P. 119 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11801-11900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnq 11801	A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
Theorem	qcn 11802	A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
Theorem	qexALT 11803	Alternate proof of qex 11800. (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 11804	Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
Theorem	qnegcl 11805	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 11806	Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
Theorem	qsubcl 11807	Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 − 𝐵) ∈ ℚ)
Theorem	qreccl 11808	Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
Theorem	qdivcl 11809	Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
Theorem	qrevaddcl 11810	Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
Theorem	nnrecq 11811	The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
Theorem	irradd 11812	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 11813	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))
5.4.14  Existence of the set of complex numbers
Theorem	rpnnen1lem2 11814*	Lemma for rpnnen1 11820. (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 11815*	Lemma for rpnnen1 11820. (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 11816*	Lemma for rpnnen1 11820. (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 11817*	Lemma for rpnnen1 11820. (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 11818*	Lemma for rpnnen1 11820. (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 11819*	Lemma for rpnnen1 11820. (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 11820	One half of rpnnen 14956, 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 11819) 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 11026 and qex 11800, or nnexALT 11022 and qexALT 11803. 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	rpnnen1lem1OLD 11821*	Lemma for rpnnen1OLD 11825. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem1 11815 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    ⇒   ⊢ (𝑥 ∈ ℝ → (𝐹‘𝑥) ∈ (ℚ ↑𝑚 ℕ))
Theorem	rpnnen1lem3OLD 11822*	Lemma for rpnnen1OLD 11825. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem3 11816 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    ⇒   ⊢ (𝑥 ∈ ℝ → ∀𝑛 ∈ ran (𝐹‘𝑥)𝑛 ≤ 𝑥)
Theorem	rpnnen1lem4OLD 11823*	Lemma for rpnnen1OLD 11825. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem4 11817 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    ⇒   ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) ∈ ℝ)
Theorem	rpnnen1lem5OLD 11824*	Lemma for rpnnen1OLD 11825. (Contributed by Mario Carneiro, 12-May-2013.) Obsolete version of rpnnen1lem5 11818 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    ⇒   ⊢ (𝑥 ∈ ℝ → sup(ran (𝐹‘𝑥), ℝ, < ) = 𝑥)
Theorem	rpnnen1OLD 11825*	One half of rpnnen 14956, 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 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. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.) Obsolete version of rpnnen1 11820 as of 13-Aug-2021. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑇 = {𝑛 ∈ ℤ ∣ (𝑛 / 𝑘) < 𝑥}    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (𝑘 ∈ ℕ ↦ (sup(𝑇, ℝ, < ) / 𝑘)))    ⇒   ⊢ ℝ ≼ (ℚ ↑𝑚 ℕ)
Theorem	reexALT 11826	Alternate proof of reex 10027. (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 11827*	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 9942), 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 11828	The set of complex numbers exists. This theorem shows that ax-cnex 9992 is redundant if we assume ax-rep 4771. See also ax-cnex 9992. (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 11829	The set of extended reals exists. (Contributed by NM, 24-Dec-2006.)
⊢ ℝ* ∈ V
Theorem	addex 11830	The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ + ∈ V
Theorem	mulex 11831	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 11832	Extend class notation to include the class of positive reals.
class ℝ+
Definition	df-rp 11833	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 11834	Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Theorem	elrpii 11835	Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ 𝐴 ∈ ℝ+
Theorem	1rp 11836	1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
⊢ 1 ∈ ℝ+
Theorem	2rp 11837	2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ 2 ∈ ℝ+
Theorem	3rp 11838	3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 3 ∈ ℝ+
Theorem	rpre 11839	A positive real is a real. (Contributed by NM, 27-Oct-2007.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ)
Theorem	rpxr 11840	A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*)
Theorem	rpcn 11841	A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ)
Theorem	nnrp 11842	A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
Theorem	rpssre 11843	The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
⊢ ℝ+ ⊆ ℝ
Theorem	rpgt0 11844	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴)
Theorem	rpge0 11845	A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴)
Theorem	rpregt0 11846	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 11847	A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
Theorem	rpne0 11848	A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0)
Theorem	rprene0 11849	A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
Theorem	rpcnne0 11850	A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
Theorem	rpcndif0 11851	A positive real number is a complex number not being 0. (Contributed by AV, 29-May-2020.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ (ℂ ∖ {0}))
Theorem	ralrp 11852	Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑))
Theorem	rexrp 11853	Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑))
Theorem	rpaddcl 11854	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 11855	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 11856	Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+)
Theorem	rpreccl 11857	Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+)
Theorem	rphalfcl 11858	Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+)
Theorem	rpgecl 11859	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 11860	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 11861	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 11862	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 11863	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 11864	Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)
⊢ (𝐴 ∈ ℝ → (-𝐴 ∈ ℝ+ ↔ 𝐴 < 0))
Theorem	0nrp 11865	Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
⊢ ¬ 0 ∈ ℝ+
Theorem	ltsubrp 11866	Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴)
Theorem	ltaddrp 11867	Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵))
Theorem	difrp 11868	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 11869	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 11870	A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ+)
Theorem	zgt1rpn0n1 11871	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.)
⊢ (𝐵 ∈ (ℤ≥‘2) → (𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
Theorem	rpred 11872	A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	rpxrd 11873	A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)
Theorem	rpcnd 11874	A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	rpgt0d 11875	A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 0 < 𝐴)
Theorem	rpge0d 11876	A positive real is greater than or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 0 ≤ 𝐴)
Theorem	rpne0d 11877	A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	rpregt0d 11878	A positive real is real and greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Theorem	rprege0d 11879	A positive real is real and greater or equal to zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
Theorem	rprene0d 11880	A positive real is a nonzero real number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
Theorem	rpcnne0d 11881	A positive real is a nonzero complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
Theorem	rpreccld 11882	Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ+)
Theorem	rprecred 11883	Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
Theorem	rphalfcld 11884	Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ+)
Theorem	reclt1d 11885	The reciprocal of a positive number less than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
Theorem	recgt1d 11886	The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 < 𝐴 ↔ (1 / 𝐴) < 1))
Theorem	rpaddcld 11887	Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ+)
Theorem	rpmulcld 11888	Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℝ+)
Theorem	rpdivcld 11889	Closure law for division of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ+)
Theorem	ltrecd 11890	The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)))
Theorem	lerecd 11891	The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴)))
Theorem	ltrec1d 11892	Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → (1 / 𝐴) < 𝐵)    ⇒   ⊢ (𝜑 → (1 / 𝐵) < 𝐴)
Theorem	lerec2d 11893	Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐴 ≤ (1 / 𝐵))    ⇒   ⊢ (𝜑 → 𝐵 ≤ (1 / 𝐴))
Theorem	lediv2ad 11894	Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴))
Theorem	ltdiv2d 11895	Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 / 𝐵) < (𝐶 / 𝐴)))
Theorem	lediv2d 11896	Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))
Theorem	ledivdivd 11897	Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐴 / 𝐵) ≤ (𝐶 / 𝐷))    ⇒   ⊢ (𝜑 → (𝐷 / 𝐶) ≤ (𝐵 / 𝐴))
Theorem	divge1 11898	The ratio of a number over a smaller positive number is larger than 1. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 1 ≤ (𝐵 / 𝐴))
Theorem	divlt1lt 11899	A real number divided by a positive real number is less than 1 iff the real number is less than the positive real number. (Contributed by AV, 25-May-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) < 1 ↔ 𝐴 < 𝐵))
Theorem	divle1le 11900	A real number divided by a positive real number is less than or equal to 1 iff the real number is less than or equal to the positive real number. (Contributed by AV, 29-Jun-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) ≤ 1 ↔ 𝐴 ≤ 𝐵))