( Q.  X.  Q. ) )
 
Definitiondf-1nq 8508 Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8711, 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.) (New usage is discouraged.)
 |- 
 1Q  =  <. 1o ,  1o >.
 
Definitiondf-rq 8509 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 df-c 8711, 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 NM, 6-Mar-1996.) (New usage is discouraged.)
 |- 
 *Q  =  ( `' 
 .Q  " { 1Q }
 )
 
Definitiondf-ltnq 8510 Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.) (New usage is discouraged.)
 |- 
 <Q  =  (  <pQ  i^i  ( Q.  X.  Q. ) )
 
Theoremenqbreq 8511 Equivalence relation for positive fractions in terms of positive integers. (Contributed by NM, 27-Aug-1995.) (New usage is discouraged.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( <. A ,  B >.  ~Q  <. C ,  D >.  <-> 
 ( A  .N  D )  =  ( B  .N  C ) ) )
 
Theoremenqbreq2 8512 Equivalence relation for positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. ) )  ->  ( A  ~Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B ) )  =  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) )
 
Theoremenqer 8513 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.) (New usage is discouraged.)
 |- 
 ~Q  Er  ( N.  X. 
 N. )
 
Theoremenqex 8514 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
 |- 
 ~Q  e.  _V
 
Theoremnqex 8515 The class of positive fractions exists. (Contributed by NM, 16-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
 |- 
 Q.  e.  _V
 
Theorem0nnq 8516 The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  Q.
 
Theoremelpqn 8517 Each positive fraction is an ordered pair of positive integers (the numerator and denominator, in "lowest terms". (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  A  e.  ( N. 
 X.  N. ) )
 
Theoremltrelnq 8518 Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
 |- 
 <Q  C_  ( Q.  X.  Q. )
 
Theorempinq 8519 The representatives of positive integers as positive fractions. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  N.  -> 
 <. A ,  1o >.  e. 
 Q. )
 
Theorem1nq 8520 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |- 
 1Q  e.  Q.
 
Theoremnqereu 8521* There is a unique element of  Q. equivalent to each element of  N.  X.  N.. (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  e.  ( N.  X.  N. )  ->  E! x  e.  Q.  x  ~Q  A )
 
Theoremnqerf 8522 Corollary of nqereu 8521: the function  /Q is actually a function. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |- 
 /Q : ( N. 
 X.  N. ) --> Q.
 
Theoremnqercl 8523 Corollary of nqereu 8521: closure of  /Q. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  ( N.  X.  N. )  ->  ( /Q `  A )  e.  Q. )
 
Theoremnqerrel 8524 Any member of  ( N.  X.  N. ) relates to the representative of its equivalence class. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  ( N.  X.  N. )  ->  A  ~Q  ( /Q `  A ) )
 
Theoremnqerid 8525 Corollary of nqereu 8521: the function  /Q acts as the identity on members of  Q.. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
 
Theoremenqeq 8526 Corollary of nqereu 8521: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q.  /\  A  ~Q  B ) 
 ->  A  =  B )
 
Theoremnqereq 8527 The function  /Q acts as a substitute for equivalence classes, and it satisfies the fundamental requirement for equivalence representatives: the representatives are equal iff the members are equivalent. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. ) )  ->  ( A  ~Q  B  <->  ( /Q `  A )  =  ( /Q `  B ) ) )
 
Theoremaddpipq2 8528 Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. ) )  ->  ( A  +pQ  B )  =  <. ( ( ( 1st `  A )  .N  ( 2nd `  B ) )  +N  (
 ( 1st `  B )  .N  ( 2nd `  A ) ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) ) >. )
 
Theoremaddpipq 8529 Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( <. A ,  B >.  +pQ  <. C ,  D >. )  =  <. ( ( A  .N  D )  +N  ( C  .N  B ) ) ,  ( B  .N  D ) >. )
 
Theoremaddpqnq 8530 Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  =  ( /Q `  ( A  +pQ  B ) ) )
 
Theoremmulpipq2 8531 Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. ) )  ->  ( A  .pQ  B )  =  <. ( ( 1st `  A )  .N  ( 1st `  B ) ) ,  ( ( 2nd `  A )  .N  ( 2nd `  B ) )
 >. )
 
Theoremmulpipq 8532 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( ( A  e.  N.  /\  B  e.  N. )  /\  ( C  e.  N.  /\  D  e.  N. ) )  ->  ( <. A ,  B >.  .pQ  <. C ,  D >. )  =  <. ( A  .N  C ) ,  ( B  .N  D ) >. )
 
Theoremmulpqnq 8533 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  =  ( /Q `  ( A  .pQ  B ) ) )
 
Theoremordpipq 8534 Ordering of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( <. A ,  B >. 
 <pQ  <. C ,  D >.  <-> 
 ( A  .N  D )  <N  ( C  .N  B ) )
 
Theoremordpinq 8535 Ordering of positive fractions in terms of positive integers. (Contributed by NM, 13-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  <Q  B  <-> 
 ( ( 1st `  A )  .N  ( 2nd `  B ) )  <N  ( ( 1st `  B )  .N  ( 2nd `  A ) ) ) )
 
Theoremaddpqf 8536 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |- 
 +pQ  : ( ( N. 
 X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
 
Theoremaddclnq 8537 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  e.  Q. )
 
Theoremmulpqf 8538 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |- 
 .pQ  : ( ( N. 
 X.  N. )  X.  ( N.  X.  N. ) ) --> ( N.  X.  N. )
 
Theoremmulclnq 8539 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  e.  Q. )
 
Theoremaddnqf 8540 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
 |- 
 +Q  : ( Q. 
 X.  Q. ) --> Q.
 
Theoremmulnqf 8541 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
 |- 
 .Q  : ( Q. 
 X.  Q. ) --> Q.
 
Theoremaddcompq 8542 Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  +pQ  B )  =  ( B  +pQ  A )
 
Theoremaddcomnq 8543 Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  +Q  B )  =  ( B  +Q  A )
 
Theoremmulcompq 8544 Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  .pQ  B )  =  ( B  .pQ  A )
 
Theoremmulcomnq 8545 Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  .Q  B )  =  ( B  .Q  A )
 
Theoremadderpqlem 8546 Lemma for adderpq 8548. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A 
 ~Q  B  <->  ( A  +pQ  C )  ~Q  ( B 
 +pQ  C ) ) )
 
Theoremmulerpqlem 8547 Lemma for mulerpq 8549. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  ( N.  X.  N. )  /\  B  e.  ( N. 
 X.  N. )  /\  C  e.  ( N.  X.  N. ) )  ->  ( A 
 ~Q  B  <->  ( A  .pQ  C )  ~Q  ( B 
 .pQ  C ) ) )
 
Theoremadderpq 8548 Addition is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( /Q `  A )  +Q  ( /Q `  B ) )  =  ( /Q `  ( A  +pQ  B ) )
 
Theoremmulerpq 8549 Multiplication is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( /Q `  A )  .Q  ( /Q `  B ) )  =  ( /Q `  ( A  .pQ  B ) )
 
Theoremaddassnq 8550 Addition of positive fractions is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( ( A  +Q  B )  +Q  C )  =  ( A  +Q  ( B  +Q  C ) )
 
Theoremmulassnq 8551 Multiplication of positive fractions is associative. (Contributed by NM, 1-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  .Q  B )  .Q  C )  =  ( A  .Q  ( B  .Q  C ) )
 
Theoremmulcanenq 8552 Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N.  /\  C  e.  N. )  -> 
 <. ( A  .N  B ) ,  ( A  .N  C ) >.  ~Q  <. B ,  C >. )
 
Theoremdistrnq 8553 Multiplication of positive fractions is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  .Q  ( B  +Q  C ) )  =  ( ( A  .Q  B )  +Q  ( A  .Q  C ) )
 
Theorem1nqenq 8554 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  e.  N.  ->  1Q  ~Q  <. A ,  A >. )
 
Theoremmulidnq 8555 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
 
Theoremrecmulnq 8556 Relationship between reciprocal and multiplication on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( ( *Q `  A )  =  B  <->  ( A  .Q  B )  =  1Q ) )
 
Theoremrecidnq 8557 A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( A  .Q  ( *Q `  A ) )  =  1Q )
 
Theoremrecclnq 8558 Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( *Q `  A )  e.  Q. )
 
Theoremrecrecnq 8559 Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  ( *Q `  ( *Q `  A ) )  =  A )
 
Theoremdmrecnq 8560 Domain of reciprocal on positive fractions. (Contributed by Mario Carneiro, 6-Mar-1996.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
 |- 
 dom  *Q  =  Q.
 
Theoremltsonq 8561 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) (New usage is discouraged.)
 |- 
 <Q  Or  Q.
 
Theoremlterpq 8562 Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)
 |-  ( A  <pQ  B  <->  ( /Q `  A )  <Q  ( /Q
 `  B ) )
 
Theoremltanq 8563 Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( C  e.  Q.  ->  ( A  <Q  B  <->  ( C  +Q  A )  <Q  ( C  +Q  B ) ) )
 
Theoremltmnq 8564 Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( C  e.  Q.  ->  ( A  <Q  B  <->  ( C  .Q  A )  <Q  ( C  .Q  B ) ) )
 
Theorem1lt2nq 8565 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |- 
 1Q  <Q  ( 1Q  +Q  1Q )
 
Theoremltaddnq 8566 The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  A  <Q  ( A  +Q  B ) )
 
Theoremltexnq 8567* Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. x ( A  +Q  x )  =  B ) )
 
Theoremhalfnq 8568* 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.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  E. x ( x  +Q  x )  =  A )
 
Theoremnsmallnq 8569* The is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  E. x  x  <Q  A )
 
Theoremltbtwnnq 8570* 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.) (New usage is discouraged.)
 |-  ( A  <Q  B  <->  E. x ( A 
 <Q  x  /\  x  <Q  B ) )
 
Theoremltrnq 8571 Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  <Q  B  <->  ( *Q `  B )  <Q  ( *Q `  A ) )
 
Theoremarchnq 8572* For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  E. x  e.  N.  A  <Q  <. x ,  1o >.
 )
 
Definitiondf-np 8573* Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other. A positive real is defined as the lower class of a Dedekind cut. Definition 9-3.1 of [Gleason] p. 121. (Note: This is a "temporary" definition used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction.) (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
 |- 
 P.  =  { x  |  ( ( (/)  C.  x  /\  x  C.  Q. )  /\  A. y  e.  x  ( A. z ( z 
 <Q  y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z ) ) }
 
Definitiondf-1p 8574 Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. Definition of [Gleason] p. 122. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
 |- 
 1P  =  { x  |  x  <Q  1Q }
 
Definitiondf-plp 8575* Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. From Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 +P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  +Q  u ) } )
 
Definitiondf-mp 8576* Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 .P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u ) } )
 
Definitiondf-ltp 8577* Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8711, and is intended to be used only by the construction. From Proposition 9-3.2 of [Gleason] p. 122. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
 |- 
 <P  =  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
 
Theoremnpex 8578 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
 |- 
 P.  e.  _V
 
Theoremelnp 8579* Membership in positive reals. (Contributed by NM, 16-Feb-1996.) (New usage is discouraged.)
 |-  ( A  e.  P.  <->  (
 ( (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  (
 A. y ( y 
 <Q  x  ->  y  e.  A )  /\  E. y  e.  A  x  <Q  y ) ) )
 
Theoremelnpi 8580* Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  P.  <->  (
 ( A  e.  _V  /\  (/)  C.  A  /\  A  C. 
 Q. )  /\  A. x  e.  A  ( A. y ( y  <Q  x 
 ->  y  e.  A )  /\  E. y  e.  A  x  <Q  y ) ) )
 
Theoremprn0 8581 A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  A  =/=  (/) )
 
Theoremprpssnq 8582 A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  A  C.  Q. )
 
Theoremelprnq 8583 A positive real is a set of positive fractions. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  B  e.  Q. )
 
Theorem0npr 8584 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  P.
 
Theoremprcdnq 8585 A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  ( C  <Q  B 
 ->  C  e.  A ) )
 
Theoremprub 8586 A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
 |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  e.  Q. )  ->  ( -.  C  e.  A  ->  B 
 <Q  C ) )
 
Theoremprnmax 8587* A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x )
 
Theoremnpomex 8588 A simplifying observation, and an indication of why any attempt to develop a theory of the real numbers without the Axiom of Infinity is doomed to failure: since every member of  P. is an infinite set, the negation of Infinity implies that  P., and hence 
RR, is empty. (Note that this proof, which used the fact that Dedekind cuts have no maximum, could just as well have used that they have no minimum, since they are downward-closed by prcdnq 8585 and nsmallnq 8569). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  om  e.  _V )
 
Theoremprnmadd 8589* A positive real has no largest member. Addition version. (Contributed by NM, 7-Apr-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  E. x ( B  +Q  x )  e.  A )
 
Theoremltrelpr 8590 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
 |- 
 <P  C_  ( P.  X.  P. )
 
Theoremgenpv 8591* Value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   =>    |-  (
 ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  =  { f  |  E. g  e.  A  E. h  e.  B  f  =  ( g G h ) } )
 
Theoremgenpelv 8592* Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   =>    |-  (
 ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( A F B )  <->  E. g  e.  A  E. h  e.  B  C  =  ( g G h ) ) )
 
Theoremgenpprecl 8593* Pre-closure law for general operation on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   =>    |-  (
 ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C G D )  e.  ( A F B ) ) )
 
Theoremgenpdm 8594* Domain of general operation on positive reals. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   =>    |-  dom  F  =  ( P.  X.  P. )
 
Theoremgenpn0 8595* The result of an operation on positive reals is not empty. (Contributed by NM, 28-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   =>    |-  (
 ( A  e.  P.  /\  B  e.  P. )  -> 
 (/)  C.  ( A F B ) )
 
Theoremgenpss 8596* The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   =>    |-  (
 ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  C_  Q. )
 
Theoremgenpnnp 8597* The result of an operation on positive reals is different from the set of positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   &    |-  (
 z  e.  Q.  ->  ( x  <Q  y  <->  ( z G x )  <Q  ( z G y ) ) )   &    |-  ( x G y )  =  ( y G x )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  -.  ( A F B )  =  Q. )
 
Theoremgenpcd 8598* Downward closure of an operation on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   &    |-  (
 ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  ( g G h )  ->  x  e.  ( A F B ) ) )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f 
 ->  x  e.  ( A F B ) ) ) )
 
Theoremgenpnmax 8599* An operation on positive reals has no largest member. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   &    |-  (
 v  e.  Q.  ->  ( z  <Q  w  <->  ( v G z )  <Q  ( v G w ) ) )   &    |-  ( z G w )  =  ( w G z )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  E. x  e.  ( A F B ) f 
 <Q  x ) )
 
Theoremgenpcl 8600* Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   &    |-  ( h  e.  Q.  ->  ( f  <Q  g  <->  ( h G f )  <Q  ( h G g ) ) )   &    |-  ( x G y )  =  ( y G x )   &    |-  ( ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e. 
 Q. )  ->  ( x  <Q  ( g G h )  ->  x  e.  ( A F B ) ) )   =>    |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e. 
 P. )
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