P. 98 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9701-9800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	zssq 9701	The integers are a subset of the rationals. (Contributed by NM, 9-Jan-2002.)
⊢ ℤ ⊆ ℚ
Theorem	nn0ssq 9702	The nonnegative integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
⊢ ℕ0 ⊆ ℚ
Theorem	nnssq 9703	The positive integers are a subset of the rationals. (Contributed by NM, 31-Jul-2004.)
⊢ ℕ ⊆ ℚ
Theorem	qssre 9704	The rationals are a subset of the reals. (Contributed by NM, 9-Jan-2002.)
⊢ ℚ ⊆ ℝ
Theorem	qsscn 9705	The rationals are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
⊢ ℚ ⊆ ℂ
Theorem	qex 9706	The set of rational numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ℚ ∈ V
Theorem	nnq 9707	A positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ)
Theorem	qcn 9708	A rational number is a complex number. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
Theorem	qaddcl 9709	Closure of addition of rationals. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ)
Theorem	qnegcl 9710	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 9711	Closure of multiplication of rationals. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
Theorem	qsubcl 9712	Closure of subtraction of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 − 𝐵) ∈ ℚ)
Theorem	qapne 9713	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 9714	Rational 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8659 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 9715	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 9716	Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
Theorem	qdivcl 9717	Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
Theorem	qrevaddcl 9718	Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
Theorem	nnrecq 9719	The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
Theorem	irradd 9720	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 9721	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 9722. (Contributed by NM, 7-Nov-2008.)
⊢ ((𝐴 ∈ (ℝ ∖ ℚ) ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 · 𝐵) ∈ (ℝ ∖ ℚ))
Theorem	irrmulap 9722*	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 9721. (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 9723*	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 9724*	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 9725*	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 7885), 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 9726	The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ + ∈ V
Theorem	mulex 9727	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 9728	Extend class notation to include the class of positive reals.
class ℝ+
Definition	df-rp 9729	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 9730	Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Theorem	elrpii 9731	Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ 𝐴 ∈ ℝ+
Theorem	1rp 9732	1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
⊢ 1 ∈ ℝ+
Theorem	2rp 9733	2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ 2 ∈ ℝ+
Theorem	3rp 9734	3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 3 ∈ ℝ+
Theorem	rpre 9735	A positive real is a real. (Contributed by NM, 27-Oct-2007.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ)
Theorem	rpxr 9736	A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*)
Theorem	rpcn 9737	A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ)
Theorem	nnrp 9738	A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
Theorem	rpssre 9739	The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
⊢ ℝ+ ⊆ ℝ
Theorem	rpgt0 9740	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴)
Theorem	rpge0 9741	A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴)
Theorem	rpregt0 9742	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 9743	A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
Theorem	rpne0 9744	A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0)
Theorem	rpap0 9745	A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 # 0)
Theorem	rprene0 9746	A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
Theorem	rpreap0 9747	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 9748	A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
Theorem	rpcnap0 9749	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 9750	Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑))
Theorem	rexrp 9751	Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑))
Theorem	rpaddcl 9752	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 9753	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 9754	Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+)
Theorem	rpreccl 9755	Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+)
Theorem	rphalfcl 9756	Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+)
Theorem	rpgecl 9757	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 9758	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 9759	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 9760	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 9761	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 9762	Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)
⊢ (𝐴 ∈ ℝ → (-𝐴 ∈ ℝ+ ↔ 𝐴 < 0))
Theorem	negelrpd 9763	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 9764	Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
⊢ ¬ 0 ∈ ℝ+
Theorem	ltsubrp 9765	Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴)
Theorem	ltaddrp 9766	Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵))
Theorem	difrp 9767	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 9768	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 9769	A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ+)
Theorem	zgt1rpn0n1 9770	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 9771	A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	rpxrd 9772	A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)
Theorem	rpcnd 9773	A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	rpgt0d 9774	A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 0 < 𝐴)
Theorem	rpge0d 9775	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 9776	A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	rpap0d 9777	A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 # 0)
Theorem	rpregt0d 9778	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 9779	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 9780	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 9781	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 9782	Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ+)
Theorem	rprecred 9783	Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
Theorem	rphalfcld 9784	Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ+)
Theorem	reclt1d 9785	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 9786	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 9787	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 9788	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 9789	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 9790	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 9791	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 9792	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 9793	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 9794	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 9795	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 9796	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 9797	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 9798	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 9799	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 9800	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))