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Theorem List for Metamath Proof Explorer - 12001-12100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremx2times 12001 Extended real version of 2times 11022. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* → (2 ·e 𝐴) = (𝐴 +𝑒 𝐴))

Theoremxnegcld 12002 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)       (𝜑 → -𝑒𝐴 ∈ ℝ*)

Theoremxaddcld 12003 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐴 +𝑒 𝐵) ∈ ℝ*)

Theoremxmulcld 12004 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)       (𝜑 → (𝐴 ·e 𝐵) ∈ ℝ*)

Theoremxadd4d 12005 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 10143. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
(𝜑 → (𝐴 ∈ ℝ*𝐴 ≠ -∞))    &   (𝜑 → (𝐵 ∈ ℝ*𝐵 ≠ -∞))    &   (𝜑 → (𝐶 ∈ ℝ*𝐶 ≠ -∞))    &   (𝜑 → (𝐷 ∈ ℝ*𝐷 ≠ -∞))       (𝜑 → ((𝐴 +𝑒 𝐵) +𝑒 (𝐶 +𝑒 𝐷)) = ((𝐴 +𝑒 𝐶) +𝑒 (𝐵 +𝑒 𝐷)))

Theoremxnn0add4d 12006 Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 12005. (Contributed by AV, 12-Dec-2020.)
(𝜑𝐴 ∈ ℕ0*)    &   (𝜑𝐵 ∈ ℕ0*)    &   (𝜑𝐶 ∈ ℕ0*)    &   (𝜑𝐷 ∈ ℕ0*)       (𝜑 → ((𝐴 +𝑒 𝐵) +𝑒 (𝐶 +𝑒 𝐷)) = ((𝐴 +𝑒 𝐶) +𝑒 (𝐵 +𝑒 𝐷)))

5.5.3  Supremum and infimum on the extended reals

Theoremxrsupexmnf 12007* Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)
(∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {-∞}) ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ (𝐴 ∪ {-∞})𝑦 < 𝑧)))

Theoremxrinfmexpnf 12008* Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)
(∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ (𝐴 ∪ {+∞}) ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ (𝐴 ∪ {+∞})𝑧 < 𝑦)))

Theoremxrsupsslem 12009* Lemma for xrsupss 12011. (Contributed by NM, 25-Oct-2005.)
((𝐴 ⊆ ℝ* ∧ (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

Theoremxrinfmsslem 12010* Lemma for xrinfmss 12012. (Contributed by NM, 19-Jan-2006.)
((𝐴 ⊆ ℝ* ∧ (𝐴 ⊆ ℝ ∨ -∞ ∈ 𝐴)) → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))

Theoremxrsupss 12011* Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)
(𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

Theoremxrinfmss 12012* Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)
(𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))

Theoremxrinfmss2 12013* Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))

Theoremxrub 12014* By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals. (Contributed by NM, 9-Apr-2006.)
((𝐴 ⊆ ℝ*𝐵 ∈ ℝ*) → (∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦𝐴 𝑥 < 𝑦) ↔ ∀𝑥 ∈ ℝ* (𝑥 < 𝐵 → ∃𝑦𝐴 𝑥 < 𝑦)))

Theoremsupxr 12015* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.) (Revised by Mario Carneiro, 21-Apr-2015.)
(((𝐴 ⊆ ℝ*𝐵 ∈ ℝ*) ∧ (∀𝑥𝐴 ¬ 𝐵 < 𝑥 ∧ ∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦𝐴 𝑥 < 𝑦))) → sup(𝐴, ℝ*, < ) = 𝐵)

Theoremsupxr2 12016* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)
(((𝐴 ⊆ ℝ*𝐵 ∈ ℝ*) ∧ (∀𝑥𝐴 𝑥𝐵 ∧ ∀𝑥 ∈ ℝ (𝑥 < 𝐵 → ∃𝑦𝐴 𝑥 < 𝑦))) → sup(𝐴, ℝ*, < ) = 𝐵)

Theoremsupxrcl 12017 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.)
(𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*)

Theoremsupxrun 12018 The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)
((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ* ∧ sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < )) → sup((𝐴𝐵), ℝ*, < ) = sup(𝐵, ℝ*, < ))

Theoremsupxrmnf 12019 Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)
(𝐴 ⊆ ℝ* → sup((𝐴 ∪ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))

Theoremsupxrpnf 12020 The supremum of a set of extended reals containing plus infinity is plus infinity. (Contributed by NM, 15-Oct-2005.)
((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)

Theoremsupxrunb1 12021* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
(𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦𝐴 𝑥𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))

Theoremsupxrunb2 12022* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
(𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦𝐴 𝑥 < 𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))

Theoremsupxrbnd1 12023* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
(𝐴 ⊆ ℝ* → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦 < 𝑥 ↔ sup(𝐴, ℝ*, < ) < +∞))

Theoremsupxrbnd2 12024* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
(𝐴 ⊆ ℝ* → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥 ↔ sup(𝐴, ℝ*, < ) < +∞))

Theoremxrsup0 12025 The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)
sup(∅, ℝ*, < ) = -∞

Theoremsupxrub 12026 A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by NM, 7-Feb-2006.)
((𝐴 ⊆ ℝ*𝐵𝐴) → 𝐵 ≤ sup(𝐴, ℝ*, < ))

Theoremsupxrlub 12027* The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by Mario Carneiro, 13-Sep-2015.)
((𝐴 ⊆ ℝ*𝐵 ∈ ℝ*) → (𝐵 < sup(𝐴, ℝ*, < ) ↔ ∃𝑥𝐴 𝐵 < 𝑥))

Theoremsupxrleub 12028* The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by NM, 22-Feb-2006.) (Revised by Mario Carneiro, 6-Sep-2014.)
((𝐴 ⊆ ℝ*𝐵 ∈ ℝ*) → (sup(𝐴, ℝ*, < ) ≤ 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵))

Theoremsupxrre 12029* The real and extended real suprema match when the real supremum exists. (Contributed by NM, 18-Oct-2005.) (Proof shortened by Mario Carneiro, 7-Sep-2014.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) → sup(𝐴, ℝ*, < ) = sup(𝐴, ℝ, < ))

Theoremsupxrbnd 12030 The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ*, < ) < +∞) → sup(𝐴, ℝ*, < ) ∈ ℝ)

Theoremsupxrgtmnf 12031 The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ*, < ))

Theoremsupxrre1 12032 The supremum of a nonempty set of reals is real iff it is less than plus infinity. (Contributed by NM, 5-Feb-2006.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) < +∞))

Theoremsupxrre2 12033 The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) ≠ +∞))

Theoremsupxrss 12034 Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴𝐵𝐵 ⊆ ℝ*) → sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < ))

Theoreminfxrcl 12035 The infimum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 19-Jan-2006.) (Revised by AV, 5-Sep-2020.)
(𝐴 ⊆ ℝ* → inf(𝐴, ℝ*, < ) ∈ ℝ*)

Theoreminfxrlb 12036 A member of a set of extended reals is greater than or equal to the set's infimum. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 5-Sep-2020.)
((𝐴 ⊆ ℝ*𝐵𝐴) → inf(𝐴, ℝ*, < ) ≤ 𝐵)

Theoreminfxrgelb 12037* The infimum of a set of extended reals is greater than or equal to a lower bound. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 5-Sep-2020.)
((𝐴 ⊆ ℝ*𝐵 ∈ ℝ*) → (𝐵 ≤ inf(𝐴, ℝ*, < ) ↔ ∀𝑥𝐴 𝐵𝑥))

Theoreminfxrre 12038* The real and extended real infima match when the real infimum exists. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 5-Sep-2020.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) → inf(𝐴, ℝ*, < ) = inf(𝐴, ℝ, < ))

Theoremxrinf0 12039 The infimum of the empty set under the extended reals is positive infinity. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 5-Sep-2020.)
inf(∅, ℝ*, < ) = +∞

Theoreminfxrss 12040 Larger sets of extended reals have smaller infima. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)
((𝐴𝐵𝐵 ⊆ ℝ*) → inf(𝐵, ℝ*, < ) ≤ inf(𝐴, ℝ*, < ))

Theoremreltre 12041* For all real numbers there is a smaller real number. (Contributed by AV, 5-Sep-2020.)
𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑦 < 𝑥

Theoremrpltrp 12042* For all positive real numbers there is a smaller positive real number. (Contributed by AV, 5-Sep-2020.)
𝑥 ∈ ℝ+𝑦 ∈ ℝ+ 𝑦 < 𝑥

Theoremreltxrnmnf 12043* For all extended real numbers not being minus infinity there is a smaller real number. (Contributed by AV, 5-Sep-2020.)
𝑥 ∈ ℝ* (-∞ < 𝑥 → ∃𝑦 ∈ ℝ 𝑦 < 𝑥)

Theoreminfmremnf 12044 The infimum of the reals is minus infinity. (Contributed by AV, 5-Sep-2020.)
inf(ℝ, ℝ*, < ) = -∞

Theoreminfmrp1 12045 The infimum of the positive reals is 0. (Contributed by AV, 5-Sep-2020.)
inf(ℝ+, ℝ, < ) = 0

5.5.4  Real number intervals

Syntaxcioo 12046 Extend class notation with the set of open intervals of extended reals.
class (,)

Syntaxcioc 12047 Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]

Syntaxcico 12048 Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)

Syntaxcicc 12049 Extend class notation with the set of closed intervals of extended reals.
class [,]

Definitiondf-ioo 12050* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})

Definitiondf-ioc 12051* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧𝑦)})

Definitiondf-ico 12052* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧 < 𝑦)})

Definitiondf-icc 12053* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑧𝑧𝑦)})

Theoremixxval 12054* Value of the interval function. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ* ∣ (𝐴𝑅𝑧𝑧𝑆𝐵)})

Theoremelixx1 12055* Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝑅𝐶𝐶𝑆𝐵)))

Theoremixxf 12056* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       𝑂:(ℝ* × ℝ*)⟶𝒫 ℝ*

Theoremixxex 12057* The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       𝑂 ∈ V

Theoremixxssxr 12058* The set of intervals of extended reals maps to subsets of extended reals. (Contributed by Mario Carneiro, 4-Jul-2014.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       (𝐴𝑂𝐵) ⊆ ℝ*

Theoremelixx3g 12059* Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show 𝐴 ∈ ℝ* and 𝐵 ∈ ℝ*. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})       (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝑅𝐶𝐶𝑆𝐵)))

Theoremixxssixx 12060* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})    &   ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴𝑅𝑤𝐴𝑇𝑤))    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝑆𝐵𝑤𝑈𝐵))       (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)

Theoremixxdisj 12061* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})    &   ((𝐵 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐵𝑇𝑤 ↔ ¬ 𝑤𝑆𝐵))       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) = ∅)

Theoremixxun 12062* Split an interval into two parts. (Contributed by Mario Carneiro, 16-Jun-2014.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})    &   ((𝐵 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐵𝑇𝑤 ↔ ¬ 𝑤𝑆𝐵))    &   𝑄 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑈𝑦)})    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝑤𝑆𝐵𝐵𝑋𝐶) → 𝑤𝑈𝐶))    &   ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵𝐵𝑇𝑤) → 𝐴𝑅𝑤))       (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝑊𝐵𝐵𝑋𝐶)) → ((𝐴𝑂𝐵) ∪ (𝐵𝑃𝐶)) = (𝐴𝑄𝐶))

Theoremixxin 12063* Intersection of two intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*𝑧 ∈ ℝ*) → (if(𝐴𝐶, 𝐶, 𝐴)𝑅𝑧 ↔ (𝐴𝑅𝑧𝐶𝑅𝑧)))    &   ((𝑧 ∈ ℝ*𝐵 ∈ ℝ*𝐷 ∈ ℝ*) → (𝑧𝑆if(𝐵𝐷, 𝐵, 𝐷) ↔ (𝑧𝑆𝐵𝑧𝑆𝐷)))       (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴𝑂𝐵) ∩ (𝐶𝑂𝐷)) = (if(𝐴𝐶, 𝐶, 𝐴)𝑂if(𝐵𝐷, 𝐵, 𝐷)))

Theoremixxss1 12064* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑆𝑦)})    &   ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵𝐵𝑇𝑤) → 𝐴𝑅𝑤))       ((𝐴 ∈ ℝ*𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶))

Theoremixxss2 12065* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑇𝑦)})    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝑤𝑇𝐵𝐵𝑊𝐶) → 𝑤𝑆𝐶))       ((𝐶 ∈ ℝ*𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))

Theoremixxss12 12066* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧𝑧𝑈𝑦)})    &   ((𝐴 ∈ ℝ*𝐶 ∈ ℝ*𝑤 ∈ ℝ*) → ((𝐴𝑊𝐶𝐶𝑇𝑤) → 𝐴𝑅𝑤))    &   ((𝑤 ∈ ℝ*𝐷 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝑤𝑈𝐷𝐷𝑋𝐵) → 𝑤𝑆𝐵))       (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝑊𝐶𝐷𝑋𝐵)) → (𝐶𝑃𝐷) ⊆ (𝐴𝑂𝐵))

Theoremixxub 12067* Extract the upper bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤 < 𝐵𝑤𝑆𝐵))    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝑆𝐵𝑤𝐵))    &   ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴 < 𝑤𝐴𝑅𝑤))    &   ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴𝑅𝑤𝐴𝑤))       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝐴𝑂𝐵) ≠ ∅) → sup((𝐴𝑂𝐵), ℝ*, < ) = 𝐵)

Theoremixxlb 12068* Extract the lower bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by AV, 12-Sep-2020.)
𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧𝑧𝑆𝑦)})    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤 < 𝐵𝑤𝑆𝐵))    &   ((𝑤 ∈ ℝ*𝐵 ∈ ℝ*) → (𝑤𝑆𝐵𝑤𝐵))    &   ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴 < 𝑤𝐴𝑅𝑤))    &   ((𝐴 ∈ ℝ*𝑤 ∈ ℝ*) → (𝐴𝑅𝑤𝐴𝑤))       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝐴𝑂𝐵) ≠ ∅) → inf((𝐴𝑂𝐵), ℝ*, < ) = 𝐴)

Theoremiooex 12069 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
(,) ∈ V

Theoremiooval 12070* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥𝑥 < 𝐵)})

Theoremioo0 12071 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵𝐴))

Theoremioon0 12072 An open interval of extended reals is nonempty iff the lower argument is less than the upper argument. (Contributed by NM, 2-Mar-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵))

Theoremndmioo 12073 The open interval function's value is empty outside of its domain. (Contributed by NM, 21-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
(¬ (𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅)

Theoremiooid 12074 An open interval with identical lower and upper bounds is empty. (Contributed by NM, 21-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝐴(,)𝐴) = ∅

Theoremelioo3g 12075 Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show 𝐴 ∈ ℝ* and 𝐵 ∈ ℝ*. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐶𝐶 < 𝐵)))

Theoremelioore 12076 A member of an open interval of reals is a real. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝐴 ∈ (𝐵(,)𝐶) → 𝐴 ∈ ℝ)

Theoremlbioo 12077 An open interval does not contain its left endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)
¬ 𝐴 ∈ (𝐴(,)𝐵)

Theoremubioo 12078 An open interval does not contain its right endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)
¬ 𝐵 ∈ (𝐴(,)𝐵)

Theoremiooval2 12079* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥𝑥 < 𝐵)})

Theoremiooin 12080 Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = (if(𝐴𝐶, 𝐶, 𝐴)(,)if(𝐵𝐷, 𝐵, 𝐷)))

Theoremiooss1 12081 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)
((𝐴 ∈ ℝ*𝐴𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))

Theoremiooss2 12082 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐶 ∈ ℝ*𝐵𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))

Theoremiocval 12083* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,]𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥𝑥𝐵)})

Theoremicoval 12084* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴𝑥𝑥 < 𝐵)})

Theoremiccval 12085* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴[,]𝐵) = {𝑥 ∈ ℝ* ∣ (𝐴𝑥𝑥𝐵)})

Theoremelioo1 12086 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ*𝐴 < 𝐶𝐶 < 𝐵)))

Theoremelioo2 12087 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶𝐶 < 𝐵)))

Theoremelioc1 12088 Membership in an open-below, closed-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴 < 𝐶𝐶𝐵)))

Theoremelico1 12089 Membership in a closed-below, open-above interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶 < 𝐵)))

Theoremelicc1 12090 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐶𝐶𝐵)))

Theoremiccid 12091 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)
(𝐴 ∈ ℝ* → (𝐴[,]𝐴) = {𝐴})

Theoremico0 12092 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵𝐴))

Theoremioc0 12093 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴(,]𝐵) = ∅ ↔ 𝐵𝐴))

Theoremicc0 12094 An empty closed interval of extended reals. (Contributed by FL, 30-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴))

Theoremelicod 12095 Membership in a left closed, right open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐴𝐶)    &   (𝜑𝐶 < 𝐵)       (𝜑𝐶 ∈ (𝐴[,)𝐵))

Theoremicogelb 12096 An element of a left closed right open interval is larger or equal to its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,)𝐵)) → 𝐴𝐶)

Theoremelicore 12097 A member of a left closed, right open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ)

Theoremubioc1 12098 The upper bound belongs to an open-below, closed-above interval. See ubicc2 12160. (Contributed by FL, 29-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵))

Theoremlbico1 12099 The lower bound belongs to a closed-below, open-above interval. See lbicc2 12159. (Contributed by FL, 29-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))

Theoremiccleub 12100 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐶𝐵)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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