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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elxr 12501 | Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.) |
⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | ||
Theorem | xrnemnf 12502 | An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) | ||
Theorem | xrnepnf 12503 | An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) | ||
Theorem | xrltnr 12504 | The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.) |
⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴) | ||
Theorem | ltpnf 12505 | Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | ||
Theorem | ltpnfd 12506 | Any (finite) real is less than plus infinity. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 < +∞) | ||
Theorem | 0ltpnf 12507 | Zero is less than plus infinity. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 0 < +∞ | ||
Theorem | mnflt 12508 | Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.) |
⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | ||
Theorem | mnfltd 12509 | Minus infinity is less than any (finite) real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → -∞ < 𝐴) | ||
Theorem | mnflt0 12510 | Minus infinity is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -∞ < 0 | ||
Theorem | mnfltpnf 12511 | Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.) |
⊢ -∞ < +∞ | ||
Theorem | mnfltxr 12512 | Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.) |
⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → -∞ < 𝐴) | ||
Theorem | pnfnlt 12513 | No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.) |
⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) | ||
Theorem | nltmnf 12514 | No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.) |
⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) | ||
Theorem | pnfge 12515 | Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.) |
⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | ||
Theorem | xnn0n0n1ge2b 12516 | An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by AV, 5-Apr-2021.) |
⊢ (𝑁 ∈ ℕ0* → ((𝑁 ≠ 0 ∧ 𝑁 ≠ 1) ↔ 2 ≤ 𝑁)) | ||
Theorem | 0lepnf 12517 | 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 0 ≤ +∞ | ||
Theorem | xnn0ge0 12518 | An extended nonnegative integer is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 10-Dec-2020.) |
⊢ (𝑁 ∈ ℕ0* → 0 ≤ 𝑁) | ||
Theorem | mnfle 12519 | Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.) |
⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | ||
Theorem | xrltnsym 12520 | Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) | ||
Theorem | xrltnsym2 12521 | 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ (𝐴 < 𝐵 ∧ 𝐵 < 𝐴)) | ||
Theorem | xrlttri 12522 | Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 10600 or axlttri 10701. (Contributed by NM, 14-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | xrlttr 12523 | Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | xrltso 12524 | 'Less than' is a strict ordering on the extended reals. (Contributed by NM, 15-Oct-2005.) |
⊢ < Or ℝ* | ||
Theorem | xrlttri2 12525 | Trichotomy law for 'less than' for extended reals. (Contributed by NM, 10-Dec-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | xrlttri3 12526 | Trichotomy law for 'less than' for extended reals. (Contributed by NM, 9-Feb-2006.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))) | ||
Theorem | xrleloe 12527 | 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | ||
Theorem | xrleltne 12528 | 'Less than or equal to' implies 'less than' is not 'equals', for extended reals. (Contributed by NM, 9-Feb-2006.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) | ||
Theorem | xrltlen 12529 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴))) | ||
Theorem | dfle2 12530 | Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.) |
⊢ ≤ = ( < ∪ ( I ↾ ℝ*)) | ||
Theorem | dflt2 12531 | Alternative definition of 'less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.) |
⊢ < = ( ≤ ∖ I ) | ||
Theorem | xrltle 12532 | 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | ||
Theorem | xrltled 12533 | 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 12532. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | xrleid 12534 | 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.) |
⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) | ||
Theorem | xrleidd 12535 | 'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 12534. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐴) | ||
Theorem | xrletri 12536 | Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | ||
Theorem | xrletri3 12537 | Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | ||
Theorem | xrletrid 12538 | Trichotomy law for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | xrlelttr 12539 | Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | xrltletr 12540 | Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 < 𝐶)) | ||
Theorem | xrletr 12541 | Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐶)) | ||
Theorem | xrlttrd 12542 | Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
Theorem | xrlelttrd 12543 | Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
Theorem | xrltletrd 12544 | Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
Theorem | xrletrd 12545 | Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) | ||
Theorem | xrltne 12546 | 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | ||
Theorem | nltpnft 12547 | An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.) |
⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞)) | ||
Theorem | xgepnf 12548 | An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
⊢ (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞)) | ||
Theorem | ngtmnft 12549 | An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.) |
⊢ (𝐴 ∈ ℝ* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴)) | ||
Theorem | xlemnf 12550 | An extended real which is less than minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) |
⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ -∞ ↔ 𝐴 = -∞)) | ||
Theorem | xrrebnd 12551 | An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.) |
⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) | ||
Theorem | xrre 12552 | A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.) |
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ) | ||
Theorem | xrre2 12553 | An extended real between two others is real. (Contributed by NM, 6-Feb-2007.) |
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ) | ||
Theorem | xrre3 12554 | A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.) |
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 ∈ ℝ) | ||
Theorem | ge0gtmnf 12555 | A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴) | ||
Theorem | ge0nemnf 12556 | A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞) | ||
Theorem | xrrege0 12557 | A nonnegative extended real that is less than a real bound is real. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ) | ||
Theorem | xrmax1 12558 | An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
Theorem | xrmax2 12559 | An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
Theorem | xrmin1 12560 | The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) | ||
Theorem | xrmin2 12561 | The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) | ||
Theorem | xrmaxeq 12562 | The maximum of two extended reals is equal to the first if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐵, 𝐴) = 𝐴) | ||
Theorem | xrmineq 12563 | The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) = 𝐵) | ||
Theorem | xrmaxlt 12564 | Two ways of saying the maximum of two extended reals is less than a third. (Contributed by NM, 7-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶))) | ||
Theorem | xrltmin 12565 | Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶))) | ||
Theorem | xrmaxle 12566 | Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶))) | ||
Theorem | xrlemin 12567 | Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) | ||
Theorem | max1 12568 | A number is less than or equal to the maximum of it and another. See also max1ALT 12569. (Contributed by NM, 3-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
Theorem | max1ALT 12569 | A number is less than or equal to the maximum of it and another. This version of max1 12568 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 12568 is preferred. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 3-Apr-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
Theorem | max2 12570 | A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
Theorem | 2resupmax 12571 | The supremum of two real numbers is the maximum of these two numbers. (Contributed by AV, 8-Jun-2021.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ, < ) = if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
Theorem | min1 12572 | The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) | ||
Theorem | min2 12573 | The minimum of two numbers is less than or equal to the second. (Contributed by NM, 3-Aug-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) | ||
Theorem | maxle 12574 | Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by NM, 29-Sep-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶))) | ||
Theorem | lemin 12575 | Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶))) | ||
Theorem | maxlt 12576 | Two ways of saying the maximum of two numbers is less than a third. (Contributed by NM, 3-Aug-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (if(𝐴 ≤ 𝐵, 𝐵, 𝐴) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶))) | ||
Theorem | ltmin 12577 | Two ways of saying a number is less than the minimum of two others. (Contributed by NM, 1-Sep-2006.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < if(𝐵 ≤ 𝐶, 𝐵, 𝐶) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶))) | ||
Theorem | lemaxle 12578 | A real number which is less than or equal to a second real number is less than or equal to the maximum/supremum of the second real number and a third real number. (Contributed by AV, 8-Jun-2021.) |
⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if(𝐶 ≤ 𝐵, 𝐵, 𝐶)) | ||
Theorem | max0sub 12579 | Decompose a real number into positive and negative parts. (Contributed by Mario Carneiro, 6-Aug-2014.) |
⊢ (𝐴 ∈ ℝ → (if(0 ≤ 𝐴, 𝐴, 0) − if(0 ≤ -𝐴, -𝐴, 0)) = 𝐴) | ||
Theorem | ifle 12580 | An if statement transforms an implication into an inequality of terms. (Contributed by Mario Carneiro, 31-Aug-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≤ 𝐴) ∧ (𝜑 → 𝜓)) → if(𝜑, 𝐴, 𝐵) ≤ if(𝜓, 𝐴, 𝐵)) | ||
Theorem | z2ge 12581* | There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ∃𝑘 ∈ ℤ (𝑀 ≤ 𝑘 ∧ 𝑁 ≤ 𝑘)) | ||
Theorem | qbtwnre 12582* | The rational numbers are dense in ℝ: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) | ||
Theorem | qbtwnxr 12583* | The rational numbers are dense in ℝ*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) | ||
Theorem | qsqueeze 12584* | If a nonnegative real is less than any positive rational, it is zero. (Contributed by NM, 6-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ ∀𝑥 ∈ ℚ (0 < 𝑥 → 𝐴 < 𝑥)) → 𝐴 = 0) | ||
Theorem | qextltlem 12585* | Lemma for qextlt 12586 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (¬ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵) ∧ ¬ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵)))) | ||
Theorem | qextlt 12586* | An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℚ (𝑥 < 𝐴 ↔ 𝑥 < 𝐵))) | ||
Theorem | qextle 12587* | An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℚ (𝑥 ≤ 𝐴 ↔ 𝑥 ≤ 𝐵))) | ||
Theorem | xralrple 12588* | Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑥))) | ||
Theorem | alrple 12589* | Show that 𝐴 is less than 𝐵 by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ+ 𝐴 ≤ (𝐵 + 𝑥))) | ||
Theorem | xnegeq 12590 | Equality of two extended numbers with -𝑒 in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) | ||
Theorem | xnegex 12591 | A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ -𝑒𝐴 ∈ V | ||
Theorem | xnegpnf 12592 | Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) |
⊢ -𝑒+∞ = -∞ | ||
Theorem | xnegmnf 12593 | Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.) |
⊢ -𝑒-∞ = +∞ | ||
Theorem | rexneg 12594 | Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | ||
Theorem | xneg0 12595 | The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ -𝑒0 = 0 | ||
Theorem | xnegcl 12596 | Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) | ||
Theorem | xnegneg 12597 | Extended real version of negneg 10925. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) | ||
Theorem | xneg11 12598 | Extended real version of neg11 10926. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | xltnegi 12599 | Forward direction of xltneg 12600. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴) | ||
Theorem | xltneg 12600 | Extended real version of ltneg 11129. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴)) |
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