Theorem List for Intuitionistic Logic Explorer - 9601-9700 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | iccleub 9601 |
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 9602 |
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 9603 |
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 9604 |
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 9605 |
An open interval is a set of reals. (Contributed by NM,
31-May-2007.)
|
⊢ (𝐴(,)𝐵) ⊆ ℝ |
|
Theorem | elioc2 9606 |
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 9607 |
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 9608 |
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 9609 |
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 9610 |
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 9611 |
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 9612 |
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 9613 |
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 9614 |
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 9615 |
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 9616 |
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 9617 |
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 9618 |
The open interval from minus to plus infinity. (Contributed by NM,
6-Feb-2007.)
|
⊢ (-∞(,)+∞) =
ℝ |
|
Theorem | iccmax 9619 |
The closed interval from minus to plus infinity. (Contributed by Mario
Carneiro, 4-Jul-2014.)
|
⊢ (-∞[,]+∞) =
ℝ* |
|
Theorem | ioopos 9620 |
The set of positive reals expressed as an open interval. (Contributed by
NM, 7-May-2007.)
|
⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥} |
|
Theorem | ioorp 9621 |
The set of positive reals expressed as an open interval. (Contributed by
Steve Rodriguez, 25-Nov-2007.)
|
⊢ (0(,)+∞) =
ℝ+ |
|
Theorem | iooshf 9622 |
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 9623 |
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 9624 |
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 9625 |
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 9626 |
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 9627 |
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 9628 |
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 9629 |
An open interval is a subset of its closure. (Contributed by Paul
Chapman, 18-Oct-2007.)
|
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
|
Theorem | icossicc 9630 |
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 9631 |
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 9632 |
An open interval is a subset of its closure-below. (Contributed by
Thierry Arnoux, 3-Mar-2017.)
|
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵) |
|
Theorem | iocssioo 9633 |
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 9634 |
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 9635 |
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 9636* |
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 9637 |
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 9638 |
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 9639 |
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 9640 |
Two ways of saying that a real number is positive. (Contributed by NM,
7-May-2007.)
|
⊢ (𝐴 ∈ (0(,)+∞) ↔ (𝐴 ∈ ℝ ∧ 0 <
𝐴)) |
|
Theorem | ioof 9641 |
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 9642 |
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 9643 |
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 9644* |
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 9645 |
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 9646 |
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 9647 |
Nonnegative real numbers are real numbers. (Contributed by Thierry
Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
|
⊢ (0[,)+∞) ⊆
ℝ |
|
Theorem | elxrge0 9648 |
Elementhood in the set of nonnegative extended reals. (Contributed by
Mario Carneiro, 28-Jun-2014.)
|
⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ*
∧ 0 ≤ 𝐴)) |
|
Theorem | 0e0icopnf 9649 |
0 is a member of (0[,)+∞) (common case).
(Contributed by David
A. Wheeler, 8-Dec-2018.)
|
⊢ 0 ∈ (0[,)+∞) |
|
Theorem | 0e0iccpnf 9650 |
0 is a member of (0[,]+∞) (common case).
(Contributed by David
A. Wheeler, 8-Dec-2018.)
|
⊢ 0 ∈ (0[,]+∞) |
|
Theorem | ge0addcl 9651 |
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 19-Jun-2014.)
|
⊢ ((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) →
(𝐴 + 𝐵) ∈ (0[,)+∞)) |
|
Theorem | ge0mulcl 9652 |
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19-Jun-2014.)
|
⊢ ((𝐴 ∈ (0[,)+∞) ∧ 𝐵 ∈ (0[,)+∞)) →
(𝐴 · 𝐵) ∈
(0[,)+∞)) |
|
Theorem | ge0xaddcl 9653 |
The nonnegative reals are closed under addition. (Contributed by Mario
Carneiro, 26-Aug-2015.)
|
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) →
(𝐴 +𝑒
𝐵) ∈
(0[,]+∞)) |
|
Theorem | lbicc2 9654 |
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 9655 |
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 9656 |
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
|
⊢ 0 ∈ (0[,]1) |
|
Theorem | 1elunit 9657 |
One is an element of the closed unit. (Contributed by Scott Fenton,
11-Jun-2013.)
|
⊢ 1 ∈ (0[,]1) |
|
Theorem | iooneg 9658 |
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 9659 |
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 9660 |
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 9661* |
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 9662 |
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 9663 |
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 9664 |
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 9665 |
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 9666 |
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 9667 |
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 9668 |
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 9669 |
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 9670 |
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 9671 |
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 9672 |
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 9673 |
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 9674* |
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 9675 |
(0[,]1) is a subset of the reals. (Contributed by
David Moews,
28-Feb-2017.)
|
⊢ (0[,]1) ⊆ ℝ |
|
Theorem | zltaddlt1le 9676 |
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)) |
|
3.5.4 Finite intervals of integers
|
|
Syntax | cfz 9677 |
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 9678* |
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 9679 for its value and additional comments.
(Contributed by NM, 6-Sep-2005.)
|
⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) |
|
Theorem | fzval 9679* |
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 9680 |
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 9681 |
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 9682 |
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 9683 |
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 9684 |
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 9685 |
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 9686 |
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 9687 |
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))) |
|
Theorem | eluzfz 9688 |
Membership in a finite set of sequential integers. (Contributed by NM,
4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
|
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → 𝐾 ∈ (𝑀...𝑁)) |
|
Theorem | elfzuz 9689 |
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 ∈ (ℤ≥‘𝑀)) |
|
Theorem | elfzuz3 9690 |
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 ∈ (ℤ≥‘𝐾)) |
|
Theorem | elfzel2 9691 |
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) |
|
Theorem | elfzel1 9692 |
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) |
|
Theorem | elfzelz 9693 |
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) |
|
Theorem | elfzle1 9694 |
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) → 𝑀 ≤ 𝐾) |
|
Theorem | elfzle2 9695 |
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) → 𝐾 ≤ 𝑁) |
|
Theorem | elfzuz2 9696 |
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 ∈ (ℤ≥‘𝑀)) |
|
Theorem | elfzle3 9697 |
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) → 𝑀 ≤ 𝑁) |
|
Theorem | eluzfz1 9698 |
Membership in a finite set of sequential integers - special case.
(Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro,
28-Apr-2015.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
|
Theorem | eluzfz2 9699 |
Membership in a finite set of sequential integers - special case.
(Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro,
28-Apr-2015.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
|
Theorem | eluzfz2b 9700 |
Membership in a finite set of sequential integers - special case.
(Contributed by NM, 14-Sep-2005.)
|
⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁)) |