Theorem List for Intuitionistic Logic Explorer - 9701-9800 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | iocval 9701* |
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 < 𝑥 ∧ 𝑥 ≤ 𝐵)}) |
|
Theorem | icoval 9702* |
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* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)}) |
|
Theorem | iccval 9703* |
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* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)}) |
|
Theorem | elioo2 9704 |
Membership in an open interval of extended reals. (Contributed by NM,
6-Feb-2007.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
|
Theorem | elioc1 9705 |
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 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
|
Theorem | elico1 9706 |
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* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
|
Theorem | elicc1 9707 |
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* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
|
Theorem | iccid 9708 |
A closed interval with identical lower and upper bounds is a singleton.
(Contributed by Jeff Hankins, 13-Jul-2009.)
|
⊢ (𝐴 ∈ ℝ* → (𝐴[,]𝐴) = {𝐴}) |
|
Theorem | icc0r 9709 |
An empty closed interval of extended reals. (Contributed by Jim
Kingdon, 30-Mar-2020.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ (𝐵 < 𝐴 → (𝐴[,]𝐵) = ∅)) |
|
Theorem | eliooxr 9710 |
An inhabited open interval spans an interval of extended reals.
(Contributed by NM, 17-Aug-2008.)
|
⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈
ℝ*)) |
|
Theorem | eliooord 9711 |
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 < 𝐴 ∧ 𝐴 < 𝐶)) |
|
Theorem | ubioc1 9712 |
The upper bound belongs to an open-below, closed-above interval. See
ubicc2 9768. (Contributed by FL, 29-May-2014.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵)) |
|
Theorem | lbico1 9713 |
The lower bound belongs to a closed-below, open-above interval. See
lbicc2 9767. (Contributed by FL, 29-May-2014.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵)) |
|
Theorem | iccleub 9714 |
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)) → 𝐶 ≤ 𝐵) |
|
Theorem | iccgelb 9715 |
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)) → 𝐴 ≤ 𝐶) |
|
Theorem | elioo5 9716 |
Membership in an open interval of extended reals. (Contributed by NM,
17-Aug-2008.)
|
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
|
Theorem | elioo4g 9717 |
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 < 𝐶 ∧ 𝐶 < 𝐵))) |
|
Theorem | ioossre 9718 |
An open interval is a set of reals. (Contributed by NM,
31-May-2007.)
|
⊢ (𝐴(,)𝐵) ⊆ ℝ |
|
Theorem | elioc2 9719 |
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 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
|
Theorem | elico2 9720 |
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 ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) |
|
Theorem | elicc2 9721 |
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 ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
|
Theorem | elicc2i 9722 |
Inference for membership in a closed interval. (Contributed by Scott
Fenton, 3-Jun-2013.)
|
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈
ℝ ⇒ ⊢ (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
|
Theorem | elicc4 9723 |
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) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
|
Theorem | iccss 9724 |
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) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵)) |
|
Theorem | iccssioo 9725 |
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 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵)) |
|
Theorem | icossico 9726 |
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*)
∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,)𝐷) ⊆ (𝐴[,)𝐵)) |
|
Theorem | iccss2 9727 |
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)) |
|
Theorem | iccssico 9728 |
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*)
∧ (𝐴 ≤ 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵)) |
|
Theorem | iccssioo2 9729 |
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)) |
|
Theorem | iccssico2 9730 |
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)) |
|
Theorem | ioomax 9731 |
The open interval from minus to plus infinity. (Contributed by NM,
6-Feb-2007.)
|
⊢ (-∞(,)+∞) =
ℝ |
|
Theorem | iccmax 9732 |
The closed interval from minus to plus infinity. (Contributed by Mario
Carneiro, 4-Jul-2014.)
|
⊢ (-∞[,]+∞) =
ℝ* |
|
Theorem | ioopos 9733 |
The set of positive reals expressed as an open interval. (Contributed by
NM, 7-May-2007.)
|
⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥} |
|
Theorem | ioorp 9734 |
The set of positive reals expressed as an open interval. (Contributed by
Steve Rodriguez, 25-Nov-2007.)
|
⊢ (0(,)+∞) =
ℝ+ |
|
Theorem | iooshf 9735 |
Shift the arguments of the open interval function. (Contributed by NM,
17-Aug-2008.)
|
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 − 𝐵) ∈ (𝐶(,)𝐷) ↔ 𝐴 ∈ ((𝐶 + 𝐵)(,)(𝐷 + 𝐵)))) |
|
Theorem | iocssre 9736 |
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) |
|
Theorem | icossre 9737 |
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) |
|
Theorem | iccssre 9738 |
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) |
|
Theorem | iccssxr 9739 |
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* |
|
Theorem | iocssxr 9740 |
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* |
|
Theorem | icossxr 9741 |
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* |
|
Theorem | ioossicc 9742 |
An open interval is a subset of its closure. (Contributed by Paul
Chapman, 18-Oct-2007.)
|
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
|
Theorem | icossicc 9743 |
A closed-below, open-above interval is a subset of its closure.
(Contributed by Thierry Arnoux, 25-Oct-2016.)
|
⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵) |
|
Theorem | iocssicc 9744 |
A closed-above, open-below interval is a subset of its closure.
(Contributed by Thierry Arnoux, 1-Apr-2017.)
|
⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) |
|
Theorem | ioossico 9745 |
An open interval is a subset of its closure-below. (Contributed by
Thierry Arnoux, 3-Mar-2017.)
|
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵) |
|
Theorem | iocssioo 9746 |
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 ≤ 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶(,]𝐷) ⊆ (𝐴(,)𝐵)) |
|
Theorem | icossioo 9747 |
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 < 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,)𝐷) ⊆ (𝐴(,)𝐵)) |
|
Theorem | ioossioo 9748 |
Condition for an open interval to be a subset of an open interval.
(Contributed by Thierry Arnoux, 26-Sep-2017.)
|
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵)) |
|
Theorem | iccsupr 9749* |
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) ∧ 𝐶 ∈ 𝑆) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥)) |
|
Theorem | elioopnf 9750 |
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18-Jun-2014.)
|
⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))) |
|
Theorem | elioomnf 9751 |
Membership in an unbounded interval of extended reals. (Contributed by
Mario Carneiro, 18-Jun-2014.)
|
⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴))) |
|
Theorem | elicopnf 9752 |
Membership in a closed unbounded interval of reals. (Contributed by
Mario Carneiro, 16-Sep-2014.)
|
⊢ (𝐴 ∈ ℝ → (𝐵 ∈ (𝐴[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵))) |
|
Theorem | repos 9753 |
Two ways of saying that a real number is positive. (Contributed by NM,
7-May-2007.)
|
⊢ (𝐴 ∈ (0(,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 <
𝐴)) |
|
Theorem | ioof 9754 |
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 |
|
Theorem | iccf 9755 |
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* |
|
Theorem | unirnioo 9756 |
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
(,) |
|
Theorem | dfioo2 9757* |
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 < 𝑤 ∧ 𝑤 < 𝑦)}) |
|
Theorem | ioorebasg 9758 |
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
(,)) |
|
Theorem | elrege0 9759 |
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)) |
|
Theorem | rge0ssre 9760 |
Nonnegative real numbers are real numbers. (Contributed by Thierry
Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
|
⊢ (0[,)+∞) ⊆
ℝ |
|
Theorem | elxrge0 9761 |
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28-Jun-2014.)
|
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ*
∧ 0 ≤ 𝐴)) |
|
Theorem | 0e0icopnf 9762 |
0 is a member of (0[,)+∞) (common case).
(Contributed by David
A. Wheeler, 8-Dec-2018.)
|
⊢ 0 ∈ (0[,)+∞) |
|
Theorem | 0e0iccpnf 9763 |
0 is a member of (0[,]+∞) (common case).
(Contributed by David
A. Wheeler, 8-Dec-2018.)
|
⊢ 0 ∈ (0[,]+∞) |
|
Theorem | ge0addcl 9764 |
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19-Jun-2014.)
|
⊢ ((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) →
(𝐴 + 𝐵) ∈ (0[,)+∞)) |
|
Theorem | ge0mulcl 9765 |
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19-Jun-2014.)
|
⊢ ((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) →
(𝐴 · 𝐵) ∈
(0[,)+∞)) |
|
Theorem | ge0xaddcl 9766 |
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 26-Aug-2015.)
|
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) →
(𝐴 +𝑒
𝐵) ∈
(0[,]+∞)) |
|
Theorem | lbicc2 9767 |
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*
∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
|
Theorem | ubicc2 9768 |
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*
∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
|
Theorem | 0elunit 9769 |
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
|
⊢ 0 ∈ (0[,]1) |
|
Theorem | 1elunit 9770 |
One is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
|
⊢ 1 ∈ (0[,]1) |
|
Theorem | iooneg 9771 |
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))) |
|
Theorem | iccneg 9772 |
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))) |
|
Theorem | icoshft 9773 |
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)))) |
|
Theorem | icoshftf1o 9774* |
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))) |
|
Theorem | icodisj 9775 |
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)) = ∅) |
|
Theorem | ioodisj 9776 |
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*)) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = ∅) |
|
Theorem | iccshftr 9777 |
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))) |
|
Theorem | iccshftri 9778 |
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)) |
|
Theorem | iccshftl 9779 |
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))) |
|
Theorem | iccshftli 9780 |
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)) |
|
Theorem | iccdil 9781 |
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))) |
|
Theorem | iccdili 9782 |
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)) |
|
Theorem | icccntr 9783 |
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))) |
|
Theorem | icccntri 9784 |
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)) |
|
Theorem | divelunit 9785 |
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) ↔ 𝐴 ≤ 𝐵)) |
|
Theorem | lincmb01cmp 9786 |
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)) |
|
Theorem | iccf1o 9787* |
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) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))))) |
|
Theorem | unitssre 9788 |
(0[,]1) is a subset of the reals. (Contributed by
David Moews,
28-Feb-2017.)
|
⊢ (0[,]1) ⊆ ℝ |
|
Theorem | zltaddlt1le 9789 |
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)) |
|
4.5.4 Finite intervals of integers
|
|
Syntax | cfz 9790 |
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 ... |
|
Definition | df-fz 9791* |
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 9792 for its value and additional comments.
(Contributed by NM, 6-Sep-2005.)
|
⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) |
|
Theorem | fzval 9792* |
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
ℕk 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 ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) |
|
Theorem | fzval2 9793 |
An alternate 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)) |
|
Theorem | fzf 9794 |
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 |
|
Theorem | elfz1 9795 |
Membership in a finite set of sequential integers. (Contributed by NM,
21-Jul-2005.)
|
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
|
Theorem | elfz 9796 |
Membership in a finite set of sequential integers. (Contributed by NM,
29-Sep-2005.)
|
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
|
Theorem | elfz2 9797 |
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) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
|
Theorem | elfz5 9798 |
Membership in a finite set of sequential integers. (Contributed by NM,
26-Dec-2005.)
|
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≤ 𝑁)) |
|
Theorem | elfz4 9799 |
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) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝐾 ∈ (𝑀...𝑁)) |
|
Theorem | elfzuzb 9800 |
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 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) |