Type  Label  Description 
Statement 

Theorem  infsupneg 9501* 
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 9500. (Contributed by Jim Kingdon,
15Jan2022.)



Theorem  supminfex 9502* 
A supremum is the negation of the infimum of that set's image under
negation. (Contributed by Jim Kingdon, 14Jan2022.)

inf 

Theorem  eluznn0 9503 
Membership in a nonnegative upper set of integers implies membership in
.
(Contributed by Paul Chapman, 22Jun2011.)



Theorem  eluznn 9504 
Membership in a positive upper set of integers implies membership in
. (Contributed
by JJ, 1Oct2018.)



Theorem  eluz2b1 9505 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)



Theorem  eluz2gt1 9506 
An integer greater than or equal to 2 is greater than 1. (Contributed by
AV, 24May2020.)



Theorem  eluz2b2 9507 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)



Theorem  eluz2b3 9508 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)



Theorem  uz2m1nn 9509 
One less than an integer greater than or equal to 2 is a positive integer.
(Contributed by Paul Chapman, 17Nov2012.)



Theorem  1nuz2 9510 
1 is not in . (Contributed by Paul Chapman,
21Nov2012.)



Theorem  elnn1uz2 9511 
A positive integer is either 1 or greater than or equal to 2.
(Contributed by Paul Chapman, 17Nov2012.)



Theorem  uz2mulcl 9512 
Closure of multiplication of integers greater than or equal to 2.
(Contributed by Paul Chapman, 26Oct2012.)



Theorem  indstr2 9513* 
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 9514 
Membership of an integer in an upper set of integers is decidable.
(Contributed by Jim Kingdon, 18Apr2020.)

DECID


Theorem  ublbneg 9515* 
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 9500. (Contributed by
Paul Chapman, 21Mar2011.)



Theorem  eqreznegel 9516* 
Two ways to express the image under negation of a set of integers.
(Contributed by Paul Chapman, 21Mar2011.)



Theorem  negm 9517* 
The image under negation of an inhabited set of reals is inhabited.
(Contributed by Jim Kingdon, 10Apr2020.)



Theorem  lbzbi 9518* 
If a set of reals is bounded below, it is bounded below by an integer.
(Contributed by Paul Chapman, 21Mar2011.)



Theorem  nn01to3 9519 
A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed
by Alexander van der Vekens, 13Sep2018.)



Theorem  nn0ge2m1nnALT 9520 
Alternate proof of nn0ge2m1nn 9144: 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 9439, 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 9144. (Contributed by Alexander van der Vekens,
1Aug2018.)
(New usage is discouraged.) (Proof modification is discouraged.)



4.4.12 Rational numbers (as a subset of complex
numbers)


Syntax  cq 9521 
Extend class notation to include the class of rationals.



Definition  dfq 9522 
Define the set of rational numbers. Based on definition of rationals in
[Apostol] p. 22. See elq 9524
for the relation "is rational." (Contributed
by NM, 8Jan2002.)



Theorem  divfnzn 9523 
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 9524* 
Membership in the set of rationals. (Contributed by NM, 8Jan2002.)
(Revised by Mario Carneiro, 28Jan2014.)



Theorem  qmulz 9525* 
If is rational, then
some integer multiple of it is an integer.
(Contributed by NM, 7Nov2008.) (Revised by Mario Carneiro,
22Jul2014.)



Theorem  znq 9526 
The ratio of an integer and a positive integer is a rational number.
(Contributed by NM, 12Jan2002.)



Theorem  qre 9527 
A rational number is a real number. (Contributed by NM,
14Nov2002.)



Theorem  zq 9528 
An integer is a rational number. (Contributed by NM, 9Jan2002.)



Theorem  zssq 9529 
The integers are a subset of the rationals. (Contributed by NM,
9Jan2002.)



Theorem  nn0ssq 9530 
The nonnegative integers are a subset of the rationals. (Contributed by
NM, 31Jul2004.)



Theorem  nnssq 9531 
The positive integers are a subset of the rationals. (Contributed by NM,
31Jul2004.)



Theorem  qssre 9532 
The rationals are a subset of the reals. (Contributed by NM,
9Jan2002.)



Theorem  qsscn 9533 
The rationals are a subset of the complex numbers. (Contributed by NM,
2Aug2004.)



Theorem  qex 9534 
The set of rational numbers exists. (Contributed by NM, 30Jul2004.)
(Revised by Mario Carneiro, 17Nov2014.)



Theorem  nnq 9535 
A positive integer is rational. (Contributed by NM, 17Nov2004.)



Theorem  qcn 9536 
A rational number is a complex number. (Contributed by NM,
2Aug2004.)



Theorem  qaddcl 9537 
Closure of addition of rationals. (Contributed by NM, 1Aug2004.)



Theorem  qnegcl 9538 
Closure law for the negative of a rational. (Contributed by NM,
2Aug2004.) (Revised by Mario Carneiro, 15Sep2014.)



Theorem  qmulcl 9539 
Closure of multiplication of rationals. (Contributed by NM,
1Aug2004.)



Theorem  qsubcl 9540 
Closure of subtraction of rationals. (Contributed by NM, 2Aug2004.)



Theorem  qapne 9541 
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 20Mar2020.)

# 

Theorem  qltlen 9542 
Rational 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 8501 which is a similar result for real numbers.
(Contributed by
Jim Kingdon, 11Oct2021.)



Theorem  qlttri2 9543 
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 9Nov2021.)



Theorem  qreccl 9544 
Closure of reciprocal of rationals. (Contributed by NM, 3Aug2004.)



Theorem  qdivcl 9545 
Closure of division of rationals. (Contributed by NM, 3Aug2004.)



Theorem  qrevaddcl 9546 
Reverse closure law for addition of rationals. (Contributed by NM,
2Aug2004.)



Theorem  nnrecq 9547 
The reciprocal of a positive integer is rational. (Contributed by NM,
17Nov2004.)



Theorem  irradd 9548 
The sum of an irrational number and a rational number is irrational.
(Contributed by NM, 7Nov2008.)



Theorem  irrmul 9549 
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.)



Theorem  elpq 9550* 
A positive rational is the quotient of two positive integers.
(Contributed by AV, 29Dec2022.)



Theorem  elpqb 9551* 
A class is a positive rational iff it is the quotient of two positive
integers. (Contributed by AV, 30Dec2022.)



4.4.13 Complex numbers as pairs of
reals


Theorem  cnref1o 9552* 
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 7732), 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.)



4.5 Order sets


4.5.1 Positive reals (as a subset of complex
numbers)


Syntax  crp 9553 
Extend class notation to include the class of positive reals.



Definition  dfrp 9554 
Define the set of positive reals. Definition of positive numbers in
[Apostol] p. 20. (Contributed by NM,
27Oct2007.)



Theorem  elrp 9555 
Membership in the set of positive reals. (Contributed by NM,
27Oct2007.)



Theorem  elrpii 9556 
Membership in the set of positive reals. (Contributed by NM,
23Feb2008.)



Theorem  1rp 9557 
1 is a positive real. (Contributed by Jeff Hankins, 23Nov2008.)



Theorem  2rp 9558 
2 is a positive real. (Contributed by Mario Carneiro, 28May2016.)



Theorem  3rp 9559 
3 is a positive real. (Contributed by Glauco Siliprandi, 11Dec2019.)



Theorem  rpre 9560 
A positive real is a real. (Contributed by NM, 27Oct2007.)



Theorem  rpxr 9561 
A positive real is an extended real. (Contributed by Mario Carneiro,
21Aug2015.)



Theorem  rpcn 9562 
A positive real is a complex number. (Contributed by NM, 11Nov2008.)



Theorem  nnrp 9563 
A positive integer is a positive real. (Contributed by NM,
28Nov2008.)



Theorem  rpssre 9564 
The positive reals are a subset of the reals. (Contributed by NM,
24Feb2008.)



Theorem  rpgt0 9565 
A positive real is greater than zero. (Contributed by FL,
27Dec2007.)



Theorem  rpge0 9566 
A positive real is greater than or equal to zero. (Contributed by NM,
22Feb2008.)



Theorem  rpregt0 9567 
A positive real is a positive real number. (Contributed by NM,
11Nov2008.) (Revised by Mario Carneiro, 31Jan2014.)



Theorem  rprege0 9568 
A positive real is a nonnegative real number. (Contributed by Mario
Carneiro, 31Jan2014.)



Theorem  rpne0 9569 
A positive real is nonzero. (Contributed by NM, 18Jul2008.)



Theorem  rpap0 9570 
A positive real is apart from zero. (Contributed by Jim Kingdon,
22Mar2020.)

# 

Theorem  rprene0 9571 
A positive real is a nonzero real number. (Contributed by NM,
11Nov2008.)



Theorem  rpreap0 9572 
A positive real is a real number apart from zero. (Contributed by Jim
Kingdon, 22Mar2020.)

# 

Theorem  rpcnne0 9573 
A positive real is a nonzero complex number. (Contributed by NM,
11Nov2008.)



Theorem  rpcnap0 9574 
A positive real is a complex number apart from zero. (Contributed by Jim
Kingdon, 22Mar2020.)

# 

Theorem  ralrp 9575 
Quantification over positive reals. (Contributed by NM, 12Feb2008.)



Theorem  rexrp 9576 
Quantification over positive reals. (Contributed by Mario Carneiro,
21May2014.)



Theorem  rpaddcl 9577 
Closure law for addition of positive reals. Part of Axiom 7 of [Apostol]
p. 20. (Contributed by NM, 27Oct2007.)



Theorem  rpmulcl 9578 
Closure law for multiplication of positive reals. Part of Axiom 7 of
[Apostol] p. 20. (Contributed by NM,
27Oct2007.)



Theorem  rpdivcl 9579 
Closure law for division of positive reals. (Contributed by FL,
27Dec2007.)



Theorem  rpreccl 9580 
Closure law for reciprocation of positive reals. (Contributed by Jeff
Hankins, 23Nov2008.)



Theorem  rphalfcl 9581 
Closure law for half of a positive real. (Contributed by Mario Carneiro,
31Jan2014.)



Theorem  rpgecl 9582 
A number greater or equal to a positive real is positive real.
(Contributed by Mario Carneiro, 28May2016.)



Theorem  rphalflt 9583 
Half of a positive real is less than the original number. (Contributed by
Mario Carneiro, 21May2014.)



Theorem  rerpdivcl 9584 
Closure law for division of a real by a positive real. (Contributed by
NM, 10Nov2008.)



Theorem  ge0p1rp 9585 
A nonnegative number plus one is a positive number. (Contributed by Mario
Carneiro, 5Oct2015.)



Theorem  rpnegap 9586 
Either a real apart from zero or its negation is a positive real, but not
both. (Contributed by Jim Kingdon, 23Mar2020.)

#


Theorem  negelrp 9587 
Elementhood of a negation in the positive real numbers. (Contributed by
Thierry Arnoux, 19Sep2018.)



Theorem  negelrpd 9588 
The negation of a negative number is in the positive real numbers.
(Contributed by Glauco Siliprandi, 26Jun2021.)



Theorem  0nrp 9589 
Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by
NM, 27Oct2007.)



Theorem  ltsubrp 9590 
Subtracting a positive real from another number decreases it.
(Contributed by FL, 27Dec2007.)



Theorem  ltaddrp 9591 
Adding a positive number to another number increases it. (Contributed by
FL, 27Dec2007.)



Theorem  difrp 9592 
Two ways to say one number is less than another. (Contributed by Mario
Carneiro, 21May2014.)



Theorem  elrpd 9593 
Membership in the set of positive reals. (Contributed by Mario
Carneiro, 28May2016.)



Theorem  nnrpd 9594 
A positive integer is a positive real. (Contributed by Mario Carneiro,
28May2016.)



Theorem  zgt1rpn0n1 9595 
An integer greater than 1 is a positive real number not equal to 0 or 1.
Useful for working with integer logarithm bases (which is a common case,
e.g., base 2, base 3, or base 10). (Contributed by Thierry Arnoux,
26Sep2017.) (Proof shortened by AV, 9Jul2022.)



Theorem  rpred 9596 
A positive real is a real. (Contributed by Mario Carneiro,
28May2016.)



Theorem  rpxrd 9597 
A positive real is an extended real. (Contributed by Mario Carneiro,
28May2016.)



Theorem  rpcnd 9598 
A positive real is a complex number. (Contributed by Mario Carneiro,
28May2016.)



Theorem  rpgt0d 9599 
A positive real is greater than zero. (Contributed by Mario Carneiro,
28May2016.)



Theorem  rpge0d 9600 
A positive real is greater than or equal to zero. (Contributed by Mario
Carneiro, 28May2016.)

