Type  Label  Description 
Statement 

Theorem  lincmb01cmp 8601 
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 8602* 
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 8603 
(0[,]1) is a subset of the reals. (Contributed by
David Moews,
28Feb2017.)

⊢ (0[,]1) ⊆ ℝ 

3.5.4 Finite intervals of integers


Syntax  cfz 8604 
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 8605* 
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 8606 for its value and additional comments.
(Contributed by NM, 6Sep2005.)

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

Theorem  fzval 8606* 
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 8607 
An alternative 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 8608 
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 8609 
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 8610 
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 8611 
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 8612 
Membership in a finite set of sequential integers. (Contributed by NM,
26Dec2005.)

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

Theorem  elfz4 8613 
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 8614 
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 8615 
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 8616 
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 8617 
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 8618 
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 8619 
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 8620 
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 8621 
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 8622 
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 8623 
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 8624 
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 8625 
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 8626 
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 8627 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 14Sep2005.)

⊢ (𝑁 ∈
(ℤ_{≥}‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁)) 

Theorem  elfz3 8628 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 21Jul2005.)

⊢ (𝑁 ∈
ℤ → 𝑁 ∈ (𝑁...𝑁)) 

Theorem  elfz1eq 8629 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 19Sep2005.)

⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) 

Theorem  elfzubelfz 8630 
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 8631 
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 8632* 
Properties of a finite interval of integers which is inhabited.
(Contributed by Jim Kingdon, 15Apr2020.)

⊢ (∃x x ∈ (𝑀...𝑁) ↔ 𝑁 ∈
(ℤ_{≥}‘𝑀)) 

Theorem  fztri3or 8633 
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 8634 
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 8635 
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 8636 
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 8637 
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 8638 
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 8639 
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 8640 
There are no integers between 1 and 0. (Contributed by Jeff Madsen,
16Jun2010.) (Proof shortened by Mario Carneiro, 28Apr2015.)

⊢ (1...0) = ∅ 

Theorem  uzsubsubfz 8641 
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 8642 
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 8643 
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 8644 
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 8645 
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 8646 
Condition for two finite intervals of integers to be disjoint.
(Contributed by Jeff Madsen, 17Jun2010.)

⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) 

Theorem  fz01en 8647 
0based and 1based finite sets of sequential integers are equinumerous.
(Contributed by Paul Chapman, 11Apr2009.)

⊢ (𝑁 ∈
ℤ → (0...(𝑁
− 1)) ≈ (1...𝑁)) 

Theorem  elfznn 8648 
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 8649 
A nonempty finite range of integers contains its end point. (Contributed
by Stefan O'Rear, 10Oct2014.)

⊢ (A ∈ ℕ ↔ A ∈ (1...A)) 

Theorem  fznn0sub 8650 
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 8651 
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 8652 
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 8653 
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 8654 
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 8655 
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 8656 
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 8657 
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 8658 
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 8659 
A finite interval of integers with one element. (Contributed by Jeff
Madsen, 2Sep2009.)

⊢ (𝑀 ∈
ℤ → (𝑀...𝑀) = {𝑀}) 

Theorem  fzssp1 8660 
Subset relationship for finite sets of sequential integers.
(Contributed by NM, 21Jul2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) 

Theorem  fzsuc 8661 
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 8662 
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 8663 
A finite set of sequential integers is disjoint with its predecessor.
(Contributed by AV, 24Aug2019.)

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

Theorem  elfzp1 8664 
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 8665 
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 8666 
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 8667 
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 8668 
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 8669 
A finite interval of integers with two elements. (Contributed by Jeff
Madsen, 2Sep2009.)

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

Theorem  fztp 8670 
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 8671 
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 8672 
(𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with
{(𝑁 +
1)}. (Contributed by Mario Carneiro, 7Mar2014.)

⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ 

Theorem  fzdifsuc 8673 
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 8674* 
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 8675* 
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 8676 
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 8677 
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 8678 
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 8679 
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 8680 
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 8681 
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 8682 
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 8683 
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 8684 
Express an upper integer set as the disjoint (see uzdisj 8685) 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 8685 
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 8686 
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 8687 
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 8688* 
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 8689 
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 8690 
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 8691 
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 8692 
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 8693 
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 8694 
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 8695 
Disjointness of the upper integers and a finite sequence. (Contributed by
Mario Carneiro, 24Aug2013.)

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

Theorem  fzp1nel 8696 
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 8697* 
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 8698* 
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 8699* 
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)) 

Theorem  fzshftral 8700* 
Shift the scanning order inside of a quantification over a finite set of
sequential integers. (Contributed by NM, 27Nov2005.)

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