P. 101 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 10001-10100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eluz2b1 10001	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 1 < 𝑁))
Theorem	eluz2gt1 10002	An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < 𝑁)
Theorem	eluz2b2 10003	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
Theorem	eluz2b3 10004	Two ways to say "an integer greater than or equal to 2". (Contributed by Paul Chapman, 23-Nov-2012.)
⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
Theorem	uz2m1nn 10005	One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ)
Theorem	1nuz2 10006	1 is not in (ℤ≥‘2). (Contributed by Paul Chapman, 21-Nov-2012.)
⊢ ¬ 1 ∈ (ℤ≥‘2)
Theorem	elnn1uz2 10007	A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)))
Theorem	uz2mulcl 10008	Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)
⊢ ((𝑀 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑀 · 𝑁) ∈ (ℤ≥‘2))
Theorem	indstr2 10009*	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.)
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜒    &   ⊢ (𝑥 ∈ (ℤ≥‘2) → (∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜓) → 𝜑))    ⇒   ⊢ (𝑥 ∈ ℕ → 𝜑)
Theorem	eluzdc 10010	Membership of an integer in an upper set of integers is decidable. (Contributed by Jim Kingdon, 18-Apr-2020.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 ∈ (ℤ≥‘𝑀))
Theorem	elnn0dc 10011	Membership of an integer in ℕ0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
⊢ (𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ0)
Theorem	elnndc 10012	Membership of an integer in ℕ is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
⊢ (𝑁 ∈ ℤ → DECID 𝑁 ∈ ℕ)
Theorem	ublbneg 10013*	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 9995. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴}𝑥 ≤ 𝑦)
Theorem	eqreznegel 10014*	Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (𝐴 ⊆ ℤ → {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴} = {𝑧 ∈ ℤ ∣ -𝑧 ∈ 𝐴})
Theorem	negm 10015*	The image under negation of an inhabited set of reals is inhabited. (Contributed by Jim Kingdon, 10-Apr-2020.)
⊢ ((𝐴 ⊆ ℝ ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑦 𝑦 ∈ {𝑧 ∈ ℝ ∣ -𝑧 ∈ 𝐴})
Theorem	lbzbi 10016*	If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦))
Theorem	nn01to3 10017	A (nonnegative) integer between 1 and 3 must be 1, 2 or 3. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
⊢ ((𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ∧ 𝑁 ≤ 3) → (𝑁 = 1 ∨ 𝑁 = 2 ∨ 𝑁 = 3))
Theorem	nn0ge2m1nnALT 10018	Alternate proof of nn0ge2m1nn 9627: 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 9927, 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 9627. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)
4.4.12  Rational numbers (as a subset of complex numbers)
Syntax	cq 10019	Extend class notation to include the class of rationals.
class ℚ
Definition	df-q 10020	Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 10022 for the relation "is rational". (Contributed by NM, 8-Jan-2002.)
⊢ ℚ = ( / “ (ℤ × ℕ))
Theorem	divfnzn 10021	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, 19-Mar-2020.)
⊢ ( / ↾ (ℤ × ℕ)) Fn (ℤ × ℕ)
Theorem	elq 10022*	Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)
⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
Theorem	qmulz 10023*	If 𝐴 is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)
⊢ (𝐴 ∈ ℚ → ∃𝑥 ∈ ℕ (𝐴 · 𝑥) ∈ ℤ)
Theorem	znq 10024	The ratio of an integer and a positive integer is a rational number. (Contributed by NM, 12-Jan-2002.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℚ)
Theorem	qre 10025	A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
Theorem	zq 10026	An integer is a rational number. (Contributed by NM, 9-Jan-2002.)
⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ)
Theorem	zssq 10027	The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
⊢ ℤ ⊆ ℚ
Theorem	nn0ssq 10028	The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
⊢ ℕ0 ⊆ ℚ
Theorem	nnssq 10029	The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
⊢ ℕ ⊆ ℚ
Theorem	qssre 10030	The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
⊢ ℚ ⊆ ℝ
Theorem	qsscn 10031	The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℚ ⊆ ℂ
Theorem	qex 10032	The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℚ ∈ V
Theorem	nnq 10033	A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
Theorem	qcn 10034	A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
Theorem	qaddcl 10035	Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
Theorem	qnegcl 10036	Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ)
Theorem	qmulcl 10037	Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
Theorem	qsubcl 10038	Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 − 𝐵) ∈ ℚ)
Theorem	qapne 10039	Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 20-Mar-2020.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 # 𝐵 ↔ 𝐴 ≠ 𝐵))
Theorem	qltlen 10040	Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8960 which is a similar result for real numbers. (Contributed by Jim Kingdon, 11-Oct-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴)))
Theorem	qlttri2 10041	Apartness is equivalent to not equal for rationals. (Contributed by Jim Kingdon, 9-Nov-2021.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
Theorem	qreccl 10042	Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
Theorem	qdivcl 10043	Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
Theorem	qrevaddcl 10044	Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
Theorem	nnrecq 10045	The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
Theorem	irradd 10046	The sum of an irrational number and a rational number is irrational. (Contributed by NM, 7-Nov-2008.)
⊢ ((𝐴 ∈ (ℝ ∖ ℚ) ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ (ℝ ∖ ℚ))
Theorem	irrmul 10047	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). For a similar theorem with irrational in place of not rational, see irrmulap 10048. (Contributed by NM, 7-Nov-2008.)
⊢ ((𝐴 ∈ (ℝ ∖ ℚ) ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 · 𝐵) ∈ (ℝ ∖ ℚ))
Theorem	irrmulap 10048*	The product of an irrational with a nonzero rational is irrational. By irrational we mean apart from any rational number. For a similar theorem with not rational in place of irrational, see irrmul 10047. (Contributed by Jim Kingdon, 25-Aug-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → ∀𝑞 ∈ ℚ 𝐴 # 𝑞)    &   ⊢ (𝜑 → 𝐵 ∈ ℚ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    &   ⊢ (𝜑 → 𝑄 ∈ ℚ)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) # 𝑄)
Theorem	elpq 10049*	A positive rational is the quotient of two positive integers. (Contributed by AV, 29-Dec-2022.)
⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
Theorem	elpqb 10050*	A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.)
⊢ ((𝐴 ∈ ℚ ∧ 0 < 𝐴) ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))
4.4.13  Complex numbers as pairs of reals
Theorem	cnref1o 10051*	There is a natural one-to-one 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 df-c 8185), 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.)
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    ⇒   ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
Theorem	addex 10052	The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ + ∈ V
Theorem	mulex 10053	The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ · ∈ V
4.5  Order sets
4.5.1  Positive reals (as a subset of complex numbers)
Syntax	crp 10054	Extend class notation to include the class of positive reals.
class ℝ+
Definition	df-rp 10055	Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥}
Theorem	elrp 10056	Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Theorem	elrpii 10057	Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ 𝐴 ∈ ℝ+
Theorem	1rp 10058	1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
⊢ 1 ∈ ℝ+
Theorem	2rp 10059	2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ 2 ∈ ℝ+
Theorem	3rp 10060	3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 3 ∈ ℝ+
Theorem	rpre 10061	A positive real is a real. (Contributed by NM, 27-Oct-2007.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ)
Theorem	rpxr 10062	A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*)
Theorem	rpcn 10063	A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ)
Theorem	nnrp 10064	A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
Theorem	rpssre 10065	The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
⊢ ℝ+ ⊆ ℝ
Theorem	rpgt0 10066	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴)
Theorem	rpge0 10067	A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴)
Theorem	rpregt0 10068	A positive real is a positive real number. (Contributed by NM, 11-Nov-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Theorem	rprege0 10069	A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
Theorem	rpne0 10070	A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0)
Theorem	rpap0 10071	A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 # 0)
Theorem	rprene0 10072	A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
Theorem	rpreap0 10073	A positive real is a real number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 # 0))
Theorem	rpcnne0 10074	A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
Theorem	rpcnap0 10075	A positive real is a complex number apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 # 0))
Theorem	ralrp 10076	Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑))
Theorem	rexrp 10077	Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑))
Theorem	rpaddcl 10078	Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 + 𝐵) ∈ ℝ+)
Theorem	rpmulcl 10079	Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 · 𝐵) ∈ ℝ+)
Theorem	rpdivcl 10080	Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+)
Theorem	rpreccl 10081	Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+)
Theorem	rphalfcl 10082	Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+)
Theorem	rpgecl 10083	A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ+)
Theorem	rphalflt 10084	Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) < 𝐴)
Theorem	rerpdivcl 10085	Closure law for division of a real by a positive real. (Contributed by NM, 10-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ)
Theorem	ge0p1rp 10086	A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ+)
Theorem	rpnegap 10087	Either a real apart from zero or its negation is a positive real, but not both. (Contributed by Jim Kingdon, 23-Mar-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0) → (𝐴 ∈ ℝ+ ⊻ -𝐴 ∈ ℝ+))
Theorem	negelrp 10088	Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)
⊢ (𝐴 ∈ ℝ → (-𝐴 ∈ ℝ+ ↔ 𝐴 < 0))
Theorem	negelrpd 10089	The negation of a negative number is in the positive real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 0)    ⇒   ⊢ (𝜑 → -𝐴 ∈ ℝ+)
Theorem	0nrp 10090	Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
⊢ ¬ 0 ∈ ℝ+
Theorem	ltsubrp 10091	Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴)
Theorem	ltaddrp 10092	Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵))
Theorem	difrp 10093	Two ways to say one number is less than another. (Contributed by Mario Carneiro, 21-May-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐵 − 𝐴) ∈ ℝ+))
Theorem	elrpd 10094	Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ+)
Theorem	nnrpd 10095	A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ+)
Theorem	zgt1rpn0n1 10096	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, 26-Sep-2017.) (Proof shortened by AV, 9-Jul-2022.)
⊢ (𝐵 ∈ (ℤ≥‘2) → (𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
Theorem	rpred 10097	A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	rpxrd 10098	A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)
Theorem	rpcnd 10099	A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	rpgt0d 10100	A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 0 < 𝐴)