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Theorem List for Intuitionistic Logic Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremixxssixx 8501* An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})    &   𝑃 = (x *, y * ↦ {z * ∣ (x𝑇z z𝑈y)})    &   ((A * w *) → (A𝑅wA𝑇w))    &   ((w * B *) → (w𝑆Bw𝑈B))       (A𝑂B) ⊆ (A𝑃B)
 
Theoremixxdisj 8502* Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})    &   𝑃 = (x *, y * ↦ {z * ∣ (x𝑇z z𝑈y)})    &   ((B * w *) → (B𝑇w ↔ ¬ w𝑆B))       ((A * B * 𝐶 *) → ((A𝑂B) ∩ (B𝑃𝐶)) = ∅)
 
Theoremixxss1 8503* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})    &   𝑃 = (x *, y * ↦ {z * ∣ (x𝑇z z𝑆y)})    &   ((A * B * w *) → ((A𝑊B B𝑇w) → A𝑅w))       ((A * A𝑊B) → (B𝑃𝐶) ⊆ (A𝑂𝐶))
 
Theoremixxss2 8504* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})    &   𝑃 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑇y)})    &   ((w * B * 𝐶 *) → ((w𝑇B B𝑊𝐶) → w𝑆𝐶))       ((𝐶 * B𝑊𝐶) → (A𝑃B) ⊆ (A𝑂𝐶))
 
Theoremixxss12 8505* Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 20-Feb-2015.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝑂 = (x *, y * ↦ {z * ∣ (x𝑅z z𝑆y)})    &   𝑃 = (x *, y * ↦ {z * ∣ (x𝑇z z𝑈y)})    &   ((A * 𝐶 * w *) → ((A𝑊𝐶 𝐶𝑇w) → A𝑅w))    &   ((w * 𝐷 * B *) → ((w𝑈𝐷 𝐷𝑋B) → w𝑆B))       (((A * B *) (A𝑊𝐶 𝐷𝑋B)) → (𝐶𝑃𝐷) ⊆ (A𝑂B))
 
Theoremiooex 8506 The set of open intervals of extended reals exists. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
(,) V
 
Theoremiooval 8507* Value of the open interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (A(,)B) = {x * ∣ (A < x x < B)})
 
Theoremiooidg 8508 An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.)
(A * → (A(,)A) = ∅)
 
Theoremelioo3g 8509 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 * and B *. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
(𝐶 (A(,)B) ↔ ((A * B * 𝐶 *) (A < 𝐶 𝐶 < B)))
 
Theoremelioo1 8510 Membership in an open interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (𝐶 (A(,)B) ↔ (𝐶 * A < 𝐶 𝐶 < B)))
 
Theoremelioore 8511 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)
 
Theoremlbioog 8512 An open interval does not contain its left endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
((A * B *) → ¬ A (A(,)B))
 
Theoremubioog 8513 An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.)
((A * B *) → ¬ B (A(,)B))
 
Theoremiooval2 8514* Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (A(,)B) = {x ℝ ∣ (A < x x < B)})
 
Theoremiooss1 8515 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)
((A * AB) → (B(,)𝐶) ⊆ (A(,)𝐶))
 
Theoremiooss2 8516 Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝐶 * B𝐶) → (A(,)B) ⊆ (A(,)𝐶))
 
Theoremiocval 8517* Value of the open-below, closed-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (A(,]B) = {x * ∣ (A < x xB)})
 
Theoremicoval 8518* Value of the closed-below, open-above interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (A[,)B) = {x * ∣ (Ax x < B)})
 
Theoremiccval 8519* Value of the closed interval function. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (A[,]B) = {x * ∣ (Ax xB)})
 
Theoremelioo2 8520 Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
((A * B *) → (𝐶 (A(,)B) ↔ (𝐶 A < 𝐶 𝐶 < B)))
 
Theoremelioc1 8521 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 * B *) → (𝐶 (A(,]B) ↔ (𝐶 * A < 𝐶 𝐶B)))
 
Theoremelico1 8522 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 * B *) → (𝐶 (A[,)B) ↔ (𝐶 * A𝐶 𝐶 < B)))
 
Theoremelicc1 8523 Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
((A * B *) → (𝐶 (A[,]B) ↔ (𝐶 * A𝐶 𝐶B)))
 
Theoremiccid 8524 A closed interval with identical lower and upper bounds is a singleton. (Contributed by Jeff Hankins, 13-Jul-2009.)
(A * → (A[,]A) = {A})
 
Theoremicc0r 8525 An empty closed interval of extended reals. (Contributed by Jim Kingdon, 30-Mar-2020.)
((A * B *) → (B < A → (A[,]B) = ∅))
 
Theoremeliooxr 8526 An inhabited open interval spans an interval of extended reals. (Contributed by NM, 17-Aug-2008.)
(A (B(,)𝐶) → (B * 𝐶 *))
 
Theoremeliooord 8527 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))
 
Theoremubioc1 8528 The upper bound belongs to an open-below, closed-above interval. See ubicc2 8583. (Contributed by FL, 29-May-2014.)
((A * B * A < B) → B (A(,]B))
 
Theoremlbico1 8529 The lower bound belongs to a closed-below, open-above interval. See lbicc2 8582. (Contributed by FL, 29-May-2014.)
((A * B * A < B) → A (A[,)B))
 
Theoremiccleub 8530 An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)
((A * B * 𝐶 (A[,]B)) → 𝐶B)
 
Theoremiccgelb 8531 An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)
((A * B * 𝐶 (A[,]B)) → A𝐶)
 
Theoremelioo5 8532 Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)
((A * B * 𝐶 *) → (𝐶 (A(,)B) ↔ (A < 𝐶 𝐶 < B)))
 
Theoremelioo4g 8533 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 * B * 𝐶 ℝ) (A < 𝐶 𝐶 < B)))
 
Theoremioossre 8534 An open interval is a set of reals. (Contributed by NM, 31-May-2007.)
(A(,)B) ⊆ ℝ
 
Theoremelioc2 8535 Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
((A * B ℝ) → (𝐶 (A(,]B) ↔ (𝐶 A < 𝐶 𝐶B)))
 
Theoremelico2 8536 Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)
((A B *) → (𝐶 (A[,)B) ↔ (𝐶 A𝐶 𝐶 < B)))
 
Theoremelicc2 8537 Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)
((A B ℝ) → (𝐶 (A[,]B) ↔ (𝐶 A𝐶 𝐶B)))
 
Theoremelicc2i 8538 Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013.)
A     &   B        (𝐶 (A[,]B) ↔ (𝐶 A𝐶 𝐶B))
 
Theoremelicc4 8539 Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
((A * B * 𝐶 *) → (𝐶 (A[,]B) ↔ (A𝐶 𝐶B)))
 
Theoremiccss 8540 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 B ℝ) (A𝐶 𝐷B)) → (𝐶[,]𝐷) ⊆ (A[,]B))
 
Theoremiccssioo 8541 Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.)
(((A * B *) (A < 𝐶 𝐷 < B)) → (𝐶[,]𝐷) ⊆ (A(,)B))
 
Theoremicossico 8542 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 * B *) (A𝐶 𝐷B)) → (𝐶[,)𝐷) ⊆ (A[,)B))
 
Theoremiccss2 8543 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 8544 Condition for a closed interval to be a subset of a half-open interval. (Contributed by Mario Carneiro, 9-Sep-2015.)
(((A * B *) (A𝐶 𝐷 < B)) → (𝐶[,]𝐷) ⊆ (A[,)B))
 
Theoremiccssioo2 8545 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 8546 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 8547 The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)
(-∞(,)+∞) = ℝ
 
Theoremiccmax 8548 The closed interval from minus to plus infinity. (Contributed by Mario Carneiro, 4-Jul-2014.)
(-∞[,]+∞) = ℝ*
 
Theoremioopos 8549 The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)
(0(,)+∞) = {x ℝ ∣ 0 < x}
 
Theoremioorp 8550 The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(0(,)+∞) = ℝ+
 
Theoremiooshf 8551 Shift the arguments of the open interval function. (Contributed by NM, 17-Aug-2008.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((AB) (𝐶(,)𝐷) ↔ A ((𝐶 + B)(,)(𝐷 + B))))
 
Theoremiocssre 8552 A closed-above interval with real upper bound is a set of reals. (Contributed by FL, 29-May-2014.)
((A * B ℝ) → (A(,]B) ⊆ ℝ)
 
Theoremicossre 8553 A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)
((A B *) → (A[,)B) ⊆ ℝ)
 
Theoremiccssre 8554 A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007.) (Proof shortened by Paul Chapman, 21-Jan-2008.)
((A B ℝ) → (A[,]B) ⊆ ℝ)
 
Theoremiccssxr 8555 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 8556 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 8557 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 8558 An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.)
(A(,)B) ⊆ (A[,]B)
 
Theoremicossicc 8559 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
(A[,)B) ⊆ (A[,]B)
 
Theoremiocssicc 8560 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
(A(,]B) ⊆ (A[,]B)
 
Theoremioossico 8561 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
(A(,)B) ⊆ (A[,)B)
 
Theoremiocssioo 8562 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
(((A * B *) (A𝐶 𝐷 < B)) → (𝐶(,]𝐷) ⊆ (A(,)B))
 
Theoremicossioo 8563 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
(((A * B *) (A < 𝐶 𝐷B)) → (𝐶[,)𝐷) ⊆ (A(,)B))
 
Theoremioossioo 8564 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
(((A * B *) (A𝐶 𝐷B)) → (𝐶(,)𝐷) ⊆ (A(,)B))
 
Theoremiccsupr 8565* 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 B ℝ) 𝑆 ⊆ (A[,]B) 𝐶 𝑆) → (𝑆 ⊆ ℝ 𝑆 ≠ ∅ x y 𝑆 yx))
 
Theoremelioopnf 8566 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
(A * → (B (A(,)+∞) ↔ (B A < B)))
 
Theoremelioomnf 8567 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)
(A * → (B (-∞(,)A) ↔ (B B < A)))
 
Theoremelicopnf 8568 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)
(A ℝ → (B (A[,)+∞) ↔ (B AB)))
 
Theoremrepos 8569 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)
(A (0(,)+∞) ↔ (A 0 < A))
 
Theoremioof 8570 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 8571 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 8572 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 8573* 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 *, y * ↦ {w ℝ ∣ (x < w w < y)})
 
Theoremioorebasg 8574 Open intervals are elements of the set of all open intervals. (Contributed by Jim Kingdon, 4-Apr-2020.)
((A * B *) → (A(,)B) ran (,))
 
Theoremelrege0 8575 The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
(A (0[,)+∞) ↔ (A 0 ≤ A))
 
Theoremrge0ssre 8576 Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.)
(0[,)+∞) ⊆ ℝ
 
Theoremelxrge0 8577 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
(A (0[,]+∞) ↔ (A * 0 ≤ A))
 
Theorem0e0icopnf 8578 0 is a member of (0[,)+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 (0[,)+∞)
 
Theorem0e0iccpnf 8579 0 is a member of (0[,]+∞) (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 (0[,]+∞)
 
Theoremge0addcl 8580 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)
((A (0[,)+∞) B (0[,)+∞)) → (A + B) (0[,)+∞))
 
Theoremge0mulcl 8581 The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)
((A (0[,)+∞) B (0[,)+∞)) → (A · B) (0[,)+∞))
 
Theoremlbicc2 8582 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 * B * AB) → A (A[,]B))
 
Theoremubicc2 8583 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 * B * AB) → B (A[,]B))
 
Theorem0elunit 8584 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
0 (0[,]1)
 
Theorem1elunit 8585 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
1 (0[,]1)
 
Theoremiooneg 8586 Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
((A B 𝐶 ℝ) → (𝐶 (A(,)B) ↔ -𝐶 (-B(,)-A)))
 
Theoremiccneg 8587 Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
((A B 𝐶 ℝ) → (𝐶 (A[,]B) ↔ -𝐶 (-B[,]-A)))
 
Theoremicoshft 8588 A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)
((A B 𝐶 ℝ) → (𝑋 (A[,)B) → (𝑋 + 𝐶) ((A + 𝐶)[,)(B + 𝐶))))
 
Theoremicoshftf1o 8589* 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 B 𝐶 ℝ) → 𝐹:(A[,)B)–1-1-onto→((A + 𝐶)[,)(B + 𝐶)))
 
Theoremicodisj 8590 End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)
((A * B * 𝐶 *) → ((A[,)B) ∩ (B[,)𝐶)) = ∅)
 
Theoremioodisj 8591 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 * B *) (𝐶 * 𝐷 *)) B𝐶) → ((A(,)B) ∩ (𝐶(,)𝐷)) = ∅)
 
Theoremiccshftr 8592 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(A + 𝑅) = 𝐶    &   (B + 𝑅) = 𝐷       (((A B ℝ) (𝑋 𝑅 ℝ)) → (𝑋 (A[,]B) ↔ (𝑋 + 𝑅) (𝐶[,]𝐷)))
 
Theoremiccshftri 8593 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
A     &   B     &   𝑅     &   (A + 𝑅) = 𝐶    &   (B + 𝑅) = 𝐷       (𝑋 (A[,]B) → (𝑋 + 𝑅) (𝐶[,]𝐷))
 
Theoremiccshftl 8594 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(A𝑅) = 𝐶    &   (B𝑅) = 𝐷       (((A B ℝ) (𝑋 𝑅 ℝ)) → (𝑋 (A[,]B) ↔ (𝑋𝑅) (𝐶[,]𝐷)))
 
Theoremiccshftli 8595 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
A     &   B     &   𝑅     &   (A𝑅) = 𝐶    &   (B𝑅) = 𝐷       (𝑋 (A[,]B) → (𝑋𝑅) (𝐶[,]𝐷))
 
Theoremiccdil 8596 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(A · 𝑅) = 𝐶    &   (B · 𝑅) = 𝐷       (((A B ℝ) (𝑋 𝑅 +)) → (𝑋 (A[,]B) ↔ (𝑋 · 𝑅) (𝐶[,]𝐷)))
 
Theoremiccdili 8597 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
A     &   B     &   𝑅 +    &   (A · 𝑅) = 𝐶    &   (B · 𝑅) = 𝐷       (𝑋 (A[,]B) → (𝑋 · 𝑅) (𝐶[,]𝐷))
 
Theoremicccntr 8598 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(A / 𝑅) = 𝐶    &   (B / 𝑅) = 𝐷       (((A B ℝ) (𝑋 𝑅 +)) → (𝑋 (A[,]B) ↔ (𝑋 / 𝑅) (𝐶[,]𝐷)))
 
Theoremicccntri 8599 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
A     &   B     &   𝑅 +    &   (A / 𝑅) = 𝐶    &   (B / 𝑅) = 𝐷       (𝑋 (A[,]B) → (𝑋 / 𝑅) (𝐶[,]𝐷))
 
Theoremdivelunit 8600 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
(((A 0 ≤ A) (B 0 < B)) → ((A / B) (0[,]1) ↔ AB))
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