HomeTheorem List (p. 132 of 503)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | xrltletrd 13101 | 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 13102 | 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 13103 | 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | ||
| Theorem | xrgtned 13104 | '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 13105 | 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 13106 | 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 13107 | 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 13108 | 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 13109 | 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 13110 | 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 13111 | 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 13112 | 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 13113 | A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴) | ||
| Theorem | ge0nemnf 13114 | A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞) | ||
| Theorem | xrrege0 13115 | 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 13116 | 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 13117 | 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 13118 | 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 13119 | 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 13120 | 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 13121 | 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 13122 | 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 13123 | 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 13124 | 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 13125 | 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 13126 | A number is less than or equal to the maximum of it and another. See also max1ALT 13127. (Contributed by NM, 3-Apr-2005.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
| Theorem | max1ALT 13127 | A number is less than or equal to the maximum of it and another. This version of max1 13126 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 13126 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 13128 | 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 13129 | 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 13130 | 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 13131 | 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 13132 | 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 13133 | 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 13134 | 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 13135 | 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 13136 | 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 13137 | 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 13138 | 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 13139* | 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 13140* | 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 13141* | 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 13142* | 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 13143* | Lemma for qextlt 13144 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 13144* | 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 13145* | 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 13146* | 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 13147* | 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 13148 | 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 13149 | A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ -𝑒𝐴 ∈ V | ||
| Theorem | xnegpnf 13150 | Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) |
| ⊢ -𝑒+∞ = -∞ | ||
| Theorem | xnegmnf 13151 | Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.) |
| ⊢ -𝑒-∞ = +∞ | ||
| Theorem | rexneg 13152 | 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 13153 | The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ -𝑒0 = 0 | ||
| Theorem | xnegcl 13154 | Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) | ||
| Theorem | xnegneg 13155 | Extended real version of negneg 11433. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) | ||
| Theorem | xneg11 13156 | Extended real version of neg11 11434. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵 ↔ 𝐴 = 𝐵)) | ||
| Theorem | xltnegi 13157 | Forward direction of xltneg 13158. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴) | ||
| Theorem | xltneg 13158 | Extended real version of ltneg 11639. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴)) | ||
| Theorem | xleneg 13159 | Extended real version of leneg 11642. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ -𝑒𝐴)) | ||
| Theorem | xlt0neg1 13160 | Extended real version of lt0neg1 11645. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴)) | ||
| Theorem | xlt0neg2 13161 | Extended real version of lt0neg2 11646. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0)) | ||
| Theorem | xle0neg1 13162 | Extended real version of le0neg1 11647. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴)) | ||
| Theorem | xle0neg2 13163 | Extended real version of le0neg2 11648. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (0 ≤ 𝐴 ↔ -𝑒𝐴 ≤ 0)) | ||
| Theorem | xaddval 13164 | 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 13165 | The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.) |
| ⊢ +𝑒 :(ℝ* × ℝ*)⟶ℝ* | ||
| Theorem | xmulval 13166 | 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 13167 | Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞) | ||
| Theorem | xaddpnf2 13168 | Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (+∞ +𝑒 𝐴) = +∞) | ||
| Theorem | xaddmnf1 13169 | Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞) | ||
| Theorem | xaddmnf2 13170 | Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞) | ||
| Theorem | pnfaddmnf 13171 | 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 13172 | 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 13173 | 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 13174 | 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 13175 | The extended real addition operation when both arguments are real. Deduction version of rexadd 13173. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) | ||
| Theorem | xnn0xaddcl 13176 | 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 13177 | 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 13178 | 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 13179 | Extended real version of negid 11430. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 -𝑒𝐴) = 0) | ||
| Theorem | xaddcl 13180 | 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 13181 | 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 13182 | Extended real version of addrid 11315. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) | ||
| Theorem | xaddlid 13183 | Extended real version of addlid 11318. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (𝐴 ∈ ℝ* → (0 +𝑒 𝐴) = 𝐴) | ||
| Theorem | xaddridd 13184 | 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 13185 | Extended nonnegative integer ordering relation. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | ||
| Theorem | xnn0lenn0nn0 13186 | 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 13187 | 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 13188 | 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 13189 | Extended real version of negdi 11440. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 𝐵) = (-𝑒𝐴 +𝑒 -𝑒𝐵)) | ||
| Theorem | xaddass 13190 | 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 13191, 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 13191 | Associativity of extended real addition. See xaddass 13190 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 13192 | Extended real version of pncan 11388. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 𝐵) +𝑒 -𝑒𝐵) = 𝐴) | ||
| Theorem | xnpcan 13193 | Extended real version of npcan 11391. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴) | ||
| Theorem | xleadd1a 13194 | Extended real version of leadd1 11607; 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)) | ||
| Theorem | xleadd2a 13195 | Commuted form of xleadd1a 13194. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐶 +𝑒 𝐴) ≤ (𝐶 +𝑒 𝐵)) | ||
| Theorem | xleadd1 13196 | Weakened version of xleadd1a 13194 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶))) | ||
| Theorem | xltadd1 13197 | Extended real version of ltadd1 11606. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +𝑒 𝐶) < (𝐵 +𝑒 𝐶))) | ||
| Theorem | xltadd2 13198 | Extended real version of ltadd2 11239. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 +𝑒 𝐴) < (𝐶 +𝑒 𝐵))) | ||
| Theorem | xaddge0 13199 | The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 +𝑒 𝐵)) | ||
| Theorem | xle2add 13200 | Extended real version of le2add 11621. (Contributed by Mario Carneiro, 23-Aug-2015.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷))) | ||
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