Type  Label  Description 
Statement 

Theorem  addcompig 7101 
Addition of positive integers is commutative. (Contributed by Jim
Kingdon, 26Aug2019.)



Theorem  addasspig 7102 
Addition of positive integers is associative. (Contributed by Jim
Kingdon, 26Aug2019.)



Theorem  mulcompig 7103 
Multiplication of positive integers is commutative. (Contributed by Jim
Kingdon, 26Aug2019.)



Theorem  mulasspig 7104 
Multiplication of positive integers is associative. (Contributed by Jim
Kingdon, 26Aug2019.)



Theorem  distrpig 7105 
Multiplication of positive integers is distributive. (Contributed by Jim
Kingdon, 26Aug2019.)



Theorem  addcanpig 7106 
Addition cancellation law for positive integers. (Contributed by Jim
Kingdon, 27Aug2019.)



Theorem  mulcanpig 7107 
Multiplication cancellation law for positive integers. (Contributed by
Jim Kingdon, 29Aug2019.)



Theorem  addnidpig 7108 
There is no identity element for addition on positive integers.
(Contributed by NM, 28Nov1995.)



Theorem  ltexpi 7109* 
Ordering on positive integers in terms of existence of sum.
(Contributed by NM, 15Mar1996.) (Revised by Mario Carneiro,
14Jun2013.)



Theorem  ltapig 7110 
Ordering property of addition for positive integers. (Contributed by Jim
Kingdon, 31Aug2019.)



Theorem  ltmpig 7111 
Ordering property of multiplication for positive integers. (Contributed
by Jim Kingdon, 31Aug2019.)



Theorem  1lt2pi 7112 
One is less than two (one plus one). (Contributed by NM, 13Mar1996.)



Theorem  nlt1pig 7113 
No positive integer is less than one. (Contributed by Jim Kingdon,
31Aug2019.)



Theorem  indpi 7114* 
Principle of Finite Induction on positive integers. (Contributed by NM,
23Mar1996.)



Theorem  nnppipi 7115 
A natural number plus a positive integer is a positive integer.
(Contributed by Jim Kingdon, 10Nov2019.)



Definition  dfplpq 7116* 
Define preaddition 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 "preaddition" operation works
directly
with ordered pairs of integers. The actual positive fraction addition
(dfplqqs 7121) works with the equivalence classes of these
ordered pairs determined by the equivalence relation
(dfenq 7119). (Analogous remarks apply to the other
"pre" operations
in the complex number construction that follows.) From Proposition
92.3 of [Gleason] p. 117. (Contributed
by NM, 28Aug1995.)



Definition  dfmpq 7117* 
Define premultiplication 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 92.4 of [Gleason]
p. 119. (Contributed by NM, 28Aug1995.)



Definition  dfltpq 7118* 
Define preordering 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,
28Aug1995.)



Definition  dfenq 7119* 
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 92.1 of
[Gleason] p. 117. (Contributed by NM,
27Aug1995.)



Definition  dfnqqs 7120 
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 92.2 of [Gleason] p. 117.
(Contributed by NM, 16Aug1995.)



Definition  dfplqqs 7121* 
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 92.3 of [Gleason] p. 117.
(Contributed by NM, 24Aug1995.)



Definition  dfmqqs 7122* 
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 92.4 of [Gleason] p. 119.
(Contributed by NM, 24Aug1995.)



Definition  df1nqqs 7123 
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 92.2 of [Gleason] p. 117.
(Contributed by NM, 29Oct1995.)



Definition  dfrq 7124* 
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 92.5 of [Gleason] p.
119, who uses an asterisk to
denote this unary operation. (Contributed by Jim Kingdon,
20Sep2019.)



Definition  dfltnqqs 7125* 
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, 13Feb1996.)



Theorem  dfplpq2 7126* 
Alternate definition of preaddition on positive fractions.
(Contributed by Jim Kingdon, 12Sep2019.)



Theorem  dfmpq2 7127* 
Alternate definition of premultiplication on positive fractions.
(Contributed by Jim Kingdon, 13Sep2019.)



Theorem  enqbreq 7128 
Equivalence relation for positive fractions in terms of positive
integers. (Contributed by NM, 27Aug1995.)



Theorem  enqbreq2 7129 
Equivalence relation for positive fractions in terms of positive integers.
(Contributed by Mario Carneiro, 8May2013.)



Theorem  enqer 7130 
The equivalence relation for positive fractions is an equivalence
relation. Proposition 92.1 of [Gleason] p. 117. (Contributed by NM,
27Aug1995.) (Revised by Mario Carneiro, 6Jul2015.)



Theorem  enqeceq 7131 
Equivalence class equality of positive fractions in terms of positive
integers. (Contributed by NM, 29Nov1995.)



Theorem  enqex 7132 
The equivalence relation for positive fractions exists. (Contributed by
NM, 3Sep1995.)



Theorem  enqdc 7133 
The equivalence relation for positive fractions is decidable.
(Contributed by Jim Kingdon, 7Sep2019.)

DECID


Theorem  enqdc1 7134 
The equivalence relation for positive fractions is decidable.
(Contributed by Jim Kingdon, 7Sep2019.)

DECID 

Theorem  nqex 7135 
The class of positive fractions exists. (Contributed by NM,
16Aug1995.) (Revised by Mario Carneiro, 27Apr2013.)



Theorem  0nnq 7136 
The empty set is not a positive fraction. (Contributed by NM,
24Aug1995.) (Revised by Mario Carneiro, 27Apr2013.)



Theorem  ltrelnq 7137 
Positive fraction 'less than' is a relation on positive fractions.
(Contributed by NM, 14Feb1996.) (Revised by Mario Carneiro,
27Apr2013.)



Theorem  1nq 7138 
The positive fraction 'one'. (Contributed by NM, 29Oct1995.)



Theorem  addcmpblnq 7139 
Lemma showing compatibility of addition. (Contributed by NM,
27Aug1995.)



Theorem  mulcmpblnq 7140 
Lemma showing compatibility of multiplication. (Contributed by NM,
27Aug1995.)



Theorem  addpipqqslem 7141 
Lemma for addpipqqs 7142. (Contributed by Jim Kingdon, 11Sep2019.)



Theorem  addpipqqs 7142 
Addition of positive fractions in terms of positive integers.
(Contributed by NM, 28Aug1995.)



Theorem  mulpipq2 7143 
Multiplication of positive fractions in terms of positive integers.
(Contributed by Mario Carneiro, 8May2013.)



Theorem  mulpipq 7144 
Multiplication of positive fractions in terms of positive integers.
(Contributed by NM, 28Aug1995.) (Revised by Mario Carneiro,
8May2013.)



Theorem  mulpipqqs 7145 
Multiplication of positive fractions in terms of positive integers.
(Contributed by NM, 28Aug1995.)



Theorem  ordpipqqs 7146 
Ordering of positive fractions in terms of positive integers.
(Contributed by Jim Kingdon, 14Sep2019.)



Theorem  addclnq 7147 
Closure of addition on positive fractions. (Contributed by NM,
29Aug1995.)



Theorem  mulclnq 7148 
Closure of multiplication on positive fractions. (Contributed by NM,
29Aug1995.)



Theorem  dmaddpqlem 7149* 
Decomposition of a positive fraction into numerator and denominator.
Lemma for dmaddpq 7151. (Contributed by Jim Kingdon, 15Sep2019.)



Theorem  nqpi 7150* 
Decomposition of a positive fraction into numerator and denominator.
Similar to dmaddpqlem 7149 but also shows that the numerator and
denominator are positive integers. (Contributed by Jim Kingdon,
20Sep2019.)



Theorem  dmaddpq 7151 
Domain of addition on positive fractions. (Contributed by NM,
24Aug1995.)



Theorem  dmmulpq 7152 
Domain of multiplication on positive fractions. (Contributed by NM,
24Aug1995.)



Theorem  addcomnqg 7153 
Addition of positive fractions is commutative. (Contributed by Jim
Kingdon, 15Sep2019.)



Theorem  addassnqg 7154 
Addition of positive fractions is associative. (Contributed by Jim
Kingdon, 16Sep2019.)



Theorem  mulcomnqg 7155 
Multiplication of positive fractions is commutative. (Contributed by
Jim Kingdon, 17Sep2019.)



Theorem  mulassnqg 7156 
Multiplication of positive fractions is associative. (Contributed by
Jim Kingdon, 17Sep2019.)



Theorem  mulcanenq 7157 
Lemma for distributive law: cancellation of common factor. (Contributed
by NM, 2Sep1995.) (Revised by Mario Carneiro, 8May2013.)



Theorem  mulcanenqec 7158 
Lemma for distributive law: cancellation of common factor. (Contributed
by Jim Kingdon, 17Sep2019.)



Theorem  distrnqg 7159 
Multiplication of positive fractions is distributive. (Contributed by
Jim Kingdon, 17Sep2019.)



Theorem  1qec 7160 
The equivalence class of ratio 1. (Contributed by NM, 4Mar1996.)



Theorem  mulidnq 7161 
Multiplication identity element for positive fractions. (Contributed by
NM, 3Mar1996.)



Theorem  recexnq 7162* 
Existence of positive fraction reciprocal. (Contributed by Jim Kingdon,
20Sep2019.)



Theorem  recmulnqg 7163 
Relationship between reciprocal and multiplication on positive
fractions. (Contributed by Jim Kingdon, 19Sep2019.)



Theorem  recclnq 7164 
Closure law for positive fraction reciprocal. (Contributed by NM,
6Mar1996.) (Revised by Mario Carneiro, 8May2013.)



Theorem  recidnq 7165 
A positive fraction times its reciprocal is 1. (Contributed by NM,
6Mar1996.) (Revised by Mario Carneiro, 8May2013.)



Theorem  recrecnq 7166 
Reciprocal of reciprocal of positive fraction. (Contributed by NM,
26Apr1996.) (Revised by Mario Carneiro, 29Apr2013.)



Theorem  rec1nq 7167 
Reciprocal of positive fraction one. (Contributed by Jim Kingdon,
29Dec2019.)



Theorem  nqtri3or 7168 
Trichotomy for positive fractions. (Contributed by Jim Kingdon,
21Sep2019.)



Theorem  ltdcnq 7169 
Lessthan for positive fractions is decidable. (Contributed by Jim
Kingdon, 12Dec2019.)

DECID 

Theorem  ltsonq 7170 
'Less than' is a strict ordering on positive fractions. (Contributed by
NM, 19Feb1996.) (Revised by Mario Carneiro, 4May2013.)



Theorem  nqtric 7171 
Trichotomy for positive fractions. (Contributed by Jim Kingdon,
21Sep2019.)



Theorem  ltanqg 7172 
Ordering property of addition for positive fractions. Proposition
92.6(ii) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
22Sep2019.)



Theorem  ltmnqg 7173 
Ordering property of multiplication for positive fractions. Proposition
92.6(iii) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
22Sep2019.)



Theorem  ltanqi 7174 
Ordering property of addition for positive fractions. One direction of
ltanqg 7172. (Contributed by Jim Kingdon, 9Dec2019.)



Theorem  ltmnqi 7175 
Ordering property of multiplication for positive fractions. One direction
of ltmnqg 7173. (Contributed by Jim Kingdon, 9Dec2019.)



Theorem  lt2addnq 7176 
Ordering property of addition for positive fractions. (Contributed by Jim
Kingdon, 7Dec2019.)



Theorem  lt2mulnq 7177 
Ordering property of multiplication for positive fractions. (Contributed
by Jim Kingdon, 18Jul2021.)



Theorem  1lt2nq 7178 
One is less than two (one plus one). (Contributed by NM, 13Mar1996.)
(Revised by Mario Carneiro, 10May2013.)



Theorem  ltaddnq 7179 
The sum of two fractions is greater than one of them. (Contributed by
NM, 14Mar1996.) (Revised by Mario Carneiro, 10May2013.)



Theorem  ltexnqq 7180* 
Ordering on positive fractions in terms of existence of sum. Definition
in Proposition 92.6 of [Gleason] p.
119. (Contributed by Jim Kingdon,
23Sep2019.)



Theorem  ltexnqi 7181* 
Ordering on positive fractions in terms of existence of sum.
(Contributed by Jim Kingdon, 30Apr2020.)



Theorem  halfnqq 7182* 
Onehalf of any positive fraction is a fraction. (Contributed by Jim
Kingdon, 23Sep2019.)



Theorem  halfnq 7183* 
Onehalf of any positive fraction exists. Lemma for Proposition
92.6(i) of [Gleason] p. 120.
(Contributed by NM, 16Mar1996.)
(Revised by Mario Carneiro, 10May2013.)



Theorem  nsmallnqq 7184* 
There is no smallest positive fraction. (Contributed by Jim Kingdon,
24Sep2019.)



Theorem  nsmallnq 7185* 
There is no smallest positive fraction. (Contributed by NM,
26Apr1996.) (Revised by Mario Carneiro, 10May2013.)



Theorem  subhalfnqq 7186* 
There is a number which is less than half of any positive fraction. The
case where 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 7182). (Contributed by Jim Kingdon,
25Nov2019.)



Theorem  ltbtwnnqq 7187* 
There exists a number between any two positive fractions. Proposition
92.6(i) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
24Sep2019.)



Theorem  ltbtwnnq 7188* 
There exists a number between any two positive fractions. Proposition
92.6(i) of [Gleason] p. 120.
(Contributed by NM, 17Mar1996.)
(Revised by Mario Carneiro, 10May2013.)



Theorem  archnqq 7189* 
For any fraction, there is an integer that is greater than it. This is
also known as the "archimedean property". (Contributed by Jim
Kingdon,
1Dec2019.)



Theorem  prarloclemarch 7190* 
A version of the Archimedean property. This variation is "stronger"
than archnqq 7189 in the sense that we provide an integer which
is larger
than a given rational even after being multiplied by a second
rational .
(Contributed by Jim Kingdon, 30Nov2019.)



Theorem  prarloclemarch2 7191* 
Like prarloclemarch 7190 but the integer must be at least two, and
there is
also added to
the right hand side. These details follow
straightforwardly but are chosen to be helpful in the proof of
prarloc 7275. (Contributed by Jim Kingdon, 25Nov2019.)



Theorem  ltrnqg 7192 
Ordering property of reciprocal for positive fractions. For a simplified
version of the forward implication, see ltrnqi 7193. (Contributed by Jim
Kingdon, 29Dec2019.)



Theorem  ltrnqi 7193 
Ordering property of reciprocal for positive fractions. For the converse,
see ltrnqg 7192. (Contributed by Jim Kingdon, 24Sep2019.)



Theorem  nnnq 7194 
The canonical embedding of positive integers into positive fractions.
(Contributed by Jim Kingdon, 26Apr2020.)



Theorem  ltnnnq 7195 
Ordering of positive integers via or is equivalent.
(Contributed by Jim Kingdon, 3Oct2020.)



Definition  dfenq0 7196* 
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, 2Nov2019.)

~_{Q0}


Definition  dfnq0 7197 
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, 2Nov2019.)

Q_{0}
~_{Q0} 

Definition  df0nq0 7198 
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, 5Nov2019.)

0_{Q0} ~_{Q0} 

Definition  dfplq0 7199* 
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, 2Nov2019.)

+_{Q0}
Q_{0}
Q_{0}
~_{Q0} ~_{Q0}
~_{Q0} 

Definition  dfmq0 7200* 
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,
2Nov2019.)

·_{Q0}
Q_{0}
Q_{0}
~_{Q0} ~_{Q0}
~_{Q0} 