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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | renfdisj 10701 | The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (ℝ ∩ {+∞, -∞}) = ∅ | ||
Theorem | ltrelxr 10702 | "Less than" is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ < ⊆ (ℝ* × ℝ*) | ||
Theorem | ltrel 10703 | "Less than" is a relation. (Contributed by NM, 14-Oct-2005.) |
⊢ Rel < | ||
Theorem | lerelxr 10704 | "Less than or equal to" is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ ≤ ⊆ (ℝ* × ℝ*) | ||
Theorem | lerel 10705 | "Less than or equal to" is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ Rel ≤ | ||
Theorem | xrlenlt 10706 | "Less than or equal to" expressed in terms of "less than", for extended reals. (Contributed by NM, 14-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | ||
Theorem | xrlenltd 10707 | "Less than or equal to" expressed in terms of "less than", for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | ||
Theorem | xrltnle 10708 | "Less than" expressed in terms of "less than or equal to", for extended reals. (Contributed by NM, 6-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | ||
Theorem | xrnltled 10709 | "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 10710 | 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 10711 | 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 10712 | 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 10611 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | axlttrn 10713 | Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. This restates ax-pre-lttrn 10612 with ordering on the extended reals. New proofs should use lttr 10717 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 10714 | 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 10613 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 10715 | 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 10614 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 10716* | 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 10615 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 10717 | Alias for axlttrn 10713, for naming consistency with lttri 10766. New proofs should generally use this instead of ax-pre-lttrn 10612. (Contributed by NM, 10-Mar-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | mulgt0 10718 | 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 10719 | '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 10720 | '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 10721 | 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.) |
⊢ < Or ℝ | ||
Theorem | gtso 10722 | 'Greater than' is a strict ordering. (Contributed by JJ, 11-Oct-2018.) |
⊢ ◡ < Or ℝ | ||
Theorem | lttri2 10723 | Consequence of trichotomy. (Contributed by NM, 9-Oct-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | lttri3 10724 | Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | ||
Theorem | lttri4 10725 | 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 10726 | Trichotomy law. (Contributed by NM, 14-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | ||
Theorem | leloe 10727 | '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 10728 | 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 10729 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | ||
Theorem | leltne 10730 | '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 10731 | Transitive law. (Contributed by NM, 23-May-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | ltletr 10732 | Transitive law. (Contributed by NM, 25-Aug-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | ltleletr 10733 | Transitive law, weaker form of ltletr 10732. (Contributed by AV, 14-Oct-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | ||
Theorem | letr 10734 | Transitive law. (Contributed by NM, 12-Nov-1999.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | ||
Theorem | ltnr 10735 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴) | ||
Theorem | leid 10736 | 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ 𝐴) | ||
Theorem | ltne 10737 | '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 10738 | 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) | ||
Theorem | ltnsym2 10739 | '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 10740 | 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 10741 | '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 10742 | Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐴 ≤ 𝐵) | ||
Theorem | eqled 10743 | 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 10744 | 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 10745 | A nonzero nonnegative number is positive. (Contributed by NM, 20-Nov-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≠ 0 ↔ 0 < 𝐴)) | ||
Theorem | lecasei 10746 | 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 10747 | Trichotomy law. (Contributed by NM, 4-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴)) | ||
Theorem | ltlecasei 10748 | 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 10749 | 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ ¬ 𝐴 < 𝐴 | ||
Theorem | eqlei 10750 | 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 10751 | Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.) |
⊢ 𝐴 ∈ ℝ ⇒ ⊢ (𝐵 = 𝐴 → 𝐵 ≤ 𝐴) | ||
Theorem | gtneii 10752 | 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐵 ≠ 𝐴 | ||
Theorem | ltneii 10753 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐴 < 𝐵 ⇒ ⊢ 𝐴 ≠ 𝐵 | ||
Theorem | lttri2i 10754 | Consequence of trichotomy. (Contributed by NM, 19-Jan-1997.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)) | ||
Theorem | lttri3i 10755 | Consequence of trichotomy. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)) | ||
Theorem | letri3i 10756 | Consequence of trichotomy. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)) | ||
Theorem | leloei 10757 | '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 10758 | '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 10759 | 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴) | ||
Theorem | lenlti 10760 | '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 10761 | '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 10762 | 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐴 ≤ 𝐵) | ||
Theorem | ltleii 10763 | '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 10764 | 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 < 𝐵 → 𝐵 ≠ 𝐴) | ||
Theorem | letrii 10765 | Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴) | ||
Theorem | lttri 10766 | '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 10767 | '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 10768 | '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 10769 | '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 10770 | 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 10771 | 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 10772 | 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 10773 | 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 10774 | 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ¬ 𝐴 < 𝐴) | ||
Theorem | gtned 10775 | 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≠ 𝐴) | ||
Theorem | ltned 10776 | 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) | ||
Theorem | ne0gt0d 10777 | 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 10778 | 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 10779 | 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 10780 | 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 10781 | 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 10782 | 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 10783 | '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 10784 | 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 10785 | '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 10786 | '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 10787 | '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 10788 | '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 10789 | '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 10790 | '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 10791 | '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 10792 | 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 10793 | '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 10794 | '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 10795 | 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 10796 | 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 10797 | 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 10798 | 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 10799 | 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 10800 | 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) |
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