Type  Label  Description 
Statement 

Theorem  eluzp1p1 9201 
Membership in the next upper set of integers. (Contributed by NM,
5Oct2005.)



Theorem  eluzaddi 9202 
Membership in a later upper set of integers. (Contributed by Paul
Chapman, 22Nov2007.)



Theorem  eluzsubi 9203 
Membership in an earlier upper set of integers. (Contributed by Paul
Chapman, 22Nov2007.)



Theorem  eluzadd 9204 
Membership in a later upper set of integers. (Contributed by Jeff Madsen,
2Sep2009.)



Theorem  eluzsub 9205 
Membership in an earlier upper set of integers. (Contributed by Jeff
Madsen, 2Sep2009.)



Theorem  uzm1 9206 
Choices for an element of an upper interval of integers. (Contributed by
Jeff Madsen, 2Sep2009.)



Theorem  uznn0sub 9207 
The nonnegative difference of integers is a nonnegative integer.
(Contributed by NM, 4Sep2005.)



Theorem  uzin 9208 
Intersection of two upper intervals of integers. (Contributed by Mario
Carneiro, 24Dec2013.)



Theorem  uzp1 9209 
Choices for an element of an upper interval of integers. (Contributed by
Jeff Madsen, 2Sep2009.)



Theorem  nn0uz 9210 
Nonnegative integers expressed as an upper set of integers. (Contributed
by NM, 2Sep2005.)



Theorem  nnuz 9211 
Positive integers expressed as an upper set of integers. (Contributed by
NM, 2Sep2005.)



Theorem  elnnuz 9212 
A positive integer expressed as a member of an upper set of integers.
(Contributed by NM, 6Jun2006.)



Theorem  elnn0uz 9213 
A nonnegative integer expressed as a member an upper set of integers.
(Contributed by NM, 6Jun2006.)



Theorem  eluz2nn 9214 
An integer is greater than or equal to 2 is a positive integer.
(Contributed by AV, 3Nov2018.)



Theorem  eluzge2nn0 9215 
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 9216 
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 9217 
If an integer is greater than 3, then it is a positive integer.
(Contributed by Alexander van der Vekens, 17Sep2018.)



Theorem  uz3m2nn 9218 
An integer greater than or equal to 3 decreased by 2 is a positive
integer. (Contributed by Alexander van der Vekens, 17Sep2018.)



Theorem  1eluzge0 9219 
1 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8Jun2018.)



Theorem  2eluzge0 9220 
2 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8Jun2018.) (Proof shortened by OpenAI, 25Mar2020.)



Theorem  2eluzge1 9221 
2 is an integer greater than or equal to 1. (Contributed by Alexander van
der Vekens, 8Jun2018.)



Theorem  uznnssnn 9222 
The upper integers starting from a natural are a subset of the naturals.
(Contributed by Scott Fenton, 29Jun2013.)



Theorem  raluz 9223* 
Restricted universal quantification in an upper set of integers.
(Contributed by NM, 9Sep2005.)



Theorem  raluz2 9224* 
Restricted universal quantification in an upper set of integers.
(Contributed by NM, 9Sep2005.)



Theorem  rexuz 9225* 
Restricted existential quantification in an upper set of integers.
(Contributed by NM, 9Sep2005.)



Theorem  rexuz2 9226* 
Restricted existential quantification in an upper set of integers.
(Contributed by NM, 9Sep2005.)



Theorem  2rexuz 9227* 
Double existential quantification in an upper set of integers.
(Contributed by NM, 3Nov2005.)



Theorem  peano2uz 9228 
Second Peano postulate for an upper set of integers. (Contributed by NM,
7Sep2005.)



Theorem  peano2uzs 9229 
Second Peano postulate for an upper set of integers. (Contributed by
Mario Carneiro, 26Dec2013.)



Theorem  peano2uzr 9230 
Reversed second Peano axiom for upper integers. (Contributed by NM,
2Jan2006.)



Theorem  uzaddcl 9231 
Addition closure law for an upper set of integers. (Contributed by NM,
4Jun2006.)



Theorem  nn0pzuz 9232 
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 9233* 
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 9234* 
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 9233 or
uzind4ALT 9234 may be used; see comment for nnind 8594. (Contributed by NM,
7Sep2005.) (New usage is discouraged.)
(Proof modification is discouraged.)



Theorem  uzind4s 9235* 
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 9236* 
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 9235 when and
must
be distinct in . (Contributed by NM,
16Nov2005.)



Theorem  uzind4i 9237* 
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 9238* 
Strong Mathematical Induction for positive integers (inference schema).
(Contributed by NM, 17Aug2001.)



Theorem  infrenegsupex 9239* 
The infimum of a set of reals is the negative of the supremum of
the negatives of its elements. (Contributed by Jim Kingdon,
14Jan2022.)

inf 

Theorem  supinfneg 9240* 
If a set of real numbers has a least upper bound, the set of the
negation of those numbers has a greatest lower bound. For a theorem
which is similar but only for the boundedness part, see ublbneg 9255.
(Contributed by Jim Kingdon, 15Jan2022.)



Theorem  infsupneg 9241* 
If a set of real numbers has a greatest lower bound, the set of the
negation of those numbers has a least upper bound. To go in the other
direction see supinfneg 9240. (Contributed by Jim Kingdon,
15Jan2022.)



Theorem  supminfex 9242* 
A supremum is the negation of the infimum of that set's image under
negation. (Contributed by Jim Kingdon, 14Jan2022.)

inf 

Theorem  eluznn0 9243 
Membership in a nonnegative upper set of integers implies membership in
.
(Contributed by Paul Chapman, 22Jun2011.)



Theorem  eluznn 9244 
Membership in a positive upper set of integers implies membership in
. (Contributed
by JJ, 1Oct2018.)



Theorem  eluz2b1 9245 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)



Theorem  eluz2gt1 9246 
An integer greater than or equal to 2 is greater than 1. (Contributed by
AV, 24May2020.)



Theorem  eluz2b2 9247 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)



Theorem  eluz2b3 9248 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)



Theorem  uz2m1nn 9249 
One less than an integer greater than or equal to 2 is a positive integer.
(Contributed by Paul Chapman, 17Nov2012.)



Theorem  1nuz2 9250 
1 is not in . (Contributed by Paul Chapman,
21Nov2012.)



Theorem  elnn1uz2 9251 
A positive integer is either 1 or greater than or equal to 2.
(Contributed by Paul Chapman, 17Nov2012.)



Theorem  uz2mulcl 9252 
Closure of multiplication of integers greater than or equal to 2.
(Contributed by Paul Chapman, 26Oct2012.)



Theorem  indstr2 9253* 
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 9254 
Membership of an integer in an upper set of integers is decidable.
(Contributed by Jim Kingdon, 18Apr2020.)

DECID


Theorem  ublbneg 9255* 
The image under negation of a boundedabove set of reals is bounded
below. For a theorem which is similar but also adds that the bounds
need to be the tightest possible, see supinfneg 9240. (Contributed by
Paul Chapman, 21Mar2011.)



Theorem  eqreznegel 9256* 
Two ways to express the image under negation of a set of integers.
(Contributed by Paul Chapman, 21Mar2011.)



Theorem  negm 9257* 
The image under negation of an inhabited set of reals is inhabited.
(Contributed by Jim Kingdon, 10Apr2020.)



Theorem  lbzbi 9258* 
If a set of reals is bounded below, it is bounded below by an integer.
(Contributed by Paul Chapman, 21Mar2011.)



Theorem  nn01to3 9259 
A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed
by Alexander van der Vekens, 13Sep2018.)



Theorem  nn0ge2m1nnALT 9260 
Alternate proof of nn0ge2m1nn 8889: 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 9182, 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 8889. (Contributed by Alexander van der Vekens,
1Aug2018.)
(New usage is discouraged.) (Proof modification is discouraged.)



3.4.12 Rational numbers (as a subset of complex
numbers)


Syntax  cq 9261 
Extend class notation to include the class of rationals.



Definition  dfq 9262 
Define the set of rational numbers. Based on definition of rationals in
[Apostol] p. 22. See elq 9264
for the relation "is rational." (Contributed
by NM, 8Jan2002.)



Theorem  divfnzn 9263 
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 9264* 
Membership in the set of rationals. (Contributed by NM, 8Jan2002.)
(Revised by Mario Carneiro, 28Jan2014.)



Theorem  qmulz 9265* 
If is rational, then
some integer multiple of it is an integer.
(Contributed by NM, 7Nov2008.) (Revised by Mario Carneiro,
22Jul2014.)



Theorem  znq 9266 
The ratio of an integer and a positive integer is a rational number.
(Contributed by NM, 12Jan2002.)



Theorem  qre 9267 
A rational number is a real number. (Contributed by NM,
14Nov2002.)



Theorem  zq 9268 
An integer is a rational number. (Contributed by NM, 9Jan2002.)



Theorem  zssq 9269 
The integers are a subset of the rationals. (Contributed by NM,
9Jan2002.)



Theorem  nn0ssq 9270 
The nonnegative integers are a subset of the rationals. (Contributed by
NM, 31Jul2004.)



Theorem  nnssq 9271 
The positive integers are a subset of the rationals. (Contributed by NM,
31Jul2004.)



Theorem  qssre 9272 
The rationals are a subset of the reals. (Contributed by NM,
9Jan2002.)



Theorem  qsscn 9273 
The rationals are a subset of the complex numbers. (Contributed by NM,
2Aug2004.)



Theorem  qex 9274 
The set of rational numbers exists. (Contributed by NM, 30Jul2004.)
(Revised by Mario Carneiro, 17Nov2014.)



Theorem  nnq 9275 
A positive integer is rational. (Contributed by NM, 17Nov2004.)



Theorem  qcn 9276 
A rational number is a complex number. (Contributed by NM,
2Aug2004.)



Theorem  qaddcl 9277 
Closure of addition of rationals. (Contributed by NM, 1Aug2004.)



Theorem  qnegcl 9278 
Closure law for the negative of a rational. (Contributed by NM,
2Aug2004.) (Revised by Mario Carneiro, 15Sep2014.)



Theorem  qmulcl 9279 
Closure of multiplication of rationals. (Contributed by NM,
1Aug2004.)



Theorem  qsubcl 9280 
Closure of subtraction of rationals. (Contributed by NM, 2Aug2004.)



Theorem  qapne 9281 
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 20Mar2020.)

# 

Theorem  qltlen 9282 
Rational 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 8259 which is a similar result for real numbers.
(Contributed by
Jim Kingdon, 11Oct2021.)



Theorem  qlttri2 9283 
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 9Nov2021.)



Theorem  qreccl 9284 
Closure of reciprocal of rationals. (Contributed by NM, 3Aug2004.)



Theorem  qdivcl 9285 
Closure of division of rationals. (Contributed by NM, 3Aug2004.)



Theorem  qrevaddcl 9286 
Reverse closure law for addition of rationals. (Contributed by NM,
2Aug2004.)



Theorem  nnrecq 9287 
The reciprocal of a positive integer is rational. (Contributed by NM,
17Nov2004.)



Theorem  irradd 9288 
The sum of an irrational number and a rational number is irrational.
(Contributed by NM, 7Nov2008.)



Theorem  irrmul 9289 
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.)



3.4.13 Complex numbers as pairs of
reals


Theorem  cnref1o 9290* 
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 7506), 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 9291 
Extend class notation to include the class of positive reals.



Definition  dfrp 9292 
Define the set of positive reals. Definition of positive numbers in
[Apostol] p. 20. (Contributed by NM,
27Oct2007.)



Theorem  elrp 9293 
Membership in the set of positive reals. (Contributed by NM,
27Oct2007.)



Theorem  elrpii 9294 
Membership in the set of positive reals. (Contributed by NM,
23Feb2008.)



Theorem  1rp 9295 
1 is a positive real. (Contributed by Jeff Hankins, 23Nov2008.)



Theorem  2rp 9296 
2 is a positive real. (Contributed by Mario Carneiro, 28May2016.)



Theorem  rpre 9297 
A positive real is a real. (Contributed by NM, 27Oct2007.)



Theorem  rpxr 9298 
A positive real is an extended real. (Contributed by Mario Carneiro,
21Aug2015.)



Theorem  rpcn 9299 
A positive real is a complex number. (Contributed by NM, 11Nov2008.)



Theorem  nnrp 9300 
A positive integer is a positive real. (Contributed by NM,
28Nov2008.)

