Home Intuitionistic Logic ExplorerTheorem List (p. 81 of 110) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltmul2 8001 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 8002 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 8003 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 8004 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 8005 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 8006 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 8007 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 8008 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 8009 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵𝐴 < (𝐴 · 𝐵)))

Theoremltmulgt12 8010 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵𝐴 < (𝐵 · 𝐴)))

Theoremlemulge11 8011 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 8012 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 8013 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 8014 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 8015 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 8016 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 8017 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 8018 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 8019 '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 8020 '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 8021 '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 8022 '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 8023 '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 8024 '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 8025 '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 8026 '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 8027 '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 8028 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 8029 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 8030 Lemma for lt2msq 8031. (Contributed by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴 · 𝐴) < (𝐵 · 𝐵))

Theoremlt2msq 8031 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 8032 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 8033 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 8034 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 8035 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 8036 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 8037 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 8038 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 8039 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 8040 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 8041 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 8042 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 8043 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 8044 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 8045 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 8046 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 8047 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 8048 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 8049* 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 8050 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)
𝐴 ∈ ℝ       𝐴 < (𝐴 + 1)

Theoremrecgt0i 8051 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 8052 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 8053 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 8054 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 8055 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 8056 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 8057 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 8058 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 8059 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 8060 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 8061 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 8062 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 8063 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 8064 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 8065 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 8066 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 8067 '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 8068 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 8069 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 8070 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 8071 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 8072 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 8073 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 8074 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 8075 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 < (𝐴 + 1))

Theoremlep1d 8076 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴 ≤ (𝐴 + 1))

Theoremltm1d 8077 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 − 1) < 𝐴)

Theoremlem1d 8078 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 8079 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))

Theoremdivgt0d 8080 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 < 𝐵)       (𝜑 → 0 < (𝐴 / 𝐵))

Theoremmulgt1d 8081 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)    &   (𝜑 → 1 < 𝐵)       (𝜑 → 1 < (𝐴 · 𝐵))

Theoremlemulge11d 8082 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 1 ≤ 𝐵)       (𝜑𝐴 ≤ (𝐴 · 𝐵))

Theoremlemulge12d 8083 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 1 ≤ 𝐵)       (𝜑𝐴 ≤ (𝐵 · 𝐴))

Theoremlemul1ad 8084 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐶)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))

Theoremlemul2ad 8085 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐶)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))

Theoremltmul12ad 8086 Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → 0 ≤ 𝐶)    &   (𝜑𝐶 < 𝐷)       (𝜑 → (𝐴 · 𝐶) < (𝐵 · 𝐷))

Theoremlemul12ad 8087 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐶)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷))

Theoremlemul12bd 8088 Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐷)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴 · 𝐶) ≤ (𝐵 · 𝐷))

Theoremmulle0r 8089 Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 0 ∧ 0 ≤ 𝐵)) → (𝐴 · 𝐵) ≤ 0)

3.3.10  Suprema

Theoremlbreu 8090* If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → ∃!𝑥𝑆𝑦𝑆 𝑥𝑦)

Theoremlbcl 8091* If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → (𝑥𝑆𝑦𝑆 𝑥𝑦) ∈ 𝑆)

Theoremlble 8092* If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦𝐴𝑆) → (𝑥𝑆𝑦𝑆 𝑥𝑦) ≤ 𝐴)

Theoremlbinf 8093* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → inf(𝑆, ℝ, < ) = (𝑥𝑆𝑦𝑆 𝑥𝑦))

Theoremlbinfcl 8094* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦) → inf(𝑆, ℝ, < ) ∈ 𝑆)

Theoremlbinfle 8095* If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
((𝑆 ⊆ ℝ ∧ ∃𝑥𝑆𝑦𝑆 𝑥𝑦𝐴𝑆) → inf(𝑆, ℝ, < ) ≤ 𝐴)

Theoremsuprubex 8096* A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵𝐴)       (𝜑𝐵 ≤ sup(𝐴, ℝ, < ))

Theoremsuprlubex 8097* The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐵 < sup(𝐴, ℝ, < ) ↔ ∃𝑧𝐴 𝐵 < 𝑧))

Theoremsuprnubex 8098* An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (¬ 𝐵 < sup(𝐴, ℝ, < ) ↔ ∀𝑧𝐴 ¬ 𝐵 < 𝑧))

Theoremsuprleubex 8099* The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
(𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))    &   (𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (sup(𝐴, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧𝐴 𝑧𝐵))

Theoremnegiso 8100 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
𝐹 = (𝑥 ∈ ℝ ↦ -𝑥)       (𝐹 Isom < , < (ℝ, ℝ) ∧ 𝐹 = 𝐹)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-10953
 Copyright terms: Public domain < Previous  Next >