Type  Label  Description 
Statement 

Theorem  uzaddcl 8501 
Addition closure law for an upper set of integers. (Contributed by NM,
4Jun2006.)

⊢ ((𝑁 ∈ (ℤ_{≥}‘𝑀) ∧ 𝐾 ∈ ℕ_{0}) → (𝑁 + 𝐾) ∈
(ℤ_{≥}‘𝑀)) 

Theorem  nn0pzuz 8502 
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.)

⊢ ((𝑁 ∈ ℕ_{0} ∧ 𝑍 ∈ ℤ) → (𝑁 + 𝑍) ∈
(ℤ_{≥}‘𝑍)) 

Theorem  uzind4 8503* 
Induction on the upper set of integers that starts at an integer
𝑀. 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.)

⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ (𝑀 ∈ ℤ → 𝜓) & ⊢ (𝑘 ∈
(ℤ_{≥}‘𝑀) → (𝜒 → 𝜃)) ⇒ ⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → 𝜏) 

Theorem  uzind4ALT 8504* 
Induction on the upper set of integers that starts at an integer
𝑀. The last four hypotheses give us
the substitution instances we
need; the first two are the basis and the induction step. Either
uzind4 8503 or uzind4ALT 8504 may be used; see comment for nnind 7906.
(Contributed by NM, 7Sep2005.) (New usage is discouraged.)
(Proof modification is discouraged.)

⊢ (𝑀 ∈ ℤ → 𝜓)
& ⊢ (𝑘 ∈ (ℤ_{≥}‘𝑀) → (𝜒 → 𝜃)) & ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) ⇒ ⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → 𝜏) 

Theorem  uzind4s 8505* 
Induction on the upper set of integers that starts at an integer 𝑀,
using explicit substitution. The hypotheses are the basis and the
induction step. (Contributed by NM, 4Nov2005.)

⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑘]𝜑)
& ⊢ (𝑘 ∈ (ℤ_{≥}‘𝑀) → (𝜑 → [(𝑘 + 1) / 𝑘]𝜑)) ⇒ ⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → [𝑁 / 𝑘]𝜑) 

Theorem  uzind4s2 8506* 
Induction on the upper set of integers that starts at an integer 𝑀,
using explicit substitution. The hypotheses are the basis and the
induction step. Use this instead of uzind4s 8505 when 𝑗 and 𝑘 must
be distinct in [(𝑘 + 1) / 𝑗]𝜑. (Contributed by NM,
16Nov2005.)

⊢ (𝑀 ∈ ℤ → [𝑀 / 𝑗]𝜑)
& ⊢ (𝑘 ∈ (ℤ_{≥}‘𝑀) → ([𝑘 / 𝑗]𝜑 → [(𝑘 + 1) / 𝑗]𝜑)) ⇒ ⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → [𝑁 / 𝑗]𝜑) 

Theorem  uzind4i 8507* 
Induction on the upper integers that start at 𝑀. 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.)

⊢ 𝑀 ∈ ℤ & ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) & ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) & ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) & ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) & ⊢ 𝜓 & ⊢ (𝑘 ∈
(ℤ_{≥}‘𝑀) → (𝜒 → 𝜃)) ⇒ ⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → 𝜏) 

Theorem  indstr 8508* 
Strong Mathematical Induction for positive integers (inference schema).
(Contributed by NM, 17Aug2001.)

⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 ∈ ℕ →
(∀𝑦 ∈ ℕ
(𝑦 < 𝑥 → 𝜓) → 𝜑)) ⇒ ⊢ (𝑥 ∈ ℕ → 𝜑) 

Theorem  eluznn0 8509 
Membership in a nonnegative upper set of integers implies membership in
ℕ_{0}. (Contributed by Paul
Chapman, 22Jun2011.)

⊢ ((𝑁 ∈ ℕ_{0} ∧ 𝑀 ∈
(ℤ_{≥}‘𝑁)) → 𝑀 ∈
ℕ_{0}) 

Theorem  eluznn 8510 
Membership in a positive upper set of integers implies membership in
ℕ. (Contributed by JJ, 1Oct2018.)

⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ_{≥}‘𝑁)) → 𝑀 ∈ ℕ) 

Theorem  eluz2b1 8511 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)

⊢ (𝑁 ∈ (ℤ_{≥}‘2)
↔ (𝑁 ∈ ℤ
∧ 1 < 𝑁)) 

Theorem  eluz2b2 8512 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)

⊢ (𝑁 ∈ (ℤ_{≥}‘2)
↔ (𝑁 ∈ ℕ
∧ 1 < 𝑁)) 

Theorem  eluz2b3 8513 
Two ways to say "an integer greater than or equal to 2."
(Contributed by
Paul Chapman, 23Nov2012.)

⊢ (𝑁 ∈ (ℤ_{≥}‘2)
↔ (𝑁 ∈ ℕ
∧ 𝑁 ≠
1)) 

Theorem  uz2m1nn 8514 
One less than an integer greater than or equal to 2 is a positive
integer. (Contributed by Paul Chapman, 17Nov2012.)

⊢ (𝑁 ∈ (ℤ_{≥}‘2)
→ (𝑁 − 1)
∈ ℕ) 

Theorem  1nuz2 8515 
1 is not in (ℤ_{≥}‘2).
(Contributed by Paul Chapman,
21Nov2012.)

⊢ ¬ 1 ∈
(ℤ_{≥}‘2) 

Theorem  elnn1uz2 8516 
A positive integer is either 1 or greater than or equal to 2.
(Contributed by Paul Chapman, 17Nov2012.)

⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈
(ℤ_{≥}‘2))) 

Theorem  uz2mulcl 8517 
Closure of multiplication of integers greater than or equal to 2.
(Contributed by Paul Chapman, 26Oct2012.)

⊢ ((𝑀 ∈ (ℤ_{≥}‘2)
∧ 𝑁 ∈
(ℤ_{≥}‘2)) → (𝑀 · 𝑁) ∈
(ℤ_{≥}‘2)) 

Theorem  indstr2 8518* 
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.)

⊢ (𝑥 = 1 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜒 & ⊢ (𝑥 ∈
(ℤ_{≥}‘2) → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑)) ⇒ ⊢ (𝑥 ∈ ℕ → 𝜑) 

Theorem  eluzdc 8519 
Membership of an integer in an upper set of integers is decidable.
(Contributed by Jim Kingdon, 18Apr2020.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
DECID 𝑁
∈ (ℤ_{≥}‘𝑀)) 

Theorem  ublbneg 8520* 
The image under negation of a boundedabove set of reals is bounded
below. (Contributed by Paul Chapman, 21Mar2011.)

⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ 𝑧 ∈ 𝐴}𝑥 ≤ 𝑦) 

Theorem  eqreznegel 8521* 
Two ways to express the image under negation of a set of integers.
(Contributed by Paul Chapman, 21Mar2011.)

⊢ (𝐴 ⊆ ℤ → {𝑧 ∈ ℝ ∣ 𝑧 ∈ 𝐴} = {𝑧 ∈ ℤ ∣ 𝑧 ∈ 𝐴}) 

Theorem  negm 8522* 
The image under negation of an inhabited set of reals is inhabited.
(Contributed by Jim Kingdon, 10Apr2020.)

⊢ ((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ {𝑧 ∈ ℝ ∣ 𝑧 ∈ 𝐴}) 

Theorem  lbzbi 8523* 
If a set of reals is bounded below, it is bounded below by an integer.
(Contributed by Paul Chapman, 21Mar2011.)

⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) 

Theorem  nn01to3 8524 
A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed
by Alexander van der Vekens, 13Sep2018.)

⊢ ((𝑁 ∈ ℕ_{0} ∧ 1 ≤
𝑁 ∧ 𝑁 ≤ 3) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3)) 

Theorem  nn0ge2m1nnALT 8525 
Alternate proof of nn0ge2m1nn 8214: 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 8451, 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 8214. (Contributed by Alexander van der Vekens,
1Aug2018.)
(New usage is discouraged.) (Proof modification is discouraged.)

⊢ ((𝑁 ∈ ℕ_{0} ∧ 2 ≤
𝑁) → (𝑁 − 1) ∈
ℕ) 

3.4.11 Rational numbers (as a subset of complex
numbers)


Syntax  cq 8526 
Extend class notation to include the class of rationals.

class ℚ 

Definition  dfq 8527 
Define the set of rational numbers. Based on definition of rationals in
[Apostol] p. 22. See elq 8529
for the relation "is rational." (Contributed
by NM, 8Jan2002.)

⊢ ℚ = ( / “ (ℤ ×
ℕ)) 

Theorem  divfnzn 8528 
Division restricted to ℤ × ℕ is a
function. Given excluded
middle, it would be easy to prove this for ℂ
× (ℂ ∖ {0}).
The key difference is that an element of ℕ
is apart from zero,
whereas being an element of ℂ ∖ {0}
implies being not equal to
zero. (Contributed by Jim Kingdon, 19Mar2020.)

⊢ ( / ↾ (ℤ × ℕ)) Fn
(ℤ × ℕ) 

Theorem  elq 8529* 
Membership in the set of rationals. (Contributed by NM, 8Jan2002.)
(Revised by Mario Carneiro, 28Jan2014.)

⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) 

Theorem  qmulz 8530* 
If 𝐴 is rational, then some integer
multiple of it is an integer.
(Contributed by NM, 7Nov2008.) (Revised by Mario Carneiro,
22Jul2014.)

⊢ (𝐴 ∈ ℚ → ∃𝑥 ∈ ℕ (𝐴 · 𝑥) ∈ ℤ) 

Theorem  znq 8531 
The ratio of an integer and a positive integer is a rational number.
(Contributed by NM, 12Jan2002.)

⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ) 

Theorem  qre 8532 
A rational number is a real number. (Contributed by NM,
14Nov2002.)

⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) 

Theorem  zq 8533 
An integer is a rational number. (Contributed by NM, 9Jan2002.)

⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) 

Theorem  zssq 8534 
The integers are a subset of the rationals. (Contributed by NM,
9Jan2002.)

⊢ ℤ ⊆ ℚ 

Theorem  nn0ssq 8535 
The nonnegative integers are a subset of the rationals. (Contributed by
NM, 31Jul2004.)

⊢ ℕ_{0} ⊆
ℚ 

Theorem  nnssq 8536 
The positive integers are a subset of the rationals. (Contributed by NM,
31Jul2004.)

⊢ ℕ ⊆ ℚ 

Theorem  qssre 8537 
The rationals are a subset of the reals. (Contributed by NM,
9Jan2002.)

⊢ ℚ ⊆ ℝ 

Theorem  qsscn 8538 
The rationals are a subset of the complex numbers. (Contributed by NM,
2Aug2004.)

⊢ ℚ ⊆ ℂ 

Theorem  qex 8539 
The set of rational numbers exists. (Contributed by NM, 30Jul2004.)
(Revised by Mario Carneiro, 17Nov2014.)

⊢ ℚ ∈ V 

Theorem  nnq 8540 
A positive integer is rational. (Contributed by NM, 17Nov2004.)

⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ) 

Theorem  qcn 8541 
A rational number is a complex number. (Contributed by NM,
2Aug2004.)

⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) 

Theorem  qaddcl 8542 
Closure of addition of rationals. (Contributed by NM, 1Aug2004.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ) 

Theorem  qnegcl 8543 
Closure law for the negative of a rational. (Contributed by NM,
2Aug2004.) (Revised by Mario Carneiro, 15Sep2014.)

⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℚ) 

Theorem  qmulcl 8544 
Closure of multiplication of rationals. (Contributed by NM,
1Aug2004.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ) 

Theorem  qsubcl 8545 
Closure of subtraction of rationals. (Contributed by NM, 2Aug2004.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 − 𝐵) ∈ ℚ) 

Theorem  qapne 8546 
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 20Mar2020.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 # 𝐵 ↔ 𝐴 ≠ 𝐵)) 

Theorem  qreccl 8547 
Closure of reciprocal of rationals. (Contributed by NM, 3Aug2004.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ) 

Theorem  qdivcl 8548 
Closure of division of rationals. (Contributed by NM, 3Aug2004.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ) 

Theorem  qrevaddcl 8549 
Reverse closure law for addition of rationals. (Contributed by NM,
2Aug2004.)

⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ)) 

Theorem  nnrecq 8550 
The reciprocal of a positive integer is rational. (Contributed by NM,
17Nov2004.)

⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈
ℚ) 

Theorem  irradd 8551 
The sum of an irrational number and a rational number is irrational.
(Contributed by NM, 7Nov2008.)

⊢ ((𝐴 ∈ (ℝ ∖ ℚ) ∧
𝐵 ∈ ℚ) →
(𝐴 + 𝐵) ∈ (ℝ ∖
ℚ)) 

Theorem  irrmul 8552 
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.)

⊢ ((𝐴 ∈ (ℝ ∖ ℚ) ∧
𝐵 ∈ ℚ ∧
𝐵 ≠ 0) → (𝐴 · 𝐵) ∈ (ℝ ∖
ℚ)) 

3.4.12 Complex numbers as pairs of
reals


Theorem  cnref1o 8553* 
There is a natural onetoone mapping from (ℝ ×
ℝ) to ℂ,
where we map ⟨𝑥, 𝑦⟩ to (𝑥 + (i · 𝑦)). In our
construction of the complex numbers, this is in fact our
definition of
ℂ (see dfc 6876), 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.)

⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ⇒ ⊢ 𝐹:(ℝ × ℝ)–11onto→ℂ 

3.5 Order sets


3.5.1 Positive reals (as a subset of complex
numbers)


Syntax  crp 8554 
Extend class notation to include the class of positive reals.

class ℝ^{+} 

Definition  dfrp 8555 
Define the set of positive reals. Definition of positive numbers in
[Apostol] p. 20. (Contributed by NM,
27Oct2007.)

⊢ ℝ^{+} = {𝑥 ∈ ℝ ∣ 0 < 𝑥} 

Theorem  elrp 8556 
Membership in the set of positive reals. (Contributed by NM,
27Oct2007.)

⊢ (𝐴 ∈ ℝ^{+} ↔ (𝐴 ∈ ℝ ∧ 0 <
𝐴)) 

Theorem  elrpii 8557 
Membership in the set of positive reals. (Contributed by NM,
23Feb2008.)

⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ 𝐴 ∈
ℝ^{+} 

Theorem  1rp 8558 
1 is a positive real. (Contributed by Jeff Hankins, 23Nov2008.)

⊢ 1 ∈
ℝ^{+} 

Theorem  2rp 8559 
2 is a positive real. (Contributed by Mario Carneiro, 28May2016.)

⊢ 2 ∈
ℝ^{+} 

Theorem  rpre 8560 
A positive real is a real. (Contributed by NM, 27Oct2007.)

⊢ (𝐴 ∈ ℝ^{+} → 𝐴 ∈
ℝ) 

Theorem  rpxr 8561 
A positive real is an extended real. (Contributed by Mario Carneiro,
21Aug2015.)

⊢ (𝐴 ∈ ℝ^{+} → 𝐴 ∈
ℝ^{*}) 

Theorem  rpcn 8562 
A positive real is a complex number. (Contributed by NM, 11Nov2008.)

⊢ (𝐴 ∈ ℝ^{+} → 𝐴 ∈
ℂ) 

Theorem  nnrp 8563 
A positive integer is a positive real. (Contributed by NM,
28Nov2008.)

⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℝ^{+}) 

Theorem  rpssre 8564 
The positive reals are a subset of the reals. (Contributed by NM,
24Feb2008.)

⊢ ℝ^{+} ⊆
ℝ 

Theorem  rpgt0 8565 
A positive real is greater than zero. (Contributed by FL,
27Dec2007.)

⊢ (𝐴 ∈ ℝ^{+} → 0 <
𝐴) 

Theorem  rpge0 8566 
A positive real is greater than or equal to zero. (Contributed by NM,
22Feb2008.)

⊢ (𝐴 ∈ ℝ^{+} → 0 ≤
𝐴) 

Theorem  rpregt0 8567 
A positive real is a positive real number. (Contributed by NM,
11Nov2008.) (Revised by Mario Carneiro, 31Jan2014.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℝ ∧ 0 <
𝐴)) 

Theorem  rprege0 8568 
A positive real is a nonnegative real number. (Contributed by Mario
Carneiro, 31Jan2014.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℝ ∧ 0 ≤
𝐴)) 

Theorem  rpne0 8569 
A positive real is nonzero. (Contributed by NM, 18Jul2008.)

⊢ (𝐴 ∈ ℝ^{+} → 𝐴 ≠ 0) 

Theorem  rpap0 8570 
A positive real is apart from zero. (Contributed by Jim Kingdon,
22Mar2020.)

⊢ (𝐴 ∈ ℝ^{+} → 𝐴 # 0) 

Theorem  rprene0 8571 
A positive real is a nonzero real number. (Contributed by NM,
11Nov2008.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)) 

Theorem  rpreap0 8572 
A positive real is a real number apart from zero. (Contributed by Jim
Kingdon, 22Mar2020.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℝ ∧ 𝐴 # 0)) 

Theorem  rpcnne0 8573 
A positive real is a nonzero complex number. (Contributed by NM,
11Nov2008.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) 

Theorem  rpcnap0 8574 
A positive real is a complex number apart from zero. (Contributed by Jim
Kingdon, 22Mar2020.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℂ ∧ 𝐴 # 0)) 

Theorem  ralrp 8575 
Quantification over positive reals. (Contributed by NM, 12Feb2008.)

⊢ (∀𝑥 ∈ ℝ^{+} 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑)) 

Theorem  rexrp 8576 
Quantification over positive reals. (Contributed by Mario Carneiro,
21May2014.)

⊢ (∃𝑥 ∈ ℝ^{+} 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑)) 

Theorem  rpaddcl 8577 
Closure law for addition of positive reals. Part of Axiom 7 of [Apostol]
p. 20. (Contributed by NM, 27Oct2007.)

⊢ ((𝐴 ∈ ℝ^{+} ∧ 𝐵 ∈ ℝ^{+})
→ (𝐴 + 𝐵) ∈
ℝ^{+}) 

Theorem  rpmulcl 8578 
Closure law for multiplication of positive reals. Part of Axiom 7 of
[Apostol] p. 20. (Contributed by NM,
27Oct2007.)

⊢ ((𝐴 ∈ ℝ^{+} ∧ 𝐵 ∈ ℝ^{+})
→ (𝐴 · 𝐵) ∈
ℝ^{+}) 

Theorem  rpdivcl 8579 
Closure law for division of positive reals. (Contributed by FL,
27Dec2007.)

⊢ ((𝐴 ∈ ℝ^{+} ∧ 𝐵 ∈ ℝ^{+})
→ (𝐴 / 𝐵) ∈
ℝ^{+}) 

Theorem  rpreccl 8580 
Closure law for reciprocation of positive reals. (Contributed by Jeff
Hankins, 23Nov2008.)

⊢ (𝐴 ∈ ℝ^{+} → (1 /
𝐴) ∈
ℝ^{+}) 

Theorem  rphalfcl 8581 
Closure law for half of a positive real. (Contributed by Mario Carneiro,
31Jan2014.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 / 2) ∈
ℝ^{+}) 

Theorem  rpgecl 8582 
A number greater or equal to a positive real is positive real.
(Contributed by Mario Carneiro, 28May2016.)

⊢ ((𝐴 ∈ ℝ^{+} ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈
ℝ^{+}) 

Theorem  rphalflt 8583 
Half of a positive real is less than the original number. (Contributed by
Mario Carneiro, 21May2014.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 / 2) < 𝐴) 

Theorem  rerpdivcl 8584 
Closure law for division of a real by a positive real. (Contributed by
NM, 10Nov2008.)

⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^{+}) → (𝐴 / 𝐵) ∈ ℝ) 

Theorem  ge0p1rp 8585 
A nonnegative number plus one is a positive number. (Contributed by Mario
Carneiro, 5Oct2015.)

⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈
ℝ^{+}) 

Theorem  rpnegap 8586 
Either a real apart from zero or its negation is a positive real, but not
both. (Contributed by Jim Kingdon, 23Mar2020.)

⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (𝐴 ∈ ℝ^{+} ⊻ 𝐴 ∈
ℝ^{+})) 

Theorem  0nrp 8587 
Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by
NM, 27Oct2007.)

⊢ ¬ 0 ∈
ℝ^{+} 

Theorem  ltsubrp 8588 
Subtracting a positive real from another number decreases it.
(Contributed by FL, 27Dec2007.)

⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^{+}) → (𝐴 − 𝐵) < 𝐴) 

Theorem  ltaddrp 8589 
Adding a positive number to another number increases it. (Contributed by
FL, 27Dec2007.)

⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^{+}) → 𝐴 < (𝐴 + 𝐵)) 

Theorem  difrp 8590 
Two ways to say one number is less than another. (Contributed by Mario
Carneiro, 21May2014.)

⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈
ℝ^{+})) 

Theorem  elrpd 8591 
Membership in the set of positive reals. (Contributed by Mario
Carneiro, 28May2016.)

⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) 

Theorem  nnrpd 8592 
A positive integer is a positive real. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈ ℕ)
⇒ ⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) 

Theorem  rpred 8593 
A positive real is a real. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) 

Theorem  rpxrd 8594 
A positive real is an extended real. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 𝐴 ∈
ℝ^{*}) 

Theorem  rpcnd 8595 
A positive real is a complex number. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) 

Theorem  rpgt0d 8596 
A positive real is greater than zero. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 0 < 𝐴) 

Theorem  rpge0d 8597 
A positive real is greater than or equal to zero. (Contributed by Mario
Carneiro, 28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴) 

Theorem  rpne0d 8598 
A positive real is nonzero. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 𝐴 ≠ 0) 

Theorem  rpap0d 8599 
A positive real is apart from zero. (Contributed by Jim Kingdon,
28Jul2021.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 𝐴 # 0) 

Theorem  rpregt0d 8600 
A positive real is real and greater than zero. (Contributed by Mario
Carneiro, 28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) 