P. 97 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9601-9700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	qreccl 9601	Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℚ)
Theorem	qdivcl 9602	Closure of division of rationals. (Contributed by NM, 3-Aug-2004.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℚ)
Theorem	qrevaddcl 9603	Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.)
⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ))
Theorem	nnrecq 9604	The reciprocal of a positive integer is rational. (Contributed by NM, 17-Nov-2004.)
⊢ (𝐴 ∈ ℕ → (1 / 𝐴) ∈ ℚ)
Theorem	irradd 9605	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 9606	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, 7-Nov-2008.)
⊢ ((𝐴 ∈ (ℝ ∖ ℚ) ∧ 𝐵 ∈ ℚ ∧ 𝐵 ≠ 0) → (𝐴 · 𝐵) ∈ (ℝ ∖ ℚ))
Theorem	elpq 9607*	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 9608*	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 9609*	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 7780), 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
4.5  Order sets
4.5.1  Positive reals (as a subset of complex numbers)
Syntax	crp 9610	Extend class notation to include the class of positive reals.
class ℝ+
Definition	df-rp 9611	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 9612	Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.)
⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
Theorem	elrpii 9613	Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ 𝐴 ∈ ℝ+
Theorem	1rp 9614	1 is a positive real. (Contributed by Jeff Hankins, 23-Nov-2008.)
⊢ 1 ∈ ℝ+
Theorem	2rp 9615	2 is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ 2 ∈ ℝ+
Theorem	3rp 9616	3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 3 ∈ ℝ+
Theorem	rpre 9617	A positive real is a real. (Contributed by NM, 27-Oct-2007.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ)
Theorem	rpxr 9618	A positive real is an extended real. (Contributed by Mario Carneiro, 21-Aug-2015.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ*)
Theorem	rpcn 9619	A positive real is a complex number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ)
Theorem	nnrp 9620	A positive integer is a positive real. (Contributed by NM, 28-Nov-2008.)
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
Theorem	rpssre 9621	The positive reals are a subset of the reals. (Contributed by NM, 24-Feb-2008.)
⊢ ℝ+ ⊆ ℝ
Theorem	rpgt0 9622	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴)
Theorem	rpge0 9623	A positive real is greater than or equal to zero. (Contributed by NM, 22-Feb-2008.)
⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴)
Theorem	rpregt0 9624	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 9625	A positive real is a nonnegative real number. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
Theorem	rpne0 9626	A positive real is nonzero. (Contributed by NM, 18-Jul-2008.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 ≠ 0)
Theorem	rpap0 9627	A positive real is apart from zero. (Contributed by Jim Kingdon, 22-Mar-2020.)
⊢ (𝐴 ∈ ℝ+ → 𝐴 # 0)
Theorem	rprene0 9628	A positive real is a nonzero real number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℝ ∧ 𝐴 ≠ 0))
Theorem	rpreap0 9629	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 9630	A positive real is a nonzero complex number. (Contributed by NM, 11-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
Theorem	rpcnap0 9631	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 9632	Quantification over positive reals. (Contributed by NM, 12-Feb-2008.)
⊢ (∀𝑥 ∈ ℝ+ 𝜑 ↔ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝜑))
Theorem	rexrp 9633	Quantification over positive reals. (Contributed by Mario Carneiro, 21-May-2014.)
⊢ (∃𝑥 ∈ ℝ+ 𝜑 ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ 𝜑))
Theorem	rpaddcl 9634	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 9635	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 9636	Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ+)
Theorem	rpreccl 9637	Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008.)
⊢ (𝐴 ∈ ℝ+ → (1 / 𝐴) ∈ ℝ+)
Theorem	rphalfcl 9638	Closure law for half of a positive real. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ (𝐴 ∈ ℝ+ → (𝐴 / 2) ∈ ℝ+)
Theorem	rpgecl 9639	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 9640	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 9641	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 9642	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 9643	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 9644	Elementhood of a negation in the positive real numbers. (Contributed by Thierry Arnoux, 19-Sep-2018.)
⊢ (𝐴 ∈ ℝ → (-𝐴 ∈ ℝ+ ↔ 𝐴 < 0))
Theorem	negelrpd 9645	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 9646	Zero is not a positive real. Axiom 9 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.)
⊢ ¬ 0 ∈ ℝ+
Theorem	ltsubrp 9647	Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴)
Theorem	ltaddrp 9648	Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵))
Theorem	difrp 9649	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 9650	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 9651	A positive integer is a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ+)
Theorem	zgt1rpn0n1 9652	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 9653	A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	rpxrd 9654	A positive real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)
Theorem	rpcnd 9655	A positive real is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℂ)
Theorem	rpgt0d 9656	A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 0 < 𝐴)
Theorem	rpge0d 9657	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 9658	A positive real is nonzero. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 0)
Theorem	rpap0d 9659	A positive real is apart from zero. (Contributed by Jim Kingdon, 28-Jul-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 # 0)
Theorem	rpregt0d 9660	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 9661	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 9662	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 9663	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 9664	Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ+)
Theorem	rprecred 9665	Closure law for reciprocation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 / 𝐴) ∈ ℝ)
Theorem	rphalfcld 9666	Closure law for half of a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 / 2) ∈ ℝ+)
Theorem	reclt1d 9667	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 9668	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 9669	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 9670	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 9671	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 9672	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 9673	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 9674	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 9675	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 9676	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 9677	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 9678	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 9679	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 9680	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 9681	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 9682	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))
Theorem	ledivge1le 9683	If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ (𝐶 ∈ ℝ+ ∧ 1 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐴 / 𝐶) ≤ 𝐵))
Theorem	ge0p1rpd 9684	A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+)
Theorem	rerpdivcld 9685	Closure law for division of a real by a positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
Theorem	ltsubrpd 9686	Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐵) < 𝐴)
Theorem	ltaddrpd 9687	Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 < (𝐴 + 𝐵))
Theorem	ltaddrp2d 9688	Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐴))
Theorem	ltmulgt11d 9689	Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 < 𝐴 ↔ 𝐵 < (𝐵 · 𝐴)))
Theorem	ltmulgt12d 9690	Multiplication by a number greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (1 < 𝐴 ↔ 𝐵 < (𝐴 · 𝐵)))
Theorem	gt0divd 9691	Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵)))
Theorem	ge0divd 9692	Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵)))
Theorem	rpgecld 9693	A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ+)
Theorem	divge0d 9694	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 / 𝐵))
Theorem	ltmul1d 9695	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
Theorem	ltmul2d 9696	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵)))
Theorem	lemul1d 9697	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
Theorem	lemul2d 9698	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))
Theorem	ltdiv1d 9699	Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)))
Theorem	lediv1d 9700	Division of both sides of a less than or equal to relation by a positive number. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐶) ≤ (𝐵 / 𝐶)))