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Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlincmb01cmp 9101 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 9102* 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 9103 (0[,]1) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)
(0[,]1) ⊆ ℝ
 
Theoremzltaddlt1le 9104 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
 
Syntaxcfz 9105 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 9106* 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 9107 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)
... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
 
Theoremfzval 9107* 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 9108 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))
 
Theoremfzf 9109 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 9110 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 9111 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 9112 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 9113 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
((𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾𝑁))
 
Theoremelfz4 9114 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 9115 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 9116 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 9117 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 9118 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 9119 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 9120 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 9121 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 9122 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 9123 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 9124 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 9125 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 9126 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 9127 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 9128 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) ↔ 𝑁 ∈ (𝑀...𝑁))
 
Theoremelfz3 9129 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
(𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁))
 
Theoremelfz1eq 9130 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
(𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁)
 
Theoremelfzubelfz 9131 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))
 
Theorempeano2fzr 9132 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
((𝐾 ∈ (ℤ𝑀) ∧ (𝐾 + 1) ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...𝑁))
 
Theoremfzm 9133* Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
(∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ𝑀))
 
Theoremfztri3or 9134 Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑀𝐾 ∈ (𝑀...𝑁) ∨ 𝑁 < 𝐾))
 
Theoremfzdcel 9135 Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀...𝑁))
 
Theoremfznlem 9136 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 → (𝑀...𝑁) = ∅))
 
Theoremfzn 9137 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
 
Theoremfzen 9138 A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀...𝑁) ≈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))
 
Theoremfz1n 9139 A 1-based finite set of sequential integers is empty iff it ends at index 0. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝑁 ∈ ℕ0 → ((1...𝑁) = ∅ ↔ 𝑁 = 0))
 
Theorem0fz1 9140 Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
((𝑁 ∈ ℕ0𝐹 Fn (1...𝑁)) → (𝐹 = ∅ ↔ 𝑁 = 0))
 
Theoremfz10 9141 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) = ∅
 
Theoremuzsubsubfz 9142 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 ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ𝐿)) → (𝑁 − (𝐿𝑀)) ∈ (𝑀...𝑁))
 
Theoremuzsubsubfz1 9143 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 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝐿)) → (𝑁 − (𝐿 − 1)) ∈ (1...𝑁))
 
Theoremige3m2fz 9144 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 ∈ (ℤ‘3) → (𝑁 − 2) ∈ (1...𝑁))
 
Theoremfzsplit2 9145 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
(((𝐾 + 1) ∈ (ℤ𝑀) ∧ 𝑁 ∈ (ℤ𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁)))
 
Theoremfzsplit 9146 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) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁)))
 
Theoremfzdisj 9147 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅)
 
Theoremfz01en 9148 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
(𝑁 ∈ ℤ → (0...(𝑁 − 1)) ≈ (1...𝑁))
 
Theoremelfznn 9149 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) → 𝐾 ∈ ℕ)
 
Theoremelfz1end 9150 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
 
Theoremfznn0sub 9151 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) → (𝑁𝐾) ∈ ℕ0)
 
Theoremfzmmmeqm 9152 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) → ((𝑁𝐿) − (𝑀𝐿)) = (𝑁𝑀))
 
Theoremfzaddel 9153 Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))))
 
Theoremfzsubel 9154 Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽𝐾) ∈ ((𝑀𝐾)...(𝑁𝐾))))
 
Theoremfzopth 9155 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 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))
 
Theoremfzass4 9156 Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐵 ∈ (𝐴...𝐷) ∧ 𝐶 ∈ (𝐵...𝐷)) ↔ (𝐵 ∈ (𝐴...𝐶) ∧ 𝐶 ∈ (𝐴...𝐷)))
 
Theoremfzss1 9157 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (ℤ𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁))
 
Theoremfzss2 9158 Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
(𝑁 ∈ (ℤ𝐾) → (𝑀...𝐾) ⊆ (𝑀...𝑁))
 
Theoremfzssuz 9159 A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)
(𝑀...𝑁) ⊆ (ℤ𝑀)
 
Theoremfzsn 9160 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
 
Theoremfzssp1 9161 Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1))
 
Theoremfzsuc 9162 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 ∈ (ℤ𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))
 
Theoremfzpred 9163 Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
(𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁)))
 
Theoremfzpreddisj 9164 A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
(𝑁 ∈ (ℤ𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅)
 
Theoremelfzp1 9165 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 ∈ (ℤ𝑀) → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1))))
 
Theoremfzp1ss 9166 Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁))
 
Theoremfzelp1 9167 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) → 𝐾 ∈ (𝑀...(𝑁 + 1)))
 
Theoremfzp1elp1 9168 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) → (𝐾 + 1) ∈ (𝑀...(𝑁 + 1)))
 
Theoremfznatpl1 9169 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (𝐼 + 1) ∈ (1...𝑁))
 
Theoremfzpr 9170 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
 
Theoremfztp 9171 A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)
(𝑀 ∈ ℤ → (𝑀...(𝑀 + 2)) = {𝑀, (𝑀 + 1), (𝑀 + 2)})
 
Theoremfzsuc2 9172 Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 − 1))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)}))
 
Theoremfzp1disj 9173 (𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with {(𝑁 + 1)}. (Contributed by Mario Carneiro, 7-Mar-2014.)
((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅
 
Theoremfzdifsuc 9174 Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)
(𝑁 ∈ (ℤ𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)}))
 
Theoremfzprval 9175* Two ways of defining the first two values of a sequence on . (Contributed by NM, 5-Sep-2011.)
(∀𝑥 ∈ (1...2)(𝐹𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵))
 
Theoremfztpval 9176* Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.)
(∀𝑥 ∈ (1...3)(𝐹𝑥) = if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵 ∧ (𝐹‘3) = 𝐶))
 
Theoremfzrev 9177 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 ∈ ((𝐽𝑁)...(𝐽𝑀)) ↔ (𝐽𝐾) ∈ (𝑀...𝑁)))
 
Theoremfzrev2 9178 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐽𝐾) ∈ ((𝐽𝑁)...(𝐽𝑀))))
 
Theoremfzrev2i 9179 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
((𝐽 ∈ ℤ ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽𝐾) ∈ ((𝐽𝑁)...(𝐽𝑀)))
 
Theoremfzrev3 9180 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
(𝐾 ∈ ℤ → (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁)))
 
Theoremfzrev3i 9181 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
(𝐾 ∈ (𝑀...𝑁) → ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁))
 
Theoremfznn 9182 Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ ℤ → (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝐾𝑁)))
 
Theoremelfz1b 9183 Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)
(𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁𝑀))
 
Theoremelfzm11 9184 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...(𝑁 − 1)) ↔ (𝐾 ∈ ℤ ∧ 𝑀𝐾𝐾 < 𝑁)))
 
Theoremuzsplit 9185 Express an upper integer set as the disjoint (see uzdisj 9186) union of the first 𝑁 values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
(𝑁 ∈ (ℤ𝑀) → (ℤ𝑀) = ((𝑀...(𝑁 − 1)) ∪ (ℤ𝑁)))
 
Theoremuzdisj 9186 The first 𝑁 elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)
((𝑀...(𝑁 − 1)) ∩ (ℤ𝑁)) = ∅
 
Theoremfseq1p1m1 9187 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), 𝐵⟩}       (𝑁 ∈ ℕ0 → ((𝐹:(1...𝑁)⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵𝐹 = (𝐺 ↾ (1...𝑁)))))
 
Theoremfseq1m1p1 9188 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)
𝐻 = {⟨𝑁, 𝐵⟩}       (𝑁 ∈ ℕ → ((𝐹:(1...(𝑁 − 1))⟶𝐴𝐵𝐴𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...𝑁)⟶𝐴 ∧ (𝐺𝑁) = 𝐵𝐹 = (𝐺 ↾ (1...(𝑁 − 1))))))
 
Theoremfz1sbc 9189* Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)
(𝑁 ∈ ℤ → (∀𝑘 ∈ (𝑁...𝑁)𝜑[𝑁 / 𝑘]𝜑))
 
Theoremelfzp1b 9190 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 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...(𝑁 − 1)) ↔ (𝐾 + 1) ∈ (1...𝑁)))
 
Theoremelfzm1b 9191 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 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (1...𝑁) ↔ (𝐾 − 1) ∈ (0...(𝑁 − 1))))
 
Theoremelfzp12 9192 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)
(𝑁 ∈ (ℤ𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀𝐾 ∈ ((𝑀 + 1)...𝑁))))
 
Theoremfzm1 9193 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑁 ∈ (ℤ𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (𝑀...(𝑁 − 1)) ∨ 𝐾 = 𝑁)))
 
Theoremfzneuz 9194 No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐾 ∈ ℤ) → ¬ (𝑀...𝑁) = (ℤ𝐾))
 
Theoremfznuz 9195 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)
(𝐾 ∈ (𝑀...𝑁) → ¬ 𝐾 ∈ (ℤ‘(𝑁 + 1)))
 
Theoremuznfz 9196 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
(𝐾 ∈ (ℤ𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1)))
 
Theoremfzp1nel 9197 One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.)
¬ (𝑁 + 1) ∈ (𝑀...𝑁)
 
Theoremfzrevral 9198* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))[(𝐾𝑘) / 𝑗]𝜑))
 
Theoremfzrevral2 9199* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀))𝜑 ↔ ∀𝑘 ∈ (𝑀...𝑁)[(𝐾𝑘) / 𝑗]𝜑))
 
Theoremfzrevral3 9200* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ (𝑀...𝑁)[((𝑀 + 𝑁) − 𝑘) / 𝑗]𝜑))
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