Home | 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 |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elfzm11 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 ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁))) | ||
Theorem | uzsplit 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 ∈ (ℤ_{≥}‘𝑀) → (ℤ_{≥}‘𝑀) = ((𝑀...(𝑁 − 1)) ∪ (ℤ_{≥}‘𝑁))) | ||
Theorem | uzdisj 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)) ∩ (ℤ_{≥}‘𝑁)) = ∅ | ||
Theorem | fseq1p1m1 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 ∈ 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 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))⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)) ↔ (𝐺:(1...𝑁)⟶𝐴 ∧ (𝐺‘𝑁) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...(𝑁 − 1)))))) | ||
Theorem | fz1sbc 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)) | ||
Theorem | elfzp1b 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))) | ||
Theorem | elfzm1b 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)))) | ||
Theorem | elfzp12 9909 | Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.) |
⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 = 𝑀 ∨ 𝐾 ∈ ((𝑀 + 1)...𝑁)))) | ||
Theorem | fzm1 9910 | Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (𝑀...(𝑁 − 1)) ∨ 𝐾 = 𝑁))) | ||
Theorem | fzneuz 9911 | No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.) |
⊢ ((𝑁 ∈ (ℤ_{≥}‘𝑀) ∧ 𝐾 ∈ ℤ) → ¬ (𝑀...𝑁) = (ℤ_{≥}‘𝐾)) | ||
Theorem | fznuz 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))) | ||
Theorem | uznfz 9913 | Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.) |
⊢ (𝐾 ∈ (ℤ_{≥}‘𝑁) → ¬ 𝐾 ∈ (𝑀...(𝑁 − 1))) | ||
Theorem | fzp1nel 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) | ||
Theorem | fzrevral 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 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))[(𝐾 − 𝑘) / 𝑗]𝜑)) | ||
Theorem | fzrevral2 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 ∈ ((𝐾 − 𝑁)...(𝐾 − 𝑀))𝜑 ↔ ∀𝑘 ∈ (𝑀...𝑁)[(𝐾 − 𝑘) / 𝑗]𝜑)) | ||
Theorem | fzrevral3 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)) | ||
Theorem | fzshftral 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))[(𝑘 − 𝐾) / 𝑗]𝜑)) | ||
Theorem | ige2m1fz1 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)) | ||
Theorem | ige2m1fz 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 ∈ ℕ_{0} ∧ 2 ≤ 𝑁) → (𝑁 − 1) ∈ (0...𝑁)) | ||
Theorem | fz01or 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)) | ||
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". | ||
Theorem | elfz2nn0 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 ∈ ℕ_{0} ∧ 𝑁 ∈ ℕ_{0} ∧ 𝐾 ≤ 𝑁)) | ||
Theorem | fznn0 9923 | Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.) |
⊢ (𝑁 ∈ ℕ_{0} → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ_{0} ∧ 𝐾 ≤ 𝑁))) | ||
Theorem | elfznn0 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 ∈ ℕ_{0}) | ||
Theorem | elfz3nn0 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 ∈ ℕ_{0}) | ||
Theorem | fz0ssnn0 9926 | Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.) |
⊢ (0...𝑁) ⊆ ℕ_{0} | ||
Theorem | fz1ssfz0 9927 | Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (1...𝑁) ⊆ (0...𝑁) | ||
Theorem | 0elfz 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 ∈ ℕ_{0} → 0 ∈ (0...𝑁)) | ||
Theorem | nn0fz0 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 ∈ ℕ_{0} ↔ 𝑁 ∈ (0...𝑁)) | ||
Theorem | elfz0add 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 ∈ ℕ_{0} ∧ 𝐵 ∈ ℕ_{0}) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵)))) | ||
Theorem | fz0tp 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} | ||
Theorem | elfz0ubfz0 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)) | ||
Theorem | elfz0fzfz0 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)) | ||
Theorem | fz0fzelfz0 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)) | ||
Theorem | fznn0sub2 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) → (𝑁 − 𝐾) ∈ (0...𝑁)) | ||
Theorem | uzsubfz0 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 ∈ ℕ_{0} ∧ 𝑁 ∈ (ℤ_{≥}‘𝐿)) → (𝑁 − 𝐿) ∈ (0...𝑁)) | ||
Theorem | fz0fzdiffz0 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)) → (𝐾 − 𝑀) ∈ (0...𝑁)) | ||
Theorem | elfzmlbm 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) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀))) | ||
Theorem | elfzmlbp 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))) → (𝐾 − 𝑀) ∈ (0...𝑁)) | ||
Theorem | fzctr 9940 | Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.) |
⊢ (𝑁 ∈ ℕ_{0} → 𝑁 ∈ (0...(2 · 𝑁))) | ||
Theorem | difelfzle 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) ∧ 𝐾 ≤ 𝑀) → (𝑀 − 𝐾) ∈ (0...𝑁)) | ||
Theorem | difelfznle 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) ∧ ¬ 𝐾 ≤ 𝑀) → ((𝑀 + 𝑁) − 𝐾) ∈ (0...𝑁)) | ||
Theorem | nn0split 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 ∈ ℕ_{0} → ℕ_{0} = ((0...𝑁) ∪ (ℤ_{≥}‘(𝑁 + 1)))) | ||
Theorem | nnsplit 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)))) | ||
Theorem | nn0disj 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))) = ∅ | ||
Theorem | 1fv 9946 | A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
⊢ ((𝑁 ∈ 𝑉 ∧ 𝑃 = {⟨0, 𝑁⟩}) → (𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁)) | ||
Theorem | 4fvwrd4 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) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑))) | ||
Theorem | 2ffzeq 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 ∈ ℕ_{0} ∧ 𝐹:(0...𝑀)⟶𝑋 ∧ 𝑃:(0...𝑁)⟶𝑌) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0...𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))) | ||
Syntax | cfzo 9949 | Syntax for half-open integer ranges. |
class ..^ | ||
Definition | df-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))) | ||
Theorem | fzof 9951 | Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | ||
Theorem | elfzoel1 9952 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐵 ∈ ℤ) | ||
Theorem | elfzoel2 9953 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐶 ∈ ℤ) | ||
Theorem | elfzoelz 9954 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ (𝐴 ∈ (𝐵..^𝐶) → 𝐴 ∈ ℤ) | ||
Theorem | fzoval 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))) | ||
Theorem | elfzo 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) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁))) | ||
Theorem | elfzo2 9957 | Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ (𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ_{≥}‘𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐾 < 𝑁)) | ||
Theorem | elfzouz 9958 | Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ_{≥}‘𝑀)) | ||
Theorem | fzodcel 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)) | ||
Theorem | fzolb 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 ∈ (ℤ_{≥}‘𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ (𝑀 ∈ (𝑀..^𝑁) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁)) | ||
Theorem | fzolb2 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 ∈ (ℤ_{≥}‘𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (𝑀..^𝑁) ↔ 𝑀 < 𝑁)) | ||
Theorem | elfzole1 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) → 𝑀 ≤ 𝐾) | ||
Theorem | elfzolt2 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) | ||
Theorem | elfzolt3 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) | ||
Theorem | elfzolt2b 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)) | ||
Theorem | elfzolt3b 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)) | ||
Theorem | fzonel 9967 | A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
⊢ ¬ 𝐵 ∈ (𝐴..^𝐵) | ||
Theorem | elfzouz2 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 ∈ (ℤ_{≥}‘𝐾)) | ||
Theorem | elfzofz 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)) | ||
Theorem | elfzo3 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 ∈ (ℤ_{≥}‘𝑀) ↔ 𝑀 ≤ 𝐾, 𝐾 ∈ (𝐾..^𝑁) ↔ 𝐾 < 𝑁. (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ (𝐾 ∈ (𝑀..^𝑁) ↔ (𝐾 ∈ (ℤ_{≥}‘𝑀) ∧ 𝐾 ∈ (𝐾..^𝑁))) | ||
Theorem | fzom 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)) | ||
Theorem | fzossfz 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) | ||
Theorem | fzon 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) → (𝑁 ≤ 𝑀 ↔ (𝑀..^𝑁) = ∅)) | ||
Theorem | fzonlt0 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) = ∅)) | ||
Theorem | fzo0 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) = ∅ | ||
Theorem | fzonnsub 9976 | If 𝐾 < 𝑁 then 𝑁 − 𝐾 is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.) |
⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑁 − 𝐾) ∈ ℕ) | ||
Theorem | fzonnsub2 9977 | If 𝑀 < 𝑁 then 𝑁 − 𝑀 is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.) |
⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝑁 − 𝑀) ∈ ℕ) | ||
Theorem | fzoss1 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 ∈ (ℤ_{≥}‘𝑀) → (𝐾..^𝑁) ⊆ (𝑀..^𝑁)) | ||
Theorem | fzoss2 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 ∈ (ℤ_{≥}‘𝐾) → (𝑀..^𝐾) ⊆ (𝑀..^𝑁)) | ||
Theorem | fzossrbm1 9980 | Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.) |
⊢ (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁)) | ||
Theorem | fzo0ss1 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) | ||
Theorem | fzossnn0 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 ∈ ℕ_{0} → (𝑀..^𝑁) ⊆ ℕ_{0}) | ||
Theorem | fzospliti 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))) | ||
Theorem | fzosplit 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))) | ||
Theorem | fzodisj 9985 | Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ ((𝐴..^𝐵) ∩ (𝐵..^𝐶)) = ∅ | ||
Theorem | fzouzsplit 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 ∈ (ℤ_{≥}‘𝐴) → (ℤ_{≥}‘𝐴) = ((𝐴..^𝐵) ∪ (ℤ_{≥}‘𝐵))) | ||
Theorem | fzouzdisj 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) ∩ (ℤ_{≥}‘𝐵)) = ∅ | ||
Theorem | lbfzo0 9988 | An integer is strictly greater than zero iff it is a member of ℕ. (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ (0 ∈ (0..^𝐴) ↔ 𝐴 ∈ ℕ) | ||
Theorem | elfzo0 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 ∈ ℕ_{0} ∧ 𝐵 ∈ ℕ ∧ 𝐴 < 𝐵)) | ||
Theorem | fzo1fzo0n0 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)) | ||
Theorem | elfzo0z 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 ∈ ℕ_{0} ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵)) | ||
Theorem | elfzo0le 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) → 𝐴 ≤ 𝐵) | ||
Theorem | elfzonn0 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 ∈ ℕ_{0}) | ||
Theorem | fzonmapblen 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 + (𝑁 − 𝐴)) < 𝑁) | ||
Theorem | fzofzim 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.) |
⊢ ((𝐾 ≠ 𝑀 ∧ 𝐾 ∈ (0...𝑀)) → 𝐾 ∈ (0..^𝑀)) | ||
Theorem | fzossnn 9996 | Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.) |
⊢ (1..^𝑁) ⊆ ℕ | ||
Theorem | elfzo1 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)) | ||
Theorem | fzo0m 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) | ||
Theorem | fzoaddel 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))) | ||
Theorem | fzoaddel2 10000 | Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ ((𝐴 ∈ (0..^(𝐵 − 𝐶)) ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 + 𝐶) ∈ (𝐶..^𝐵)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |