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Theorem List for Metamath Proof Explorer - 12201-12300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfzf 12201 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 12202 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 12203 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 12204 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 12205 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
((𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾𝑁))

Theoremelfz4 12206 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 12207 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 12208 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 12209 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 12210 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 12211 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 12212 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 12213 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)

Theoremfzssz 12214 A finite sequence of integers is a set of integers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝑀...𝑁) ⊆ ℤ

Theoremelfzle1 12215 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 12216 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 12217 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 12218 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 12219 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 12220 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 12221 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) ↔ 𝑁 ∈ (𝑀...𝑁))

Theoremelfz3 12222 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
(𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁))

Theoremelfz1eq 12223 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
(𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁)

Theoremelfzubelfz 12224 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 12225 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))

Theoremfzn0 12226 Properties of a finite interval of integers which is nonempty. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝑀...𝑁) ≠ ∅ ↔ 𝑁 ∈ (ℤ𝑀))

Theoremfz0 12227 A finite set of sequential integers is empty if its bounds are not integers. (Contributed by AV, 13-Oct-2018.)
((𝑀 ∉ ℤ ∨ 𝑁 ∉ ℤ) → (𝑀...𝑁) = ∅)

Theoremfzn 12228 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 12229 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 12230 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))

Theorem0nelfz1 12231 0 is not an element of a finite interval of integers starting at 1. (Contributed by AV, 27-Aug-2020.)
0 ∉ (1...𝑁)

Theorem0fz1 12232 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 12233 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 12234 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 12235 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 12236 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 12237 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 12238 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 12239 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅)

Theoremfz01en 12240 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 12241 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 12242 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))

Theoremfz1ssnn 12243 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(1...𝐴) ⊆ ℕ

Theoremfznn0sub 12244 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 12245 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 12246 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))))

Theoremfzadd2 12247 Membership of a sum in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
(((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑂 ∈ ℤ ∧ 𝑃 ∈ ℤ)) → ((𝐽 ∈ (𝑀...𝑁) ∧ 𝐾 ∈ (𝑂...𝑃)) → (𝐽 + 𝐾) ∈ ((𝑀 + 𝑂)...(𝑁 + 𝑃))))

Theoremfzsubel 12248 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 12249 A finite set of sequential integers has the ordered pair property (compare opth 4871) under certain conditions. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽𝑁 = 𝐾)))

Theoremfzass4 12250 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 12251 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 12252 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 12253 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 12254 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})

Theoremfzssp1 12255 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))

Theoremfzssnn 12256 Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
(𝑀 ∈ ℕ → (𝑀...𝑁) ⊆ ℕ)

Theoremssfzunsn 12257 A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 8-Jun-2021.)
((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (𝑀...if(𝐼𝑁, 𝑁, 𝐼)))

Theoremfzsuc 12258 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 12259 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 12260 A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
(𝑁 ∈ (ℤ𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅)

Theoremelfzp1 12261 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 12262 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 12263 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 12264 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 12265 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 12266 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})

Theoremfztp 12267 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 12268 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 12269 (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 12270 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 12271* 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 12272* 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 12273 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 12274 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 12275 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 12276 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 12277 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 12278 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 12279 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 12280 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 12281 Express an upper integer set as the disjoint (see uzdisj 12282) union of the first 𝑁 values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
(𝑁 ∈ (ℤ𝑀) → (ℤ𝑀) = ((𝑀...(𝑁 − 1)) ∪ (ℤ𝑁)))

Theoremuzdisj 12282 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 12283 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 12284 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 12285* 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 12286 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 12287 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 12288 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 12289 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 12290 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 12291 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 12292 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
(𝐾 ∈ (ℤ𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1)))

Theoremfzp1nel 12293 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 12294* 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 12295* 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 12296* 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))

Theoremfzshftral 12297* Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))[(𝑘𝐾) / 𝑗]𝜑))

Theoremige2m1fz1 12298 Membership of an integer greater than 1 decreased by 1 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
(𝑁 ∈ (ℤ‘2) → (𝑁 − 1) ∈ (1...𝑁))

Theoremige2m1fz 12299 Membership in a 0 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander van der Vekens, 15-Sep-2018.)
((𝑁 ∈ ℕ0 ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ (0...𝑁))

5.5.6  Finite intervals of nonnegative integers

Finite intervals of nonnegative integers (or "finite sets of sequential nonnegative integers") are finite intervals of integers with 0 as lower bound: (0...𝑁), usually abbreviated by "fz0".

Theoremelfz2nn0 12300 Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0𝑁 ∈ ℕ0𝐾𝑁))

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