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Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulcanenqec 7201 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 7202 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 7203 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.)
 |-  ( A  e.  N.  ->  1Q  =  [ <. A ,  A >. ]  ~Q  )
 
Theoremmulidnq 7204 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.)
 |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
 
Theoremrecexnq 7205* 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 7206 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 7207 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 7208 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 7209 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 7210 Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.)
 |-  ( *Q `  1Q )  =  1Q
 
Theoremnqtri3or 7211 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 7212 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 7213 '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 7214 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 7215 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 7216 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 7217 Ordering property of addition for positive fractions. One direction of ltanqg 7215. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  ( C  +Q  A )  <Q  ( C  +Q  B ) )
 
Theoremltmnqi 7218 Ordering property of multiplication for positive fractions. One direction of ltmnqg 7216. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  ( C  .Q  A )  <Q  ( C  .Q  B ) )
 
Theoremlt2addnq 7219 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 7220 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 7221 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 7222 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 7223* 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 7224* 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 7225* 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 7226* 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 7227* There is no smallest positive fraction. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  x  <Q  A )
 
Theoremnsmallnq 7228* 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 7229* 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 7225). (Contributed by Jim Kingdon, 25-Nov-2019.)
 |-  ( A  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  A )
 
Theoremltbtwnnqq 7230* 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 7231* 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 7232* 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 7233* A version of the Archimedean property. This variation is "stronger" than archnqq 7232 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 7234* Like prarloclemarch 7233 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 7318. (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 7235 Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7236. (Contributed by Jim Kingdon, 29-Dec-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  <Q  B  <-> 
 ( *Q `  B )  <Q  ( *Q `  A ) ) )
 
Theoremltrnqi 7236 Ordering property of reciprocal for positive fractions. For the converse, see ltrnqg 7235. (Contributed by Jim Kingdon, 24-Sep-2019.)
 |-  ( A  <Q  B  ->  ( *Q `  B ) 
 <Q  ( *Q `  A ) )
 
Theoremnnnq 7237 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 7238 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 7239* Define equivalence relation for nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
 |- ~Q0  =  { <. x ,  y >.  |  ( ( x  e.  ( om  X.  N. )  /\  y  e.  ( om  X.  N. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  /\  ( z  .o  u )  =  ( w  .o  v ) ) ) }
 
Definitiondf-nq0 7240 Define class of nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
 |- Q0  =  ( ( om  X.  N. ) /. ~Q0  )
 
Definitiondf-0nq0 7241 Define nonnegative fraction constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 5-Nov-2019.)
 |- 0Q0  =  [ <. (/) ,  1o >. ] ~Q0
 
Definitiondf-plq0 7242* Define addition on nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
 |- +Q0  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e. Q0  /\  y  e. Q0 )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( ( w  .o  f )  +o  (
 v  .o  u )
 ) ,  ( v  .o  f ) >. ] ~Q0  )
 ) }
 
Definitiondf-mq0 7243* Define multiplication on nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
 |- ·Q0  =  { <.
 <. x ,  y >. ,  z >.  |  (
 ( x  e. Q0  /\  y  e. Q0 )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] ~Q0  /\  y  =  [ <. u ,  f >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u ) ,  ( v  .o  f ) >. ] ~Q0  ) ) }
 
Theoremdfmq0qs 7244* Multiplication on nonnegative fractions. This definition is similar to df-mq0 7243 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 7245* Addition on nonnegative fractions. This definition is similar to df-plq0 7242 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 7246 Equivalence on positive fractions in terms of equivalence on nonnegative fractions. (Contributed by Jim Kingdon, 12-Nov-2019.)
 |- 
 ~Q  =  ( ~Q0  i^i  ( ( N. 
 X.  N. )  X.  ( N.  X.  N. ) ) )
 
Theoremenq0sym 7247 The equivalence relation for nonnegative fractions is symmetric. Lemma for enq0er 7250. (Contributed by Jim Kingdon, 14-Nov-2019.)
 |-  ( f ~Q0  g  ->  g ~Q0  f )
 
Theoremenq0ref 7248 The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7250. (Contributed by Jim Kingdon, 14-Nov-2019.)
 |-  ( f  e.  ( om  X.  N. )  <->  f ~Q0  f )
 
Theoremenq0tr 7249 The equivalence relation for nonnegative fractions is transitive. Lemma for enq0er 7250. (Contributed by Jim Kingdon, 14-Nov-2019.)
 |-  ( ( f ~Q0  g  /\  g ~Q0  h )  ->  f ~Q0  h )
 
Theoremenq0er 7250 The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.)
 |- ~Q0  Er  ( om  X.  N. )
 
Theoremenq0breq 7251 Equivalence relation for nonnegative 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 7252 Equivalence class equality of nonnegative 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 7253 A nonnegative 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 7254 The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
 |- ~Q0  e.  _V
 
Theoremnq0ex 7255 The class of positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.)
 |- Q0  e.  _V
 
Theoremnqnq0 7256 A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)
 |- 
 Q.  C_ Q0
 
Theoremnq0nn 7257* Decomposition of a nonnegative 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 7258 Lemma showing compatibility of addition on nonnegative 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 7259 Lemma showing compatibility of multiplication on nonnegative 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 ) >. ) )
 
Theoremmulcanenq0ec 7260 Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e.  N. 
 /\  B  e.  om  /\  C  e.  N. )  ->  [ <. ( A  .o  B ) ,  ( A  .o  C ) >. ] ~Q0  =  [ <. B ,  C >. ] ~Q0  )
 
Theoremnnnq0lem1 7261* Decomposing nonnegative fractions into natural numbers. Lemma for addnnnq0 7264 and mulnnnq0 7265. (Contributed by Jim Kingdon, 23-Nov-2019.)
 |-  ( ( ( A  e.  ( ( om  X. 
 N. ) /. ~Q0  )  /\  B  e.  ( ( om  X.  N. ) /. ~Q0  ) )  /\  (
 ( ( A  =  [ <. w ,  v >. ] ~Q0  /\  B  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ C ] ~Q0  )  /\  ( ( A  =  [ <. s ,  f >. ] ~Q0  /\  B  =  [ <. g ,  h >. ] ~Q0  )  /\  q  =  [ D ] ~Q0  ) ) )  ->  ( ( ( ( w  e.  om  /\  v  e.  N. )  /\  ( s  e.  om  /\  f  e.  N. )
 )  /\  ( ( u  e.  om  /\  t  e.  N. )  /\  (
 g  e.  om  /\  h  e.  N. )
 ) )  /\  (
 ( w  .o  f
 )  =  ( v  .o  s )  /\  ( u  .o  h )  =  ( t  .o  g ) ) ) )
 
Theoremaddnq0mo 7262* There is at most one result from adding nonnegative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
 |-  ( ( A  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  B  e.  ( ( om  X.  N. ) /. ~Q0  ) )  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ] ~Q0  /\  B  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ <. ( ( w  .o  t )  +o  (
 v  .o  u )
 ) ,  ( v  .o  t ) >. ] ~Q0  )
 )
 
Theoremmulnq0mo 7263* There is at most one result from multiplying nonnegative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
 |-  ( ( A  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  B  e.  ( ( om  X.  N. ) /. ~Q0  ) )  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ] ~Q0  /\  B  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u ) ,  ( v  .o  t ) >. ] ~Q0  ) )
 
Theoremaddnnnq0 7264 Addition of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
 |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ] ~Q0 +Q0  [ <. C ,  D >. ] ~Q0  )  =  [ <. ( ( A  .o  D )  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )
 
Theoremmulnnnq0 7265 Multiplication of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)
 |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ] ~Q0 ·Q0  [ <. C ,  D >. ] ~Q0  )  =  [ <. ( A  .o  C ) ,  ( B  .o  D ) >. ] ~Q0  )
 
Theoremaddclnq0 7266 Closure of addition on nonnegative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0 ) 
 ->  ( A +Q0  B )  e. Q0 )
 
Theoremmulclnq0 7267 Closure of multiplication on nonnegative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0 ) 
 ->  ( A ·Q0  B )  e. Q0 )
 
Theoremnqpnq0nq 7268 A positive fraction plus a nonnegative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e. Q0 )  ->  ( A +Q0  B )  e.  Q. )
 
Theoremnqnq0a 7269 Addition of positive fractions is equal with  +Q or +Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  =  ( A +Q0  B ) )
 
Theoremnqnq0m 7270 Multiplication of positive fractions is equal with  .Q or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  =  ( A ·Q0  B ) )
 
Theoremnq0m0r 7271 Multiplication with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
 |-  ( A  e. Q0  ->  (0Q0 ·Q0  A )  = 0Q0 )
 
Theoremnq0a0 7272 Addition with zero for nonnegative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
 |-  ( A  e. Q0  ->  ( A +Q0 0Q0 )  =  A )
 
Theoremnnanq0 7273 Addition of nonnegative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.)
 |-  ( ( N  e.  om 
 /\  M  e.  om  /\  A  e.  N. )  ->  [ <. ( N  +o  M ) ,  A >. ] ~Q0  =  ( [ <. N ,  A >. ] ~Q0 +Q0  [ <. M ,  A >. ] ~Q0  ) )
 
Theoremdistrnq0 7274 Multiplication of nonnegative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0  /\  C  e. Q0 )  ->  ( A ·Q0  ( B +Q0  C ) )  =  ( ( A ·Q0  B ) +Q0  ( A ·Q0  C ) ) )
 
Theoremmulcomnq0 7275 Multiplication of nonnegative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0 ) 
 ->  ( A ·Q0  B )  =  ( B ·Q0  A ) )
 
Theoremaddassnq0lemcl 7276 A natural number closure law. Lemma for addassnq0 7277. (Contributed by Jim Kingdon, 3-Dec-2019.)
 |-  ( ( ( I  e.  om  /\  J  e.  N. )  /\  ( K  e.  om  /\  L  e.  N. ) )  ->  ( ( ( I  .o  L )  +o  ( J  .o  K ) )  e.  om  /\  ( J  .o  L )  e.  N. ) )
 
Theoremaddassnq0 7277 Addition of nonnegative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0  /\  C  e. Q0 )  ->  ( ( A +Q0  B ) +Q0  C )  =  ( A +Q0  ( B +Q0  C ) ) )
 
Theoremdistnq0r 7278 Multiplication of nonnegative fractions is distributive. Version of distrnq0 7274 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0  /\  C  e. Q0 )  ->  ( ( B +Q0  C ) ·Q0  A )  =  ( ( B ·Q0  A ) +Q0  ( C ·Q0  A ) ) )
 
Theoremaddpinq1 7279 Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
 |-  ( A  e.  N.  ->  [ <. ( A  +N  1o ) ,  1o >. ] 
 ~Q  =  ( [ <. A ,  1o >. ] 
 ~Q  +Q  1Q )
 )
 
Theoremnq02m 7280 Multiply a nonnegative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( A  e. Q0  ->  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  A )  =  ( A +Q0  A ) )
 
Definitiondf-inp 7281* 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.

Here we follow the definition of a Dedekind cut from Definition 11.2.1 of [HoTT], p. (varies) with the one exception that we define it over positive rational numbers rather than all rational numbers.

A Dedekind cut is an ordered pair of a lower set  l and an upper set  u which is inhabited ( E. q  e. 
Q. q  e.  l  /\  E. r  e. 
Q. r  e.  u), rounded ( A. q  e.  Q. ( q  e.  l  <->  E. r  e.  Q. ( q  <Q  r  /\  r  e.  l
) ) and likewise for  u), disjoint ( A. q  e. 
Q. -.  ( q  e.  l  /\  q  e.  u )) and located ( A. q  e. 
Q. A. r  e.  Q. ( q  <Q  r  ->  ( q  e.  l  \/  r  e.  u
) )). See HoTT for more discussion of those terms and different ways of defining Dedekind cuts.

(Note: This is a "temporary" definition used in the construction of complex numbers, and is intended to be used only by the construction.) (Contributed by Jim Kingdon, 25-Sep-2019.)

 |- 
 P.  =  { <. l ,  u >.  |  ( ( ( l  C_  Q. 
 /\  u  C_  Q. )  /\  ( E. q  e. 
 Q.  q  e.  l  /\  E. r  e.  Q.  r  e.  u )
 )  /\  ( ( A. q  e.  Q.  ( q  e.  l  <->  E. r  e.  Q.  (
 q  <Q  r  /\  r  e.  l ) )  /\  A. r  e.  Q.  (
 r  e.  u  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  u )
 ) )  /\  A. q  e.  Q.  -.  (
 q  e.  l  /\  q  e.  u )  /\  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  l  \/  r  e.  u ) ) ) ) }
 
Definitiondf-i1p 7282* Define the positive real constant 1. 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, 25-Sep-2019.)
 |- 
 1P  =  <. { l  |  l  <Q  1Q } ,  { u  |  1Q  <Q  u } >.
 
Definitiondf-iplp 7283* Define addition on positive reals. From Section 11.2.1 of [HoTT], p. (varies). We write this definition to closely resemble the definition in HoTT although some of the conditions are redundant (for example,  r  e.  ( 1st `  x ) implies 
r  e.  Q.) and can be simplified as shown at genpdf 7323.

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, 26-Sep-2019.)

 |- 
 +P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  <. { q  e. 
 Q.  |  E. r  e.  Q.  E. s  e. 
 Q.  ( r  e.  ( 1st `  x )  /\  s  e.  ( 1st `  y )  /\  q  =  ( r  +Q  s ) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
 r  e.  ( 2nd `  x )  /\  s  e.  ( 2nd `  y
 )  /\  q  =  ( r  +Q  s
 ) ) } >. )
 
Definitiondf-imp 7284* Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 7283 or the definition of multiplication on positive reals in Metamath Proof Explorer. This is as opposed to the more complicated definition of multiplication given in Section 11.2.1 of [HoTT], p. (varies), which appears to be motivated by handling negative numbers or handling modified Dedekind cuts in which locatedness is omitted.

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, 29-Sep-2019.)

 |- 
 .P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  <. { q  e. 
 Q.  |  E. r  e.  Q.  E. s  e. 
 Q.  ( r  e.  ( 1st `  x )  /\  s  e.  ( 1st `  y )  /\  q  =  ( r  .Q  s ) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
 r  e.  ( 2nd `  x )  /\  s  e.  ( 2nd `  y
 )  /\  q  =  ( r  .Q  s
 ) ) } >. )
 
Definitiondf-iltp 7285* Define ordering on positive reals. We define  x 
<P  y if there is a positive fraction  q which is an element of the upper cut of  x and the lower cut of  y. From the definition of < in Section 11.2.1 of [HoTT], p. (varies).

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, 29-Sep-2019.)

 |- 
 <P  =  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
 
Theoremnpsspw 7286 Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |- 
 P.  C_  ( ~P Q.  X. 
 ~P Q. )
 
Theorempreqlu 7287 Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  =  B 
 <->  ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B ) ) ) )
 
Theoremnpex 7288 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
 |- 
 P.  e.  _V
 
Theoremelinp 7289* Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  <->  ( ( ( L  C_  Q.  /\  U  C_ 
 Q. )  /\  ( E. q  e.  Q.  q  e.  L  /\  E. r  e.  Q.  r  e.  U ) )  /\  ( ( A. q  e.  Q.  ( q  e.  L  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  L )
 )  /\  A. r  e. 
 Q.  ( r  e.  U  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  U )
 ) )  /\  A. q  e.  Q.  -.  (
 q  e.  L  /\  q  e.  U )  /\  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  L  \/  r  e.  U )
 ) ) ) )
 
Theoremprop 7290 A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( A  e.  P.  -> 
 <. ( 1st `  A ) ,  ( 2nd `  A ) >.  e.  P. )
 
Theoremelnp1st2nd 7291* Membership in positive reals, using  1st and  2nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)
 |-  ( A  e.  P.  <->  (
 ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A ) ) )  /\  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  A ) 
 <-> 
 E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) ) 
 /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  ( q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) 
 /\  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  ( 1st `  A )  \/  r  e.  ( 2nd `  A ) ) ) ) ) )
 
Theoremprml 7292* A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  E. x  e.  Q.  x  e.  L )
 
Theoremprmu 7293* A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  E. x  e.  Q.  x  e.  U )
 
Theoremprssnql 7294 The lower cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  L  C_ 
 Q. )
 
Theoremprssnqu 7295 The upper cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  U  C_ 
 Q. )
 
Theoremelprnql 7296 An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L )  ->  B  e.  Q. )
 
Theoremelprnqu 7297 An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  U )  ->  B  e.  Q. )
 
Theorem0npr 7298 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
 |- 
 -.  (/)  e.  P.
 
Theoremprcdnql 7299 A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L )  ->  ( C  <Q  B  ->  C  e.  L ) )
 
Theoremprcunqu 7300 An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  C  e.  U )  ->  ( C  <Q  B  ->  B  e.  U ) )
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