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Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremen2other2 6801 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { U. ( P  \  { X } ) }
 )  =  X )
 
Theoremdju1p1e2 6802 Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  ( 1o 1o )  ~~  2o
 
Theoreminfpwfidom 6803 The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption 
( ~P A  i^i  Fin )  e.  _V because this theorem also implies that  A is a set if  ~P A  i^i  Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
 
Theoremexmidfodomrlemeldju 6804 Lemma for exmidfodomr 6809. A variant of djur 6736. (Contributed by Jim Kingdon, 2-Jul-2022.)
 |-  ( ph  ->  A  C_ 
 1o )   &    |-  ( ph  ->  B  e.  ( A 1o )
 )   =>    |-  ( ph  ->  ( B  =  (inl `  (/) )  \/  B  =  (inr `  (/) ) ) )
 
Theoremexmidfodomrlemreseldju 6805 Lemma for exmidfodomrlemrALT 6808. A variant of eldju 6738. (Contributed by Jim Kingdon, 9-Jul-2022.)
 |-  ( ph  ->  A  C_ 
 1o )   &    |-  ( ph  ->  B  e.  ( A 1o )
 )   =>    |-  ( ph  ->  (
 ( (/)  e.  A  /\  B  =  ( (inl  |`  A ) `  (/) ) )  \/  B  =  ( (inr  |`  1o ) `  (/) ) ) )
 
Theoremexmidfodomrlemim 6806* Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  (EXMID 
 ->  A. x A. y
 ( ( E. z  z  e.  y  /\  y 
 ~<_  x )  ->  E. f  f : x -onto-> y ) )
 
Theoremexmidfodomrlemr 6807* The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  ( A. x A. y ( ( E. z  z  e.  y  /\  y  ~<_  x )  ->  E. f  f : x -onto-> y )  -> EXMID )
 
TheoremexmidfodomrlemrALT 6808* The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. An alternative proof of exmidfodomrlemr 6807. In particular, this proof uses eldju 6738 instead of djur 6736 and avoids djulclb 6726. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.)
 |-  ( A. x A. y ( ( E. z  z  e.  y  /\  y  ~<_  x )  ->  E. f  f : x -onto-> y )  -> EXMID )
 
Theoremexmidfodomr 6809* Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  (EXMID  <->  A. x A. y ( ( E. z  z  e.  y  /\  y  ~<_  x )  ->  E. f  f : x -onto-> y ) )
 
PART 3  REAL AND COMPLEX NUMBERS

This section derives the basics of real and complex numbers.

To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 6217 and similar theorems ), going from there to positive integers (df-ni 6842) and then positive rational numbers (df-nqqs 6886) does not involve a major change in approach compared with the Metamath Proof Explorer.

It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle. With excluded middle, it is natural to define the cut as the lower set only (as Metamath Proof Explorer does), but we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero".

 
3.1  Construction and axiomatization of real and complex numbers
 
3.1.1  Dedekind-cut construction of real and complex numbers
 
Syntaxcnpi 6810 The set of positive integers, which is the set of natural numbers  om with 0 removed.

Note: This is the start of the Dedekind-cut construction of real and _complex numbers.

 class  N.
 
Syntaxcpli 6811 Positive integer addition.
 class  +N
 
Syntaxcmi 6812 Positive integer multiplication.
 class  .N
 
Syntaxclti 6813 Positive integer ordering relation.
 class  <N
 
Syntaxcplpq 6814 Positive pre-fraction addition.
 class  +pQ
 
Syntaxcmpq 6815 Positive pre-fraction multiplication.
 class  .pQ
 
Syntaxcltpq 6816 Positive pre-fraction ordering relation.
 class  <pQ
 
Syntaxceq 6817 Equivalence class used to construct positive fractions.
 class  ~Q
 
Syntaxcnq 6818 Set of positive fractions.
 class  Q.
 
Syntaxc1q 6819 The positive fraction constant 1.
 class  1Q
 
Syntaxcplq 6820 Positive fraction addition.
 class  +Q
 
Syntaxcmq 6821 Positive fraction multiplication.
 class  .Q
 
Syntaxcrq 6822 Positive fraction reciprocal operation.
 class  *Q
 
Syntaxcltq 6823 Positive fraction ordering relation.
 class  <Q
 
Syntaxceq0 6824 Equivalence class used to construct nonnegative fractions.
 class ~Q0
 
Syntaxcnq0 6825 Set of nonnegative fractions.
 class Q0
 
Syntaxc0q0 6826 The nonnegative fraction constant 0.
 class 0Q0
 
Syntaxcplq0 6827 Nonnegative fraction addition.
 class +Q0
 
Syntaxcmq0 6828 Nonnegative fraction multiplication.
 class ·Q0
 
Syntaxcnp 6829 Set of positive reals.
 class  P.
 
Syntaxc1p 6830 Positive real constant 1.
 class  1P
 
Syntaxcpp 6831 Positive real addition.
 class  +P.
 
Syntaxcmp 6832 Positive real multiplication.
 class  .P.
 
Syntaxcltp 6833 Positive real ordering relation.
 class  <P
 
Syntaxcer 6834 Equivalence class used to construct signed reals.
 class  ~R
 
Syntaxcnr 6835 Set of signed reals.
 class  R.
 
Syntaxc0r 6836 The signed real constant 0.
 class  0R
 
Syntaxc1r 6837 The signed real constant 1.
 class  1R
 
Syntaxcm1r 6838 The signed real constant -1.
 class  -1R
 
Syntaxcplr 6839 Signed real addition.
 class  +R
 
Syntaxcmr 6840 Signed real multiplication.
 class  .R
 
Syntaxcltr 6841 Signed real ordering relation.
 class  <R
 
Definitiondf-ni 6842 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 6843 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 6844 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 6845 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 6846 Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
 |-  ( A  e.  N.  <->  ( A  e.  om  /\  A  =/= 
 (/) ) )
 
Theorempinn 6847 A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.)
 |-  ( A  e.  N.  ->  A  e.  om )
 
Theorempion 6848 A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.)
 |-  ( A  e.  N.  ->  A  e.  On )
 
Theorempiord 6849 A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.)
 |-  ( A  e.  N.  ->  Ord  A )
 
Theoremniex 6850 The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.)
 |- 
 N.  e.  _V
 
Theorem0npi 6851 The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.)
 |- 
 -.  (/)  e.  N.
 
Theoremelni2 6852 Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.)
 |-  ( A  e.  N.  <->  ( A  e.  om  /\  (/)  e.  A ) )
 
Theorem1pi 6853 Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.)
 |- 
 1o  e.  N.
 
Theoremaddpiord 6854 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 6855 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 6856 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 6857 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 6858 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 6859 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 6860 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 6861 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 6862 Positive integer 'less than' is a relation on positive integers. (Contributed by NM, 8-Feb-1996.)
 |- 
 <N  C_  ( N.  X.  N. )
 
Theoremdmaddpi 6863 Domain of addition on positive integers. (Contributed by NM, 26-Aug-1995.)
 |- 
 dom  +N  =  ( N.  X.  N. )
 
Theoremdmmulpi 6864 Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
 |- 
 dom  .N  =  ( N.  X.  N. )
 
Theoremaddclpi 6865 Closure of addition of positive integers. (Contributed by NM, 18-Oct-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  +N  B )  e.  N. )
 
Theoremmulclpi 6866 Closure of multiplication of positive integers. (Contributed by NM, 18-Oct-1995.)
 |-  ( ( A  e.  N. 
 /\  B  e.  N. )  ->  ( A  .N  B )  e.  N. )
 
Theoremaddcompig 6867 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 6868 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 6869 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 6870 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 6871 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 6872 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 6873 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 6874 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 6875* 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 6876 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 6877 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 6878 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.)
 |- 
 1o  <N  ( 1o  +N  1o )
 
Theoremnlt1pig 6879 No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.)
 |-  ( A  e.  N.  ->  -.  A  <N  1o )
 
Theoremindpi 6880* 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 6881 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 6882* 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 6887) works with the equivalence classes of these ordered pairs determined by the equivalence relation  ~Q (df-enq 6885). (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 6883* 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 6884* 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 6885* 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 6886 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 6887* 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 6888* 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 6889 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 6890* 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 6891* 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 6892* 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 6893* 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 6894 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 6895 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 6896 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 6897 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 6898 The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)
 |- 
 ~Q  e.  _V
 
Theoremenqdc 6899 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 6900 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 )
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