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Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiccgelb 8801 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 8802 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)))

Theoremelioo4g 8803 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 8804 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
(𝐴(,)𝐵) ⊆ ℝ

Theoremelioc2 8805 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 8806 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 8807 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 8808 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 8809 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 8810 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 8811 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 8812 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 8813 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 8814 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 8815 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 8816 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 8817 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
(-∞(,)+∞) = ℝ

Theoremiccmax 8818 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
(-∞[,]+∞) = ℝ*

Theoremioopos 8819 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
(0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥}

Theoremioorp 8820 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(0(,)+∞) = ℝ+

Theoremiooshf 8821 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 8822 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 8823 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 8824 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 8825 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 8826 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 8827 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 8828 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
(𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)

Theoremicossicc 8829 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
(𝐴[,)𝐵) ⊆ (𝐴[,]𝐵)

Theoremiocssicc 8830 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
(𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)

Theoremioossico 8831 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
(𝐴(,)𝐵) ⊆ (𝐴[,)𝐵)

Theoremiocssioo 8832 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 8833 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 8834 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 8835* A nonempty subset of a closed real interval satisfies the conditions for the existence of its supremum. To be useful without excluded middle, we'll probably need to change not equal to apart, and perhaps make other changes, but the theorem does hold as stated here. (Contributed by Paul Chapman, 21-Jan-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑆 ⊆ (𝐴[,]𝐵) ∧ 𝐶𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑦𝑥))

Theoremelioopnf 8836 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))

Theoremelioomnf 8837 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐴 ∈ ℝ* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴)))

Theoremelicopnf 8838 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
(𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴𝐵)))

Theoremrepos 8839 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
(𝐴 ∈ (0(,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))

Theoremioof 8840 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 8841 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 8842 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 8843* 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)})

Theoremioorebasg 8844 Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ ran (,))

Theoremelrege0 8845 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))

Theoremrge0ssre 8846 Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
(0[,)+∞) ⊆ ℝ

Theoremelxrge0 8847 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
(𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴))

Theorem0e0icopnf 8848 0 is a member of (0[,)+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ∈ (0[,)+∞)

Theorem0e0iccpnf 8849 0 is a member of (0[,]+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ∈ (0[,]+∞)

Theoremge0addcl 8850 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) → (𝐴 + 𝐵) ∈ (0[,)+∞))

Theoremge0mulcl 8851 The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) → (𝐴 · 𝐵) ∈ (0[,)+∞))

Theoremlbicc2 8852 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 8853 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 8854 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
0 ∈ (0[,]1)

Theorem1elunit 8855 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
1 ∈ (0[,]1)

Theoremiooneg 8856 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 8857 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 8858 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 8859* 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)))

Theoremicodisj 8860 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)) = ∅)

Theoremioodisj 8861 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)) = ∅)

Theoremiccshftr 8862 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 8863 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 8864 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 8865 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 8866 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 8867 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 8868 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 8869 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 8870 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 8871 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 8872* 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)))))

Theoremunitssre 8873 (0[,]1) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
(0[,]1) ⊆ ℝ

3.5.4  Finite intervals of integers

Syntaxcfz 8874 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 8875* 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 8876 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})

Theoremfzval 8876* 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 8877 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))

Theoremfzf 8878 Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
...:(ℤ × ℤ)⟶𝒫 ℤ

Theoremelfz1 8879 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ ℤ ∧ 𝑀𝐾𝐾𝑁)))

Theoremelfz 8880 Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀𝐾𝐾𝑁)))

Theoremelfz2 8881 Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show 𝑀 ∈ ℤ and 𝑁 ∈ ℤ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀𝐾𝐾𝑁)))

Theoremelfz5 8882 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
((𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾𝑁))

Theoremelfz4 8883 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀𝐾𝐾𝑁)) → 𝐾 ∈ (𝑀...𝑁))

Theoremelfzuzb 8884 Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ𝐾)))

Theoremeluzfz 8885 Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ𝐾)) → 𝐾 ∈ (𝑀...𝑁))

Theoremelfzuz 8886 A member of a finite set of sequential integers belongs to an upper set of integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))

Theoremelfzuz3 8887 Membership in a finite set of sequential integers implies membership in an upper set of integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝐾))

Theoremelfzel2 8888 Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)

Theoremelfzel1 8889 Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)

Theoremelfzelz 8890 A member of a finite set of sequential integer is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)

Theoremelfzle1 8891 A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) → 𝑀𝐾)

Theoremelfzle2 8892 A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) → 𝐾𝑁)

Theoremelfzuz2 8893 Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ𝑀))

Theoremelfzle3 8894 Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) → 𝑀𝑁)

Theoremeluzfz1 8895 Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))

Theoremeluzfz2 8896 Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))

Theoremeluzfz2b 8897 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) ↔ 𝑁 ∈ (𝑀...𝑁))

Theoremelfz3 8898 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
(𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁))

Theoremelfz1eq 8899 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
(𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁)

Theoremelfzubelfz 8900 If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018.)
(𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (𝑀...𝑁))

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