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Theorem List for Metamath Proof Explorer - 12501-12600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeluzfz 12501 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 12502 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 12503 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 12504 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 12505 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 12506 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 12507 A finite sequence of integers is a set of integers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝑀...𝑁) ⊆ ℤ
 
Theoremelfzle1 12508 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 12509 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 12510 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 12511 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 12512 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 12513 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 12514 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
(𝑁 ∈ (ℤ𝑀) ↔ 𝑁 ∈ (𝑀...𝑁))
 
Theoremelfz3 12515 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
(𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁))
 
Theoremelfz1eq 12516 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
(𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁)
 
Theoremelfzubelfz 12517 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 12518 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 12519 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 12520 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 12521 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 12522 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 12523 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 12524 0 is not an element of a finite interval of integers starting at 1. (Contributed by AV, 27-Aug-2020.)
0 ∉ (1...𝑁)
 
Theorem0fz1 12525 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 12526 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 12527 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 12528 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 12529 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 12530 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 12531 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 12532 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
(𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅)
 
Theoremfz01en 12533 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 12534 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 12535 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
(𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
 
Theoremfz1ssnn 12536 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(1...𝐴) ⊆ ℕ
 
Theoremfznn0sub 12537 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 12538 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 12539 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 12540 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 12541 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 12542 A finite set of sequential integers has the ordered pair property (compare opth 5081) 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 12543 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 12544 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 12545 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 12546 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 12547 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
 
Theoremfzssp1 12548 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 12549 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)
 
Theoremssfzunsnext 12550 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, 13-Nov-2021.)
((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑆 ∪ {𝐼}) ⊆ (if(𝐼𝑀, 𝐼, 𝑀)...if(𝐼𝑁, 𝑁, 𝐼)))
 
Theoremssfzunsn 12551 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.) (Proof shortened by AV, 13-Nov-2021.)
((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (𝑀...if(𝐼𝑁, 𝑁, 𝐼)))
 
Theoremfzsuc 12552 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 12553 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 12554 A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
(𝑁 ∈ (ℤ𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅)
 
Theoremelfzp1 12555 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 12556 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 12557 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 12558 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 12559 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 12560 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
 
Theoremfztp 12561 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 12562 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 12563 (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 12564 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 12565* 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 12566* 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 12567 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 12568 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 12569 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 12570 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 12571 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 12572 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 12573 Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.) (Proof shortened by AV, 23-Jan-2022.)
(𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁𝑀))
 
Theoremelfz1uz 12574 Membership in a 1 based finite set of sequential integers with an upper integer. (Contributed by AV, 23-Jan-2022.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) → 𝑁 ∈ (1...𝑀))
 
Theoremelfzm11 12575 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 12576 Express an upper integer set as the disjoint (see uzdisj 12577) union of the first 𝑁 values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
(𝑁 ∈ (ℤ𝑀) → (ℤ𝑀) = ((𝑀...(𝑁 − 1)) ∪ (ℤ𝑁)))
 
Theoremuzdisj 12577 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 12578 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 12579 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 12580* 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 12581 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 12582 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 12583 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 12584 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 12585 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 12586 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 12587 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
(𝐾 ∈ (ℤ𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1)))
 
Theoremfzp1nel 12588 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 12589* 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 12590* 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 12591* 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 12592* 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 12593 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 12594 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 12595 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))
 
Theoremfznn0 12596 Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)
(𝑁 ∈ ℕ0 → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0𝐾𝑁)))
 
Theoremelfznn0 12597 A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (0...𝑁) → 𝐾 ∈ ℕ0)
 
Theoremelfz3nn0 12598 The upper bound of a nonempty finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
 
Theoremfz0ssnn0 12599 Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)
(0...𝑁) ⊆ ℕ0
 
Theoremfz1ssfz0 12600 Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(1...𝑁) ⊆ (0...𝑁)
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