Home | Intuitionistic Logic Explorer Theorem List (p. 88 of 94) | < 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 | ige2m1fz1 8701 | 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 8702 | 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)) | ||
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 8703 | 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 8704 | 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 8705 | 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 8706 | 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 | 0elfz 8707 | 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 8708 | 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 8709 | 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 | elfz0addOLD 8710 | 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 8709 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((A ∈ ℕ_{0} ∧ B ∈ ℕ_{0}) → (𝑁 ∈ (0...A) → 𝑁 ∈ (0...(A + B)))) | ||
Theorem | fz0tp 8711 | 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 8712 | 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 8713 | 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 8714 | 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 8715 | 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 8716 | 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 8717 | 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 8718 | 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 | elfzmlbmOLD 8719 | 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 8718 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀))) | ||
Theorem | elfzmlbp 8720 | 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 8721 | 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 8722 | 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 8723 | 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 8724 | Express the set of nonnegative integers as the disjoint (see nn0disj 8725) 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 | nn0disj 8725 | 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 8726 | 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)) | ||
Theorem | 4fvwrd4 8727* | 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 8728* | 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 8729 | Syntax for half-open integer ranges. |
class ..^ | ||
Definition | df-fzo 8730* | 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 8605, which includes 𝑁. Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 8763 with fzsplit 8645, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | ||
Theorem | fzof 8731 | Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | ||
Theorem | elfzoel1 8732 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ (A ∈ (B..^𝐶) → B ∈ ℤ) | ||
Theorem | elfzoel2 8733 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ (A ∈ (B..^𝐶) → 𝐶 ∈ ℤ) | ||
Theorem | elfzoelz 8734 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ (A ∈ (B..^𝐶) → A ∈ ℤ) | ||
Theorem | fzoval 8735 | 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 8736 | 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 8737 | 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 8738 | Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ_{≥}‘𝑀)) | ||
Theorem | fzolb 8739 | 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 ∈ (ℤ_{≥}‘𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ (𝑀 ∈ (𝑀..^𝑁) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁)) | ||
Theorem | fzolb2 8740 | 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 ∈ (ℤ_{≥}‘𝑁). (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ (𝑀..^𝑁) ↔ 𝑀 < 𝑁)) | ||
Theorem | elfzole1 8741 | 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 8742 | 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 8743 | 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 8744 | 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 8745 | 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 8746 | A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
⊢ ¬ B ∈ (A..^B) | ||
Theorem | elfzouz2 8747 | 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 8748 | 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 8749 | 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 8750* | 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 8751 | 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 8752 | 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 8753 | 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 8754 | 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 8755 | 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 8756 | 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 8757 | 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 8758 | 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 8759 | Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.) |
⊢ (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁)) | ||
Theorem | fzo0ss1 8760 | 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 8761 | 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 8762 | 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 8763 | 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 8764 | Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ ((A..^B) ∩ (B..^𝐶)) = ∅ | ||
Theorem | fzouzsplit 8765 | 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 8766 | 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 8767 | 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 8768 | 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 8769 | 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 8770 | Membership in a half-open range of nonnegative integers, generalization of elfzo0 8768 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 8771 | 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 8772 | 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 8773 | 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 8774 | 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 8775 | Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.) |
⊢ (1..^𝑁) ⊆ ℕ | ||
Theorem | elfzo1 8776 | 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 8777* | 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 8778 | 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 8779 | 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)) | ||
Theorem | fzosubel 8780 | 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 | fzosubel2 8781 | Membership in a translated half-open integer range implies translated membership in the original range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ ((A ∈ ((B + 𝐶)..^(B + 𝐷)) ∧ (B ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (A − B) ∈ (𝐶..^𝐷)) | ||
Theorem | fzosubel3 8782 | Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ ((A ∈ (B..^(B + 𝐷)) ∧ 𝐷 ∈ ℤ) → (A − B) ∈ (0..^𝐷)) | ||
Theorem | eluzgtdifelfzo 8783 | Membership of the difference of integers in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
⊢ ((A ∈ ℤ ∧ B ∈ ℤ) → ((𝑁 ∈ (ℤ_{≥}‘A) ∧ B < A) → (𝑁 − A) ∈ (0..^(𝑁 − B)))) | ||
Theorem | ige2m2fzo 8784 | Membership of an integer greater than 1 decreased by 2 in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
⊢ (𝑁 ∈ (ℤ_{≥}‘2) → (𝑁 − 2) ∈ (0..^(𝑁 − 1))) | ||
Theorem | fzocatel 8785 | Translate membership in a half-open integer range. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
⊢ (((A ∈ (0..^(B + 𝐶)) ∧ ¬ A ∈ (0..^B)) ∧ (B ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (A − B) ∈ (0..^𝐶)) | ||
Theorem | ubmelfzo 8786 | If an integer in a 1 based finite set of sequential integers is subtracted from the upper bound of this finite set of sequential integers, the result is contained in a half-open range of nonnegative integers with the same upper bound. (Contributed by AV, 18-Mar-2018.) (Revised by AV, 30-Oct-2018.) |
⊢ (𝐾 ∈ (1...𝑁) → (𝑁 − 𝐾) ∈ (0..^𝑁)) | ||
Theorem | elfzodifsumelfzo 8787 | If an integer is in a half-open range of nonnegative integers with a difference as upper bound, the sum of the integer with the subtrahend of the difference is in the a half-open range of nonnegative integers containing the minuend of the difference. (Contributed by AV, 13-Nov-2018.) |
⊢ ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...𝑃)) → (𝐼 ∈ (0..^(𝑁 − 𝑀)) → (𝐼 + 𝑀) ∈ (0..^𝑃))) | ||
Theorem | elfzom1elp1fzo 8788 | Membership of an integer incremented by one in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Proof shortened by AV, 5-Jan-2020.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → (𝐼 + 1) ∈ (0..^𝑁)) | ||
Theorem | elfzom1elfzo 8789 | Membership in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(𝑁 − 1))) → 𝐼 ∈ (0..^𝑁)) | ||
Theorem | fzval3 8790 | Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ (𝑁 ∈ ℤ → (𝑀...𝑁) = (𝑀..^(𝑁 + 1))) | ||
Theorem | fzosn 8791 | Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
⊢ (A ∈ ℤ → (A..^(A + 1)) = {A}) | ||
Theorem | elfzomin 8792 | Membership of an integer in the smallest open range of integers. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
⊢ (𝑍 ∈ ℤ → 𝑍 ∈ (𝑍..^(𝑍 + 1))) | ||
Theorem | zpnn0elfzo 8793 | Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ_{0}) → (𝑍 + 𝑁) ∈ (𝑍..^((𝑍 + 𝑁) + 1))) | ||
Theorem | zpnn0elfzo1 8794 | Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ_{0}) → (𝑍 + 𝑁) ∈ (𝑍..^(𝑍 + (𝑁 + 1)))) | ||
Theorem | fzosplitsnm1 8795 | Removing a singleton from a half-open integer range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
⊢ ((A ∈ ℤ ∧ B ∈ (ℤ_{≥}‘(A + 1))) → (A..^B) = ((A..^(B − 1)) ∪ {(B − 1)})) | ||
Theorem | elfzonlteqm1 8796 | If an element of a half-open integer range is not less than the upper bound of the range decreased by 1, it must be equal to the upper bound of the range decreased by 1. (Contributed by AV, 3-Nov-2018.) |
⊢ ((A ∈ (0..^B) ∧ ¬ A < (B − 1)) → A = (B − 1)) | ||
Theorem | fzonn0p1 8797 | A nonnegative integer is element of the half-open range of nonnegative integers with the element increased by one as an upper bound. (Contributed by Alexander van der Vekens, 5-Aug-2018.) |
⊢ (𝑁 ∈ ℕ_{0} → 𝑁 ∈ (0..^(𝑁 + 1))) | ||
Theorem | fzossfzop1 8798 | A half-open range of nonnegative integers is a subset of a half-open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.) |
⊢ (𝑁 ∈ ℕ_{0} → (0..^𝑁) ⊆ (0..^(𝑁 + 1))) | ||
Theorem | fzonn0p1p1 8799 | If a nonnegative integer is element of a half-open range of nonnegative integers, increasing this integer by one results in an element of a half- open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.) |
⊢ (𝐼 ∈ (0..^𝑁) → (𝐼 + 1) ∈ (0..^(𝑁 + 1))) | ||
Theorem | elfzom1p1elfzo 8800 | Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, 1-Aug-2018.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (0..^(𝑁 − 1))) → (𝑋 + 1) ∈ (0..^𝑁)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |