P. 131 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13001-13100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lbzbi 13001*	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	zsupss 13002*	Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 11262.) (Contributed by Mario Carneiro, 21-Apr-2015.)
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	suprzcl2 13003*	The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 12723 avoids ax-pre-sup 11262.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → sup(𝐴, ℝ, < ) ∈ 𝐴)
Theorem	suprzub 13004*	The supremum of a bounded-above set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.)
⊢ ((𝐴 ⊆ ℤ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ, < ))
Theorem	uzsupss 13005*	Any bounded subset of an upper set of integers has a supremum. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 21-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ 𝑍 ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ 𝑍 (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝑍 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	nn01to3 13006	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 13007	Alternate proof of nn0ge2m1nn 12622: 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 12909, 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 12622. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ ℕ)
5.4.12  Well-ordering principle for bounded-below sets of integers
Theorem	uzwo3 13008*	Well-ordering principle: any nonempty subset of an upper set of integers has a unique least element. This generalization of uzwo2 12977 allows the lower bound 𝐵 to be any real number. See also nnwo 12978 and nnwos 12980. (Contributed by NM, 12-Nov-2004.) (Proof shortened by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 27-Sep-2020.)
⊢ ((𝐵 ∈ ℝ ∧ (𝐴 ⊆ {𝑧 ∈ ℤ ∣ 𝐵 ≤ 𝑧} ∧ 𝐴 ≠ ∅)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)
Theorem	zmin 13009*	There is a unique smallest integer greater than or equal to a given real number. (Contributed by NM, 12-Nov-2004.) (Revised by Mario Carneiro, 13-Jun-2014.)
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝐴 ≤ 𝑥 ∧ ∀𝑦 ∈ ℤ (𝐴 ≤ 𝑦 → 𝑥 ≤ 𝑦)))
Theorem	zmax 13010*	There is a unique largest integer less than or equal to a given real number. (Contributed by NM, 15-Nov-2004.)
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ ∀𝑦 ∈ ℤ (𝑦 ≤ 𝐴 → 𝑦 ≤ 𝑥)))
Theorem	zbtwnre 13011*	There is a unique integer between a real number and the number plus one. Exercise 5 of [Apostol] p. 28. (Contributed by NM, 13-Nov-2004.)
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝐴 ≤ 𝑥 ∧ 𝑥 < (𝐴 + 1)))
Theorem	rebtwnz 13012*	There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28. (Contributed by NM, 15-Nov-2004.)
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1)))
5.4.13  Rational numbers (as a subset of complex numbers)
Syntax	cq 13013	Extend class notation to include the class of rationals.
class ℚ
Definition	df-q 13014	Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 13015 for the relation "is rational". (Contributed by NM, 8-Jan-2002.)
⊢ ℚ = ( / “ (ℤ × ℕ))
Theorem	elq 13015*	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 13016*	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 13017	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 13018	A rational number is a real number. (Contributed by NM, 14-Nov-2002.)
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
Theorem	zq 13019	An integer is a rational number. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Steven Nguyen, 23-Mar-2023.)
⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ)
Theorem	qred 13020	A rational number is a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℚ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	zssq 13021	The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
⊢ ℤ ⊆ ℚ
Theorem	nn0ssq 13022	The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
⊢ ℕ0 ⊆ ℚ
Theorem	nnssq 13023	The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
⊢ ℕ ⊆ ℚ
Theorem	qssre 13024	The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
⊢ ℚ ⊆ ℝ
Theorem	qsscn 13025	The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℚ ⊆ ℂ
Theorem	qex 13026	The set of rational numbers exists. See also qexALT 13029. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℚ ∈ V
Theorem	nnq 13027	A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
Theorem	qcn 13028	A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
Theorem	qexALT 13029	Alternate proof of qex 13026. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℚ ∈ V
Theorem	qaddcl 13030	Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
Theorem	qnegcl 13031	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 13032	Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
Theorem	qsubcl 13033	Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 − 𝐵) ∈ ℚ)
Theorem	qreccl 13034	Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
Theorem	qdivcl 13035	Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
Theorem	qrevaddcl 13036	Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
Theorem	nnrecq 13037	The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
Theorem	irradd 13038	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 13039	The product of an irrational with a nonzero rational is irrational. (Contributed by NM, 7-Nov-2008.)
⊢ ((𝐴 ∈ (ℝ ∖ ℚ) ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 · 𝐵) ∈ (ℝ ∖ ℚ))
Theorem	elpq 13040*	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 13041*	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))
5.4.14  Existence of the set of complex numbers
Theorem	rpnnen1lem2 13042*	Lemma for rpnnen1 13048. (Contributed by Mario Carneiro, 12-May-2013.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    ⇒   ⊢ ((𝑥 ∈ ℝ ∧ 𝑘 ∈ ℕ) → sup(𝑇, ℝ, < ) ∈ ℤ)
Theorem	rpnnen1lem1 13043*	Lemma for rpnnen1 13048. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    &   ⊢ ℕ ∈ V    &   ⊢ ℚ ∈ V    ⇒   ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑m ℕ))
Theorem	rpnnen1lem3 13044*	Lemma for rpnnen1 13048. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    &   ⊢ ℕ ∈ V    &   ⊢ ℚ ∈ V    ⇒   ⊢ (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥)
Theorem	rpnnen1lem4 13045*	Lemma for rpnnen1 13048. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    &   ⊢ ℕ ∈ V    &   ⊢ ℚ ∈ V    ⇒   ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
Theorem	rpnnen1lem5 13046*	Lemma for rpnnen1 13048. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    &   ⊢ ℕ ∈ V    &   ⊢ ℚ ∈ V    ⇒   ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥)
Theorem	rpnnen1lem6 13047*	Lemma for rpnnen1 13048. (Contributed by Mario Carneiro, 12-May-2013.) (Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    &   ⊢ ℕ ∈ V    &   ⊢ ℚ ∈ V    ⇒   ⊢ ℝ ≼ (ℚ ↑m ℕ)
Theorem	rpnnen1 13048	One half of rpnnen 16275, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number 𝑥 to the sequence (𝐹‘𝑥):ℕ⟶ℚ (see rpnnen1lem6 13047) such that ((𝐹‘𝑥)‘𝑘) is the largest rational number with denominator 𝑘 that is strictly less than 𝑥. In this manner, we get a monotonically increasing sequence that converges to 𝑥, and since each sequence converges to a unique real number, this mapping from reals to sequences of rational numbers is injective. Note: The ℕ and ℚ existence hypotheses provide for use with either nnex 12299 and qex 13026, or nnexALT 12295 and qexALT 13029. The proof should not be modified to use any of those 4 theorems. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.) (Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.)
⊢ ℕ ∈ V    &   ⊢ ℚ ∈ V    ⇒   ⊢ ℝ ≼ (ℚ ↑m ℕ)
Theorem	reexALT 13049	Alternate proof of reex 11275. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℝ ∈ V
Theorem	cnref1o 13050*	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 11190), 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	cnexALT 13051	The set of complex numbers exists. This theorem shows that ax-cnex 11240 is redundant if we assume ax-rep 5303. See also ax-cnex 11240. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ℂ ∈ V
Theorem	xrex 13052	The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
⊢ ℝ* ∈ V
Theorem	mpoaddex 13053*	The addition operation is a set. Version of addex 13054 using maps-to notation , which does not require ax-addf 11263. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) ∈ V
Theorem	addex 13054	The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ + ∈ V
Theorem	mpomulex 13055*	The multiplication operation is a set. Version of mulex 13056 using maps-to notation , which does not require ax-mulf 11264. (Contributed by GG, 16-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ∈ V
Theorem	mulex 13056	The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ · ∈ V
5.5  Order sets
5.5.1  Positive reals (as a subset of complex numbers)
Syntax	crp 13057	Extend class notation to include the class of positive reals.
class ℝ+
Definition	df-rp 13058	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 13059	Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Theorem	elrpii 13060	Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ 𝐴 ∈ ℝ+
Theorem	1rp 13061	1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
⊢ 1 ∈ ℝ+
Theorem	2rp 13062	2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ 2 ∈ ℝ+
Theorem	3rp 13063	3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 3 ∈ ℝ+
Theorem	rpssre 13064	The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
⊢ ℝ+ ⊆ ℝ
Theorem	rpre 13065	A positive real is a real. (Contributed by NM, 27-Oct-2007.) (Proof shortened by Steven Nguyen, 8-Oct-2022.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ)
Theorem	rpxr 13066	A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*)
Theorem	rpcn 13067	A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ)
Theorem	nnrp 13068	A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
Theorem	rpgt0 13069	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴)
Theorem	rpge0 13070	A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴)
Theorem	rpregt0 13071	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 13072	A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
Theorem	rpne0 13073	A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0)
Theorem	rprene0 13074	A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
Theorem	rpcnne0 13075	A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
Theorem	rpcndif0 13076	A positive real number is a complex number not being 0. (Contributed by AV, 29-May-2020.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ (ℂ ∖ {0}))
Theorem	ralrp 13077	Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑))
Theorem	rexrp 13078	Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑))
Theorem	rpaddcl 13079	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 13080	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	rpmtmip 13081	"Minus times minus is plus", see also nnmtmip 12319, holds for positive reals, too (formalized to "The product of two negative reals is a positive real"). "The reason for this" in this case is that (-𝐴 · -𝐵) = (𝐴 · 𝐵) for all complex numbers 𝐴 and 𝐵 because of mul2neg 11729, 𝐴 and 𝐵 are complex numbers because of rpcn 13067, and (𝐴 · 𝐵) ∈ ℝ+ because of rpmulcl 13080. Note that the opposites -𝐴 and -𝐵 of the positive reals 𝐴 and 𝐵 are negative reals. (Contributed by AV, 23-Dec-2022.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (-𝐴 · -𝐵) ∈ ℝ+)
Theorem	rpdivcl 13082	Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+)
Theorem	rpreccl 13083	Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+)
Theorem	rphalfcl 13084	Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+)
Theorem	rpgecl 13085	A number greater than 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 13086	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 13087	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 13088	A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ+)
Theorem	rpneg 13089	Either a nonzero real or its negation is a positive real, but not both. Axiom 8 of [Apostol] p. 20. (Contributed by NM, 7-Nov-2008.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴 ∈ ℝ+ ↔ ¬ -𝐴 ∈ ℝ+))
Theorem	negelrp 13090	Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)
⊢ (𝐴 ∈ ℝ → (-𝐴 ∈ ℝ+ ↔ 𝐴 < 0))
Theorem	negelrpd 13091	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 13092	Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
⊢ ¬ 0 ∈ ℝ+
Theorem	ltsubrp 13093	Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴)
Theorem	ltaddrp 13094	Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵))
Theorem	difrp 13095	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 13096	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 13097	A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ+)
Theorem	zgt1rpn0n1 13098	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 13099	A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	rpxrd 13100	A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)