P. 120 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 11901-12000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mvllmuli 11901	Move the left term in a product on the LHS to the RHS, inference form. Uses divcan4i 11815. (Contributed by David A. Wheeler, 11-Oct-2018.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    &   ⊢ (𝐴 · 𝐵) = 𝐶    ⇒   ⊢ 𝐵 = (𝐶 / 𝐴)
Theorem	ldiv 11902	Left-division. (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) = 𝐶 ↔ 𝐴 = (𝐶 / 𝐵)))
Theorem	rdiv 11903	Right-division. (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) = 𝐶 ↔ 𝐵 = (𝐶 / 𝐴)))
Theorem	mdiv 11904	A division law. (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 = (𝐶 / 𝐵) ↔ 𝐵 = (𝐶 / 𝐴)))
Theorem	lineq 11905	Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑌 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (((𝐴 · 𝑋) + 𝐵) = 𝑌 ↔ 𝑋 = ((𝑌 − 𝐵) / 𝐴)))
5.3.7  Ordering on reals (cont.)
Theorem	elimgt0 11906	Hypothesis for weak deduction theorem to eliminate 0 < 𝐴. (Contributed by NM, 15-May-1999.)
⊢ 0 < if(0 < 𝐴, 𝐴, 1)
Theorem	elimge0 11907	Hypothesis for weak deduction theorem to eliminate 0 ≤ 𝐴. (Contributed by NM, 30-Jul-1999.)
⊢ 0 ≤ if(0 ≤ 𝐴, 𝐴, 0)
Theorem	ltp1 11908	A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
Theorem	lep1 11909	A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1))
Theorem	ltm1 11910	A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴)
Theorem	lem1 11911	A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴)
Theorem	letrp1 11912	A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ (𝐵 + 1))
Theorem	p1le 11913	A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴 ≤ 𝐵)
Theorem	recgt0 11914	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 0 < (1 / 𝐴))
Theorem	prodgt0 11915	Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐵)
Theorem	prodgt02 11916	Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐵 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐴)
Theorem	ltmul1a 11917	Lemma for ltmul1 11918. Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) ∧ 𝐴 < 𝐵) → (𝐴 · 𝐶) < (𝐵 · 𝐶))
Theorem	ltmul1 11918	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
Theorem	ltmul2 11919	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵)))
Theorem	lemul1 11920	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
Theorem	lemul2 11921	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))
Theorem	lemul1a 11922	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))
Theorem	lemul2a 11923	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))
Theorem	ltmul12a 11924	Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 < 𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 ≤ 𝐶 ∧ 𝐶 < 𝐷))) → (𝐴 · 𝐶) < (𝐵 · 𝐷))
Theorem	lemul12b 11925	Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 ≤ 𝐷))) → ((𝐴 ≤ 𝐵 ∧ 𝐶 ≤ 𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)))
Theorem	lemul12a 11926	Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)
⊢ ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ) ∧ ((𝐶 ∈ ℝ ∧ 0 ≤ 𝐶) ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐵 ∧ 𝐶 ≤ 𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)))
Theorem	mulgt1 11927	The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (1 < 𝐴 ∧ 1 < 𝐵)) → 1 < (𝐴 · 𝐵))
Theorem	ltmulgt11 11928	Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ 𝐴 < (𝐴 · 𝐵)))
Theorem	ltmulgt12 11929	Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ 𝐴 < (𝐵 · 𝐴)))
Theorem	lemulge11 11930	Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 1 ≤ 𝐵)) → 𝐴 ≤ (𝐴 · 𝐵))
Theorem	lemulge12 11931	Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 1 ≤ 𝐵)) → 𝐴 ≤ (𝐵 · 𝐴))
Theorem	ltdiv1 11932	Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)))
Theorem	lediv1 11933	Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐶) ≤ (𝐵 / 𝐶)))
Theorem	gt0div 11934	Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 < 𝐴 ↔ 0 < (𝐴 / 𝐵)))
Theorem	ge0div 11935	Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐵) → (0 ≤ 𝐴 ↔ 0 ≤ (𝐴 / 𝐵)))
Theorem	divgt0 11936	The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 / 𝐵))
Theorem	divge0 11937	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵))
Theorem	mulge0b 11938	A condition for multiplication to be nonnegative. (Contributed by Scott Fenton, 25-Jun-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 · 𝐵) ↔ ((𝐴 ≤ 0 ∧ 𝐵 ≤ 0) ∨ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵))))
Theorem	mulle0b 11939	A condition for multiplication to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 𝐵) ≤ 0 ↔ ((𝐴 ≤ 0 ∧ 0 ≤ 𝐵) ∨ (0 ≤ 𝐴 ∧ 𝐵 ≤ 0))))
Theorem	mulsuble0b 11940	A condition for multiplication of subtraction to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (((𝐴 − 𝐵) · (𝐶 − 𝐵)) ≤ 0 ↔ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) ∨ (𝐶 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))))
Theorem	ltmuldiv 11941	'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) < 𝐵 ↔ 𝐴 < (𝐵 / 𝐶)))
Theorem	ltmuldiv2 11942	'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝐴) < 𝐵 ↔ 𝐴 < (𝐵 / 𝐶)))
Theorem	ltdivmul 11943	'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) < 𝐵 ↔ 𝐴 < (𝐶 · 𝐵)))
Theorem	ledivmul 11944	'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐶 · 𝐵)))
Theorem	ltdivmul2 11945	'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) < 𝐵 ↔ 𝐴 < (𝐵 · 𝐶)))
Theorem	lt2mul2div 11946	'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
⊢ (((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷))) → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵)))
Theorem	ledivmul2 11947	'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 · 𝐶)))
Theorem	lemuldiv 11948	'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶)))
Theorem	lemuldiv2 11949	'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝐴) ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 / 𝐶)))
Theorem	ltrec 11950	The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)))
Theorem	lerec 11951	The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴)))
Theorem	lt2msq1 11952	Lemma for lt2msq 11953. (Contributed by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴 · 𝐴) < (𝐵 · 𝐵))
Theorem	lt2msq 11953	Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐴) < (𝐵 · 𝐵)))
Theorem	ltdiv2 11954	Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐶 / 𝐵) < (𝐶 / 𝐴)))
Theorem	ltrec1 11955	Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((1 / 𝐴) < 𝐵 ↔ (1 / 𝐵) < 𝐴))
Theorem	lerec2 11956	Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ≤ (1 / 𝐵) ↔ 𝐵 ≤ (1 / 𝐴)))
Theorem	ledivdiv 11957	Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)
⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 0 < 𝐶) ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷))) → ((𝐴 / 𝐵) ≤ (𝐶 / 𝐷) ↔ (𝐷 / 𝐶) ≤ (𝐵 / 𝐴)))
Theorem	lediv2 11958	Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))
Theorem	ltdiv23 11959	Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵))
Theorem	lediv23 11960	Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐵) ≤ 𝐶 ↔ (𝐴 / 𝐶) ≤ 𝐵))
Theorem	lediv12a 11961	Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)
⊢ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 < 𝐶 ∧ 𝐶 ≤ 𝐷))) → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶))
Theorem	lediv2a 11962	Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴 ≤ 𝐵) → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴))
Theorem	reclt1 11963	The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
Theorem	recgt1 11964	The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐴 ↔ (1 / 𝐴) < 1))
Theorem	recgt1i 11965	The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (0 < (1 / 𝐴) ∧ (1 / 𝐴) < 1))
Theorem	recp1lt1 11966	Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 / (1 + 𝐴)) < 1)
Theorem	recreclt 11967	Given a positive number 𝐴, construct a new positive number less than both 𝐴 and 1. (Contributed by NM, 28-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ((1 / (1 + (1 / 𝐴))) < 1 ∧ (1 / (1 + (1 / 𝐴))) < 𝐴))
Theorem	le2msq 11968	The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))
Theorem	msq11 11969	The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 · 𝐴) = (𝐵 · 𝐵) ↔ 𝐴 = 𝐵))
Theorem	ledivp1 11970	"Less than or equal to" and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 / (𝐵 + 1)) · 𝐵) ≤ 𝐴)
Theorem	squeeze0 11971*	If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℝ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 = 0)
Theorem	ltp1i 11972	A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 < (𝐴 + 1)
Theorem	recgt0i 11973	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 < 𝐴 → 0 < (1 / 𝐴))
Theorem	recgt0ii 11974	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ 0 < (1 / 𝐴)
Theorem	prodgt0i 11975	Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 < (𝐴 · 𝐵)) → 0 < 𝐵)
Theorem	divgt0i 11976	The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 / 𝐵))
Theorem	divge0i 11977	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 < 𝐵) → 0 ≤ (𝐴 / 𝐵))
Theorem	ltreci 11978	The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)))
Theorem	lereci 11979	The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴)))
Theorem	lt2msqi 11980	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐴) < (𝐵 · 𝐵)))
Theorem	le2msqi 11981	The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))
Theorem	msq11i 11982	The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((𝐴 · 𝐴) = (𝐵 · 𝐵) ↔ 𝐴 = 𝐵))
Theorem	divgt0i2i 11983	The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 0 < 𝐵    ⇒   ⊢ (0 < 𝐴 → 0 < (𝐴 / 𝐵))
Theorem	ltrecii 11984	The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 0 < 𝐴    &   ⊢ 0 < 𝐵    ⇒   ⊢ (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴))
Theorem	divgt0ii 11985	The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 0 < 𝐴    &   ⊢ 0 < 𝐵    ⇒   ⊢ 0 < (𝐴 / 𝐵)
Theorem	ltmul1i 11986	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶)))
Theorem	ltdiv1i 11987	Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶)))
Theorem	ltmuldivi 11988	'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → ((𝐴 · 𝐶) < 𝐵 ↔ 𝐴 < (𝐵 / 𝐶)))
Theorem	ltmul2i 11989	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵)))
Theorem	lemul1i 11990	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
Theorem	lemul2i 11991	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (0 < 𝐶 → (𝐴 ≤ 𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))
Theorem	ltdiv23i 11992	Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((0 < 𝐵 ∧ 0 < 𝐶) → ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵))
Theorem	ledivp1i 11993	"Less than or equal to" and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐶 ∧ 𝐴 ≤ (𝐵 / (𝐶 + 1))) → (𝐴 · 𝐶) ≤ 𝐵)
Theorem	ltdivp1i 11994	Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐶 ∧ 𝐴 < (𝐵 / (𝐶 + 1))) → (𝐴 · 𝐶) < 𝐵)
Theorem	ltdiv23ii 11995	Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 0 < 𝐵    &   ⊢ 0 < 𝐶    ⇒   ⊢ ((𝐴 / 𝐵) < 𝐶 ↔ (𝐴 / 𝐶) < 𝐵)
Theorem	ltmul1ii 11996	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 0 < 𝐶    ⇒   ⊢ (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶))
Theorem	ltdiv1ii 11997	Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 0 < 𝐶    ⇒   ⊢ (𝐴 < 𝐵 ↔ (𝐴 / 𝐶) < (𝐵 / 𝐶))
Theorem	ltp1d 11998	A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 < (𝐴 + 1))
Theorem	lep1d 11999	A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ≤ (𝐴 + 1))
Theorem	ltm1d 12000	A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 − 1) < 𝐴)