Type  Label  Description 
Statement 

Theorem  uzin 8601 
Intersection of two upper intervals of integers. (Contributed by Mario
Carneiro, 24Dec2013.)



Theorem  uzp1 8602 
Choices for an element of an upper interval of integers. (Contributed by
Jeff Madsen, 2Sep2009.)



Theorem  nn0uz 8603 
Nonnegative integers expressed as an upper set of integers. (Contributed
by NM, 2Sep2005.)



Theorem  nnuz 8604 
Positive integers expressed as an upper set of integers. (Contributed by
NM, 2Sep2005.)



Theorem  elnnuz 8605 
A positive integer expressed as a member of an upper set of integers.
(Contributed by NM, 6Jun2006.)



Theorem  elnn0uz 8606 
A nonnegative integer expressed as a member an upper set of integers.
(Contributed by NM, 6Jun2006.)



Theorem  eluz2nn 8607 
An integer is greater than or equal to 2 is a positive integer.
(Contributed by AV, 3Nov2018.)



Theorem  eluzge2nn0 8608 
If an integer is greater than or equal to 2, then it is a nonnegative
integer. (Contributed by AV, 27Aug2018.) (Proof shortened by AV,
3Nov2018.)



Theorem  uzuzle23 8609 
An integer in the upper set of integers starting at 3 is element of the
upper set of integers starting at 2. (Contributed by Alexander van der
Vekens, 17Sep2018.)



Theorem  eluzge3nn 8610 
If an integer is greater than 3, then it is a positive integer.
(Contributed by Alexander van der Vekens, 17Sep2018.)



Theorem  uz3m2nn 8611 
An integer greater than or equal to 3 decreased by 2 is a positive
integer. (Contributed by Alexander van der Vekens, 17Sep2018.)



Theorem  1eluzge0 8612 
1 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8Jun2018.)



Theorem  2eluzge0 8613 
2 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8Jun2018.) (Proof shortened by OpenAI, 25Mar2020.)



Theorem  2eluzge0OLD 8614 
2 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8Jun2018.) Obsolete version of 2eluzge0 8613 as of
25Mar2020. (New usage is discouraged.)
(Proof modification is discouraged.)



Theorem  2eluzge1 8615 
2 is an integer greater than or equal to 1. (Contributed by Alexander van
der Vekens, 8Jun2018.)



Theorem  uznnssnn 8616 
The upper integers starting from a natural are a subset of the naturals.
(Contributed by Scott Fenton, 29Jun2013.)



Theorem  raluz 8617* 
Restricted universal quantification in an upper set of integers.
(Contributed by NM, 9Sep2005.)



Theorem  raluz2 8618* 
Restricted universal quantification in an upper set of integers.
(Contributed by NM, 9Sep2005.)



Theorem  rexuz 8619* 
Restricted existential quantification in an upper set of integers.
(Contributed by NM, 9Sep2005.)



Theorem  rexuz2 8620* 
Restricted existential quantification in an upper set of integers.
(Contributed by NM, 9Sep2005.)



Theorem  2rexuz 8621* 
Double existential quantification in an upper set of integers.
(Contributed by NM, 3Nov2005.)



Theorem  peano2uz 8622 
Second Peano postulate for an upper set of integers. (Contributed by NM,
7Sep2005.)



Theorem  peano2uzs 8623 
Second Peano postulate for an upper set of integers. (Contributed by
Mario Carneiro, 26Dec2013.)



Theorem  peano2uzr 8624 
Reversed second Peano axiom for upper integers. (Contributed by NM,
2Jan2006.)



Theorem  uzaddcl 8625 
Addition closure law for an upper set of integers. (Contributed by NM,
4Jun2006.)



Theorem  nn0pzuz 8626 
The sum of a nonnegative integer and an integer is an integer greater than
or equal to that integer. (Contributed by Alexander van der Vekens,
3Oct2018.)



Theorem  uzind4 8627* 
Induction on the upper set of integers that starts at an integer .
The first four hypotheses give us the substitution instances we need,
and the last two are the basis and the induction step. (Contributed by
NM, 7Sep2005.)



Theorem  uzind4ALT 8628* 
Induction on the upper set of integers that starts at an integer .
The last four hypotheses give us the substitution instances we need; the
first two are the basis and the induction step. Either uzind4 8627 or
uzind4ALT 8628 may be used; see comment for nnind 8006. (Contributed by NM,
7Sep2005.) (New usage is discouraged.)
(Proof modification is discouraged.)



Theorem  uzind4s 8629* 
Induction on the upper set of integers that starts at an integer ,
using explicit substitution. The hypotheses are the basis and the
induction step. (Contributed by NM, 4Nov2005.)



Theorem  uzind4s2 8630* 
Induction on the upper set of integers that starts at an integer ,
using explicit substitution. The hypotheses are the basis and the
induction step. Use this instead of uzind4s 8629 when and
must
be distinct in . (Contributed by NM,
16Nov2005.)



Theorem  uzind4i 8631* 
Induction on the upper integers that start at . The first
hypothesis specifies the lower bound, the next four give us the
substitution instances we need, and the last two are the basis and the
induction step. (Contributed by NM, 4Sep2005.)



Theorem  indstr 8632* 
Strong Mathematical Induction for positive integers (inference schema).
(Contributed by NM, 17Aug2001.)



Theorem  eluznn0 8633 
Membership in a nonnegative upper set of integers implies membership in
.
(Contributed by Paul Chapman, 22Jun2011.)



Theorem  eluznn 8634 
Membership in a positive upper set of integers implies membership in
. (Contributed
by JJ, 1Oct2018.)



Theorem  eluz2b1 8635 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)



Theorem  eluz2gt1 8636 
An integer greater than or equal to 2 is greater than 1. (Contributed by
AV, 24May2020.)



Theorem  eluz2b2 8637 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)



Theorem  eluz2b3 8638 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)



Theorem  uz2m1nn 8639 
One less than an integer greater than or equal to 2 is a positive integer.
(Contributed by Paul Chapman, 17Nov2012.)



Theorem  1nuz2 8640 
1 is not in . (Contributed by Paul Chapman,
21Nov2012.)



Theorem  elnn1uz2 8641 
A positive integer is either 1 or greater than or equal to 2.
(Contributed by Paul Chapman, 17Nov2012.)



Theorem  uz2mulcl 8642 
Closure of multiplication of integers greater than or equal to 2.
(Contributed by Paul Chapman, 26Oct2012.)



Theorem  indstr2 8643* 
Strong Mathematical Induction for positive integers (inference schema).
The first two hypotheses give us the substitution instances we need; the
last two are the basis and the induction step. (Contributed by Paul
Chapman, 21Nov2012.)



Theorem  eluzdc 8644 
Membership of an integer in an upper set of integers is decidable.
(Contributed by Jim Kingdon, 18Apr2020.)

DECID


Theorem  ublbneg 8645* 
The image under negation of a boundedabove set of reals is bounded
below. (Contributed by Paul Chapman, 21Mar2011.)



Theorem  eqreznegel 8646* 
Two ways to express the image under negation of a set of integers.
(Contributed by Paul Chapman, 21Mar2011.)



Theorem  negm 8647* 
The image under negation of an inhabited set of reals is inhabited.
(Contributed by Jim Kingdon, 10Apr2020.)



Theorem  lbzbi 8648* 
If a set of reals is bounded below, it is bounded below by an integer.
(Contributed by Paul Chapman, 21Mar2011.)



Theorem  nn01to3 8649 
A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed
by Alexander van der Vekens, 13Sep2018.)



Theorem  nn0ge2m1nnALT 8650 
Alternate proof of nn0ge2m1nn 8299: If a nonnegative integer is greater
than or equal to two, the integer decreased by 1 is a positive integer.
This version is proved using eluz2 8575, a theorem for upper sets of
integers, which are defined later than the positive and nonnegative
integers. This proof is, however, much shorter than the proof of
nn0ge2m1nn 8299. (Contributed by Alexander van der Vekens,
1Aug2018.)
(New usage is discouraged.) (Proof modification is discouraged.)



3.4.11 Rational numbers (as a subset of complex
numbers)


Syntax  cq 8651 
Extend class notation to include the class of rationals.



Definition  dfq 8652 
Define the set of rational numbers. Based on definition of rationals in
[Apostol] p. 22. See elq 8654
for the relation "is rational." (Contributed
by NM, 8Jan2002.)



Theorem  divfnzn 8653 
Division restricted to is a function. Given
excluded
middle, it would be easy to prove this for .
The key difference is that an element of is apart from zero,
whereas being an element of
implies being not equal to
zero. (Contributed by Jim Kingdon, 19Mar2020.)



Theorem  elq 8654* 
Membership in the set of rationals. (Contributed by NM, 8Jan2002.)
(Revised by Mario Carneiro, 28Jan2014.)



Theorem  qmulz 8655* 
If is rational, then
some integer multiple of it is an integer.
(Contributed by NM, 7Nov2008.) (Revised by Mario Carneiro,
22Jul2014.)



Theorem  znq 8656 
The ratio of an integer and a positive integer is a rational number.
(Contributed by NM, 12Jan2002.)



Theorem  qre 8657 
A rational number is a real number. (Contributed by NM,
14Nov2002.)



Theorem  zq 8658 
An integer is a rational number. (Contributed by NM, 9Jan2002.)



Theorem  zssq 8659 
The integers are a subset of the rationals. (Contributed by NM,
9Jan2002.)



Theorem  nn0ssq 8660 
The nonnegative integers are a subset of the rationals. (Contributed by
NM, 31Jul2004.)



Theorem  nnssq 8661 
The positive integers are a subset of the rationals. (Contributed by NM,
31Jul2004.)



Theorem  qssre 8662 
The rationals are a subset of the reals. (Contributed by NM,
9Jan2002.)



Theorem  qsscn 8663 
The rationals are a subset of the complex numbers. (Contributed by NM,
2Aug2004.)



Theorem  qex 8664 
The set of rational numbers exists. (Contributed by NM, 30Jul2004.)
(Revised by Mario Carneiro, 17Nov2014.)



Theorem  nnq 8665 
A positive integer is rational. (Contributed by NM, 17Nov2004.)



Theorem  qcn 8666 
A rational number is a complex number. (Contributed by NM,
2Aug2004.)



Theorem  qaddcl 8667 
Closure of addition of rationals. (Contributed by NM, 1Aug2004.)



Theorem  qnegcl 8668 
Closure law for the negative of a rational. (Contributed by NM,
2Aug2004.) (Revised by Mario Carneiro, 15Sep2014.)



Theorem  qmulcl 8669 
Closure of multiplication of rationals. (Contributed by NM,
1Aug2004.)



Theorem  qsubcl 8670 
Closure of subtraction of rationals. (Contributed by NM, 2Aug2004.)



Theorem  qapne 8671 
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 20Mar2020.)

# 

Theorem  qltlen 8672 
Rational 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 7695 which is a similar result for real numbers.
(Contributed by
Jim Kingdon, 11Oct2021.)



Theorem  qlttri2 8673 
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 9Nov2021.)



Theorem  qreccl 8674 
Closure of reciprocal of rationals. (Contributed by NM, 3Aug2004.)



Theorem  qdivcl 8675 
Closure of division of rationals. (Contributed by NM, 3Aug2004.)



Theorem  qrevaddcl 8676 
Reverse closure law for addition of rationals. (Contributed by NM,
2Aug2004.)



Theorem  nnrecq 8677 
The reciprocal of a positive integer is rational. (Contributed by NM,
17Nov2004.)



Theorem  irradd 8678 
The sum of an irrational number and a rational number is irrational.
(Contributed by NM, 7Nov2008.)



Theorem  irrmul 8679 
The product of a real which is not rational with a nonzero rational is not
rational. Note that by "not rational" we mean the negation of
"is
rational" (whereas "irrational" is often defined to mean
apart from any
rational number  given excluded middle these two definitions would be
equivalent). (Contributed by NM, 7Nov2008.)



3.4.12 Complex numbers as pairs of
reals


Theorem  cnref1o 8680* 
There is a natural onetoone mapping from
to ,
where we map to . In our
construction of the complex numbers, this is in fact our
definition of
(see dfc 6953), but in the axiomatic treatment we can only
show
that there is the expected mapping between these two sets. (Contributed
by Mario Carneiro, 16Jun2013.) (Revised by Mario Carneiro,
17Feb2014.)



3.5 Order sets


3.5.1 Positive reals (as a subset of complex
numbers)


Syntax  crp 8681 
Extend class notation to include the class of positive reals.



Definition  dfrp 8682 
Define the set of positive reals. Definition of positive numbers in
[Apostol] p. 20. (Contributed by NM,
27Oct2007.)



Theorem  elrp 8683 
Membership in the set of positive reals. (Contributed by NM,
27Oct2007.)



Theorem  elrpii 8684 
Membership in the set of positive reals. (Contributed by NM,
23Feb2008.)



Theorem  1rp 8685 
1 is a positive real. (Contributed by Jeff Hankins, 23Nov2008.)



Theorem  2rp 8686 
2 is a positive real. (Contributed by Mario Carneiro, 28May2016.)



Theorem  rpre 8687 
A positive real is a real. (Contributed by NM, 27Oct2007.)



Theorem  rpxr 8688 
A positive real is an extended real. (Contributed by Mario Carneiro,
21Aug2015.)



Theorem  rpcn 8689 
A positive real is a complex number. (Contributed by NM, 11Nov2008.)



Theorem  nnrp 8690 
A positive integer is a positive real. (Contributed by NM,
28Nov2008.)



Theorem  rpssre 8691 
The positive reals are a subset of the reals. (Contributed by NM,
24Feb2008.)



Theorem  rpgt0 8692 
A positive real is greater than zero. (Contributed by FL,
27Dec2007.)



Theorem  rpge0 8693 
A positive real is greater than or equal to zero. (Contributed by NM,
22Feb2008.)



Theorem  rpregt0 8694 
A positive real is a positive real number. (Contributed by NM,
11Nov2008.) (Revised by Mario Carneiro, 31Jan2014.)



Theorem  rprege0 8695 
A positive real is a nonnegative real number. (Contributed by Mario
Carneiro, 31Jan2014.)



Theorem  rpne0 8696 
A positive real is nonzero. (Contributed by NM, 18Jul2008.)



Theorem  rpap0 8697 
A positive real is apart from zero. (Contributed by Jim Kingdon,
22Mar2020.)

# 

Theorem  rprene0 8698 
A positive real is a nonzero real number. (Contributed by NM,
11Nov2008.)



Theorem  rpreap0 8699 
A positive real is a real number apart from zero. (Contributed by Jim
Kingdon, 22Mar2020.)

# 

Theorem  rpcnne0 8700 
A positive real is a nonzero complex number. (Contributed by NM,
11Nov2008.)

