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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiccgelb 12101 An element of a closed interval is more than or equal to its lower bound. (Contributed by Thierry Arnoux, 23-Dec-2016.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ (𝐴[,]𝐵)) → 𝐴𝐶)

Theoremelioo5 12102 Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶𝐶 < 𝐵)))

Theoremeliooxr 12103 A nonempty open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
(𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ*𝐶 ∈ ℝ*))

Theoremeliooord 12104 Ordering implied by a member of an open interval of reals. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 9-May-2014.)
(𝐴 ∈ (𝐵(,)𝐶) → (𝐵 < 𝐴𝐴 < 𝐶))

Theoremelioo4g 12105 Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶𝐶 < 𝐵)))

Theoremioossre 12106 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
(𝐴(,)𝐵) ⊆ ℝ

Theoremelioc2 12107 Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶𝐶𝐵)))

Theoremelico2 12108 Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴𝐶𝐶 < 𝐵)))

Theoremelicc2 12109 Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴𝐶𝐶𝐵)))

Theoremelicc2i 12110 Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴𝐶𝐶𝐵))

Theoremelicc4 12111 Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴𝐶𝐶𝐵)))

Theoremiccss 12112 Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 20-Feb-2015.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))

Theoremiccssioo 12113 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴 < 𝐶𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))

Theoremicossico 12114 Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Thierry Arnoux, 21-Sep-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶[,)𝐷) ⊆ (𝐴[,)𝐵))

Theoremiccss2 12115 Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵))

Theoremiccssico 12116 Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝐶𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))

Theoremiccssioo2 12117 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))

Theoremiccssico2 12118 Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))

Theoremioomax 12119 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
(-∞(,)+∞) = ℝ

Theoremiccmax 12120 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
(-∞[,]+∞) = ℝ*

Theoremioopos 12121 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
(0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥}

Theoremioorp 12122 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(0(,)+∞) = ℝ+

Theoremiooshf 12123 Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴𝐵) ∈ (𝐶(,)𝐷) ↔ 𝐴 ∈ ((𝐶 + 𝐵)(,)(𝐷 + 𝐵))))

Theoremiocssre 12124 A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)

Theoremicossre 12125 A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) ⊆ ℝ)

Theoremiccssre 12126 A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)

Theoremiccssxr 12127 A closed interval is a set of extended reals. (Contributed by FL, 28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)
(𝐴[,]𝐵) ⊆ ℝ*

Theoremiocssxr 12128 An open-below, closed-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
(𝐴(,]𝐵) ⊆ ℝ*

Theoremicossxr 12129 A closed-below, open-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
(𝐴[,)𝐵) ⊆ ℝ*

Theoremioossicc 12130 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
(𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)

Theoremicossicc 12131 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
(𝐴[,)𝐵) ⊆ (𝐴[,]𝐵)

Theoremiocssicc 12132 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
(𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)

Theoremioossico 12133 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
(𝐴(,)𝐵) ⊆ (𝐴[,)𝐵)

Theoremiocssioo 12134 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝐶𝐷 < 𝐵)) → (𝐶(,]𝐷) ⊆ (𝐴(,)𝐵))

Theoremicossioo 12135 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴 < 𝐶𝐷𝐵)) → (𝐶[,)𝐷) ⊆ (𝐴(,)𝐵))

Theoremioossioo 12136 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴𝐶𝐷𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))

Theoremiccsupr 12137* A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum (see suprcl 10862). (Contributed by Paul Chapman, 21-Jan-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥))

Theoremelioopnf 12138 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))

Theoremelioomnf 12139 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐴 ∈ ℝ* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴)))

Theoremelicopnf 12140 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴𝐵)))

Theoremrepos 12141 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
(𝐴 ∈ (0(,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))

Theoremioof 12142 The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
(,):(ℝ* × ℝ*)⟶𝒫 ℝ

Theoremiccf 12143 The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
[,]:(ℝ* × ℝ*)⟶𝒫 ℝ*

Theoremunirnioo 12144 The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
ℝ = ran (,)

Theoremdfioo2 12145* Alternate definition of the set of open intervals of extended reals. (Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro, 1-Sep-2015.)
(,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑤 ∈ ℝ ∣ (𝑥 < 𝑤𝑤 < 𝑦)})

Theoremioorebas 12146 Open intervals are elements of the set of all open intervals. (Contributed by Mario Carneiro, 26-Mar-2015.)
(𝐴(,)𝐵) ∈ ran (,)

Theoremxrge0neqmnf 12147 An extended nonnegative real cannot be minus infinity. (Contributed by Thierry Arnoux, 9-Jun-2017.)
(𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞)

Theoremxrge0nre 12148 An extended real which is not a real is plus infinity. (Contributed by Thierry Arnoux, 16-Oct-2017.)
((𝐴 ∈ (0[,]+∞) ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 = +∞)

Theoremelrege0 12149 The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
(𝐴 ∈ (0[,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))

Theoremnn0rp0 12150 A nonnegative integer is a nonnegative real number. (Contributed by AV, 24-May-2020.)
(𝑁 ∈ ℕ0𝑁 ∈ (0[,)+∞))

Theoremrge0ssre 12151 Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
(0[,)+∞) ⊆ ℝ

Theoremelxrge0 12152 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴))

Theorem0e0icopnf 12153 0 is a member of (0[,)+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ∈ (0[,)+∞)

Theorem0e0iccpnf 12154 0 is a member of (0[,]+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ∈ (0[,]+∞)

Theoremge0addcl 12155 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) → (𝐴 + 𝐵) ∈ (0[,)+∞))

Theoremge0mulcl 12156 The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) → (𝐴 · 𝐵) ∈ (0[,)+∞))

Theoremge0xaddcl 12157 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞))

Theoremge0xmulcl 12158 The nonnegative extended reals are closed under multiplication. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴 ·e 𝐵) ∈ (0[,]+∞))

Theoremlbicc2 12159 The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))

Theoremubicc2 12160 The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐵 ∈ (𝐴[,]𝐵))

Theorem0elunit 12161 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
0 ∈ (0[,]1)

Theorem1elunit 12162 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
1 ∈ (0[,]1)

Theoremiooneg 12163 Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ -𝐶 ∈ (-𝐵(,)-𝐴)))

Theoremiccneg 12164 Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ -𝐶 ∈ (-𝐵[,]-𝐴)))

Theoremicoshft 12165 A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝑋 ∈ (𝐴[,)𝐵) → (𝑋 + 𝐶) ∈ ((𝐴 + 𝐶)[,)(𝐵 + 𝐶))))

Theoremicoshftf1o 12166* Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
𝐹 = (𝑥 ∈ (𝐴[,)𝐵) ↦ (𝑥 + 𝐶))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐹:(𝐴[,)𝐵)–1-1-onto→((𝐴 + 𝐶)[,)(𝐵 + 𝐶)))

Theoremicoun 12167 The union of end-to-end closed-below, open-above real intervals. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴𝐵𝐵𝐶)) → ((𝐴[,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴[,)𝐶))

Theoremicodisj 12168 End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) → ((𝐴[,)𝐵) ∩ (𝐵[,)𝐶)) = ∅)

Theoremsnunioo 12169 The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))

Theoremsnunico 12170 The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → ((𝐴[,)𝐵) ∪ {𝐵}) = (𝐴[,]𝐵))

Theoremsnunioc 12171 The closure of the open end of a left-open real interval. (Contributed by Thierry Arnoux, 28-Mar-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → ({𝐴} ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))

Theoremprunioo 12172 The closure of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵))

Theoremioodisj 12173 If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.)
((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ*𝐷 ∈ ℝ*)) ∧ 𝐵𝐶) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = ∅)

Theoremioojoin 12174 Join two open intervals to create a third. (Contributed by NM, 11-Aug-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵 < 𝐶)) → (((𝐴(,)𝐵) ∪ {𝐵}) ∪ (𝐵(,)𝐶)) = (𝐴(,)𝐶))

Theoremdifreicc 12175 The class difference of and a closed interval. (Contributed by FL, 18-Jun-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ ∖ (𝐴[,]𝐵)) = ((-∞(,)𝐴) ∪ (𝐵(,)+∞)))

Theoremiccsplit 12176 Split a closed interval into the union of two closed intervals. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝐴[,]𝐶) ∪ (𝐶[,]𝐵)))

Theoremiccshftr 12177 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 + 𝑅) = 𝐶    &   (𝐵 + 𝑅) = 𝐷       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷)))

Theoremiccshftri 12178 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝑅 ∈ ℝ    &   (𝐴 + 𝑅) = 𝐶    &   (𝐵 + 𝑅) = 𝐷       (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))

Theoremiccshftl 12179 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝑅) = 𝐶    &   (𝐵𝑅) = 𝐷       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋𝑅) ∈ (𝐶[,]𝐷)))

Theoremiccshftli 12180 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝑅 ∈ ℝ    &   (𝐴𝑅) = 𝐶    &   (𝐵𝑅) = 𝐷       (𝑋 ∈ (𝐴[,]𝐵) → (𝑋𝑅) ∈ (𝐶[,]𝐷))

Theoremiccdil 12181 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 · 𝑅) = 𝐶    &   (𝐵 · 𝑅) = 𝐷       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 · 𝑅) ∈ (𝐶[,]𝐷)))

Theoremiccdili 12182 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝑅 ∈ ℝ+    &   (𝐴 · 𝑅) = 𝐶    &   (𝐵 · 𝑅) = 𝐷       (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 · 𝑅) ∈ (𝐶[,]𝐷))

Theoremicccntr 12183 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴 / 𝑅) = 𝐶    &   (𝐵 / 𝑅) = 𝐷       (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 / 𝑅) ∈ (𝐶[,]𝐷)))

Theoremicccntri 12184 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝑅 ∈ ℝ+    &   (𝐴 / 𝑅) = 𝐶    &   (𝐵 / 𝑅) = 𝐷       (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 / 𝑅) ∈ (𝐶[,]𝐷))

Theoremdivelunit 12185 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ∈ (0[,]1) ↔ 𝐴𝐵))

Theoremlincmb01cmp 12186 A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵)) ∈ (𝐴[,]𝐵))

Theoremiccf1o 12187* Describe a bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵]. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) ∧ 𝐹 = (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦𝐴) / (𝐵𝐴)))))

Theoremiccen 12188 Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (0[,]1) ≈ (𝐴[,]𝐵))

Theoremxov1plusxeqvd 12189 A complex number 𝑋 is positive real iff 𝑋 / (1 + 𝑋) is in (0(,)1). Deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ -1)       (𝜑 → (𝑋 ∈ ℝ+ ↔ (𝑋 / (1 + 𝑋)) ∈ (0(,)1)))

Theoremunitssre 12190 (0[,]1) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
(0[,]1) ⊆ ℝ

Theoremsupicc 12191 Supremum of a bounded set of real numbers. (Contributed by Thierry Arnoux, 17-May-2019.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑𝐴 ≠ ∅)       (𝜑 → sup(𝐴, ℝ, < ) ∈ (𝐵[,]𝐶))

Theoremsupiccub 12192 The supremum of a bounded set of real numbers is an upper bound. (Contributed by Thierry Arnoux, 20-May-2019.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐷𝐴)       (𝜑𝐷 ≤ sup(𝐴, ℝ, < ))

Theoremsupicclub 12193* The supremum of a bounded set of real numbers is the least upper bound. (Contributed by Thierry Arnoux, 23-May-2019.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐷𝐴)       (𝜑 → (𝐷 < sup(𝐴, ℝ, < ) ↔ ∃𝑧𝐴 𝐷 < 𝑧))

Theoremsupicclub2 12194* The supremum of a bounded set of real numbers is the least upper bound. (Contributed by Thierry Arnoux, 23-May-2019.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐷𝐴)    &   ((𝜑𝑧𝐴) → 𝑧𝐷)       (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐷)

Theoremzltaddlt1le 12195 The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ (0(,)1)) → ((𝑀 + 𝐴) < 𝑁 ↔ (𝑀 + 𝐴) ≤ 𝑁))

Theoremxnn0xrge0 12196 An extended nonnegative integer is an extended nonnegative real. (Contributed by AV, 10-Dec-2020.)
(𝐴 ∈ ℕ0*𝐴 ∈ (0[,]+∞))

5.5.5  Finite intervals of integers

Syntaxcfz 12197 Extend class notation to include the notation for a contiguous finite set of integers. Read "𝑀...𝑁 " as "the set of integers from 𝑀 to 𝑁 inclusive."
class ...

Definitiondf-fz 12198* Define an operation that produces a finite set of sequential integers. Read "𝑀...𝑁 " as "the set of integers from 𝑀 to 𝑁 inclusive." See fzval 12199 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})

Theoremfzval 12199* The value of a finite set of sequential integers. E.g., 2...5 means the set {2, 3, 4, 5}. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where _k means our 1...𝑘; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})

Theoremfzval2 12200 An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))

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