Type  Label  Description 
Statement 

Theorem  ge0mulcl 8601 
The nonnegative reals are closed under multiplication. (Contributed by
Mario Carneiro, 19Jun2014.)

⊢ ((A ∈ (0[,)+∞) ∧ B ∈ (0[,)+∞)) → (A · B)
∈ (0[,)+∞)) 

Theorem  lbicc2 8602 
The lower bound of a closed interval is a member of it. (Contributed by
Paul Chapman, 26Nov2007.) (Revised by FL, 29May2014.) (Revised by
Mario Carneiro, 9Sep2015.)

⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*} ∧ A ≤
B) → A ∈ (A[,]B)) 

Theorem  ubicc2 8603 
The upper bound of a closed interval is a member of it. (Contributed by
Paul Chapman, 26Nov2007.) (Revised by FL, 29May2014.)

⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*} ∧ A ≤
B) → B ∈ (A[,]B)) 

Theorem  0elunit 8604 
Zero is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)

⊢ 0 ∈
(0[,]1) 

Theorem  1elunit 8605 
One is an element of the closed unit. (Contributed by Scott Fenton,
11Jun2013.)

⊢ 1 ∈
(0[,]1) 

Theorem  iooneg 8606 
Membership in a negated open real interval. (Contributed by Paul Chapman,
26Nov2007.)

⊢ ((A ∈ ℝ ∧
B ∈
ℝ ∧ 𝐶 ∈
ℝ) → (𝐶 ∈ (A(,)B) ↔
𝐶 ∈ (B(,)A))) 

Theorem  iccneg 8607 
Membership in a negated closed real interval. (Contributed by Paul
Chapman, 26Nov2007.)

⊢ ((A ∈ ℝ ∧
B ∈
ℝ ∧ 𝐶 ∈
ℝ) → (𝐶 ∈ (A[,]B) ↔
𝐶 ∈ (B[,]A))) 

Theorem  icoshft 8608 
A shifted real is a member of a shifted, closedbelow, openabove real
interval. (Contributed by Paul Chapman, 25Mar2008.)

⊢ ((A ∈ ℝ ∧
B ∈
ℝ ∧ 𝐶 ∈
ℝ) → (𝑋 ∈ (A[,)B) →
(𝑋 + 𝐶) ∈
((A + 𝐶)[,)(B
+ 𝐶)))) 

Theorem  icoshftf1o 8609* 
Shifting a closedbelow, openabove interval is onetoone onto.
(Contributed by Paul Chapman, 25Mar2008.) (Proof shortened by Mario
Carneiro, 1Sep2015.)

⊢ 𝐹 = (x
∈ (A[,)B) ↦
(x + 𝐶)) ⇒ ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐹:(A[,)B)–11onto→((A +
𝐶)[,)(B + 𝐶))) 

Theorem  icodisj 8610 
Endtoend closedbelow, openabove real intervals are disjoint.
(Contributed by Mario Carneiro, 16Jun2014.)

⊢ ((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*} ∧ 𝐶 ∈
ℝ^{*}) → ((A[,)B) ∩ (B[,)𝐶)) = ∅) 

Theorem  ioodisj 8611 
If the upper bound of one open interval is less than or equal to the
lower bound of the other, the intervals are disjoint. (Contributed by
Jeff Hankins, 13Jul2009.)

⊢ ((((A ∈ ℝ^{*} ∧ B ∈ ℝ^{*}) ∧ (𝐶 ∈
ℝ^{*} ∧ 𝐷 ∈
ℝ^{*})) ∧ B ≤ 𝐶) → ((A(,)B) ∩
(𝐶(,)𝐷)) = ∅) 

Theorem  iccshftr 8612 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ (A + 𝑅) = 𝐶
& ⊢ (B + 𝑅) = 𝐷 ⇒ ⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧
(𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈
(A[,]B) ↔ (𝑋 + 𝑅) ∈
(𝐶[,]𝐷))) 

Theorem  iccshftri 8613 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ A ∈ ℝ & ⊢ B ∈
ℝ
& ⊢ 𝑅 ∈
ℝ
& ⊢ (A + 𝑅) = 𝐶
& ⊢ (B + 𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈
(A[,]B) → (𝑋 + 𝑅) ∈
(𝐶[,]𝐷)) 

Theorem  iccshftl 8614 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ (A −
𝑅) = 𝐶
& ⊢ (B −
𝑅) = 𝐷 ⇒ ⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧
(𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈
(A[,]B) ↔ (𝑋 − 𝑅) ∈
(𝐶[,]𝐷))) 

Theorem  iccshftli 8615 
Membership in a shifted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ A ∈ ℝ & ⊢ B ∈
ℝ
& ⊢ 𝑅 ∈
ℝ
& ⊢ (A −
𝑅) = 𝐶
& ⊢ (B −
𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈
(A[,]B) → (𝑋 − 𝑅) ∈
(𝐶[,]𝐷)) 

Theorem  iccdil 8616 
Membership in a dilated interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ (A ·
𝑅) = 𝐶
& ⊢ (B ·
𝑅) = 𝐷 ⇒ ⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧
(𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ^{+})) → (𝑋 ∈
(A[,]B) ↔ (𝑋 · 𝑅) ∈
(𝐶[,]𝐷))) 

Theorem  iccdili 8617 
Membership in a dilated interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ A ∈ ℝ & ⊢ B ∈
ℝ
& ⊢ 𝑅 ∈
ℝ^{+}
& ⊢ (A ·
𝑅) = 𝐶
& ⊢ (B ·
𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈
(A[,]B) → (𝑋 · 𝑅) ∈
(𝐶[,]𝐷)) 

Theorem  icccntr 8618 
Membership in a contracted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ (A / 𝑅) = 𝐶
& ⊢ (B / 𝑅) = 𝐷 ⇒ ⊢ (((A ∈ ℝ ∧ B ∈ ℝ) ∧
(𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ^{+})) → (𝑋 ∈
(A[,]B) ↔ (𝑋 / 𝑅) ∈
(𝐶[,]𝐷))) 

Theorem  icccntri 8619 
Membership in a contracted interval. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ A ∈ ℝ & ⊢ B ∈
ℝ
& ⊢ 𝑅 ∈
ℝ^{+}
& ⊢ (A / 𝑅) = 𝐶
& ⊢ (B / 𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈
(A[,]B) → (𝑋 / 𝑅) ∈
(𝐶[,]𝐷)) 

Theorem  divelunit 8620 
A condition for a ratio to be a member of the closed unit. (Contributed
by Scott Fenton, 11Jun2013.)

⊢ (((A ∈ ℝ ∧ 0
≤ A) ∧
(B ∈
ℝ ∧ 0 < B)) → ((A
/ B) ∈
(0[,]1) ↔ A ≤ B)) 

Theorem  lincmb01cmp 8621 
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 8622* 
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 8623 
(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 8624 
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 8625* 
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 8626 for its value and additional comments.
(Contributed by NM, 6Sep2005.)

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

Theorem  fzval 8626* 
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 8627 
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 8628 
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 8629 
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 8630 
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 8631 
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 8632 
Membership in a finite set of sequential integers. (Contributed by NM,
26Dec2005.)

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

Theorem  elfz4 8633 
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 8634 
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 8635 
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 8636 
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 8637 
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 8638 
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 8639 
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 8640 
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 8641 
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 8642 
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 8643 
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 8644 
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 8645 
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 8646 
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 8647 
Membership in a finite set of sequential integers  special case.
(Contributed by NM, 14Sep2005.)

⊢ (𝑁 ∈
(ℤ_{≥}‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁)) 

Theorem  elfz3 8648 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 21Jul2005.)

⊢ (𝑁 ∈
ℤ → 𝑁 ∈ (𝑁...𝑁)) 

Theorem  elfz1eq 8649 
Membership in a finite set of sequential integers containing one integer.
(Contributed by NM, 19Sep2005.)

⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) 

Theorem  elfzubelfz 8650 
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 8651 
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 8652* 
Properties of a finite interval of integers which is inhabited.
(Contributed by Jim Kingdon, 15Apr2020.)

⊢ (∃x x ∈ (𝑀...𝑁) ↔ 𝑁 ∈
(ℤ_{≥}‘𝑀)) 

Theorem  fztri3or 8653 
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 8654 
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 8655 
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 8656 
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 8657 
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 8658 
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 8659 
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 8660 
There are no integers between 1 and 0. (Contributed by Jeff Madsen,
16Jun2010.) (Proof shortened by Mario Carneiro, 28Apr2015.)

⊢ (1...0) = ∅ 

Theorem  uzsubsubfz 8661 
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 8662 
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 8663 
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 8664 
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 8665 
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 8666 
Condition for two finite intervals of integers to be disjoint.
(Contributed by Jeff Madsen, 17Jun2010.)

⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) 

Theorem  fz01en 8667 
0based and 1based finite sets of sequential integers are equinumerous.
(Contributed by Paul Chapman, 11Apr2009.)

⊢ (𝑁 ∈
ℤ → (0...(𝑁
− 1)) ≈ (1...𝑁)) 

Theorem  elfznn 8668 
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 8669 
A nonempty finite range of integers contains its end point. (Contributed
by Stefan O'Rear, 10Oct2014.)

⊢ (A ∈ ℕ ↔ A ∈ (1...A)) 

Theorem  fznn0sub 8670 
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 8671 
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 8672 
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 8673 
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 8674 
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 8675 
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 8676 
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 8677 
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 8678 
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 8679 
A finite interval of integers with one element. (Contributed by Jeff
Madsen, 2Sep2009.)

⊢ (𝑀 ∈
ℤ → (𝑀...𝑀) = {𝑀}) 

Theorem  fzssp1 8680 
Subset relationship for finite sets of sequential integers.
(Contributed by NM, 21Jul2005.) (Revised by Mario Carneiro,
28Apr2015.)

⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) 

Theorem  fzsuc 8681 
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 8682 
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 8683 
A finite set of sequential integers is disjoint with its predecessor.
(Contributed by AV, 24Aug2019.)

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

Theorem  elfzp1 8684 
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 8685 
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 8686 
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 8687 
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 8688 
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 8689 
A finite interval of integers with two elements. (Contributed by Jeff
Madsen, 2Sep2009.)

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

Theorem  fztp 8690 
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 8691 
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 8692 
(𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with
{(𝑁 +
1)}. (Contributed by Mario Carneiro, 7Mar2014.)

⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ 

Theorem  fzdifsuc 8693 
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 8694* 
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 8695* 
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 8696 
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 8697 
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 8698 
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 8699 
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 8700 
The "complement" of a member of a finite set of sequential integers.
(Contributed by NM, 20Nov2005.)

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