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Theorem List for Intuitionistic Logic Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdivelunit 10001 A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
(((𝐴 ∈ ℝ ∧ 0 ≀ 𝐴) ∧ (𝐡 ∈ ℝ ∧ 0 < 𝐡)) β†’ ((𝐴 / 𝐡) ∈ (0[,]1) ↔ 𝐴 ≀ 𝐡))
 
Theoremlincmb01cmp 10002 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 ∧ 𝐡 ∈ ℝ ∧ 𝐴 < 𝐡) ∧ 𝑇 ∈ (0[,]1)) β†’ (((1 βˆ’ 𝑇) Β· 𝐴) + (𝑇 Β· 𝐡)) ∈ (𝐴[,]𝐡))
 
Theoremiccf1o 10003* Describe a bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐡]. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐹 = (π‘₯ ∈ (0[,]1) ↦ ((π‘₯ Β· 𝐡) + ((1 βˆ’ π‘₯) Β· 𝐴)))    β‡’   ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐴 < 𝐡) β†’ (𝐹:(0[,]1)–1-1-ontoβ†’(𝐴[,]𝐡) ∧ ◑𝐹 = (𝑦 ∈ (𝐴[,]𝐡) ↦ ((𝑦 βˆ’ 𝐴) / (𝐡 βˆ’ 𝐴)))))
 
Theoremunitssre 10004 (0[,]1) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
(0[,]1) βŠ† ℝ
 
Theoremiccen 10005 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 ∧ 𝐡 ∈ ℝ ∧ 𝐴 < 𝐡) β†’ (0[,]1) β‰ˆ (𝐴[,]𝐡))
 
Theoremzltaddlt1le 10006 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 ∈ β„€ ∧ 𝑁 ∈ β„€ ∧ 𝐴 ∈ (0(,)1)) β†’ ((𝑀 + 𝐴) < 𝑁 ↔ (𝑀 + 𝐴) ≀ 𝑁))
 
4.5.4  Finite intervals of integers
 
Syntaxcfz 10007 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".

This symbol is also used informally in some comments to denote an ellipsis, e.g., 𝐴 + 𝐴↑2 + ... + 𝐴↑(𝑁 βˆ’ 1).

class ...
 
Definitiondf-fz 10008* 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 10009 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
... = (π‘š ∈ β„€, 𝑛 ∈ β„€ ↦ {π‘˜ ∈ β„€ ∣ (π‘š ≀ π‘˜ ∧ π‘˜ ≀ 𝑛)})
 
Theoremfzval 10009* 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 β„•k means our 1...π‘˜; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀...𝑁) = {π‘˜ ∈ β„€ ∣ (𝑀 ≀ π‘˜ ∧ π‘˜ ≀ 𝑁)})
 
Theoremfzval2 10010 An alternate way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ β„€))
 
Theoremfzf 10011 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.)
...:(β„€ Γ— β„€)βŸΆπ’« β„€
 
Theoremelfz1 10012 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ β„€ ∧ 𝑀 ≀ 𝐾 ∧ 𝐾 ≀ 𝑁)))
 
Theoremelfz 10013 Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)
((𝐾 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≀ 𝐾 ∧ 𝐾 ≀ 𝑁)))
 
Theoremelfz2 10014 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 ∈ β„€ and 𝑁 ∈ β„€. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€ ∧ 𝐾 ∈ β„€) ∧ (𝑀 ≀ 𝐾 ∧ 𝐾 ≀ 𝑁)))
 
Theoremelfzd 10015 Membership in a finite set of sequential integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ≀ 𝐾)    &   (πœ‘ β†’ 𝐾 ≀ 𝑁)    β‡’   (πœ‘ β†’ 𝐾 ∈ (𝑀...𝑁))
 
Theoremelfz5 10016 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
((𝐾 ∈ (β„€β‰₯β€˜π‘€) ∧ 𝑁 ∈ β„€) β†’ (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≀ 𝑁))
 
Theoremelfz4 10017 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€ ∧ 𝐾 ∈ β„€) ∧ (𝑀 ≀ 𝐾 ∧ 𝐾 ≀ 𝑁)) β†’ 𝐾 ∈ (𝑀...𝑁))
 
Theoremelfzuzb 10018 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 ∈ (β„€β‰₯β€˜π‘€) ∧ 𝑁 ∈ (β„€β‰₯β€˜πΎ)))
 
Theoremeluzfz 10019 Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝐾 ∈ (β„€β‰₯β€˜π‘€) ∧ 𝑁 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝐾 ∈ (𝑀...𝑁))
 
Theoremelfzuz 10020 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) β†’ 𝐾 ∈ (β„€β‰₯β€˜π‘€))
 
Theoremelfzuz3 10021 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) β†’ 𝑁 ∈ (β„€β‰₯β€˜πΎ))
 
Theoremelfzel2 10022 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) β†’ 𝑁 ∈ β„€)
 
Theoremelfzel1 10023 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) β†’ 𝑀 ∈ β„€)
 
Theoremelfzelz 10024 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) β†’ 𝐾 ∈ β„€)
 
Theoremelfzelzd 10025 A member of a finite set of sequential integers is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(πœ‘ β†’ 𝐾 ∈ (𝑀...𝑁))    β‡’   (πœ‘ β†’ 𝐾 ∈ β„€)
 
Theoremelfzle1 10026 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) β†’ 𝑀 ≀ 𝐾)
 
Theoremelfzle2 10027 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) β†’ 𝐾 ≀ 𝑁)
 
Theoremelfzuz2 10028 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) β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
 
Theoremelfzle3 10029 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) β†’ 𝑀 ≀ 𝑁)
 
Theoremeluzfz1 10030 Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑀 ∈ (𝑀...𝑁))
 
Theoremeluzfz2 10031 Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑁 ∈ (𝑀...𝑁))
 
Theoremeluzfz2b 10032 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) ↔ 𝑁 ∈ (𝑀...𝑁))
 
Theoremelfz3 10033 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
(𝑁 ∈ β„€ β†’ 𝑁 ∈ (𝑁...𝑁))
 
Theoremelfz1eq 10034 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
(𝐾 ∈ (𝑁...𝑁) β†’ 𝐾 = 𝑁)
 
Theoremelfzubelfz 10035 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) β†’ 𝑁 ∈ (𝑀...𝑁))
 
Theorempeano2fzr 10036 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
((𝐾 ∈ (β„€β‰₯β€˜π‘€) ∧ (𝐾 + 1) ∈ (𝑀...𝑁)) β†’ 𝐾 ∈ (𝑀...𝑁))
 
Theoremfzm 10037* Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
(βˆƒπ‘₯ π‘₯ ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
 
Theoremfztri3or 10038 Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐾 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝐾 < 𝑀 ∨ 𝐾 ∈ (𝑀...𝑁) ∨ 𝑁 < 𝐾))
 
Theoremfzdcel 10039 Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐾 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ DECID 𝐾 ∈ (𝑀...𝑁))
 
Theoremfznlem 10040 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑁 < 𝑀 β†’ (𝑀...𝑁) = βˆ…))
 
Theoremfzn 10041 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑁 < 𝑀 ↔ (𝑀...𝑁) = βˆ…))
 
Theoremfzen 10042 A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€ ∧ 𝐾 ∈ β„€) β†’ (𝑀...𝑁) β‰ˆ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))
 
Theoremfz1n 10043 A 1-based finite set of sequential integers is empty iff it ends at index 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ β„•0 β†’ ((1...𝑁) = βˆ… ↔ 𝑁 = 0))
 
Theorem0fz1 10044 Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
((𝑁 ∈ β„•0 ∧ 𝐹 Fn (1...𝑁)) β†’ (𝐹 = βˆ… ↔ 𝑁 = 0))
 
Theoremfz10 10045 There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
(1...0) = βˆ…
 
Theoremuzsubsubfz 10046 Membership of an integer greater than L decreased by ( L - M ) in an M based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
((𝐿 ∈ (β„€β‰₯β€˜π‘€) ∧ 𝑁 ∈ (β„€β‰₯β€˜πΏ)) β†’ (𝑁 βˆ’ (𝐿 βˆ’ 𝑀)) ∈ (𝑀...𝑁))
 
Theoremuzsubsubfz1 10047 Membership of an integer greater than L decreased by ( L - 1 ) in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
((𝐿 ∈ β„• ∧ 𝑁 ∈ (β„€β‰₯β€˜πΏ)) β†’ (𝑁 βˆ’ (𝐿 βˆ’ 1)) ∈ (1...𝑁))
 
Theoremige3m2fz 10048 Membership of an integer greater than 2 decreased by 2 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
(𝑁 ∈ (β„€β‰₯β€˜3) β†’ (𝑁 βˆ’ 2) ∈ (1...𝑁))
 
Theoremfzsplit2 10049 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
(((𝐾 + 1) ∈ (β„€β‰₯β€˜π‘€) ∧ 𝑁 ∈ (β„€β‰₯β€˜πΎ)) β†’ (𝑀...𝑁) = ((𝑀...𝐾) βˆͺ ((𝐾 + 1)...𝑁)))
 
Theoremfzsplit 10050 Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
(𝐾 ∈ (𝑀...𝑁) β†’ (𝑀...𝑁) = ((𝑀...𝐾) βˆͺ ((𝐾 + 1)...𝑁)))
 
Theoremfzdisj 10051 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝐾 < 𝑀 β†’ ((𝐽...𝐾) ∩ (𝑀...𝑁)) = βˆ…)
 
Theoremfz01en 10052 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
(𝑁 ∈ β„€ β†’ (0...(𝑁 βˆ’ 1)) β‰ˆ (1...𝑁))
 
Theoremelfznn 10053 A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.)
(𝐾 ∈ (1...𝑁) β†’ 𝐾 ∈ β„•)
 
Theoremelfz1end 10054 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ β„• ↔ 𝐴 ∈ (1...𝐴))
 
Theoremfz1ssnn 10055 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(1...𝐴) βŠ† β„•
 
Theoremfznn0sub 10056 Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) β†’ (𝑁 βˆ’ 𝐾) ∈ β„•0)
 
Theoremfzmmmeqm 10057 Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
(𝑀 ∈ (𝐿...𝑁) β†’ ((𝑁 βˆ’ 𝐿) βˆ’ (𝑀 βˆ’ 𝐿)) = (𝑁 βˆ’ 𝑀))
 
Theoremfzaddel 10058 Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
(((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ (𝐽 ∈ β„€ ∧ 𝐾 ∈ β„€)) β†’ (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))))
 
Theoremfzsubel 10059 Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
(((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ (𝐽 ∈ β„€ ∧ 𝐾 ∈ β„€)) β†’ (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 βˆ’ 𝐾) ∈ ((𝑀 βˆ’ 𝐾)...(𝑁 βˆ’ 𝐾))))
 
Theoremfzopth 10060 A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
 
Theoremfzass4 10061 Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐡 ∈ (𝐴...𝐷) ∧ 𝐢 ∈ (𝐡...𝐷)) ↔ (𝐡 ∈ (𝐴...𝐢) ∧ 𝐢 ∈ (𝐴...𝐷)))
 
Theoremfzss1 10062 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝐾...𝑁) βŠ† (𝑀...𝑁))
 
Theoremfzss2 10063 Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝑁 ∈ (β„€β‰₯β€˜πΎ) β†’ (𝑀...𝐾) βŠ† (𝑀...𝑁))
 
Theoremfzssuz 10064 A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)
(𝑀...𝑁) βŠ† (β„€β‰₯β€˜π‘€)
 
Theoremfzsn 10065 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ β„€ β†’ (𝑀...𝑀) = {𝑀})
 
Theoremfzssp1 10066 Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑀...𝑁) βŠ† (𝑀...(𝑁 + 1))
 
Theoremfzssnn 10067 Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
(𝑀 ∈ β„• β†’ (𝑀...𝑁) βŠ† β„•)
 
Theoremfzsuc 10068 Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) βˆͺ {(𝑁 + 1)}))
 
Theoremfzpred 10069 Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝑀...𝑁) = ({𝑀} βˆͺ ((𝑀 + 1)...𝑁)))
 
Theoremfzpreddisj 10070 A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = βˆ…)
 
Theoremelfzp1 10071 Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1))))
 
Theoremfzp1ss 10072 Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑀 ∈ β„€ β†’ ((𝑀 + 1)...𝑁) βŠ† (𝑀...𝑁))
 
Theoremfzelp1 10073 Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) β†’ 𝐾 ∈ (𝑀...(𝑁 + 1)))
 
Theoremfzp1elp1 10074 Add one to an element of a finite set of integers. (Contributed by Jeff Madsen, 6-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (𝑀...𝑁) β†’ (𝐾 + 1) ∈ (𝑀...(𝑁 + 1)))
 
Theoremfznatpl1 10075 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
((𝑁 ∈ β„• ∧ 𝐼 ∈ (1...(𝑁 βˆ’ 1))) β†’ (𝐼 + 1) ∈ (1...𝑁))
 
Theoremfzpr 10076 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ β„€ β†’ (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
 
Theoremfztp 10077 A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)
(𝑀 ∈ β„€ β†’ (𝑀...(𝑀 + 2)) = {𝑀, (𝑀 + 1), (𝑀 + 2)})
 
Theoremfzsuc2 10078 Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ (β„€β‰₯β€˜(𝑀 βˆ’ 1))) β†’ (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) βˆͺ {(𝑁 + 1)}))
 
Theoremfzp1disj 10079 (𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with {(𝑁 + 1)}. (Contributed by Mario Carneiro, 7-Mar-2014.)
((𝑀...𝑁) ∩ {(𝑁 + 1)}) = βˆ…
 
Theoremfzdifsuc 10080 Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) βˆ– {(𝑁 + 1)}))
 
Theoremfzprval 10081* Two ways of defining the first two values of a sequence on β„•. (Contributed by NM, 5-Sep-2011.)
(βˆ€π‘₯ ∈ (1...2)(πΉβ€˜π‘₯) = if(π‘₯ = 1, 𝐴, 𝐡) ↔ ((πΉβ€˜1) = 𝐴 ∧ (πΉβ€˜2) = 𝐡))
 
Theoremfztpval 10082* Two ways of defining the first three values of a sequence on β„•. (Contributed by NM, 13-Sep-2011.)
(βˆ€π‘₯ ∈ (1...3)(πΉβ€˜π‘₯) = if(π‘₯ = 1, 𝐴, if(π‘₯ = 2, 𝐡, 𝐢)) ↔ ((πΉβ€˜1) = 𝐴 ∧ (πΉβ€˜2) = 𝐡 ∧ (πΉβ€˜3) = 𝐢))
 
Theoremfzrev 10083 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
(((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ (𝐽 ∈ β„€ ∧ 𝐾 ∈ β„€)) β†’ (𝐾 ∈ ((𝐽 βˆ’ 𝑁)...(𝐽 βˆ’ 𝑀)) ↔ (𝐽 βˆ’ 𝐾) ∈ (𝑀...𝑁)))
 
Theoremfzrev2 10084 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
(((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ (𝐽 ∈ β„€ ∧ 𝐾 ∈ β„€)) β†’ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐽 βˆ’ 𝐾) ∈ ((𝐽 βˆ’ 𝑁)...(𝐽 βˆ’ 𝑀))))
 
Theoremfzrev2i 10085 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
((𝐽 ∈ β„€ ∧ 𝐾 ∈ (𝑀...𝑁)) β†’ (𝐽 βˆ’ 𝐾) ∈ ((𝐽 βˆ’ 𝑁)...(𝐽 βˆ’ 𝑀)))
 
Theoremfzrev3 10086 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
(𝐾 ∈ β„€ β†’ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 + 𝑁) βˆ’ 𝐾) ∈ (𝑀...𝑁)))
 
Theoremfzrev3i 10087 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
(𝐾 ∈ (𝑀...𝑁) β†’ ((𝑀 + 𝑁) βˆ’ 𝐾) ∈ (𝑀...𝑁))
 
Theoremfznn 10088 Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ β„€ β†’ (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ β„• ∧ 𝐾 ≀ 𝑁)))
 
Theoremelfz1b 10089 Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)
(𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ β„• ∧ 𝑀 ∈ β„• ∧ 𝑁 ≀ 𝑀))
 
Theoremelfzm11 10090 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝐾 ∈ (𝑀...(𝑁 βˆ’ 1)) ↔ (𝐾 ∈ β„€ ∧ 𝑀 ≀ 𝐾 ∧ 𝐾 < 𝑁)))
 
Theoremuzsplit 10091 Express an upper integer set as the disjoint (see uzdisj 10092) union of the first 𝑁 values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (β„€β‰₯β€˜π‘€) = ((𝑀...(𝑁 βˆ’ 1)) βˆͺ (β„€β‰₯β€˜π‘)))
 
Theoremuzdisj 10092 The first 𝑁 elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)
((𝑀...(𝑁 βˆ’ 1)) ∩ (β„€β‰₯β€˜π‘)) = βˆ…
 
Theoremfseq1p1m1 10093 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.)
𝐻 = {⟨(𝑁 + 1), 𝐡⟩}    β‡’   (𝑁 ∈ β„•0 β†’ ((𝐹:(1...𝑁)⟢𝐴 ∧ 𝐡 ∈ 𝐴 ∧ 𝐺 = (𝐹 βˆͺ 𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟢𝐴 ∧ (πΊβ€˜(𝑁 + 1)) = 𝐡 ∧ 𝐹 = (𝐺 β†Ύ (1...𝑁)))))
 
Theoremfseq1m1p1 10094 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)
𝐻 = {βŸ¨π‘, 𝐡⟩}    β‡’   (𝑁 ∈ β„• β†’ ((𝐹:(1...(𝑁 βˆ’ 1))⟢𝐴 ∧ 𝐡 ∈ 𝐴 ∧ 𝐺 = (𝐹 βˆͺ 𝐻)) ↔ (𝐺:(1...𝑁)⟢𝐴 ∧ (πΊβ€˜π‘) = 𝐡 ∧ 𝐹 = (𝐺 β†Ύ (1...(𝑁 βˆ’ 1))))))
 
Theoremfz1sbc 10095* Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)
(𝑁 ∈ β„€ β†’ (βˆ€π‘˜ ∈ (𝑁...𝑁)πœ‘ ↔ [𝑁 / π‘˜]πœ‘))
 
Theoremelfzp1b 10096 An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝐾 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝐾 ∈ (0...(𝑁 βˆ’ 1)) ↔ (𝐾 + 1) ∈ (1...𝑁)))
 
Theoremelfzm1b 10097 An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝐾 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝐾 ∈ (1...𝑁) ↔ (𝐾 βˆ’ 1) ∈ (0...(𝑁 βˆ’ 1))))
 
Theoremelfzp12 10098 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁))))
 
Theoremfzm1 10099 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (𝑀...(𝑁 βˆ’ 1)) ∨ 𝐾 = 𝑁)))
 
Theoremfzneuz 10100 No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.)
((𝑁 ∈ (β„€β‰₯β€˜π‘€) ∧ 𝐾 ∈ β„€) β†’ Β¬ (𝑀...𝑁) = (β„€β‰₯β€˜πΎ))
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