HomeHome Intuitionistic Logic Explorer
Theorem List (p. 100 of 135)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelfzm11 9901 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 9902 Express an upper integer set as the disjoint (see uzdisj 9903) union of the first 𝑁 values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
(𝑁 ∈ (ℤ𝑀) → (ℤ𝑀) = ((𝑀...(𝑁 − 1)) ∪ (ℤ𝑁)))
 
Theoremuzdisj 9903 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 9904 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 9905 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 9906* 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 9907 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 9908 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 9909 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 9910 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 9911 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 9912 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 9913 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
(𝐾 ∈ (ℤ𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1)))
 
Theoremfzp1nel 9914 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 9915* 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 9916* 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 9917* 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 9918* 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 9919 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 9920 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))
 
Theoremfz01or 9921 An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.)
(𝐴 ∈ (0...1) ↔ (𝐴 = 0 ∨ 𝐴 = 1))
 
4.5.5  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 9922 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 9923 Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)
(𝑁 ∈ ℕ0 → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0𝐾𝑁)))
 
Theoremelfznn0 9924 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 9925 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 9926 Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)
(0...𝑁) ⊆ ℕ0
 
Theoremfz1ssfz0 9927 Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
(1...𝑁) ⊆ (0...𝑁)
 
Theorem0elfz 9928 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 ∈ ℕ0 → 0 ∈ (0...𝑁))
 
Theoremnn0fz0 9929 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 ∈ ℕ0𝑁 ∈ (0...𝑁))
 
Theoremelfz0add 9930 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 ∈ ℕ0𝐵 ∈ ℕ0) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵))))
 
Theoremfz0tp 9931 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 9932 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 9933 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 9934 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 9935 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 9936 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 ∈ ℕ0𝑁 ∈ (ℤ𝐿)) → (𝑁𝐿) ∈ (0...𝑁))
 
Theoremfz0fzdiffz0 9937 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 9938 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)))
 
Theoremelfzmlbp 9939 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 ∈ ℤ ∧ 𝐾 ∈ (𝑀...(𝑀 + 𝑁))) → (𝐾𝑀) ∈ (0...𝑁))
 
Theoremfzctr 9940 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)
(𝑁 ∈ ℕ0𝑁 ∈ (0...(2 · 𝑁)))
 
Theoremdifelfzle 9941 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 9942 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 9943 Express the set of nonnegative integers as the disjoint (see nn0disj 9945) union of the first 𝑁 + 1 values and the rest. (Contributed by AV, 8-Nov-2019.)
(𝑁 ∈ ℕ0 → ℕ0 = ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))))
 
Theoremnnsplit 9944 Express the set of positive integers as the disjoint union of the first 𝑁 values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝑁 ∈ ℕ → ℕ = ((1...𝑁) ∪ (ℤ‘(𝑁 + 1))))
 
Theoremnn0disj 9945 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 9946 A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
((𝑁𝑉𝑃 = {⟨0, 𝑁⟩}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁))
 
Theorem4fvwrd4 9947* 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) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
 
Theorem2ffzeq 9948* 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 ∈ ℕ0𝐹:(0...𝑀)⟶𝑋𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹𝑖) = (𝑃𝑖))))
 
4.5.6  Half-open integer ranges
 
Syntaxcfzo 9949 Syntax for half-open integer ranges.
class ..^
 
Definitiondf-fzo 9950* 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 9821, which includes 𝑁. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 9984 with fzsplit 9861, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1)))
 
Theoremfzof 9951 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^:(ℤ × ℤ)⟶𝒫 ℤ
 
Theoremelfzoel1 9952 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ)
 
Theoremelfzoel2 9953 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ)
 
Theoremelfzoelz 9954 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ)
 
Theoremfzoval 9955 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 9956 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀..^𝑁) ↔ (𝑀𝐾𝐾 < 𝑁)))
 
Theoremelfzo2 9957 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐾 < 𝑁))
 
Theoremelfzouz 9958 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
(𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ𝑀))
 
Theoremfzodcel 9959 Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐾 ∈ (𝑀..^𝑁))
 
Theoremfzolb 9960 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 alternate 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 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁))
 
Theoremfzolb2 9961 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 alternate 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 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (𝑀..^𝑁) ↔ 𝑀 < 𝑁))
 
Theoremelfzole1 9962 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 9963 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 9964 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 9965 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 9966 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 9967 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
¬ 𝐵 ∈ (𝐴..^𝐵)
 
Theoremelfzouz2 9968 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 9969 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 9970 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 9971* 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 9972 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 9973 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 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁𝑀 ↔ (𝑀..^𝑁) = ∅))
 
Theoremfzonlt0 9974 A half-open integer range is empty if the bounds are equal or reversed. (Contributed by AV, 20-Oct-2018.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ 𝑀 < 𝑁 ↔ (𝑀..^𝑁) = ∅))
 
Theoremfzo0 9975 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 9976 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 9977 If 𝑀 < 𝑁 then 𝑁𝑀 is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
(𝐾 ∈ (𝑀..^𝑁) → (𝑁𝑀) ∈ ℕ)
 
Theoremfzoss1 9978 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 9979 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 9980 Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
(𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
 
Theoremfzo0ss1 9981 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 9982 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 ∈ ℕ0 → (𝑀..^𝑁) ⊆ ℕ0)
 
Theoremfzospliti 9983 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 9984 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 9985 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅
 
Theoremfzouzsplit 9986 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 9987 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 9988 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 9989 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 ∈ ℕ0𝐵 ∈ ℕ ∧ 𝐴 < 𝐵))
 
Theoremfzo1fzo0n0 9990 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 9991 Membership in a half-open range of nonnegative integers, generalization of elfzo0 9989 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
(𝐴 ∈ (0..^𝐵) ↔ (𝐴 ∈ ℕ0𝐵 ∈ ℤ ∧ 𝐴 < 𝐵))
 
Theoremelfzo0le 9992 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 9993 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 ∈ ℕ0)
 
Theoremfzonmapblen 9994 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 9995 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.)
((𝐾𝑀𝐾 ∈ (0...𝑀)) → 𝐾 ∈ (0..^𝑀))
 
Theoremfzossnn 9996 Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
(1..^𝑁) ⊆ ℕ
 
Theoremelfzo1 9997 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝑁 ∈ (1..^𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 < 𝑀))
 
Theoremfzo0m 9998* 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 9999 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 10000 Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐴 ∈ (0..^(𝐵𝐶)) ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 + 𝐶) ∈ (𝐶..^𝐵))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13439
  Copyright terms: Public domain < Previous  Next >