Type  Label  Description 
Statement 

Definition  dfmpq 6501* 
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 6502* 
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 6503* 
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 6504 
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 6505* 
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 6506* 
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 6507 
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 6508* 
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 6509* 
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 6510* 
Alternative definition of preaddition on positive fractions.
(Contributed by Jim Kingdon, 12Sep2019.)



Theorem  dfmpq2 6511* 
Alternative definition of premultiplication on positive fractions.
(Contributed by Jim Kingdon, 13Sep2019.)



Theorem  enqbreq 6512 
Equivalence relation for positive fractions in terms of positive
integers. (Contributed by NM, 27Aug1995.)



Theorem  enqbreq2 6513 
Equivalence relation for positive fractions in terms of positive
integers. (Contributed by Mario Carneiro, 8May2013.)



Theorem  enqer 6514 
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 6515 
Equivalence class equality of positive fractions in terms of positive
integers. (Contributed by NM, 29Nov1995.)



Theorem  enqex 6516 
The equivalence relation for positive fractions exists. (Contributed by
NM, 3Sep1995.)



Theorem  enqdc 6517 
The equivalence relation for positive fractions is decidable.
(Contributed by Jim Kingdon, 7Sep2019.)

DECID


Theorem  enqdc1 6518 
The equivalence relation for positive fractions is decidable.
(Contributed by Jim Kingdon, 7Sep2019.)

DECID 

Theorem  nqex 6519 
The class of positive fractions exists. (Contributed by NM,
16Aug1995.) (Revised by Mario Carneiro, 27Apr2013.)



Theorem  0nnq 6520 
The empty set is not a positive fraction. (Contributed by NM,
24Aug1995.) (Revised by Mario Carneiro, 27Apr2013.)



Theorem  ltrelnq 6521 
Positive fraction 'less than' is a relation on positive fractions.
(Contributed by NM, 14Feb1996.) (Revised by Mario Carneiro,
27Apr2013.)



Theorem  1nq 6522 
The positive fraction 'one'. (Contributed by NM, 29Oct1995.)



Theorem  addcmpblnq 6523 
Lemma showing compatibility of addition. (Contributed by NM,
27Aug1995.)



Theorem  mulcmpblnq 6524 
Lemma showing compatibility of multiplication. (Contributed by NM,
27Aug1995.)



Theorem  addpipqqslem 6525 
Lemma for addpipqqs 6526. (Contributed by Jim Kingdon, 11Sep2019.)



Theorem  addpipqqs 6526 
Addition of positive fractions in terms of positive integers.
(Contributed by NM, 28Aug1995.)



Theorem  mulpipq2 6527 
Multiplication of positive fractions in terms of positive integers.
(Contributed by Mario Carneiro, 8May2013.)



Theorem  mulpipq 6528 
Multiplication of positive fractions in terms of positive integers.
(Contributed by NM, 28Aug1995.) (Revised by Mario Carneiro,
8May2013.)



Theorem  mulpipqqs 6529 
Multiplication of positive fractions in terms of positive integers.
(Contributed by NM, 28Aug1995.)



Theorem  ordpipqqs 6530 
Ordering of positive fractions in terms of positive integers.
(Contributed by Jim Kingdon, 14Sep2019.)



Theorem  addclnq 6531 
Closure of addition on positive fractions. (Contributed by NM,
29Aug1995.)



Theorem  mulclnq 6532 
Closure of multiplication on positive fractions. (Contributed by NM,
29Aug1995.)



Theorem  dmaddpqlem 6533* 
Decomposition of a positive fraction into numerator and denominator.
Lemma for dmaddpq 6535. (Contributed by Jim Kingdon, 15Sep2019.)



Theorem  nqpi 6534* 
Decomposition of a positive fraction into numerator and denominator.
Similar to dmaddpqlem 6533 but also shows that the numerator and
denominator are positive integers. (Contributed by Jim Kingdon,
20Sep2019.)



Theorem  dmaddpq 6535 
Domain of addition on positive fractions. (Contributed by NM,
24Aug1995.)



Theorem  dmmulpq 6536 
Domain of multiplication on positive fractions. (Contributed by NM,
24Aug1995.)



Theorem  addcomnqg 6537 
Addition of positive fractions is commutative. (Contributed by Jim
Kingdon, 15Sep2019.)



Theorem  addassnqg 6538 
Addition of positive fractions is associative. (Contributed by Jim
Kingdon, 16Sep2019.)



Theorem  mulcomnqg 6539 
Multiplication of positive fractions is commutative. (Contributed by
Jim Kingdon, 17Sep2019.)



Theorem  mulassnqg 6540 
Multiplication of positive fractions is associative. (Contributed by
Jim Kingdon, 17Sep2019.)



Theorem  mulcanenq 6541 
Lemma for distributive law: cancellation of common factor. (Contributed
by NM, 2Sep1995.) (Revised by Mario Carneiro, 8May2013.)



Theorem  mulcanenqec 6542 
Lemma for distributive law: cancellation of common factor. (Contributed
by Jim Kingdon, 17Sep2019.)



Theorem  distrnqg 6543 
Multiplication of positive fractions is distributive. (Contributed by
Jim Kingdon, 17Sep2019.)



Theorem  1qec 6544 
The equivalence class of ratio 1. (Contributed by NM, 4Mar1996.)



Theorem  mulidnq 6545 
Multiplication identity element for positive fractions. (Contributed by
NM, 3Mar1996.)



Theorem  recexnq 6546* 
Existence of positive fraction reciprocal. (Contributed by Jim Kingdon,
20Sep2019.)



Theorem  recmulnqg 6547 
Relationship between reciprocal and multiplication on positive
fractions. (Contributed by Jim Kingdon, 19Sep2019.)



Theorem  recclnq 6548 
Closure law for positive fraction reciprocal. (Contributed by NM,
6Mar1996.) (Revised by Mario Carneiro, 8May2013.)



Theorem  recidnq 6549 
A positive fraction times its reciprocal is 1. (Contributed by NM,
6Mar1996.) (Revised by Mario Carneiro, 8May2013.)



Theorem  recrecnq 6550 
Reciprocal of reciprocal of positive fraction. (Contributed by NM,
26Apr1996.) (Revised by Mario Carneiro, 29Apr2013.)



Theorem  rec1nq 6551 
Reciprocal of positive fraction one. (Contributed by Jim Kingdon,
29Dec2019.)



Theorem  nqtri3or 6552 
Trichotomy for positive fractions. (Contributed by Jim Kingdon,
21Sep2019.)



Theorem  ltdcnq 6553 
Lessthan for positive fractions is decidable. (Contributed by Jim
Kingdon, 12Dec2019.)

DECID 

Theorem  ltsonq 6554 
'Less than' is a strict ordering on positive fractions. (Contributed by
NM, 19Feb1996.) (Revised by Mario Carneiro, 4May2013.)



Theorem  nqtric 6555 
Trichotomy for positive fractions. (Contributed by Jim Kingdon,
21Sep2019.)



Theorem  ltanqg 6556 
Ordering property of addition for positive fractions. Proposition
92.6(ii) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
22Sep2019.)



Theorem  ltmnqg 6557 
Ordering property of multiplication for positive fractions. Proposition
92.6(iii) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
22Sep2019.)



Theorem  ltanqi 6558 
Ordering property of addition for positive fractions. One direction of
ltanqg 6556. (Contributed by Jim Kingdon, 9Dec2019.)



Theorem  ltmnqi 6559 
Ordering property of multiplication for positive fractions. One direction
of ltmnqg 6557. (Contributed by Jim Kingdon, 9Dec2019.)



Theorem  lt2addnq 6560 
Ordering property of addition for positive fractions. (Contributed by Jim
Kingdon, 7Dec2019.)



Theorem  lt2mulnq 6561 
Ordering property of multiplication for positive fractions. (Contributed
by Jim Kingdon, 18Jul2021.)



Theorem  1lt2nq 6562 
One is less than two (one plus one). (Contributed by NM, 13Mar1996.)
(Revised by Mario Carneiro, 10May2013.)



Theorem  ltaddnq 6563 
The sum of two fractions is greater than one of them. (Contributed by
NM, 14Mar1996.) (Revised by Mario Carneiro, 10May2013.)



Theorem  ltexnqq 6564* 
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 6565* 
Ordering on positive fractions in terms of existence of sum.
(Contributed by Jim Kingdon, 30Apr2020.)



Theorem  halfnqq 6566* 
Onehalf of any positive fraction is a fraction. (Contributed by Jim
Kingdon, 23Sep2019.)



Theorem  halfnq 6567* 
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 6568* 
There is no smallest positive fraction. (Contributed by Jim Kingdon,
24Sep2019.)



Theorem  nsmallnq 6569* 
There is no smallest positive fraction. (Contributed by NM,
26Apr1996.) (Revised by Mario Carneiro, 10May2013.)



Theorem  subhalfnqq 6570* 
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 6566). (Contributed by Jim Kingdon,
25Nov2019.)



Theorem  ltbtwnnqq 6571* 
There exists a number between any two positive fractions. Proposition
92.6(i) of [Gleason] p. 120.
(Contributed by Jim Kingdon,
24Sep2019.)



Theorem  ltbtwnnq 6572* 
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 6573* 
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 6574* 
A version of the Archimedean property. This variation is "stronger"
than archnqq 6573 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 6575* 
Like prarloclemarch 6574 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 6659. (Contributed by Jim Kingdon, 25Nov2019.)



Theorem  ltrnqg 6576 
Ordering property of reciprocal for positive fractions. For a simplified
version of the forward implication, see ltrnqi 6577. (Contributed by Jim
Kingdon, 29Dec2019.)



Theorem  ltrnqi 6577 
Ordering property of reciprocal for positive fractions. For the converse,
see ltrnqg 6576. (Contributed by Jim Kingdon, 24Sep2019.)



Theorem  nnnq 6578 
The canonical embedding of positive integers into positive fractions.
(Contributed by Jim Kingdon, 26Apr2020.)



Theorem  ltnnnq 6579 
Ordering of positive integers via or is equivalent.
(Contributed by Jim Kingdon, 3Oct2020.)



Definition  dfenq0 6580* 
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 6581 
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 6582 
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 6583* 
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 6584* 
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} 

Theorem  dfmq0qs 6585* 
Multiplication on nonnegative fractions. This definition is similar to
dfmq0 6584 but expands Q_{0}
(Contributed by Jim Kingdon,
22Nov2019.)

·_{Q0}
~_{Q0}
~_{Q0}
~_{Q0} ~_{Q0}
~_{Q0} 

Theorem  dfplq0qs 6586* 
Addition on nonnegative fractions. This definition is similar to
dfplq0 6583 but expands Q_{0}
(Contributed by Jim Kingdon,
24Nov2019.)

+_{Q0}
~_{Q0}
~_{Q0}
~_{Q0} ~_{Q0}
~_{Q0} 

Theorem  enq0enq 6587 
Equivalence on positive fractions in terms of equivalence on
nonnegative fractions. (Contributed by Jim Kingdon, 12Nov2019.)

~_{Q0} 

Theorem  enq0sym 6588 
The equivalence relation for nonnegative fractions is symmetric. Lemma
for enq0er 6591. (Contributed by Jim Kingdon, 14Nov2019.)

~_{Q0}
~_{Q0} 

Theorem  enq0ref 6589 
The equivalence relation for nonnegative fractions is reflexive. Lemma
for enq0er 6591. (Contributed by Jim Kingdon, 14Nov2019.)

~_{Q0} 

Theorem  enq0tr 6590 
The equivalence relation for nonnegative fractions is transitive.
Lemma for enq0er 6591. (Contributed by Jim Kingdon, 14Nov2019.)

~_{Q0}
~_{Q0} ~_{Q0} 

Theorem  enq0er 6591 
The equivalence relation for nonnegative fractions is an equivalence
relation. (Contributed by Jim Kingdon, 12Nov2019.)

~_{Q0} 

Theorem  enq0breq 6592 
Equivalence relation for nonnegative fractions in terms of natural
numbers. (Contributed by NM, 27Aug1995.)

~_{Q0}


Theorem  enq0eceq 6593 
Equivalence class equality of nonnegative fractions in terms of natural
numbers. (Contributed by Jim Kingdon, 24Nov2019.)

~_{Q0} ~_{Q0} 

Theorem  nqnq0pi 6594 
A nonnegative fraction is a positive fraction if its numerator and
denominator are positive integers. (Contributed by Jim Kingdon,
10Nov2019.)

~_{Q0} 

Theorem  enq0ex 6595 
The equivalence relation for positive fractions exists. (Contributed by
Jim Kingdon, 18Nov2019.)

~_{Q0} 

Theorem  nq0ex 6596 
The class of positive fractions exists. (Contributed by Jim Kingdon,
18Nov2019.)

Q_{0} 

Theorem  nqnq0 6597 
A positive fraction is a nonnegative fraction. (Contributed by Jim
Kingdon, 18Nov2019.)

Q_{0} 

Theorem  nq0nn 6598* 
Decomposition of a nonnegative fraction into numerator and denominator.
(Contributed by Jim Kingdon, 24Nov2019.)

Q_{0}
~_{Q0} 

Theorem  addcmpblnq0 6599 
Lemma showing compatibility of addition on nonnegative fractions.
(Contributed by Jim Kingdon, 23Nov2019.)

~_{Q0}


Theorem  mulcmpblnq0 6600 
Lemma showing compatibility of multiplication on nonnegative fractions.
(Contributed by Jim Kingdon, 20Nov2019.)

~_{Q0}
