Type  Label  Description 
Statement 

Theorem  uzind4i 8601* 
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 8602* 
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 8603 
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 8604 
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 8605 
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  eluz2gt1 8606 
An integer greater than or equal to 2 is greater than 1. (Contributed by
AV, 24May2020.)

⊢ (𝑁 ∈ (ℤ_{≥}‘2)
→ 1 < 𝑁) 

Theorem  eluz2b2 8607 
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 8608 
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 8609 
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 8610 
1 is not in (ℤ_{≥}‘2).
(Contributed by Paul Chapman,
21Nov2012.)

⊢ ¬ 1 ∈
(ℤ_{≥}‘2) 

Theorem  elnn1uz2 8611 
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 8612 
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 8613* 
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 8614 
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 8615* 
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 8616* 
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 8617* 
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 8618* 
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 8619 
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 8620 
Alternate proof of nn0ge2m1nn 8269: 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 8545, 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 8269. (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 8621 
Extend class notation to include the class of rationals.

class ℚ 

Definition  dfq 8622 
Define the set of rational numbers. Based on definition of rationals in
[Apostol] p. 22. See elq 8624
for the relation "is rational." (Contributed
by NM, 8Jan2002.)

⊢ ℚ = ( / “ (ℤ ×
ℕ)) 

Theorem  divfnzn 8623 
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 8624* 
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 8625* 
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 8626 
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 8627 
A rational number is a real number. (Contributed by NM,
14Nov2002.)

⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) 

Theorem  zq 8628 
An integer is a rational number. (Contributed by NM, 9Jan2002.)

⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) 

Theorem  zssq 8629 
The integers are a subset of the rationals. (Contributed by NM,
9Jan2002.)

⊢ ℤ ⊆ ℚ 

Theorem  nn0ssq 8630 
The nonnegative integers are a subset of the rationals. (Contributed by
NM, 31Jul2004.)

⊢ ℕ_{0} ⊆
ℚ 

Theorem  nnssq 8631 
The positive integers are a subset of the rationals. (Contributed by NM,
31Jul2004.)

⊢ ℕ ⊆ ℚ 

Theorem  qssre 8632 
The rationals are a subset of the reals. (Contributed by NM,
9Jan2002.)

⊢ ℚ ⊆ ℝ 

Theorem  qsscn 8633 
The rationals are a subset of the complex numbers. (Contributed by NM,
2Aug2004.)

⊢ ℚ ⊆ ℂ 

Theorem  qex 8634 
The set of rational numbers exists. (Contributed by NM, 30Jul2004.)
(Revised by Mario Carneiro, 17Nov2014.)

⊢ ℚ ∈ V 

Theorem  nnq 8635 
A positive integer is rational. (Contributed by NM, 17Nov2004.)

⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ) 

Theorem  qcn 8636 
A rational number is a complex number. (Contributed by NM,
2Aug2004.)

⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) 

Theorem  qaddcl 8637 
Closure of addition of rationals. (Contributed by NM, 1Aug2004.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ) 

Theorem  qnegcl 8638 
Closure law for the negative of a rational. (Contributed by NM,
2Aug2004.) (Revised by Mario Carneiro, 15Sep2014.)

⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℚ) 

Theorem  qmulcl 8639 
Closure of multiplication of rationals. (Contributed by NM,
1Aug2004.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ) 

Theorem  qsubcl 8640 
Closure of subtraction of rationals. (Contributed by NM, 2Aug2004.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 − 𝐵) ∈ ℚ) 

Theorem  qapne 8641 
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 20Mar2020.)

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

Theorem  qltlen 8642 
Rational 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 7665 which is a similar result for real numbers.
(Contributed by
Jim Kingdon, 11Oct2021.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴))) 

Theorem  qlttri2 8643 
Apartness is equivalent to not equal for rationals. (Contributed by Jim
Kingdon, 9Nov2021.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) 

Theorem  qreccl 8644 
Closure of reciprocal of rationals. (Contributed by NM, 3Aug2004.)

⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ) 

Theorem  qdivcl 8645 
Closure of division of rationals. (Contributed by NM, 3Aug2004.)

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

Theorem  qrevaddcl 8646 
Reverse closure law for addition of rationals. (Contributed by NM,
2Aug2004.)

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

Theorem  nnrecq 8647 
The reciprocal of a positive integer is rational. (Contributed by NM,
17Nov2004.)

⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈
ℚ) 

Theorem  irradd 8648 
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 8649 
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 8650* 
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 6923), 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 8651 
Extend class notation to include the class of positive reals.

class ℝ^{+} 

Definition  dfrp 8652 
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 8653 
Membership in the set of positive reals. (Contributed by NM,
27Oct2007.)

⊢ (𝐴 ∈ ℝ^{+} ↔ (𝐴 ∈ ℝ ∧ 0 <
𝐴)) 

Theorem  elrpii 8654 
Membership in the set of positive reals. (Contributed by NM,
23Feb2008.)

⊢ 𝐴 ∈ ℝ & ⊢ 0 < 𝐴 ⇒ ⊢ 𝐴 ∈
ℝ^{+} 

Theorem  1rp 8655 
1 is a positive real. (Contributed by Jeff Hankins, 23Nov2008.)

⊢ 1 ∈
ℝ^{+} 

Theorem  2rp 8656 
2 is a positive real. (Contributed by Mario Carneiro, 28May2016.)

⊢ 2 ∈
ℝ^{+} 

Theorem  rpre 8657 
A positive real is a real. (Contributed by NM, 27Oct2007.)

⊢ (𝐴 ∈ ℝ^{+} → 𝐴 ∈
ℝ) 

Theorem  rpxr 8658 
A positive real is an extended real. (Contributed by Mario Carneiro,
21Aug2015.)

⊢ (𝐴 ∈ ℝ^{+} → 𝐴 ∈
ℝ^{*}) 

Theorem  rpcn 8659 
A positive real is a complex number. (Contributed by NM, 11Nov2008.)

⊢ (𝐴 ∈ ℝ^{+} → 𝐴 ∈
ℂ) 

Theorem  nnrp 8660 
A positive integer is a positive real. (Contributed by NM,
28Nov2008.)

⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℝ^{+}) 

Theorem  rpssre 8661 
The positive reals are a subset of the reals. (Contributed by NM,
24Feb2008.)

⊢ ℝ^{+} ⊆
ℝ 

Theorem  rpgt0 8662 
A positive real is greater than zero. (Contributed by FL,
27Dec2007.)

⊢ (𝐴 ∈ ℝ^{+} → 0 <
𝐴) 

Theorem  rpge0 8663 
A positive real is greater than or equal to zero. (Contributed by NM,
22Feb2008.)

⊢ (𝐴 ∈ ℝ^{+} → 0 ≤
𝐴) 

Theorem  rpregt0 8664 
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 8665 
A positive real is a nonnegative real number. (Contributed by Mario
Carneiro, 31Jan2014.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℝ ∧ 0 ≤
𝐴)) 

Theorem  rpne0 8666 
A positive real is nonzero. (Contributed by NM, 18Jul2008.)

⊢ (𝐴 ∈ ℝ^{+} → 𝐴 ≠ 0) 

Theorem  rpap0 8667 
A positive real is apart from zero. (Contributed by Jim Kingdon,
22Mar2020.)

⊢ (𝐴 ∈ ℝ^{+} → 𝐴 # 0) 

Theorem  rprene0 8668 
A positive real is a nonzero real number. (Contributed by NM,
11Nov2008.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0)) 

Theorem  rpreap0 8669 
A positive real is a real number apart from zero. (Contributed by Jim
Kingdon, 22Mar2020.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℝ ∧ 𝐴 # 0)) 

Theorem  rpcnne0 8670 
A positive real is a nonzero complex number. (Contributed by NM,
11Nov2008.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) 

Theorem  rpcnap0 8671 
A positive real is a complex number apart from zero. (Contributed by Jim
Kingdon, 22Mar2020.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 ∈ ℂ ∧ 𝐴 # 0)) 

Theorem  ralrp 8672 
Quantification over positive reals. (Contributed by NM, 12Feb2008.)

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

Theorem  rexrp 8673 
Quantification over positive reals. (Contributed by Mario Carneiro,
21May2014.)

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

Theorem  rpaddcl 8674 
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 8675 
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 8676 
Closure law for division of positive reals. (Contributed by FL,
27Dec2007.)

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

Theorem  rpreccl 8677 
Closure law for reciprocation of positive reals. (Contributed by Jeff
Hankins, 23Nov2008.)

⊢ (𝐴 ∈ ℝ^{+} → (1 /
𝐴) ∈
ℝ^{+}) 

Theorem  rphalfcl 8678 
Closure law for half of a positive real. (Contributed by Mario Carneiro,
31Jan2014.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 / 2) ∈
ℝ^{+}) 

Theorem  rpgecl 8679 
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 8680 
Half of a positive real is less than the original number. (Contributed by
Mario Carneiro, 21May2014.)

⊢ (𝐴 ∈ ℝ^{+} → (𝐴 / 2) < 𝐴) 

Theorem  rerpdivcl 8681 
Closure law for division of a real by a positive real. (Contributed by
NM, 10Nov2008.)

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

Theorem  ge0p1rp 8682 
A nonnegative number plus one is a positive number. (Contributed by Mario
Carneiro, 5Oct2015.)

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

Theorem  rpnegap 8683 
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 8684 
Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by
NM, 27Oct2007.)

⊢ ¬ 0 ∈
ℝ^{+} 

Theorem  ltsubrp 8685 
Subtracting a positive real from another number decreases it.
(Contributed by FL, 27Dec2007.)

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

Theorem  ltaddrp 8686 
Adding a positive number to another number increases it. (Contributed by
FL, 27Dec2007.)

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

Theorem  difrp 8687 
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 8688 
Membership in the set of positive reals. (Contributed by Mario
Carneiro, 28May2016.)

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

Theorem  nnrpd 8689 
A positive integer is a positive real. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈ ℕ)
⇒ ⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) 

Theorem  rpred 8690 
A positive real is a real. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) 

Theorem  rpxrd 8691 
A positive real is an extended real. (Contributed by Mario Carneiro,
28May2016.)

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

Theorem  rpcnd 8692 
A positive real is a complex number. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℂ) 

Theorem  rpgt0d 8693 
A positive real is greater than zero. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 0 < 𝐴) 

Theorem  rpge0d 8694 
A positive real is greater than or equal to zero. (Contributed by Mario
Carneiro, 28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 0 ≤ 𝐴) 

Theorem  rpne0d 8695 
A positive real is nonzero. (Contributed by Mario Carneiro,
28May2016.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 𝐴 ≠ 0) 

Theorem  rpap0d 8696 
A positive real is apart from zero. (Contributed by Jim Kingdon,
28Jul2021.)

⊢ (𝜑 → 𝐴 ∈
ℝ^{+}) ⇒ ⊢ (𝜑 → 𝐴 # 0) 

Theorem  rpregt0d 8697 
A positive real is real and greater than zero. (Contributed by Mario
Carneiro, 28May2016.)

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

Theorem  rprege0d 8698 
A positive real is real and greater or equal to zero. (Contributed by
Mario Carneiro, 28May2016.)

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

Theorem  rprene0d 8699 
A positive real is a nonzero real number. (Contributed by Mario
Carneiro, 28May2016.)

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

Theorem  rpcnne0d 8700 
A positive real is a nonzero complex number. (Contributed by Mario
Carneiro, 28May2016.)

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