HomeTheorem List (p. 114 of 497)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | xrnltled 11301 | "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 | ssxr 11302 | The three (non-exclusive) possibilities implied by a subset of extended reals. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| ⊢ (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴)) | ||
| Theorem | ltxrlt 11303 | The standard less-than <ℝ and the extended real less-than < are identical when restricted to the non-extended reals ℝ. (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵)) | ||
| Theorem | axlttri 11304 | Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttri 11201 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) | ||
| Theorem | axlttrn 11305 | Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. This restates ax-pre-lttrn 11202 with ordering on the extended reals. New proofs should use lttr 11309 instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
| Theorem | axltadd 11306 | Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-ltadd 11203 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐶 + 𝐴) < (𝐶 + 𝐵))) | ||
| Theorem | axmulgt0 11307 | The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-mulgt0 11204 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 · 𝐵))) | ||
| Theorem | axsup 11308* | A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-sup 11205 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
| ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | ||
| Theorem | lttr 11309 | Alias for axlttrn 11305, for naming consistency with lttri 11359. New proofs should generally use this instead of ax-pre-lttrn 11202. (Contributed by NM, 10-Mar-2008.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
| Theorem | mulgt0 11310 | The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.) |
| ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 < (𝐴 · 𝐵)) | ||
| Theorem | lenlt 11311 | 'Less than or equal to' expressed in terms of 'less than'. (Contributed by NM, 13-May-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | ||
| Theorem | ltnle 11312 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | ||
| Theorem | ltso 11313 | 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) |
| ⊢ < Or ℝ | ||
| Theorem | gtso 11314 | 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) |
| ⊢ ◡ < Or ℝ | ||
| Theorem | lttri2 11315 | Consequence of trichotomy. (Contributed by NM, 9-Oct-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | ||
| Theorem | lttri3 11316 | Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | ||
| Theorem | lttri4 11317 | 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 11318 | Trichotomy law. (Contributed by NM, 14-May-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | ||
| Theorem | leloe 11319 | '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 11320 | 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 11321 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | ||
| Theorem | leltne 11322 | '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 11323 | Transitive law. (Contributed by NM, 23-May-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
| Theorem | leltletr 11324 | Transitive law, weaker form of lelttr 11323. (Contributed by AV, 14-Oct-2018.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 ≤ 𝐶)) | ||
| Theorem | ltletr 11325 | Transitive law. (Contributed by NM, 25-Aug-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) | ||
| Theorem | ltleletr 11326 | Transitive law, weaker form of ltletr 11325. (Contributed by AV, 14-Oct-2018.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | ||
| Theorem | letr 11327 | Transitive law. (Contributed by NM, 12-Nov-1999.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | ||
| Theorem | ltnr 11328 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
| ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | ||
| Theorem | leid 11329 | 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
| ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | ||
| Theorem | ltne 11330 | '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 11331 | 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) | ||
| Theorem | ltnsym2 11332 | '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 11333 | 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 11334 | '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 11335 | Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | ||
| Theorem | eqled 11336 | 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 11337 | 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 11338 | A nonzero nonnegative number is positive. (Contributed by NM, 20-Nov-2007.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≠ 0 ↔ 0 < 𝐴)) | ||
| Theorem | lecasei 11339 | 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 11340 | Trichotomy law. (Contributed by NM, 4-Apr-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) | ||
| Theorem | ltlecasei 11341 | 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 11342 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ ¬ 𝐴 < 𝐴 | ||
| Theorem | eqlei 11343 | 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 11344 | Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.) |
| ⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐵 = 𝐴 → 𝐵 ≤ 𝐴) | ||
| Theorem | gtneii 11345 | 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐵 ≠ 𝐴 | ||
| Theorem | ltneii 11346 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐴 ≠ 𝐵 | ||
| Theorem | lttri2i 11347 | Consequence of trichotomy. (Contributed by NM, 19-Jan-1997.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) | ||
| Theorem | lttri3i 11348 | Consequence of trichotomy. (Contributed by NM, 14-May-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | ||
| Theorem | letri3i 11349 | Consequence of trichotomy. (Contributed by NM, 14-May-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)) | ||
| Theorem | leloei 11350 | '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 11351 | '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 11352 | 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴) | ||
| Theorem | lenlti 11353 | '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 11354 | '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 11355 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐴 ≤ 𝐵) | ||
| Theorem | ltleii 11356 | '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 11357 | 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐵 ≠ 𝐴) | ||
| Theorem | letrii 11358 | Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴) | ||
| Theorem | lttri 11359 | '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 11360 | '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 11361 | '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 11362 | '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 11363 | 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 11364 | 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 11365 | 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 11366 | 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 11367 | 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ¬ 𝐴 < 𝐴) | ||
| Theorem | gtned 11368 | 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≠ 𝐴) | ||
| Theorem | ltned 11369 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
| Theorem | ne0gt0d 11370 | 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 11371 | 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 11372 | 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 11373 | 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 11374 | 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 11375 | 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 11376 | '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 11377 | 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 11378 | '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 11379 | '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 11380 | '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 11381 | '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 11382 | '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 11383 | '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 11384 | '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 11385 | 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 11386 | '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 11387 | '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 11388 | 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 11389 | 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 11390 | 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 11391 | 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 11392 | 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 11393 | 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 11394 | 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 11395 | 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 11396* | The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 11205 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 11397* | 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 11398 | 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 11399 | Commutative/associative law. (Contributed by NM, 8-Oct-1999.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐴 · 𝐶) · 𝐵)) | ||
| Theorem | mul31 11400 | Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.) |
| ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = ((𝐶 · 𝐵) · 𝐴)) | ||
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