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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | uzsscn 43701 | An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (ℤ≥‘𝑀) ⊆ ℂ | ||
Theorem | absimnre 43702 | 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 43703 | 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 43704 | 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 43705 | 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 43706 | The positive reals are a subset of the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ ℝ+ ⊆ ℝ* | ||
Theorem | monoordxrv 43707* | 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 43708* | 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 43709* | 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 43710* | 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 43711* | 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 43712 | Extended real version of lenegcon1 11659. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 ≤ 𝐵 ↔ -𝑒𝐵 ≤ 𝐴)) | ||
Theorem | xlenegcon2 43713 | Extended real version of lenegcon2 11660. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ -𝑒𝐵 ↔ 𝐵 ≤ -𝑒𝐴)) | ||
Theorem | pimxrneun 43714 | 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 43715* | 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 43716* | 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 43717* | 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 43718* | A simple counterexample related to theorem rexanuz2 15234, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15234 and similar theorems would not hold (see rexanre 15231 and rexanuz 15230). (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 43719 | 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 43720 | An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝐴(,)𝐵) ⊆ (𝐴(,]𝐵) | ||
Theorem | ioondisj2 43721 | 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 43722 | 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 43723 | 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 43724* | 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 43725 | 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 43726 | 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 43727 | 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 43728 | 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 43729 | An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ | ||
Theorem | iocgtlb 43730 | 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 43731 | 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 43732 | 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 43733 | 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 43734 | 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 43735 | 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 43736 | 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 43737 | 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 43738 | 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 43739 | 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 43740 | 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 43741 | A left-open right-closed interval does not contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ ¬ 𝐴 ∈ (𝐴(,]𝐵) | ||
Theorem | ioomidp 43742 | 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 43743 | 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 43744 | 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 43745 | Biconditional form of ioossioo 13358. (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 43746* | 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 43747 | 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 43748 | 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 43749 | 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 43750* | 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 43751 | 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 43752 | 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 43753 | 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 43754 | A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝐴 ∈ ℝ* → ¬ 𝐵 ∈ (𝐴[,)𝐵)) | ||
Theorem | eliccxrd 43755 | 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 | pnfel0pnf 43756 | +∞ is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ +∞ ∈ (0[,]+∞) | ||
Theorem | eliccnelico 43757 | An element of a closed interval that is not a member of the left-closed right-open interval, must be the upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,)𝐵)) ⇒ ⊢ (𝜑 → 𝐶 = 𝐵) | ||
Theorem | eliccelicod 43758 | A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝐶 ≠ 𝐵) ⇒ ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) | ||
Theorem | ge0xrre 43759 | A nonnegative extended real that is not +∞ is a real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ) | ||
Theorem | ge0lere 43760 | A nonnegative extended Real number smaller than or equal to a Real number, is a Real number. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐵 ∈ ℝ) | ||
Theorem | elicores 43761* | Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝐴 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥[,)𝑦)) | ||
Theorem | inficc 43762 | The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝑆 ⊆ (𝐴[,]𝐵)) & ⊢ (𝜑 → 𝑆 ≠ ∅) ⇒ ⊢ (𝜑 → inf(𝑆, ℝ*, < ) ∈ (𝐴[,]𝐵)) | ||
Theorem | qinioo 43763 | The rational numbers are dense in ℝ. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → ((ℚ ∩ (𝐴(,)𝐵)) = ∅ ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | lenelioc 43764 | A real number smaller than or equal to the lower bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ≤ 𝐴) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) | ||
Theorem | ioonct 43765 | A nonempty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝐶 = (𝐴(,)𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐶 ≼ ω) | ||
Theorem | xrgtnelicc 43766 | A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 < 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵)) | ||
Theorem | iccdificc 43767 | The difference of two closed intervals with the same lower bound. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → ((𝐴[,]𝐶) ∖ (𝐴[,]𝐵)) = (𝐵(,]𝐶)) | ||
Theorem | iocnct 43768 | A nonempty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝐶 = (𝐴(,]𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐶 ≼ ω) | ||
Theorem | iccnct 43769 | A closed interval, with more than one element is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ 𝐶 = (𝐴[,]𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐶 ≼ ω) | ||
Theorem | iooiinicc 43770* | A closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ ((𝐴 − (1 / 𝑛))(,)(𝐵 + (1 / 𝑛))) = (𝐴[,]𝐵)) | ||
Theorem | iccgelbd 43771 | An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) | ||
Theorem | iooltubd 43772 | An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → 𝐶 < 𝐵) | ||
Theorem | icoltubd 43773 | An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) ⇒ ⊢ (𝜑 → 𝐶 < 𝐵) | ||
Theorem | qelioo 43774* | The rational numbers are dense in ℝ*: any two extended real numbers have a rational between them. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℚ 𝑥 ∈ (𝐴(,)𝐵)) | ||
Theorem | tgqioo2 43775* | Every open set of reals is the (countable) union of open interval with rational bounds. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ 𝐽 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) ⇒ ⊢ (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = ∪ 𝑞)) | ||
Theorem | iccleubd 43776 | An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) ⇒ ⊢ (𝜑 → 𝐶 ≤ 𝐵) | ||
Theorem | elioored 43777 | A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ (𝐵(,)𝐶)) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
Theorem | ioogtlbd 43778 | An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
Theorem | ioofun 43779 | (,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ Fun (,) | ||
Theorem | icomnfinre 43780 | A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴)) | ||
Theorem | sqrlearg 43781 | The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴↑2) ≤ 𝐴 ↔ 𝐴 ∈ (0[,]1))) | ||
Theorem | ressiocsup 43782 | If the supremum belongs to a set of reals, the set is a subset of the unbounded below, right-closed interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑆 = sup(𝐴, ℝ*, < ) & ⊢ (𝜑 → 𝑆 ∈ 𝐴) & ⊢ 𝐼 = (-∞(,]𝑆) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐼) | ||
Theorem | ressioosup 43783 | If the supremum does not belong to a set of reals, the set is a subset of the unbounded below, right-open interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑆 = sup(𝐴, ℝ*, < ) & ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) & ⊢ 𝐼 = (-∞(,)𝑆) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐼) | ||
Theorem | iooiinioc 43784* | A left-open, right-closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵)) | ||
Theorem | ressiooinf 43785 | If the infimum does not belong to a set of reals, the set is a subset of the unbounded above, left-open interval, with lower bound equal to the infimum. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ 𝑆 = inf(𝐴, ℝ*, < ) & ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴) & ⊢ 𝐼 = (𝑆(,)+∞) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐼) | ||
Theorem | icogelbd 43786 | An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) | ||
Theorem | iocleubd 43787 | An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) ⇒ ⊢ (𝜑 → 𝐶 ≤ 𝐵) | ||
Theorem | uzinico 43788 | An upper interval of integers is the intersection of the integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → 𝑍 = (ℤ ∩ (𝑀[,)+∞))) | ||
Theorem | preimaiocmnf 43789* | Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (◡𝐹 “ (-∞(,]𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≤ 𝐵}) | ||
Theorem | uzinico2 43790 | An upper interval of integers is the intersection of a larger upper interval of integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞))) | ||
Theorem | uzinico3 43791 | An upper interval of integers doesn't change when it's intersected with a left-closed, unbounded above interval, with the same lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) | ||
Theorem | icossico2 43792 | Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶)) | ||
Theorem | dmico 43793 | The domain of the closed-below, open-above interval function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ dom [,) = (ℝ* × ℝ*) | ||
Theorem | ndmico 43794 | The closed-below, open-above interval function's value is empty outside of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ∅) | ||
Theorem | uzubioo 43795* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) | ||
Theorem | uzubico 43796* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ (𝑋[,)+∞)𝑘 ∈ 𝑍) | ||
Theorem | uzubioo2 43797* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥(,)+∞)𝑘 ∈ 𝑍) | ||
Theorem | uzubico2 43798* | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥[,)+∞)𝑘 ∈ 𝑍) | ||
Theorem | iocgtlbd 43799 | An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
Theorem | xrtgioo2 43800 | The topology on the extended reals coincides with the standard topology on the reals, when restricted to ℝ. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾t ℝ) |
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