Theorem List for Intuitionistic Logic Explorer - 9201-9300 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | eluzp1p1 9201 |
Membership in the next upper set of integers. (Contributed by NM,
5-Oct-2005.)
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Theorem | eluzaddi 9202 |
Membership in a later upper set of integers. (Contributed by Paul
Chapman, 22-Nov-2007.)
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Theorem | eluzsubi 9203 |
Membership in an earlier upper set of integers. (Contributed by Paul
Chapman, 22-Nov-2007.)
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Theorem | eluzadd 9204 |
Membership in a later upper set of integers. (Contributed by Jeff Madsen,
2-Sep-2009.)
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Theorem | eluzsub 9205 |
Membership in an earlier upper set of integers. (Contributed by Jeff
Madsen, 2-Sep-2009.)
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Theorem | uzm1 9206 |
Choices for an element of an upper interval of integers. (Contributed by
Jeff Madsen, 2-Sep-2009.)
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Theorem | uznn0sub 9207 |
The nonnegative difference of integers is a nonnegative integer.
(Contributed by NM, 4-Sep-2005.)
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Theorem | uzin 9208 |
Intersection of two upper intervals of integers. (Contributed by Mario
Carneiro, 24-Dec-2013.)
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Theorem | uzp1 9209 |
Choices for an element of an upper interval of integers. (Contributed by
Jeff Madsen, 2-Sep-2009.)
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Theorem | nn0uz 9210 |
Nonnegative integers expressed as an upper set of integers. (Contributed
by NM, 2-Sep-2005.)
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Theorem | nnuz 9211 |
Positive integers expressed as an upper set of integers. (Contributed by
NM, 2-Sep-2005.)
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Theorem | elnnuz 9212 |
A positive integer expressed as a member of an upper set of integers.
(Contributed by NM, 6-Jun-2006.)
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Theorem | elnn0uz 9213 |
A nonnegative integer expressed as a member an upper set of integers.
(Contributed by NM, 6-Jun-2006.)
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Theorem | eluz2nn 9214 |
An integer is greater than or equal to 2 is a positive integer.
(Contributed by AV, 3-Nov-2018.)
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Theorem | eluzge2nn0 9215 |
If an integer is greater than or equal to 2, then it is a nonnegative
integer. (Contributed by AV, 27-Aug-2018.) (Proof shortened by AV,
3-Nov-2018.)
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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, 17-Sep-2018.)
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Theorem | eluzge3nn 9217 |
If an integer is greater than 3, then it is a positive integer.
(Contributed by Alexander van der Vekens, 17-Sep-2018.)
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Theorem | uz3m2nn 9218 |
An integer greater than or equal to 3 decreased by 2 is a positive
integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
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Theorem | 1eluzge0 9219 |
1 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8-Jun-2018.)
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Theorem | 2eluzge0 9220 |
2 is an integer greater than or equal to 0. (Contributed by Alexander van
der Vekens, 8-Jun-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
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Theorem | 2eluzge1 9221 |
2 is an integer greater than or equal to 1. (Contributed by Alexander van
der Vekens, 8-Jun-2018.)
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Theorem | uznnssnn 9222 |
The upper integers starting from a natural are a subset of the naturals.
(Contributed by Scott Fenton, 29-Jun-2013.)
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Theorem | raluz 9223* |
Restricted universal quantification in an upper set of integers.
(Contributed by NM, 9-Sep-2005.)
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Theorem | raluz2 9224* |
Restricted universal quantification in an upper set of integers.
(Contributed by NM, 9-Sep-2005.)
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Theorem | rexuz 9225* |
Restricted existential quantification in an upper set of integers.
(Contributed by NM, 9-Sep-2005.)
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Theorem | rexuz2 9226* |
Restricted existential quantification in an upper set of integers.
(Contributed by NM, 9-Sep-2005.)
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Theorem | 2rexuz 9227* |
Double existential quantification in an upper set of integers.
(Contributed by NM, 3-Nov-2005.)
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Theorem | peano2uz 9228 |
Second Peano postulate for an upper set of integers. (Contributed by NM,
7-Sep-2005.)
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Theorem | peano2uzs 9229 |
Second Peano postulate for an upper set of integers. (Contributed by
Mario Carneiro, 26-Dec-2013.)
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Theorem | peano2uzr 9230 |
Reversed second Peano axiom for upper integers. (Contributed by NM,
2-Jan-2006.)
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Theorem | uzaddcl 9231 |
Addition closure law for an upper set of integers. (Contributed by NM,
4-Jun-2006.)
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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,
3-Oct-2018.)
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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, 7-Sep-2005.)
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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,
7-Sep-2005.) (New usage is discouraged.)
(Proof modification is discouraged.)
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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, 4-Nov-2005.)
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   ![]. ].](_drbrack.gif)            ![]. ].](_drbrack.gif)          ![]. ].](_drbrack.gif)   |
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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     ![]. ].](_drbrack.gif) . (Contributed by NM,
16-Nov-2005.)
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   ![]. ].](_drbrack.gif)          ![]. ].](_drbrack.gif)
    ![]. ].](_drbrack.gif)          ![]. ].](_drbrack.gif)   |
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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, 4-Sep-2005.)
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Theorem | indstr 9238* |
Strong Mathematical Induction for positive integers (inference schema).
(Contributed by NM, 17-Aug-2001.)
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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,
14-Jan-2022.)
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       inf             |
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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, 15-Jan-2022.)
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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,
15-Jan-2022.)
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Theorem | supminfex 9242* |
A supremum is the negation of the infimum of that set's image under
negation. (Contributed by Jim Kingdon, 14-Jan-2022.)
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Theorem | eluznn0 9243 |
Membership in a nonnegative upper set of integers implies membership in
.
(Contributed by Paul Chapman, 22-Jun-2011.)
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Theorem | eluznn 9244 |
Membership in a positive upper set of integers implies membership in
. (Contributed
by JJ, 1-Oct-2018.)
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Theorem | eluz2b1 9245 |
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23-Nov-2012.)
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Theorem | eluz2gt1 9246 |
An integer greater than or equal to 2 is greater than 1. (Contributed by
AV, 24-May-2020.)
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Theorem | eluz2b2 9247 |
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23-Nov-2012.)
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Theorem | eluz2b3 9248 |
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23-Nov-2012.)
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Theorem | uz2m1nn 9249 |
One less than an integer greater than or equal to 2 is a positive integer.
(Contributed by Paul Chapman, 17-Nov-2012.)
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Theorem | 1nuz2 9250 |
1 is not in     . (Contributed by Paul Chapman,
21-Nov-2012.)
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Theorem | elnn1uz2 9251 |
A positive integer is either 1 or greater than or equal to 2.
(Contributed by Paul Chapman, 17-Nov-2012.)
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Theorem | uz2mulcl 9252 |
Closure of multiplication of integers greater than or equal to 2.
(Contributed by Paul Chapman, 26-Oct-2012.)
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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, 21-Nov-2012.)
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Theorem | eluzdc 9254 |
Membership of an integer in an upper set of integers is decidable.
(Contributed by Jim Kingdon, 18-Apr-2020.)
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   DECID
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Theorem | ublbneg 9255* |
The image under negation of a bounded-above 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, 21-Mar-2011.)
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Theorem | eqreznegel 9256* |
Two ways to express the image under negation of a set of integers.
(Contributed by Paul Chapman, 21-Mar-2011.)
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Theorem | negm 9257* |
The image under negation of an inhabited set of reals is inhabited.
(Contributed by Jim Kingdon, 10-Apr-2020.)
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Theorem | lbzbi 9258* |
If a set of reals is bounded below, it is bounded below by an integer.
(Contributed by Paul Chapman, 21-Mar-2011.)
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Theorem | nn01to3 9259 |
A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed
by Alexander van der Vekens, 13-Sep-2018.)
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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,
1-Aug-2018.)
(New usage is discouraged.) (Proof modification is discouraged.)
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3.4.12 Rational numbers (as a subset of complex
numbers)
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Syntax | cq 9261 |
Extend class notation to include the class of rationals.
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Definition | df-q 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, 8-Jan-2002.)
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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, 19-Mar-2020.)
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Theorem | elq 9264* |
Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.)
(Revised by Mario Carneiro, 28-Jan-2014.)
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Theorem | qmulz 9265* |
If is rational, then
some integer multiple of it is an integer.
(Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro,
22-Jul-2014.)
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Theorem | znq 9266 |
The ratio of an integer and a positive integer is a rational number.
(Contributed by NM, 12-Jan-2002.)
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Theorem | qre 9267 |
A rational number is a real number. (Contributed by NM,
14-Nov-2002.)
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Theorem | zq 9268 |
An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
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Theorem | zssq 9269 |
The integers are a subset of the rationals. (Contributed by NM,
9-Jan-2002.)
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Theorem | nn0ssq 9270 |
The nonnegative integers are a subset of the rationals. (Contributed by
NM, 31-Jul-2004.)
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Theorem | nnssq 9271 |
The positive integers are a subset of the rationals. (Contributed by NM,
31-Jul-2004.)
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Theorem | qssre 9272 |
The rationals are a subset of the reals. (Contributed by NM,
9-Jan-2002.)
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Theorem | qsscn 9273 |
The rationals are a subset of the complex numbers. (Contributed by NM,
2-Aug-2004.)
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Theorem | qex 9274 |
The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.)
(Revised by Mario Carneiro, 17-Nov-2014.)
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Theorem | nnq 9275 |
A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
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Theorem | qcn 9276 |
A rational number is a complex number. (Contributed by NM,
2-Aug-2004.)
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Theorem | qaddcl 9277 |
Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
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Theorem | qnegcl 9278 |
Closure law for the negative of a rational. (Contributed by NM,
2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)
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Theorem | qmulcl 9279 |
Closure of multiplication of rationals. (Contributed by NM,
1-Aug-2004.)
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Theorem | qsubcl 9280 |
Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
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Theorem | qapne 9281 |
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 20-Mar-2020.)
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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, 11-Oct-2021.)
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Theorem | qlttri2 9283 |
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 9-Nov-2021.)
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Theorem | qreccl 9284 |
Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
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Theorem | qdivcl 9285 |
Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
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Theorem | qrevaddcl 9286 |
Reverse closure law for addition of rationals. (Contributed by NM,
2-Aug-2004.)
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Theorem | nnrecq 9287 |
The reciprocal of a positive integer is rational. (Contributed by NM,
17-Nov-2004.)
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Theorem | irradd 9288 |
The sum of an irrational number and a rational number is irrational.
(Contributed by NM, 7-Nov-2008.)
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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, 7-Nov-2008.)
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3.4.13 Complex numbers as pairs of
reals
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Theorem | cnref1o 9290* |
There is a natural one-to-one mapping from 
 to ,
where we map    to     . In our
construction of the complex numbers, this is in fact our
definition of
(see df-c 7506), but in the axiomatic treatment we can only
show
that there is the expected mapping between these two sets. (Contributed
by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro,
17-Feb-2014.)
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3.5 Order sets
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3.5.1 Positive reals (as a subset of complex
numbers)
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Syntax | crp 9291 |
Extend class notation to include the class of positive reals.
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Definition | df-rp 9292 |
Define the set of positive reals. Definition of positive numbers in
[Apostol] p. 20. (Contributed by NM,
27-Oct-2007.)
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Theorem | elrp 9293 |
Membership in the set of positive reals. (Contributed by NM,
27-Oct-2007.)
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Theorem | elrpii 9294 |
Membership in the set of positive reals. (Contributed by NM,
23-Feb-2008.)
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Theorem | 1rp 9295 |
1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
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Theorem | 2rp 9296 |
2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | rpre 9297 |
A positive real is a real. (Contributed by NM, 27-Oct-2007.)
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Theorem | rpxr 9298 |
A positive real is an extended real. (Contributed by Mario Carneiro,
21-Aug-2015.)
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Theorem | rpcn 9299 |
A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
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Theorem | nnrp 9300 |
A positive integer is a positive real. (Contributed by NM,
28-Nov-2008.)
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