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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | xrnemnf 13101 | 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 13102 | 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 13103 | The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.) |
⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴) | ||
Theorem | ltpnf 13104 | Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.) |
⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | ||
Theorem | ltpnfd 13105 | Any (finite) real is less than plus infinity. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 < +∞) | ||
Theorem | 0ltpnf 13106 | Zero is less than plus infinity. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 0 < +∞ | ||
Theorem | mnflt 13107 | Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.) |
⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | ||
Theorem | mnfltd 13108 | Minus infinity is less than any (finite) real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → -∞ < 𝐴) | ||
Theorem | mnflt0 13109 | Minus infinity is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -∞ < 0 | ||
Theorem | mnfltpnf 13110 | Minus infinity is less than plus infinity. (Contributed by NM, 14-Oct-2005.) |
⊢ -∞ < +∞ | ||
Theorem | mnfltxr 13111 | 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 13112 | No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.) |
⊢ (𝐴 ∈ ℝ* → ¬ +∞ < 𝐴) | ||
Theorem | nltmnf 13113 | No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.) |
⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) | ||
Theorem | pnfge 13114 | Plus infinity is an upper bound for extended reals. (Contributed by NM, 30-Jan-2006.) |
⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ +∞) | ||
Theorem | xnn0n0n1ge2b 13115 | 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 13116 | 0 less than or equal to positive infinity. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 0 ≤ +∞ | ||
Theorem | xnn0ge0 13117 | 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 13118 | Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.) |
⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) | ||
Theorem | mnfled 13119 | Minus infinity is less than or equal to any extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → -∞ ≤ 𝐴) | ||
Theorem | xrltnsym 13120 | Ordering on the extended reals is not symmetric. (Contributed by NM, 15-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) | ||
Theorem | xrltnsym2 13121 | '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 13122 | Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 11186 or axlttri 11289. (Contributed by NM, 14-Oct-2005.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))) | ||
Theorem | xrlttr 13123 | 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 13124 | 'Less than' is a strict ordering on the extended reals. (Contributed by NM, 15-Oct-2005.) |
⊢ < Or ℝ* | ||
Theorem | xrlttri2 13125 | 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 13126 | 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 13127 | '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 13128 | '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 13129 | '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 13130 | Alternative definition of 'less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 6-Nov-2015.) |
⊢ ≤ = ( < ∪ ( I ↾ ℝ*)) | ||
Theorem | dflt2 13131 | Alternative definition of 'less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015.) |
⊢ < = ( ≤ ∖ I ) | ||
Theorem | xrltle 13132 | '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 13133 | 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle 13132. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | xrleid 13134 | 'Less than or equal to' is reflexive for extended reals. (Contributed by NM, 7-Feb-2007.) |
⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) | ||
Theorem | xrleidd 13135 | 'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 13134. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐴) | ||
Theorem | xrletri 13136 | Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)) | ||
Theorem | xrletri3 13137 | Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | ||
Theorem | xrletrid 13138 | 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 13139 | 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 13140 | 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 13141 | 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 13142 | 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 13143 | 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 13144 | 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 13145 | 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 13146 | 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | ||
Theorem | nltpnft 13147 | 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 13148 | 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 13149 | 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 13150 | 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 13151 | 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 13152 | 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 13153 | 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 13154 | 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 13155 | A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴) | ||
Theorem | ge0nemnf 13156 | A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞) | ||
Theorem | xrrege0 13157 | 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 13158 | 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 13159 | 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 13160 | 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 13161 | 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 13162 | 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 13163 | 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 13164 | 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 13165 | 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 13166 | 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 13167 | 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 13168 | A number is less than or equal to the maximum of it and another. See also max1ALT 13169. (Contributed by NM, 3-Apr-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
Theorem | max1ALT 13169 | A number is less than or equal to the maximum of it and another. This version of max1 13168 omits the 𝐵 ∈ ℝ antecedent. Although it doesn't exploit undefined behavior, it is still considered poor style, and the use of max1 13168 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 13170 | 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 13171 | 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 13172 | 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 13173 | 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 13174 | 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 13175 | 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 13176 | 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 13177 | 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 13178 | 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 13179 | 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 13180 | 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 13181* | 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 13182* | 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 13183* | 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 13184* | 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 13185* | Lemma for qextlt 13186 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 13186* | 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 13187* | 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 13188* | 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 13189* | 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 13190 | 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 13191 | A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ -𝑒𝐴 ∈ V | ||
Theorem | xnegpnf 13192 | Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) |
⊢ -𝑒+∞ = -∞ | ||
Theorem | xnegmnf 13193 | Minus -∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.) |
⊢ -𝑒-∞ = +∞ | ||
Theorem | rexneg 13194 | 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 13195 | The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ -𝑒0 = 0 | ||
Theorem | xnegcl 13196 | Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) | ||
Theorem | xnegneg 13197 | Extended real version of negneg 11514. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ (𝐴 ∈ ℝ* → -𝑒-𝑒𝐴 = 𝐴) | ||
Theorem | xneg11 13198 | Extended real version of neg11 11515. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 = -𝑒𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | xltnegi 13199 | Forward direction of xltneg 13200. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → -𝑒𝐵 < -𝑒𝐴) | ||
Theorem | xltneg 13200 | Extended real version of ltneg 11718. (Contributed by Mario Carneiro, 20-Aug-2015.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ -𝑒𝐵 < -𝑒𝐴)) |
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