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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lttri4 10001 | Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴)) | ||
Theorem | letri3 10002 | Trichotomy law. (Contributed by NM, 14-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | ||
Theorem | leloe 10003 | 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by NM, 13-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | ||
Theorem | eqlelt 10004 | Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵))) | ||
Theorem | ltle 10005 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | ||
Theorem | leltne 10006 | 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by NM, 27-Jul-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) | ||
Theorem | lelttr 10007 | Transitive law. (Contributed by NM, 23-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | ltletr 10008 | Transitive law. (Contributed by NM, 25-Aug-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | ltleletr 10009 | Transitive law, weaker form of ltletr 10008. (Contributed by AV, 14-Oct-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | ||
Theorem | letr 10010 | Transitive law. (Contributed by NM, 12-Nov-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | ||
Theorem | ltnr 10011 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | ||
Theorem | leid 10012 | 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | ||
Theorem | ltne 10013 | 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | ||
Theorem | ltnsym 10014 | 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) | ||
Theorem | ltnsym2 10015 | 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (𝐴 < 𝐵 ∧ 𝐵 < 𝐴)) | ||
Theorem | letric 10016 | Trichotomy law. (Contributed by NM, 18-Aug-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | ||
Theorem | ltlen 10017 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴))) | ||
Theorem | eqle 10018 | Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | ||
Theorem | eqled 10019 | Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | ltadd2 10020 | Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))) | ||
Theorem | ne0gt0 10021 | A nonzero nonnegative number is positive. (Contributed by NM, 20-Nov-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≠ 0 ↔ 0 < 𝐴)) | ||
Theorem | lecasei 10022 | Ordering elimination by cases. (Contributed by NM, 6-Jul-2007.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝐴 ≤ 𝐵) → 𝜓) & ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | lelttric 10023 | Trichotomy law. (Contributed by NM, 4-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) | ||
Theorem | ltlecasei 10024 | Ordering elimination by cases. (Contributed by NM, 1-Jul-2007.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝜓) & ⊢ ((𝜑 ∧ 𝐵 ≤ 𝐴) → 𝜓) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | ltnri 10025 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ ¬ 𝐴 < 𝐴 | ||
Theorem | eqlei 10026 | Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐴 = 𝐵 → 𝐴 ≤ 𝐵) | ||
Theorem | eqlei2 10027 | Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐵 = 𝐴 → 𝐵 ≤ 𝐴) | ||
Theorem | gtneii 10028 | 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐵 ≠ 𝐴 | ||
Theorem | ltneii 10029 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐴 ≠ 𝐵 | ||
Theorem | lttri2i 10030 | Consequence of trichotomy. (Contributed by NM, 19-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) | ||
Theorem | lttri3i 10031 | Consequence of trichotomy. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | ||
Theorem | letri3i 10032 | Consequence of trichotomy. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)) | ||
Theorem | leloei 10033 | 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | ltleni 10034 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴)) | ||
Theorem | ltnsymi 10035 | 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴) | ||
Theorem | lenlti 10036 | 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴) | ||
Theorem | ltnlei 10037 | 'Less than' in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴) | ||
Theorem | ltlei 10038 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐴 ≤ 𝐵) | ||
Theorem | ltleii 10039 | 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐴 ≤ 𝐵 | ||
Theorem | ltnei 10040 | 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐵 ≠ 𝐴) | ||
Theorem | letrii 10041 | Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴) | ||
Theorem | lttri 10042 | 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶) | ||
Theorem | lelttri 10043 | 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶) | ||
Theorem | ltletri 10044 | 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶) | ||
Theorem | letri 10045 | 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶) | ||
Theorem | le2tri3i 10046 | Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶 ∧ 𝐶 = 𝐴)) | ||
Theorem | ltadd2i 10047 | Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by OpenAI, 25-Mar-2020.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)) | ||
Theorem | mulgt0i 10048 | The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵)) | ||
Theorem | mulgt0ii 10049 | The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 0 < 𝐴 & ⊢ 0 < 𝐵 ⇒ ⊢ 0 < (𝐴 · 𝐵) | ||
Theorem | ltnrd 10050 | 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ¬ 𝐴 < 𝐴) | ||
Theorem | gtned 10051 | 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≠ 𝐴) | ||
Theorem | ltned 10052 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | ne0gt0d 10053 | A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → 0 < 𝐴) | ||
Theorem | lttrid 10054 | Ordering on reals satisfies strict trichotomy. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | lttri2d 10055 | Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | lttri3d 10056 | Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | ||
Theorem | lttri4d 10057 | Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴)) | ||
Theorem | letri3d 10058 | Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | ||
Theorem | leloed 10059 | 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | ||
Theorem | eqleltd 10060 | Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵))) | ||
Theorem | ltlend 10061 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴))) | ||
Theorem | lenltd 10062 | 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | ||
Theorem | ltnled 10063 | 'Less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | ||
Theorem | ltled 10064 | 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | ltnsymd 10065 | 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | ||
Theorem | nltled 10066 | 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ¬ 𝐵 < 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | lensymd 10067 | 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | ||
Theorem | letrid 10068 | Trichotomy law for 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | ||
Theorem | leltned 10069 | 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) | ||
Theorem | leneltd 10070 | 'Less than or equal to' and 'not equals' implies 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 < 𝐵) | ||
Theorem | mulgt0d 10071 | The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 · 𝐵)) | ||
Theorem | ltadd2d 10072 | Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))) | ||
Theorem | letrd 10073 | Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) | ||
Theorem | lelttrd 10074 | Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
Theorem | ltadd2dd 10075 | Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (𝐶 + 𝐴) < (𝐶 + 𝐵)) | ||
Theorem | ltletrd 10076 | Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
Theorem | lttrd 10077 | Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
Theorem | lelttrdi 10078 | If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.) |
⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 < 𝐶)) | ||
Theorem | dedekind 10079* | The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 9893 with appropriate adjustments, states that, if 𝐴 completely preceeds 𝐵, then there is some number separating the two of them. (Contributed by Scott Fenton, 13-Jun-2013.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 < 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)) | ||
Theorem | dedekindle 10080* | The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.) |
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 ≤ 𝑦) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)) | ||
Theorem | mul12 10081 | Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶))) | ||
Theorem | mul32 10082 | Commutative/associative law. (Contributed by NM, 8-Oct-1999.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) | ||
Theorem | mul31 10083 | Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴)) | ||
Theorem | mul4 10084 | Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))) | ||
Theorem | muladd11 10085 | A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((1 + 𝐴) · (1 + 𝐵)) = ((1 + 𝐴) + (𝐵 + (𝐴 · 𝐵)))) | ||
Theorem | 1p1times 10086 | Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℂ → ((1 + 1) · 𝐴) = (𝐴 + 𝐴)) | ||
Theorem | peano2cn 10087 | A theorem for complex numbers analogous the second Peano postulate peano2nn 10909. (Contributed by NM, 17-Aug-2005.) |
⊢ (𝐴 ∈ ℂ → (𝐴 + 1) ∈ ℂ) | ||
Theorem | peano2re 10088 | A theorem for reals analogous the second Peano postulate peano2nn 10909. (Contributed by NM, 5-Jul-2005.) |
⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | ||
Theorem | readdcan 10089 | Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 + 𝐴) = (𝐶 + 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | 00id 10090 | 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.) |
⊢ (0 + 0) = 0 | ||
Theorem | mul02lem1 10091 | Lemma for mul02 10093. If any real does not produce 0 when multiplied by 0, then any complex is equal to double itself. (Contributed by Scott Fenton, 3-Jan-2013.) |
⊢ (((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) ∧ 𝐵 ∈ ℂ) → 𝐵 = (𝐵 + 𝐵)) | ||
Theorem | mul02lem2 10092 | Lemma for mul02 10093. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.) |
⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0) | ||
Theorem | mul02 10093 | Multiplication by 0. Theorem I.6 of [Apostol] p. 18. Based on ideas by Eric Schmidt. (Contributed by NM, 10-Aug-1999.) (Revised by Scott Fenton, 3-Jan-2013.) |
⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | ||
Theorem | mul01 10094 | Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.) |
⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | ||
Theorem | addid1 10095 | 0 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | ||
Theorem | cnegex 10096* | Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) | ||
Theorem | cnegex2 10097* | Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.) |
⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) | ||
Theorem | addid2 10098 | 0 is a left identity for addition. This used to be one of our complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) | ||
Theorem | addcan 10099 | Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | addcan2 10100 | Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) |
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