HomeTheorem List (p. 461 of 505)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | supxrre3rnmpt 46001* | The indexed supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)) | ||
| Theorem | uzublem 46002* | A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝑋 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ) & ⊢ 𝑋 = if(𝑊 ≤ 𝑌, 𝑌, 𝑊) & ⊢ (𝜑 → 𝐾 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝐵 ≤ 𝑌) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥) | ||
| Theorem | uzub 46003* | A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)) | ||
| Theorem | ssrexr 46004 | A subset of the reals is a subset of the extended reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ⊆ ℝ*) | ||
| Theorem | supxrmnf2 46005 | Removing minus infinity from a set does not affect its supremum. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ⊆ ℝ* → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) | ||
| Theorem | supxrcli 46006 | The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐴 ⊆ ℝ* ⇒ ⊢ sup(𝐴, ℝ*, < ) ∈ ℝ* | ||
| Theorem | uzid3 46007 | Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑁)) | ||
| Theorem | infxrlesupxr 46008 | The supremum of a nonempty set is greater than or equal to the infimum. The second condition is needed, see supxrltinfxr 46021. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐴 ≠ ∅) ⇒ ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) | ||
| Theorem | xnegeqd 46009 | Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → -𝑒𝐴 = -𝑒𝐵) | ||
| Theorem | xnegrecl 46010 | The extended real negative of a real number is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ) | ||
| Theorem | xnegnegi 46011 | Extended real version of negneg 11496. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐴 ∈ ℝ* ⇒ ⊢ -𝑒-𝑒𝐴 = 𝐴 | ||
| Theorem | xnegeqi 46012 | Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐴 = 𝐵 ⇒ ⊢ -𝑒𝐴 = -𝑒𝐵 | ||
| Theorem | nfxnegd 46013 | Deduction version of nfxneg 46033. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴) | ||
| Theorem | xnegnegd 46014 | Extended real version of negnegd 11548. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → -𝑒-𝑒𝐴 = 𝐴) | ||
| Theorem | uzred 46015 | An upper integer is a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
| Theorem | xnegcli 46016 | Closure of extended real negative. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐴 ∈ ℝ* ⇒ ⊢ -𝑒𝐴 ∈ ℝ* | ||
| Theorem | supminfrnmpt 46017* | The indexed supremum of a bounded-above set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ⇒ ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < )) | ||
| Theorem | infxrpnf 46018 | Adding plus infinity to a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ⊆ ℝ* → inf((𝐴 ∪ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < )) | ||
| Theorem | infxrrnmptcl 46019* | The infimum of an arbitrary indexed set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ∈ ℝ*) | ||
| Theorem | leneg2d 46020 | Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ -𝐵 ↔ 𝐵 ≤ -𝐴)) | ||
| Theorem | supxrltinfxr 46021 | The supremum of the empty set is strictly smaller than the infimum of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ sup(∅, ℝ*, < ) < inf(∅, ℝ*, < ) | ||
| Theorem | max1d 46022 | A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
| Theorem | supxrleubrnmptf 46023 | The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) ⇒ ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) | ||
| Theorem | nleltd 46024 | 'Not less than or equal to' implies 'grater than'. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ¬ 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 < 𝐵) | ||
| Theorem | zxrd 46025 | An integer is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ*) | ||
| Theorem | infxrgelbrnmpt 46026* | The infimum of an indexed set of extended reals is greater than or equal to a lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵)) | ||
| Theorem | rphalfltd 46027 | Half of a positive real is less than the original number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 / 2) < 𝐴) | ||
| Theorem | uzssz2 46028 | An upper set of integers is a subset of all integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ 𝑍 ⊆ ℤ | ||
| Theorem | leneg3d 46029 | Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ 𝐴)) | ||
| Theorem | max2d 46030 | A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
| Theorem | uzn0bi 46031 | The upper integers function needs to be applied to an integer, in order to return a nonempty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ ((ℤ≥‘𝑀) ≠ ∅ ↔ 𝑀 ∈ ℤ) | ||
| Theorem | xnegrecl2 46032 | If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ -𝑒𝐴 ∈ ℝ) → 𝐴 ∈ ℝ) | ||
| Theorem | nfxneg 46033 | Bound-variable hypothesis builder for the negative of an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥-𝑒𝐴 | ||
| Theorem | uzxrd 46034 | An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ*) | ||
| Theorem | infxrpnf2 46035 | Removing plus infinity from a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ⊆ ℝ* → inf((𝐴 ∖ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < )) | ||
| Theorem | supminfxr 46036* | The extended real suprema of a set of reals is the extended real negative of the extended real infima of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ*, < )) | ||
| Theorem | infrpgernmpt 46037* | The infimum of a nonempty, bounded below, indexed subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) | ||
| Theorem | xnegre 46038 | An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ)) | ||
| Theorem | xnegrecl2d 46039 | If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → -𝑒𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
| Theorem | uzxr 46040 | An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ∈ (ℤ≥‘𝑀) → 𝐴 ∈ ℝ*) | ||
| Theorem | supminfxr2 46041* | The extended real suprema of a set of extended reals is the extended real negative of the extended real infima of that set's image under extended real negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ*) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥 ∈ 𝐴}, ℝ*, < )) | ||
| Theorem | xnegred 46042 | An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ)) | ||
| Theorem | supminfxrrnmpt 46043* | The indexed supremum of a set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < )) | ||
| Theorem | min1d 46044 | The minimum of two numbers is less than or equal to the first. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) | ||
| Theorem | min2d 46045 | The minimum of two numbers is less than or equal to the second. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) | ||
| Theorem | xrnpnfmnf 46046 | An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ +∞) ⇒ ⊢ (𝜑 → 𝐴 = -∞) | ||
| Theorem | uzsscn 46047 | An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (ℤ≥‘𝑀) ⊆ ℂ | ||
| Theorem | absimnre 46048 | The absolute value of the imaginary part of a non-real, complex number, is strictly positive. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+) | ||
| Theorem | uzsscn2 46049 | An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ 𝑍 ⊆ ℂ | ||
| Theorem | xrtgcntopre 46050 | The standard topologies on the extended reals and on the complex numbers, coincide when restricted to the reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ ((ordTop‘ ≤ ) ↾t ℝ) = ((TopOpen‘ℂfld) ↾t ℝ) | ||
| Theorem | absimlere 46051 | The absolute value of the imaginary part of a complex number is a lower bound of the distance to any real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝐵 − 𝐴))) | ||
| Theorem | rpssxr 46052 | The positive reals are a subset of the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ ℝ+ ⊆ ℝ* | ||
| Theorem | monoordxrv 46053* | Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
| ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) ⇒ ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁)) | ||
| Theorem | monoordxr 46054* | Ordering relation for a monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) ⇒ ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁)) | ||
| Theorem | monoord2xrv 46055* | Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
| ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀)) | ||
| Theorem | monoord2xr 46056* | Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀)) | ||
| Theorem | xrpnf 46057* | An extended real is plus infinity iff it's larger than all real numbers. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ∀𝑥 ∈ ℝ 𝑥 ≤ 𝐴)) | ||
| Theorem | xlenegcon1 46058 | Extended real version of lenegcon1 11706. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ 𝐴)) | ||
| Theorem | xlenegcon2 46059 | Extended real version of lenegcon2 11707. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ -𝑒𝐵 ↔ 𝐵 ≤ -𝑒𝐴)) | ||
| Theorem | pimxrneun 46060 | The preimage of a set of extended reals that does not contain a value 𝐶 is the union of the preimage of the elements smaller than 𝐶 and the preimage of the subset of elements larger than 𝐶. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵})) | ||
| Theorem | caucvgbf 46061* | A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))) | ||
| Theorem | cvgcau 46062* | A convergent function is Cauchy. (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ 𝑍) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋)) | ||
| Theorem | cvgcaule 46063* | A convergent function is Cauchy. (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ 𝑍) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ≤ 𝑋)) | ||
| Theorem | rexanuz2nf 46064* | A simple counterexample related to theorem rexanuz2 15391, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15391 and similar theorems would not hold (see rexanre 15388 and rexanuz 15387). (Contributed by Glauco Siliprandi, 15-Feb-2025.) |
| ⊢ 𝑍 = ℕ0 & ⊢ (𝜑 ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑘)) & ⊢ (𝜓 ↔ 0 < 𝑘) ⇒ ⊢ ¬ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)) | ||
| Theorem | gtnelioc 46065 | A real number larger than the upper bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) | ||
| Theorem | ioossioc 46066 | An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝐴(,)𝐵) ⊆ (𝐴(,]𝐵) | ||
| Theorem | ioondisj2 46067 | A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐶 < 𝐷)) ∧ (𝐴 < 𝐷 ∧ 𝐷 ≤ 𝐵)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ≠ ∅) | ||
| Theorem | ioondisj1 46068 | A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐶 < 𝐷)) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ≠ ∅) | ||
| Theorem | ioogtlb 46069 | An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶) | ||
| Theorem | evthiccabs 46070* | Extreme Value Theorem on y closed interval, for the absolute value of y continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑦)) ≤ (abs‘(𝐹‘𝑥)) ∧ ∃𝑧 ∈ (𝐴[,]𝐵)∀𝑤 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑤)))) | ||
| Theorem | ltnelicc 46071 | A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 < 𝐴) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) | ||
| Theorem | eliood 46072 | Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐶) & ⊢ (𝜑 → 𝐶 < 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | ||
| Theorem | iooabslt 46073 | An upper bound for the distance from the center of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) ⇒ ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < 𝐵) | ||
| Theorem | gtnelicc 46074 | A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) | ||
| Theorem | iooinlbub 46075 | An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ | ||
| Theorem | iocgtlb 46076 | An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 < 𝐶) | ||
| Theorem | iocleub 46077 | An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵) | ||
| Theorem | eliccd 46078 | Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐶 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) | ||
| Theorem | eliccre 46079 | A member of a closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℝ) | ||
| Theorem | eliooshift 46080 | Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 + 𝐷) ∈ ((𝐵 + 𝐷)(,)(𝐶 + 𝐷)))) | ||
| Theorem | eliocd 46081 | Membership in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐶) & ⊢ (𝜑 → 𝐶 ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) | ||
| Theorem | icoltub 46082 | An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵) | ||
| Theorem | eliocre 46083 | A member of a left-open right-closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ) | ||
| Theorem | iooltub 46084 | An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 < 𝐵) | ||
| Theorem | ioontr 46085 | The interior of an interval in the standard topology on ℝ is the open interval itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵) | ||
| Theorem | snunioo1 46086 | The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵)) | ||
| Theorem | lbioc 46087 | A left-open right-closed interval does not contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ¬ 𝐴 ∈ (𝐴(,]𝐵) | ||
| Theorem | ioomidp 46088 | The midpoint is an element of the open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵)) | ||
| Theorem | iccdifioo 46089 | If the open inverval is removed from the closed interval, only the bounds are left. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ (𝐴(,)𝐵)) = {𝐴, 𝐵}) | ||
| Theorem | iccdifprioo 46090 | An open interval is the closed interval without the bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵)) | ||
| Theorem | ioossioobi 46091 | Biconditional form of ioossioo 13459. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐷 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 < 𝐷) ⇒ ⊢ (𝜑 → ((𝐶(,)𝐷) ⊆ (𝐴(,)𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵))) | ||
| Theorem | iccshift 46092* | A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑇 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴[,]𝐵)𝑤 = (𝑧 + 𝑇)}) | ||
| Theorem | iccsuble 46093 | An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵)) ⇒ ⊢ (𝜑 → (𝐶 − 𝐷) ≤ (𝐵 − 𝐴)) | ||
| Theorem | iocopn 46094 | A left-open right-closed interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝐽 = (𝐾 ↾t (𝐴(,]𝐵)) & ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐶(,]𝐵) ∈ 𝐽) | ||
| Theorem | eliccelioc 46095 | Membership in a closed interval and in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐶 ≠ 𝐴))) | ||
| Theorem | iooshift 46096* | An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝑇 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) | ||
| Theorem | iccintsng 46097 | Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) = {𝐵}) | ||
| Theorem | icoiccdif 46098 | Left-closed right-open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ((𝐴[,]𝐵) ∖ {𝐵})) | ||
| Theorem | icoopn 46099 | A left-closed right-open interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝐽 = (𝐾 ↾t (𝐴[,)𝐵)) & ⊢ (𝜑 → 𝐶 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴[,)𝐶) ∈ 𝐽) | ||
| Theorem | icoub 46100 | A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| ⊢ (𝐴 ∈ ℝ* → ¬ 𝐵 ∈ (𝐴[,)𝐵)) | ||
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