![]() |
Metamath
Proof Explorer Theorem List (p. 110 of 429) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-27903) |
![]() (27904-29428) |
![]() (29429-42879) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ltm1 10901 | A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.) |
⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴) | ||
Theorem | lem1 10902 | 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 10903 | 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 10904 | 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 10905 | 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 10906 | 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 10907 | 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 | prodge0 10908 | 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) | ||
Theorem | prodge02 10909 | 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) | ||
Theorem | ltmul1a 10910 | Lemma for ltmul1 10911. 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 10911 | 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 10912 | 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 10913 | 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 10914 | 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 10915 | 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 10916 | 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 10917 | 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 10918 | 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 10919 | 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 10920 | 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 10921 | 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 10922 | 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 10923 | 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 10924 | 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 10925 | 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 10926 | 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 10927 | 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 10928 | 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 10929 | 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 10930 | 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 10931 | 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 10932 | 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 10933 | 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 10934 | '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 10935 | '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 10936 | '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 10937 | '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 10938 | '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 10939 | '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 10940 | '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 10941 | '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 10942 | '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 10943 | 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 10944 | 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 10945 | Lemma for lt2msq 10946. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐴 · 𝐴) < (𝐵 · 𝐵)) | ||
Theorem | lt2msq 10946 | 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 10947 | 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 10948 | 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 10949 | 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 10950 | 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 10951 | 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 10952 | 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 10953 | 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 10954 | 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 10955 | 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 10956 | 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 10957 | 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 10958 | 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 10959 | 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 10960 | 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 10961 | 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 10962 | 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 10963 | 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 10964* | 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 10965 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ 𝐴 < (𝐴 + 1) | ||
Theorem | recgt0i 10966 | 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 10967 | 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 10968 | 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 | prodge0i 10969 | 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) | ||
Theorem | divgt0i 10970 | 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 10971 | 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 10972 | 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 10973 | 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 10974 | 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 10975 | 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 10976 | 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 10977 | 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 10978 | 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 10979 | 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 10980 | 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 10981 | 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 10982 | '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 10983 | 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 10984 | 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 10985 | 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 10986 | 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 10987 | 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 10988 | 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 10989 | 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 10990 | 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 10991 | 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 10992 | A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 < (𝐴 + 1)) | ||
Theorem | lep1d 10993 | 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 10994 | A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 − 1) < 𝐴) | ||
Theorem | lem1d 10995 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 − 1) ≤ 𝐴) | ||
Theorem | recgt0d 10996 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) ⇒ ⊢ (𝜑 → 0 < (1 / 𝐴)) | ||
Theorem | divgt0d 10997 | The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 / 𝐵)) | ||
Theorem | mulgt1d 10998 | The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 1 < 𝐴) & ⊢ (𝜑 → 1 < 𝐵) ⇒ ⊢ (𝜑 → 1 < (𝐴 · 𝐵)) | ||
Theorem | lemulge11d 10999 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 1 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ (𝐴 · 𝐵)) | ||
Theorem | lemulge12d 11000 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 1 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ (𝐵 · 𝐴)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |