Type  Label  Description 
Statement 

Theorem  lincmb01cmp 9101 
A linear combination of two reals which lies in the interval between them.
(Contributed by Jeff Madsen, 2Sep2009.) (Proof shortened by Mario
Carneiro, 8Sep2015.)

⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵)) ∈ (𝐴[,]𝐵)) 

Theorem  iccf1o 9102* 
Describe a bijection from [0, 1] to an arbitrary
nontrivial
closed interval [𝐴, 𝐵]. (Contributed by Mario Carneiro,
8Sep2015.)

⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴))) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐹:(0[,]1)–11onto→(𝐴[,]𝐵) ∧ ^{◡}𝐹 = (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))))) 

Theorem  unitssre 9103 
(0[,]1) is a subset of the reals. (Contributed by
David Moews,
28Feb2017.)

⊢ (0[,]1) ⊆ ℝ 

Theorem  zltaddlt1le 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, 1Jul2021.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ (0(,)1)) → ((𝑀 + 𝐴) < 𝑁 ↔ (𝑀 + 𝐴) ≤ 𝑁)) 

3.5.4 Finite intervals of integers


Syntax  cfz 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 ... 

Definition  dffz 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, 6Sep2005.)

⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) 

Theorem  fzval 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 112.1 of [Gleason] p. 141, where
ℕ_k means our 1...𝑘; he calls these sets
segments of the
integers. (Contributed by NM, 6Sep2005.) (Revised by Mario Carneiro,
3Nov2013.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) 

Theorem  fzval2 9108 
An alternate way of expressing a finite set of sequential integers.
(Contributed by Mario Carneiro, 3Nov2013.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ)) 

Theorem  fzf 9109 
Establish the domain and codomain of the finite integer sequence
function. (Contributed by Scott Fenton, 8Aug2013.) (Revised by Mario
Carneiro, 16Nov2013.)

⊢ ...:(ℤ ×
ℤ)⟶𝒫 ℤ 

Theorem  elfz1 9110 
Membership in a finite set of sequential integers. (Contributed by NM,
21Jul2005.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) 

Theorem  elfz 9111 
Membership in a finite set of sequential integers. (Contributed by NM,
29Sep2005.)

⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) 

Theorem  elfz2 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,
6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) 

Theorem  elfz5 9113 
Membership in a finite set of sequential integers. (Contributed by NM,
26Dec2005.)

⊢ ((𝐾 ∈ (ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≤ 𝑁)) 

Theorem  elfz4 9114 
Membership in a finite set of sequential integers. (Contributed by NM,
21Jul2005.) (Revised by Mario Carneiro, 28Apr2015.)

⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝐾 ∈ (𝑀...𝑁)) 

Theorem  elfzuzb 9115 
Membership in a finite set of sequential integers in terms of sets of
upper integers. (Contributed by NM, 18Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ (ℤ_{≥}‘𝐾))) 

Theorem  eluzfz 9116 
Membership in a finite set of sequential integers. (Contributed by NM,
4Oct2005.) (Revised by Mario Carneiro, 28Apr2015.)

⊢ ((𝐾 ∈ (ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ (ℤ_{≥}‘𝐾)) → 𝐾 ∈ (𝑀...𝑁)) 

Theorem  elfzuz 9117 
A member of a finite set of sequential integers belongs to an upper set of
integers. (Contributed by NM, 17Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ_{≥}‘𝑀)) 

Theorem  elfzuz3 9118 
Membership in a finite set of sequential integers implies membership in an
upper set of integers. (Contributed by NM, 28Sep2005.) (Revised by
Mario Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_{≥}‘𝐾)) 

Theorem  elfzel2 9119 
Membership in a finite set of sequential integer implies the upper bound
is an integer. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) 

Theorem  elfzel1 9120 
Membership in a finite set of sequential integer implies the lower bound
is an integer. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) 

Theorem  elfzelz 9121 
A member of a finite set of sequential integer is an integer.
(Contributed by NM, 6Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ) 

Theorem  elfzle1 9122 
A member of a finite set of sequential integer is greater than or equal to
the lower bound. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾) 

Theorem  elfzle2 9123 
A member of a finite set of sequential integer is less than or equal to
the upper bound. (Contributed by NM, 6Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁) 

Theorem  elfzuz2 9124 
Implication of membership in a finite set of sequential integers.
(Contributed by NM, 20Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_{≥}‘𝑀)) 

Theorem  elfzle3 9125 
Membership in a finite set of sequential integer implies the bounds are
comparable. (Contributed by NM, 18Sep2005.) (Revised by Mario
Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑁) 

Theorem  eluzfz1 9126 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 21Jul2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) 

Theorem  eluzfz2 9127 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 13Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) 

Theorem  eluzfz2b 9128 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 14Sep2005.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁)) 

Theorem  elfz3 9129 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 21Jul2005.)

⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁)) 

Theorem  elfz1eq 9130 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 19Sep2005.)

⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) 

Theorem  elfzubelfz 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, 31May2018.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (𝑀...𝑁)) 

Theorem  peano2fzr 9132 
A Peanopostulatelike theorem for downward closure of a finite set of
sequential integers. (Contributed by Mario Carneiro, 27May2014.)

⊢ ((𝐾 ∈ (ℤ_{≥}‘𝑀) ∧ (𝐾 + 1) ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...𝑁)) 

Theorem  fzm 9133* 
Properties of a finite interval of integers which is inhabited.
(Contributed by Jim Kingdon, 15Apr2020.)

⊢ (∃𝑥 𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (ℤ_{≥}‘𝑀)) 

Theorem  fztri3or 9134 
Trichotomy in terms of a finite interval of integers. (Contributed by Jim
Kingdon, 1Jun2020.)

⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑀 ∨ 𝐾 ∈ (𝑀...𝑁) ∨ 𝑁 < 𝐾)) 

Theorem  fzdcel 9135 
Decidability of membership in a finite interval of integers. (Contributed
by Jim Kingdon, 1Jun2020.)

⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) →
DECID 𝐾
∈ (𝑀...𝑁)) 

Theorem  fznlem 9136 
A finite set of sequential integers is empty if the bounds are reversed.
(Contributed by Jim Kingdon, 16Apr2020.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 → (𝑀...𝑁) = ∅)) 

Theorem  fzn 9137 
A finite set of sequential integers is empty if the bounds are reversed.
(Contributed by NM, 22Aug2005.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅)) 

Theorem  fzen 9138 
A shifted finite set of sequential integers is equinumerous to the
original set. (Contributed by Paul Chapman, 11Apr2009.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀...𝑁) ≈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) 

Theorem  fz1n 9139 
A 1based finite set of sequential integers is empty iff it ends at index
0. (Contributed by Paul Chapman, 22Jun2011.)

⊢ (𝑁 ∈ ℕ_{0} →
((1...𝑁) = ∅ ↔
𝑁 = 0)) 

Theorem  0fz1 9140 
Two ways to say a finite 1based sequence is empty. (Contributed by Paul
Chapman, 26Oct2012.)

⊢ ((𝑁 ∈ ℕ_{0} ∧ 𝐹 Fn (1...𝑁)) → (𝐹 = ∅ ↔ 𝑁 = 0)) 

Theorem  fz10 9141 
There are no integers between 1 and 0. (Contributed by Jeff Madsen,
16Jun2010.) (Proof shortened by Mario Carneiro, 28Apr2015.)

⊢ (1...0) = ∅ 

Theorem  uzsubsubfz 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, 14Sep2018.)

⊢ ((𝐿 ∈ (ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ (ℤ_{≥}‘𝐿)) → (𝑁 − (𝐿 − 𝑀)) ∈ (𝑀...𝑁)) 

Theorem  uzsubsubfz1 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, 14Sep2018.)

⊢ ((𝐿 ∈ ℕ ∧ 𝑁 ∈ (ℤ_{≥}‘𝐿)) → (𝑁 − (𝐿 − 1)) ∈ (1...𝑁)) 

Theorem  ige3m2fz 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,
14Sep2018.)

⊢ (𝑁 ∈ (ℤ_{≥}‘3)
→ (𝑁 − 2)
∈ (1...𝑁)) 

Theorem  fzsplit2 9145 
Split a finite interval of integers into two parts. (Contributed by
Mario Carneiro, 13Apr2016.)

⊢ (((𝐾 + 1) ∈
(ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ (ℤ_{≥}‘𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) 

Theorem  fzsplit 9146 
Split a finite interval of integers into two parts. (Contributed by
Jeff Madsen, 17Jun2010.) (Revised by Mario Carneiro, 13Apr2016.)

⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) 

Theorem  fzdisj 9147 
Condition for two finite intervals of integers to be disjoint.
(Contributed by Jeff Madsen, 17Jun2010.)

⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) 

Theorem  fz01en 9148 
0based and 1based finite sets of sequential integers are equinumerous.
(Contributed by Paul Chapman, 11Apr2009.)

⊢ (𝑁 ∈ ℤ → (0...(𝑁 − 1)) ≈ (1...𝑁)) 

Theorem  elfznn 9149 
A member of a finite set of sequential integers starting at 1 is a
positive integer. (Contributed by NM, 24Aug2005.)

⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℕ) 

Theorem  elfz1end 9150 
A nonempty finite range of integers contains its end point. (Contributed
by Stefan O'Rear, 10Oct2014.)

⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴)) 

Theorem  fznn0sub 9151 
Subtraction closure for a member of a finite set of sequential integers.
(Contributed by NM, 16Sep2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑁 − 𝐾) ∈
ℕ_{0}) 

Theorem  fzmmmeqm 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,
27May2018.)

⊢ (𝑀 ∈ (𝐿...𝑁) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀)) 

Theorem  fzaddel 9153 
Membership of a sum in a finite set of sequential integers. (Contributed
by NM, 30Jul2005.)

⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))) 

Theorem  fzsubel 9154 
Membership of a difference in a finite set of sequential integers.
(Contributed by NM, 30Jul2005.)

⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 − 𝐾) ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)))) 

Theorem  fzopth 9155 
A finite set of sequential integers can represent an ordered pair.
(Contributed by NM, 31Oct2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))) 

Theorem  fzass4 9156 
Two ways to express a nondecreasing sequence of four integers.
(Contributed by Stefan O'Rear, 15Aug2015.)

⊢ ((𝐵 ∈ (𝐴...𝐷) ∧ 𝐶 ∈ (𝐵...𝐷)) ↔ (𝐵 ∈ (𝐴...𝐶) ∧ 𝐶 ∈ (𝐴...𝐷))) 

Theorem  fzss1 9157 
Subset relationship for finite sets of sequential integers.
(Contributed by NM, 28Sep2005.) (Proof shortened by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈ (ℤ_{≥}‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁)) 

Theorem  fzss2 9158 
Subset relationship for finite sets of sequential integers.
(Contributed by NM, 4Oct2005.) (Revised by Mario Carneiro,
30Apr2015.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝐾) → (𝑀...𝐾) ⊆ (𝑀...𝑁)) 

Theorem  fzssuz 9159 
A finite set of sequential integers is a subset of an upper set of
integers. (Contributed by NM, 28Oct2005.)

⊢ (𝑀...𝑁) ⊆
(ℤ_{≥}‘𝑀) 

Theorem  fzsn 9160 
A finite interval of integers with one element. (Contributed by Jeff
Madsen, 2Sep2009.)

⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) 

Theorem  fzssp1 9161 
Subset relationship for finite sets of sequential integers.
(Contributed by NM, 21Jul2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) 

Theorem  fzsuc 9162 
Join a successor to the end of a finite set of sequential integers.
(Contributed by NM, 19Jul2008.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) 

Theorem  fzpred 9163 
Join a predecessor to the beginning of a finite set of sequential
integers. (Contributed by AV, 24Aug2019.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) 

Theorem  fzpreddisj 9164 
A finite set of sequential integers is disjoint with its predecessor.
(Contributed by AV, 24Aug2019.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅) 

Theorem  elfzp1 9165 
Append an element to a finite set of sequential integers. (Contributed by
NM, 19Sep2005.) (Proof shortened by Mario Carneiro, 28Apr2015.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1)))) 

Theorem  fzp1ss 9166 
Subset relationship for finite sets of sequential integers. (Contributed
by NM, 26Jul2005.) (Revised by Mario Carneiro, 28Apr2015.)

⊢ (𝑀 ∈ ℤ → ((𝑀 + 1)...𝑁) ⊆ (𝑀...𝑁)) 

Theorem  fzelp1 9167 
Membership in a set of sequential integers with an appended element.
(Contributed by NM, 7Dec2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (𝑀...(𝑁 + 1))) 

Theorem  fzp1elp1 9168 
Add one to an element of a finite set of integers. (Contributed by Jeff
Madsen, 6Jun2010.) (Revised by Mario Carneiro, 28Apr2015.)

⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 + 1) ∈ (𝑀...(𝑁 + 1))) 

Theorem  fznatpl1 9169 
Shift membership in a finite sequence of naturals. (Contributed by Scott
Fenton, 17Jul2013.)

⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (𝐼 + 1) ∈ (1...𝑁)) 

Theorem  fzpr 9170 
A finite interval of integers with two elements. (Contributed by Jeff
Madsen, 2Sep2009.)

⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)}) 

Theorem  fztp 9171 
A finite interval of integers with three elements. (Contributed by NM,
13Sep2011.) (Revised by Mario Carneiro, 7Mar2014.)

⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 2)) = {𝑀, (𝑀 + 1), (𝑀 + 2)}) 

Theorem  fzsuc2 9172 
Join a successor to the end of a finite set of sequential integers.
(Contributed by Mario Carneiro, 7Mar2014.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈
(ℤ_{≥}‘(𝑀 − 1))) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) 

Theorem  fzp1disj 9173 
(𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with
{(𝑁 +
1)}. (Contributed by Mario Carneiro, 7Mar2014.)

⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ 

Theorem  fzdifsuc 9174 
Remove a successor from the end of a finite set of sequential integers.
(Contributed by AV, 4Sep2019.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)})) 

Theorem  fzprval 9175* 
Two ways of defining the first two values of a sequence on ℕ.
(Contributed by NM, 5Sep2011.)

⊢ (∀𝑥 ∈ (1...2)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, 𝐵) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵)) 

Theorem  fztpval 9176* 
Two ways of defining the first three values of a sequence on ℕ.
(Contributed by NM, 13Sep2011.)

⊢ (∀𝑥 ∈ (1...3)(𝐹‘𝑥) = if(𝑥 = 1, 𝐴, if(𝑥 = 2, 𝐵, 𝐶)) ↔ ((𝐹‘1) = 𝐴 ∧ (𝐹‘2) = 𝐵 ∧ (𝐹‘3) = 𝐶)) 

Theorem  fzrev 9177 
Reversal of start and end of a finite set of sequential integers.
(Contributed by NM, 25Nov2005.)

⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀)) ↔ (𝐽 − 𝐾) ∈ (𝑀...𝑁))) 

Theorem  fzrev2 9178 
Reversal of start and end of a finite set of sequential integers.
(Contributed by NM, 25Nov2005.)

⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐽 − 𝐾) ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀)))) 

Theorem  fzrev2i 9179 
Reversal of start and end of a finite set of sequential integers.
(Contributed by NM, 25Nov2005.)

⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽 − 𝐾) ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀))) 

Theorem  fzrev3 9180 
The "complement" of a member of a finite set of sequential integers.
(Contributed by NM, 20Nov2005.)

⊢ (𝐾 ∈ ℤ → (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁))) 

Theorem  fzrev3i 9181 
The "complement" of a member of a finite set of sequential integers.
(Contributed by NM, 20Nov2005.)

⊢ (𝐾 ∈ (𝑀...𝑁) → ((𝑀 + 𝑁) − 𝐾) ∈ (𝑀...𝑁)) 

Theorem  fznn 9182 
Finite set of sequential integers starting at 1. (Contributed by NM,
31Aug2011.) (Revised by Mario Carneiro, 18Jun2015.)

⊢ (𝑁 ∈ ℤ → (𝐾 ∈ (1...𝑁) ↔ (𝐾 ∈ ℕ ∧ 𝐾 ≤ 𝑁))) 

Theorem  elfz1b 9183 
Membership in a 1 based finite set of sequential integers. (Contributed
by AV, 30Oct2018.)

⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀)) 

Theorem  elfzm11 9184 
Membership in a finite set of sequential integers. (Contributed by Paul
Chapman, 21Mar2011.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...(𝑁 − 1)) ↔ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁))) 

Theorem  uzsplit 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,
24Apr2014.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) →
(ℤ_{≥}‘𝑀) = ((𝑀...(𝑁 − 1)) ∪
(ℤ_{≥}‘𝑁))) 

Theorem  uzdisj 9186 
The first 𝑁 elements of an upper integer set are
distinct from any
later members. (Contributed by Mario Carneiro, 24Apr2014.)

⊢ ((𝑀...(𝑁 − 1)) ∩
(ℤ_{≥}‘𝑁)) = ∅ 

Theorem  fseq1p1m1 9187 
Add/remove an item to/from the end of a finite sequence. (Contributed
by Paul Chapman, 17Nov2012.) (Revised by Mario Carneiro,
7Mar2014.)

⊢ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩} ⇒ ⊢ (𝑁 ∈ ℕ_{0} → ((𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁))))) 

Theorem  fseq1m1p1 9188 
Add/remove an item to/from the end of a finite sequence. (Contributed
by Paul Chapman, 17Nov2012.)

⊢ 𝐻 = {⟨𝑁, 𝐵⟩} ⇒ ⊢ (𝑁 ∈ ℕ → ((𝐹:(1...(𝑁 − 1))⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)) ↔ (𝐺:(1...𝑁)⟶𝐴 ∧ (𝐺‘𝑁) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...(𝑁 − 1)))))) 

Theorem  fz1sbc 9189* 
Quantification over a onemember finite set of sequential integers in
terms of substitution. (Contributed by NM, 28Nov2005.)

⊢ (𝑁 ∈ ℤ → (∀𝑘 ∈ (𝑁...𝑁)𝜑 ↔ [𝑁 / 𝑘]𝜑)) 

Theorem  elfzp1b 9190 
An integer is a member of a 0based finite set of sequential integers iff
its successor is a member of the corresponding 1based set. (Contributed
by Paul Chapman, 22Jun2011.)

⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...(𝑁 − 1)) ↔ (𝐾 + 1) ∈ (1...𝑁))) 

Theorem  elfzm1b 9191 
An integer is a member of a 1based finite set of sequential integers iff
its predecessor is a member of the corresponding 0based set.
(Contributed by Paul Chapman, 22Jun2011.)

⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (1...𝑁) ↔ (𝐾 − 1) ∈ (0...(𝑁 − 1)))) 

Theorem  elfzp12 9192 
Options for membership in a finite interval of integers. (Contributed by
Jeff Madsen, 18Jun2010.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) 

Theorem  fzm1 9193 
Choices for an element of a finite interval of integers. (Contributed by
Jeff Madsen, 2Sep2009.)

⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (𝑀...(𝑁 − 1)) ∨ 𝐾 = 𝑁))) 

Theorem  fzneuz 9194 
No finite set of sequential integers equals an upper set of integers.
(Contributed by NM, 11Dec2005.)

⊢ ((𝑁 ∈ (ℤ_{≥}‘𝑀) ∧ 𝐾 ∈ ℤ) → ¬ (𝑀...𝑁) = (ℤ_{≥}‘𝐾)) 

Theorem  fznuz 9195 
Disjointness of the upper integers and a finite sequence. (Contributed by
Mario Carneiro, 30Jun2013.) (Revised by Mario Carneiro,
24Aug2013.)

⊢ (𝐾 ∈ (𝑀...𝑁) → ¬ 𝐾 ∈
(ℤ_{≥}‘(𝑁 + 1))) 

Theorem  uznfz 9196 
Disjointness of the upper integers and a finite sequence. (Contributed by
Mario Carneiro, 24Aug2013.)

⊢ (𝐾 ∈ (ℤ_{≥}‘𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1))) 

Theorem  fzp1nel 9197 
One plus the upper bound of a finite set of integers is not a member of
that set. (Contributed by Scott Fenton, 16Dec2017.)

⊢ ¬ (𝑁 + 1) ∈ (𝑀...𝑁) 

Theorem  fzrevral 9198* 
Reversal of scanning order inside of a quantification over a finite set
of sequential integers. (Contributed by NM, 25Nov2005.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))[(𝐾 − 𝑘) / 𝑗]𝜑)) 

Theorem  fzrevral2 9199* 
Reversal of scanning order inside of a quantification over a finite set
of sequential integers. (Contributed by NM, 25Nov2005.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (∀𝑗 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))𝜑 ↔ ∀𝑘 ∈ (𝑀...𝑁)[(𝐾 − 𝑘) / 𝑗]𝜑)) 

Theorem  fzrevral3 9200* 
Reversal of scanning order inside of a quantification over a finite set
of sequential integers. (Contributed by NM, 20Nov2005.)

⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (∀𝑗 ∈ (𝑀...𝑁)𝜑 ↔ ∀𝑘 ∈ (𝑀...𝑁)[((𝑀 + 𝑁) − 𝑘) / 𝑗]𝜑)) 