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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdiv12apd 7801 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶)))
 
Theoremdiv23apd 7802 A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵))
 
Theoremdivdirapd 7803 Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
 
Theoremdivsubdirapd 7804 Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)       (𝜑 → ((𝐴𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶)))
 
Theoremdiv11apd 7805 One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐶 # 0)    &   (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶))       (𝜑𝐴 = 𝐵)
 
Theoremdivmuldivapd 7806 Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐵 # 0)    &   (𝜑𝐷 # 0)       (𝜑 → ((𝐴 / 𝐵) · (𝐶 / 𝐷)) = ((𝐴 · 𝐶) / (𝐵 · 𝐷)))
 
Theoremrerecclapd 7807 Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 # 0)       (𝜑 → (1 / 𝐴) ∈ ℝ)
 
Theoremredivclapd 7808 Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐵 # 0)       (𝜑 → (𝐴 / 𝐵) ∈ ℝ)
 
Theoremmvllmulapd 7809 Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑 → (𝐴 · 𝐵) = 𝐶)       (𝜑𝐵 = (𝐶 / 𝐴))
 
3.3.9  Ordering on reals (cont.)
 
Theoremltp1 7810 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
(𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
 
Theoremlep1 7811 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)
(𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1))
 
Theoremltm1 7812 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)
(𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴)
 
Theoremlem1 7813 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴)
 
Theoremletrp1 7814 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → 𝐴 ≤ (𝐵 + 1))
 
Theoremp1le 7815 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴𝐵)
 
Theoremrecgt0 7816 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))
 
Theoremprodgt0gt0 7817 Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 7818 for the same theorem with 0 < 𝐴 replaced by the weaker condition 0 ≤ 𝐴. (Contributed by Jim Kingdon, 29-Feb-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < (𝐴 · 𝐵))) → 0 < 𝐵)
 
Theoremprodgt0 7818 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)
 
Theoremprodgt02 7819 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)
 
Theoremprodge0 7820 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 ≤ (𝐴 · 𝐵))) → 0 ≤ 𝐵)
 
Theoremprodge02 7821 Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐵 ∧ 0 ≤ (𝐴 · 𝐵))) → 0 ≤ 𝐴)
 
Theoremltmul2 7822 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)))
 
Theoremlemul2 7823 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)) → (𝐴𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))
 
Theoremlemul1a 7824 Multiplication of both sides of 'less than or equal to' by a nonnegative number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 21-Feb-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ 𝐴𝐵) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))
 
Theoremlemul2a 7825 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)) ∧ 𝐴𝐵) → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))
 
Theoremltmul12a 7826 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴𝐴 < 𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 ≤ 𝐶𝐶 < 𝐷))) → (𝐴 · 𝐶) < (𝐵 · 𝐷))
 
Theoremlemul12b 7827 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))) → ((𝐴𝐵𝐶𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)))
 
Theoremlemul12a 7828 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)) → ((𝐴𝐵𝐶𝐷) → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷)))
 
Theoremmulgt1 7829 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))
 
Theoremltmulgt11 7830 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵𝐴 < (𝐴 · 𝐵)))
 
Theoremltmulgt12 7831 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵𝐴 < (𝐵 · 𝐴)))
 
Theoremlemulge11 7832 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))
 
Theoremlemulge12 7833 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))
 
Theoremltdiv1 7834 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)))
 
Theoremlediv1 7835 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)) → (𝐴𝐵 ↔ (𝐴 / 𝐶) ≤ (𝐵 / 𝐶)))
 
Theoremgt0div 7836 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)))
 
Theoremge0div 7837 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)))
 
Theoremdivgt0 7838 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))
 
Theoremdivge0 7839 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))
 
Theoremltmuldiv 7840 '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) < 𝐵𝐴 < (𝐵 / 𝐶)))
 
Theoremltmuldiv2 7841 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐶 · 𝐴) < 𝐵𝐴 < (𝐵 / 𝐶)))
 
Theoremltdivmul 7842 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) < 𝐵𝐴 < (𝐶 · 𝐵)))
 
Theoremledivmul 7843 '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) ≤ 𝐵𝐴 ≤ (𝐶 · 𝐵)))
 
Theoremltdivmul2 7844 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 / 𝐶) < 𝐵𝐴 < (𝐵 · 𝐶)))
 
Theoremlt2mul2div 7845 '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)))
 
Theoremledivmul2 7846 '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) ≤ 𝐵𝐴 ≤ (𝐵 · 𝐶)))
 
Theoremlemuldiv 7847 '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) ≤ 𝐵𝐴 ≤ (𝐵 / 𝐶)))
 
Theoremlemuldiv2 7848 '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) ≤ 𝐵𝐴 ≤ (𝐵 / 𝐶)))
 
Theoremltrec 7849 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)))
 
Theoremlerec 7850 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)) → (𝐴𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴)))
 
Theoremlt2msq1 7851 Lemma for lt2msq 7852. (Contributed by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴 · 𝐴) < (𝐵 · 𝐵))
 
Theoremlt2msq 7852 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)))
 
Theoremltdiv2 7853 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)))
 
Theoremltrec1 7854 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))
 
Theoremlerec2 7855 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)))
 
Theoremledivdiv 7856 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)))
 
Theoremlediv2 7857 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)) → (𝐴𝐵 ↔ (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))
 
Theoremltdiv23 7858 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))
 
Theoremlediv23 7859 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))
 
Theoremlediv12a 7860 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)
((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴𝐴𝐵)) ∧ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (0 < 𝐶𝐶𝐷))) → (𝐴 / 𝐷) ≤ (𝐵 / 𝐶))
 
Theoremlediv2a 7861 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)) ∧ 𝐴𝐵) → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴))
 
Theoremreclt1 7862 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)))
 
Theoremrecgt1 7863 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))
 
Theoremrecgt1i 7864 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))
 
Theoremrecp1lt1 7865 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)
((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 / (1 + 𝐴)) < 1)
 
Theoremrecreclt 7866 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))
 
Theoremle2msq 7867 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)) → (𝐴𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))
 
Theoremmsq11 7868 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))
 
Theoremledivp1 7869 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)
 
Theoremsqueeze0 7870* 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 < 𝑥𝐴 < 𝑥)) → 𝐴 = 0)
 
Theoremltp1i 7871 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
𝐴 ∈ ℝ       𝐴 < (𝐴 + 1)
 
Theoremrecgt0i 7872 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))
 
Theoremrecgt0ii 7873 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)
 
Theoremprodgt0i 7874 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)
 
Theoremprodge0i 7875 Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 ≤ (𝐴 · 𝐵)) → 0 ≤ 𝐵)
 
Theoremdivgt0i 7876 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))
 
Theoremdivge0i 7877 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))
 
Theoremltreci 7878 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)))
 
Theoremlereci 7879 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) → (𝐴𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴)))
 
Theoremlt2msqi 7880 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)))
 
Theoremle2msqi 7881 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵)))
 
Theoremmsq11i 7882 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))
 
Theoremdivgt0i2i 7883 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))
 
Theoremltrecii 7884 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))
 
Theoremdivgt0ii 7885 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)
 
Theoremltmul1i 7886 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)))
 
Theoremltdiv1i 7887 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)))
 
Theoremltmuldivi 7888 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (0 < 𝐶 → ((𝐴 · 𝐶) < 𝐵𝐴 < (𝐵 / 𝐶)))
 
Theoremltmul2i 7889 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)))
 
Theoremlemul1i 7890 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 → (𝐴𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶)))
 
Theoremlemul2i 7891 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 → (𝐴𝐵 ↔ (𝐶 · 𝐴) ≤ (𝐶 · 𝐵)))
 
Theoremltdiv23i 7892 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))
 
Theoremltdiv23ii 7893 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)
 
Theoremltmul1ii 7894 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))
 
Theoremltdiv1ii 7895 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))
 
Theoremltp1d 7896 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 < (𝐴 + 1))
 
Theoremlep1d 7897 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≤ (𝐴 + 1))
 
Theoremltm1d 7898 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 − 1) < 𝐴)
 
Theoremlem1d 7899 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 − 1) ≤ 𝐴)
 
Theoremrecgt0d 7900 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)       (𝜑 → 0 < (1 / 𝐴))
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