HomeTheorem List (p. 132 of 504)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | xrletrid 13101 | 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 13102 | 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 13103 | 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 13104 | 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 13105 | 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 13106 | 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 13107 | 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 13108 | 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 13109 | 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | ||
| Theorem | xrgtned 13110 | 'Greater than' implies not equal. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≠ 𝐴) | ||
| Theorem | nltpnft 13111 | 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 13112 | 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 13113 | 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 13114 | 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 13115 | 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 13116 | 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 13117 | 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 13118 | 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 13119 | A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴) | ||
| Theorem | ge0nemnf 13120 | A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞) | ||
| Theorem | xrrege0 13121 | 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 13122 | 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 13123 | 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 13124 | 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 13125 | 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 13126 | 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 13127 | 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 13128 | 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 13129 | 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 13130 | 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 13131 | 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 13132 | A number is less than or equal to the maximum of it and another. See also max1ALT 13133. (Contributed by NM, 3-Apr-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
| Theorem | max1ALT 13133 | A number is less than or equal to the maximum of it and another. This version of max1 13132 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 13132 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 13134 | 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 13135 | 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 13136 | 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 13137 | 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 13138 | 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 13139 | 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 13140 | 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 13141 | 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 13142 | 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 13143 | 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 13144 | 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 13145* | 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 13146* | 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 13147* | 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 13148* | 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 13149* | Lemma for qextlt 13150 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 13150* | 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 13151* | 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 13152* | 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 13153* | 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 13154 | 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 13155 | A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ -𝑒𝐴 ∈ V | ||
| Theorem | xnegpnf 13156 | Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) |
| ⊢ -𝑒+∞ = -∞ | ||
| Theorem | xnegmnf 13157 | Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.) |
| ⊢ -𝑒-∞ = +∞ | ||
| Theorem | rexneg 13158 | 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 13159 | The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ -𝑒0 = 0 | ||
| Theorem | xnegcl 13160 | Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) | ||
| Theorem | xnegneg 13161 | Extended real version of negneg 11440. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) | ||
| Theorem | xneg11 13162 | Extended real version of neg11 11441. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵 ↔ 𝐴 = 𝐵)) | ||
| Theorem | xltnegi 13163 | Forward direction of xltneg 13164. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴) | ||
| Theorem | xltneg 13164 | Extended real version of ltneg 11646. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴)) | ||
| Theorem | xleneg 13165 | Extended real version of leneg 11649. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴)) | ||
| Theorem | xlt0neg1 13166 | Extended real version of lt0neg1 11652. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴)) | ||
| Theorem | xlt0neg2 13167 | Extended real version of lt0neg2 11653. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0)) | ||
| Theorem | xle0neg1 13168 | Extended real version of le0neg1 11654. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴)) | ||
| Theorem | xle0neg2 13169 | Extended real version of le0neg2 11655. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0)) | ||
| Theorem | xaddval 13170 | Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))) | ||
| Theorem | xaddf 13171 | The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.) |
| ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ* | ||
| Theorem | xmulval 13172 | Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))))) | ||
| Theorem | xaddpnf1 13173 | Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞) | ||
| Theorem | xaddpnf2 13174 | Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞) | ||
| Theorem | xaddmnf1 13175 | Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞) | ||
| Theorem | xaddmnf2 13176 | Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞) | ||
| Theorem | pnfaddmnf 13177 | Addition of positive and negative infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (+∞ +𝑒 -∞) = 0 | ||
| Theorem | mnfaddpnf 13178 | Addition of negative and positive infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (-∞ +𝑒 +∞) = 0 | ||
| Theorem | rexadd 13179 | The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) | ||
| Theorem | rexsub 13180 | Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴 − 𝐵)) | ||
| Theorem | rexaddd 13181 | The extended real addition operation when both arguments are real. Deduction version of rexadd 13179. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) | ||
| Theorem | xnn0xaddcl 13182 | The extended nonnegative integers are closed under extended addition. (Contributed by AV, 10-Dec-2020.) |
| ⊢ ((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → (𝐴 +𝑒 𝐵) ∈ ℕ0*) | ||
| Theorem | xaddnemnf 13183 | Closure of extended real addition in the subset ℝ* / {-∞}. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞)) → (𝐴 +𝑒 𝐵) ≠ -∞) | ||
| Theorem | xaddnepnf 13184 | Closure of extended real addition in the subset ℝ* / {+∞}. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞)) → (𝐴 +𝑒 𝐵) ≠ +∞) | ||
| Theorem | xnegid 13185 | Extended real version of negid 11437. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) | ||
| Theorem | xaddcl 13186 | The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) ∈ ℝ*) | ||
| Theorem | xaddcom 13187 | The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = (𝐵 +𝑒 𝐴)) | ||
| Theorem | xaddrid 13188 | Extended real version of addrid 11322. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) | ||
| Theorem | xaddlid 13189 | Extended real version of addlid 11325. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴) | ||
| Theorem | xaddridd 13190 | 0 is a right identity for extended real addition. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 0) = 𝐴) | ||
| Theorem | xnn0lem1lt 13191 | Extended nonnegative integer ordering relation. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
| Theorem | xnn0lenn0nn0 13192 | An extended nonnegative integer which is less than or equal to a nonnegative integer is a nonnegative integer. (Contributed by AV, 24-Nov-2021.) |
| ⊢ ((𝑀 ∈ ℕ0* ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℕ0) | ||
| Theorem | xnn0le2is012 13193 | An extended nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 24-Nov-2021.) |
| ⊢ ((𝑁 ∈ ℕ0* ∧ 𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2)) | ||
| Theorem | xnn0xadd0 13194 | The sum of two extended nonnegative integers is 0 iff each of the two extended nonnegative integers is 0. (Contributed by AV, 14-Dec-2020.) |
| ⊢ ((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) → ((𝐴 +𝑒 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0))) | ||
| Theorem | xnegdi 13195 | Extended real version of negdi 11447. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵)) | ||
| Theorem | xaddass 13196 | Associativity of extended real addition. The correct condition here is "it is not the case that both +∞ and -∞ appear as one of 𝐴, 𝐵, 𝐶, i.e. ¬ {+∞, -∞} ⊆ {𝐴, 𝐵, 𝐶}", but this condition is difficult to work with, so we break the theorem into two parts: this one, where -∞ is not present in 𝐴, 𝐵, 𝐶, and xaddass2 13197, where +∞ is not present. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) ∧ (𝐶 ∈ ℝ* ∧ 𝐶 ≠ -∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶))) | ||
| Theorem | xaddass2 13197 | Associativity of extended real addition. See xaddass 13196 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞) ∧ (𝐶 ∈ ℝ* ∧ 𝐶 ≠ +∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶))) | ||
| Theorem | xpncan 13198 | Extended real version of pncan 11395. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴) | ||
| Theorem | xnpcan 13199 | Extended real version of npcan 11398. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴) | ||
| Theorem | xleadd1a 13200 | Extended real version of leadd1 11614; note that the converse implication is not true, unlike the real version (for example 0 < 1 but (1 +𝑒 +∞) ≤ (0 +𝑒 +∞)). (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶)) | ||
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