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Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfznn 8701 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 𝐾𝑁)))

Theoremelfz1b 8702 Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)
(𝑁 (1...𝑀) ↔ (𝑁 𝑀 𝑁𝑀))

Theoremelfzm11 8703 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 𝑁 ℤ) → (𝐾 (𝑀...(𝑁 − 1)) ↔ (𝐾 𝑀𝐾 𝐾 < 𝑁)))

Theoremuzsplit 8704 Express an upper integer set as the disjoint (see uzdisj 8705) union of the first 𝑁 values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
(𝑁 (ℤ𝑀) → (ℤ𝑀) = ((𝑀...(𝑁 − 1)) ∪ (ℤ𝑁)))

Theoremuzdisj 8705 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 8706 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 0 → ((𝐹:(1...𝑁)⟶A B A 𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...(𝑁 + 1))⟶A (𝐺‘(𝑁 + 1)) = B 𝐹 = (𝐺 ↾ (1...𝑁)))))

Theoremfseq1m1p1 8707 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))⟶A B A 𝐺 = (𝐹𝐻)) ↔ (𝐺:(1...𝑁)⟶A (𝐺𝑁) = B 𝐹 = (𝐺 ↾ (1...(𝑁 − 1))))))

Theoremfz1sbc 8708* 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 8709 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 𝑁 ℤ) → (𝐾 (0...(𝑁 − 1)) ↔ (𝐾 + 1) (1...𝑁)))

Theoremelfzm1b 8710 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 𝑁 ℤ) → (𝐾 (1...𝑁) ↔ (𝐾 − 1) (0...(𝑁 − 1))))

Theoremelfzp12 8711 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)
(𝑁 (ℤ𝑀) → (𝐾 (𝑀...𝑁) ↔ (𝐾 = 𝑀 𝐾 ((𝑀 + 1)...𝑁))))

Theoremfzm1 8712 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 8713 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 8714 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 8715 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
(𝐾 (ℤ𝑁) → ¬ 𝐾 (𝑀...(𝑁 − 1)))

Theoremfzp1nel 8716 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 8717* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
((𝑀 𝑁 𝐾 ℤ) → (𝑗 (𝑀...𝑁)φ𝑘 ((𝐾𝑁)...(𝐾𝑀))[(𝐾𝑘) / 𝑗]φ))

Theoremfzrevral2 8718* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
((𝑀 𝑁 𝐾 ℤ) → (𝑗 ((𝐾𝑁)...(𝐾𝑀))φ𝑘 (𝑀...𝑁)[(𝐾𝑘) / 𝑗]φ))

Theoremfzrevral3 8719* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
((𝑀 𝑁 ℤ) → (𝑗 (𝑀...𝑁)φ𝑘 (𝑀...𝑁)[((𝑀 + 𝑁) − 𝑘) / 𝑗]φ))

Theoremfzshftral 8720* Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)
((𝑀 𝑁 𝐾 ℤ) → (𝑗 (𝑀...𝑁)φ𝑘 ((𝑀 + 𝐾)...(𝑁 + 𝐾))[(𝑘𝐾) / 𝑗]φ))

Theoremige2m1fz1 8721 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 8722 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 0 2 ≤ 𝑁) → (𝑁 − 1) (0...𝑁))

3.5.5  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 8723 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 0 𝑁 0 𝐾𝑁))

Theoremfznn0 8724 Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)
(𝑁 0 → (𝐾 (0...𝑁) ↔ (𝐾 0 𝐾𝑁)))

Theoremelfznn0 8725 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 0)

Theoremelfz3nn0 8726 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 0)

Theorem0elfz 8727 0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.)
(𝑁 0 → 0 (0...𝑁))

Theoremnn0fz0 8728 A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.)
(𝑁 0𝑁 (0...𝑁))

Theoremelfz0add 8729 An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
((A 0 B 0) → (𝑁 (0...A) → 𝑁 (0...(A + B))))

Theoremelfz0addOLD 8730 An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) Obsolete version of elfz0add 8729 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
((A 0 B 0) → (𝑁 (0...A) → 𝑁 (0...(A + B))))

Theoremfz0tp 8731 An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
(0...2) = {0, 1, 2}

Theoremelfz0ubfz0 8732 An element of a finite set of sequential nonnegative integers is an element of a finite set of sequential nonnegative integers with the upper bound being an element of the finite set of sequential nonnegative integers with the same lower bound as for the first interval and the element under consideration as upper bound. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
((𝐾 (0...𝑁) 𝐿 (𝐾...𝑁)) → 𝐾 (0...𝐿))

Theoremelfz0fzfz0 8733 A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.)
((𝑀 (0...𝐿) 𝑁 (𝐿...𝑋)) → 𝑀 (0...𝑁))

Theoremfz0fzelfz0 8734 If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.)
((𝑁 (0...𝑅) 𝑀 (𝑁...𝑅)) → 𝑀 (0...𝑅))

Theoremfznn0sub2 8735 Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐾 (0...𝑁) → (𝑁𝐾) (0...𝑁))

Theoremuzsubfz0 8736 Membership of an integer greater than L decreased by L in a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
((𝐿 0 𝑁 (ℤ𝐿)) → (𝑁𝐿) (0...𝑁))

Theoremfz0fzdiffz0 8737 The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.)
((𝑀 (0...𝑁) 𝐾 (𝑀...𝑁)) → (𝐾𝑀) (0...𝑁))

Theoremelfzmlbm 8738 Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝐾 (𝑀...𝑁) → (𝐾𝑀) (0...(𝑁𝑀)))

TheoremelfzmlbmOLD 8739 Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) Obsolete version of elfzmlbm 8738 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐾 (𝑀...𝑁) → (𝐾𝑀) (0...(𝑁𝑀)))

Theoremelfzmlbp 8740 Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
((𝑁 𝐾 (𝑀...(𝑀 + 𝑁))) → (𝐾𝑀) (0...𝑁))

Theoremfzctr 8741 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)
(𝑁 0𝑁 (0...(2 · 𝑁)))

Theoremdifelfzle 8742 The difference of two integers from a finite set of sequential nonnegative integers is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
((𝐾 (0...𝑁) 𝑀 (0...𝑁) 𝐾𝑀) → (𝑀𝐾) (0...𝑁))

Theoremdifelfznle 8743 The difference of two integers from a finite set of sequential nonnegative integers increased by the upper bound is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
((𝐾 (0...𝑁) 𝑀 (0...𝑁) ¬ 𝐾𝑀) → ((𝑀 + 𝑁) − 𝐾) (0...𝑁))

Theoremnn0split 8744 Express the set of nonnegative integers as the disjoint (see nn0disj 8745) union of the first 𝑁 + 1 values and the rest. (Contributed by AV, 8-Nov-2019.)
(𝑁 0 → ℕ0 = ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))))

Theoremnn0disj 8745 The first 𝑁 + 1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)
((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅

Theorem1fv 8746 A one value function. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
((𝑁 𝑉 𝑃 = {⟨0, 𝑁⟩}) → (𝑃:(0...0)⟶𝑉 (𝑃‘0) = 𝑁))

Theorem4fvwrd4 8747* The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
((𝐿 (ℤ‘3) 𝑃:(0...𝐿)⟶𝑉) → 𝑎 𝑉 𝑏 𝑉 𝑐 𝑉 𝑑 𝑉 (((𝑃‘0) = 𝑎 (𝑃‘1) = 𝑏) ((𝑃‘2) = 𝑐 (𝑃‘3) = 𝑑)))

Theorem2ffzeq 8748* Two functions over 0 based finite set of sequential integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
((𝑀 0 𝐹:(0...𝑀)⟶𝑋 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 𝑖 (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))

3.5.6  Half-open integer ranges

Syntaxcfzo 8749 Syntax for half-open integer ranges.
class ..^

Definitiondf-fzo 8750* Define a function generating sets of integers using a half-open range. Read (𝑀..^𝑁) as the integers from 𝑀 up to, but not including, 𝑁; contrast with (𝑀...𝑁) df-fz 8625, which includes 𝑁. Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 8783 with fzsplit 8665, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ = (𝑚 ℤ, 𝑛 ℤ ↦ (𝑚...(𝑛 − 1)))

Theoremfzof 8751 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^:(ℤ × ℤ)⟶𝒫 ℤ

Theoremelfzoel1 8752 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(A (B..^𝐶) → B ℤ)

Theoremelfzoel2 8753 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(A (B..^𝐶) → 𝐶 ℤ)

Theoremelfzoelz 8754 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(A (B..^𝐶) → A ℤ)

Theoremfzoval 8755 Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝑁 ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))

Theoremelfzo 8756 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐾 𝑀 𝑁 ℤ) → (𝐾 (𝑀..^𝑁) ↔ (𝑀𝐾 𝐾 < 𝑁)))

Theoremelfzo2 8757 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 (𝑀..^𝑁) ↔ (𝐾 (ℤ𝑀) 𝑁 𝐾 < 𝑁))

Theoremelfzouz 8758 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 (𝑀..^𝑁) → 𝐾 (ℤ𝑀))

Theoremfzolb 8759 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with 𝑀 < 𝑁. This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 𝑀 (ℤ𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝑀 (𝑀..^𝑁) ↔ (𝑀 𝑁 𝑀 < 𝑁))

Theoremfzolb2 8760 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with 𝑀 < 𝑁. This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 𝑀 (ℤ𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.)
((𝑀 𝑁 ℤ) → (𝑀 (𝑀..^𝑁) ↔ 𝑀 < 𝑁))

Theoremelfzole1 8761 A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐾 (𝑀..^𝑁) → 𝑀𝐾)

Theoremelfzolt2 8762 A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐾 (𝑀..^𝑁) → 𝐾 < 𝑁)

Theoremelfzolt3 8763 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝐾 (𝑀..^𝑁) → 𝑀 < 𝑁)

Theoremelfzolt2b 8764 A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 (𝑀..^𝑁) → 𝐾 (𝐾..^𝑁))

Theoremelfzolt3b 8765 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 (𝑀..^𝑁) → 𝑀 (𝑀..^𝑁))

Theoremfzonel 8766 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
¬ B (A..^B)

Theoremelfzouz2 8767 The upper bound of a half-open range is greater or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 (𝑀..^𝑁) → 𝑁 (ℤ𝐾))

Theoremelfzofz 8768 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝐾 (𝑀..^𝑁) → 𝐾 (𝑀...𝑁))

Theoremelfzo3 8769 Express membership in a half-open integer interval in terms of the "less than or equal" and "less than" predicates on integers, resp. 𝐾 (ℤ𝑀) ↔ 𝑀𝐾, 𝐾 (𝐾..^𝑁) ↔ 𝐾 < 𝑁. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 (𝑀..^𝑁) ↔ (𝐾 (ℤ𝑀) 𝐾 (𝐾..^𝑁)))

Theoremfzom 8770* A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.)
(x x (𝑀..^𝑁) ↔ 𝑀 (𝑀..^𝑁))

Theoremfzossfz 8771 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(A..^B) ⊆ (A...B)

Theoremfzon 8772 A half-open set of sequential integers is empty if the bounds are equal or reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
((𝑀 𝑁 ℤ) → (𝑁𝑀 ↔ (𝑀..^𝑁) = ∅))

Theoremfzonlt0 8773 A half-open integer range is empty if the bounds are equal or reversed. (Contributed by AV, 20-Oct-2018.)
((𝑀 𝑁 ℤ) → (¬ 𝑀 < 𝑁 ↔ (𝑀..^𝑁) = ∅))

Theoremfzo0 8774 Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(A..^A) = ∅

Theoremfzonnsub 8775 If 𝐾 < 𝑁 then 𝑁𝐾 is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
(𝐾 (𝑀..^𝑁) → (𝑁𝐾) ℕ)

Theoremfzonnsub2 8776 If 𝑀 < 𝑁 then 𝑁𝑀 is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝐾 (𝑀..^𝑁) → (𝑁𝑀) ℕ)

Theoremfzoss1 8777 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝐾 (ℤ𝑀) → (𝐾..^𝑁) ⊆ (𝑀..^𝑁))

Theoremfzoss2 8778 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(𝑁 (ℤ𝐾) → (𝑀..^𝐾) ⊆ (𝑀..^𝑁))

Theoremfzossrbm1 8779 Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
(𝑁 ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))

Theoremfzo0ss1 8780 Subset relationship for half-open integer ranges with lower bounds 0 and 1. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
(1..^𝑁) ⊆ (0..^𝑁)

Theoremfzossnn0 8781 A half-open integer range starting at a nonnegative integer is a subset of the nonnegative integers. (Contributed by Alexander van der Vekens, 13-May-2018.)
(𝑀 0 → (𝑀..^𝑁) ⊆ ℕ0)

Theoremfzospliti 8782 One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
((A (B..^𝐶) 𝐷 ℤ) → (A (B..^𝐷) A (𝐷..^𝐶)))

Theoremfzosplit 8783 Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐷 (B...𝐶) → (B..^𝐶) = ((B..^𝐷) ∪ (𝐷..^𝐶)))

Theoremfzodisj 8784 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
((A..^B) ∩ (B..^𝐶)) = ∅

Theoremfzouzsplit 8785 Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
(B (ℤA) → (ℤA) = ((A..^B) ∪ (ℤB)))

Theoremfzouzdisj 8786 A half-open integer range does not overlap the upper integer range starting at the endpoint of the first range. (Contributed by Mario Carneiro, 21-Sep-2016.)
((A..^B) ∩ (ℤB)) = ∅

Theoremlbfzo0 8787 An integer is strictly greater than zero iff it is a member of . (Contributed by Mario Carneiro, 29-Sep-2015.)
(0 (0..^A) ↔ A ℕ)

Theoremelfzo0 8788 Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(A (0..^B) ↔ (A 0 B A < B))

Theoremfzo1fzo0n0 8789 An integer between 1 and an upper bound of a half-open integer range is not 0 and between 0 and the upper bound of the half-open integer range. (Contributed by Alexander van der Vekens, 21-Mar-2018.)
(𝐾 (1..^𝑁) ↔ (𝐾 (0..^𝑁) 𝐾 ≠ 0))

Theoremelfzo0z 8790 Membership in a half-open range of nonnegative integers, generalization of elfzo0 8788 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
(A (0..^B) ↔ (A 0 B A < B))

Theoremelfzo0le 8791 A member in a half-open range of nonnegative integers is less than or equal to the upper bound of the range. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
(A (0..^B) → AB)

Theoremelfzonn0 8792 A member of a half-open range of nonnegative integers is a nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
(𝐾 (0..^𝑁) → 𝐾 0)

Theoremfzonmapblen 8793 The result of subtracting a nonnegative integer from a positive integer and adding another nonnegative integer which is less than the first one is less then the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.)
((A (0..^𝑁) B (0..^𝑁) B < A) → (B + (𝑁A)) < 𝑁)

Theoremfzofzim 8794 If a nonnegative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open integer range. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
((𝐾𝑀 𝐾 (0...𝑀)) → 𝐾 (0..^𝑀))

Theoremfzossnn 8795 Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
(1..^𝑁) ⊆ ℕ

Theoremelfzo1 8796 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝑁 (1..^𝑀) ↔ (𝑁 𝑀 𝑁 < 𝑀))

Theoremfzo0m 8797* A half-open integer range based at 0 is inhabited precisely if the upper bound is a positive integer. (Contributed by Jim Kingdon, 20-Apr-2020.)
(x x (0..^A) ↔ A ℕ)

Theoremfzoaddel 8798 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((A (B..^𝐶) 𝐷 ℤ) → (A + 𝐷) ((B + 𝐷)..^(𝐶 + 𝐷)))

Theoremfzoaddel2 8799 Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((A (0..^(B𝐶)) B 𝐶 ℤ) → (A + 𝐶) (𝐶..^B))

Theoremfzosubel 8800 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((A (B..^𝐶) 𝐷 ℤ) → (A𝐷) ((B𝐷)..^(𝐶𝐷)))

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