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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fzsuc 8701 | 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 ∈ (ℤ_{≥}‘𝑀) → (𝑀...(𝑁 + 1)) = ((𝑀...𝑁) ∪ {(𝑁 + 1)})) | ||
Theorem | fzpred 8702 | Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.) |
⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝑀...𝑁) = ({𝑀} ∪ ((𝑀 + 1)...𝑁))) | ||
Theorem | fzpreddisj 8703 | A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.) |
⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → ({𝑀} ∩ ((𝑀 + 1)...𝑁)) = ∅) | ||
Theorem | elfzp1 8704 | 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 ∈ (ℤ_{≥}‘𝑀) → (𝐾 ∈ (𝑀...(𝑁 + 1)) ↔ (𝐾 ∈ (𝑀...𝑁) ∨ 𝐾 = (𝑁 + 1)))) | ||
Theorem | fzp1ss 8705 | 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)) | ||
Theorem | fzelp1 8706 | 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))) | ||
Theorem | fzp1elp1 8707 | 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))) | ||
Theorem | fznatpl1 8708 | 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)) | ||
Theorem | fzpr 8709 | A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)}) | ||
Theorem | fztp 8710 | 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)}) | ||
Theorem | fzsuc2 8711 | 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)})) | ||
Theorem | fzp1disj 8712 | (𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with {(𝑁 + 1)}. (Contributed by Mario Carneiro, 7-Mar-2014.) |
⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ | ||
Theorem | fzdifsuc 8713 | Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.) |
⊢ (𝑁 ∈ (ℤ_{≥}‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 + 1)) ∖ {(𝑁 + 1)})) | ||
Theorem | fzprval 8714* | Two ways of defining the first two values of a sequence on ℕ. (Contributed by NM, 5-Sep-2011.) |
⊢ (∀x ∈ (1...2)(𝐹‘x) = if(x = 1, A, B) ↔ ((𝐹‘1) = A ∧ (𝐹‘2) = B)) | ||
Theorem | fztpval 8715* | Two ways of defining the first three values of a sequence on ℕ. (Contributed by NM, 13-Sep-2011.) |
⊢ (∀x ∈ (1...3)(𝐹‘x) = if(x = 1, A, if(x = 2, B, 𝐶)) ↔ ((𝐹‘1) = A ∧ (𝐹‘2) = B ∧ (𝐹‘3) = 𝐶)) | ||
Theorem | fzrev 8716 | 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 ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀)) ↔ (𝐽 − 𝐾) ∈ (𝑀...𝑁))) | ||
Theorem | fzrev2 8717 | 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) ↔ (𝐽 − 𝐾) ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀)))) | ||
Theorem | fzrev2i 8718 | Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
⊢ ((𝐽 ∈ ℤ ∧ 𝐾 ∈ (𝑀...𝑁)) → (𝐽 − 𝐾) ∈ ((𝐽 − 𝑁)...(𝐽 − 𝑀))) | ||
Theorem | fzrev3 8719 | 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))) | ||
Theorem | fzrev3i 8720 | 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)) | ||
Theorem | fznn 8721 | 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 ∧ 𝐾 ≤ 𝑁))) | ||
Theorem | elfz1b 8722 | Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.) |
⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀)) | ||
Theorem | elfzm11 8723 | 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 8724 | Express an upper integer set as the disjoint (see uzdisj 8725) 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 8725 | 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 8726 | 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 8727 | 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 8728* | 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 8729 | 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 8730 | 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 8731 | 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 8732 | 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 8733 | 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 8734 | 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 8735 | 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 8736 | 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 8737* | 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 8738* | 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 8739* | 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 8740* | 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 8741 | 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 8742 | 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 8743 | 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 8744 | 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 8745 | 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 8746 | 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 8747 | 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 8748 | 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 8749 | 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 8750 | 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 8749 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 8751 | 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 8752 | 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 8753 | 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 8754 | 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 8755 | 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 8756 | 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 8757 | 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 8758 | 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 8759 | 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 8758 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀))) | ||
Theorem | elfzmlbp 8760 | 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 8761 | 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 8762 | 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 8763 | 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 8764 | Express the set of nonnegative integers as the disjoint (see nn0disj 8765) 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 8765 | 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 8766 | 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 8767* | 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 8768* | 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 8769 | Syntax for half-open integer ranges. |
class ..^ | ||
Definition | df-fzo 8770* | 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 8645, which includes 𝑁. Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 8803 with fzsplit 8685, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | ||
Theorem | fzof 8771 | Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | ||
Theorem | elfzoel1 8772 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ (A ∈ (B..^𝐶) → B ∈ ℤ) | ||
Theorem | elfzoel2 8773 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ (A ∈ (B..^𝐶) → 𝐶 ∈ ℤ) | ||
Theorem | elfzoelz 8774 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
⊢ (A ∈ (B..^𝐶) → A ∈ ℤ) | ||
Theorem | fzoval 8775 | 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 8776 | 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 8777 | 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 8778 | Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ (ℤ_{≥}‘𝑀)) | ||
Theorem | fzolb 8779 | 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 8780 | 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 8781 | 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 8782 | 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 8783 | 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 8784 | 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 8785 | 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 8786 | A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
⊢ ¬ B ∈ (A..^B) | ||
Theorem | elfzouz2 8787 | 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 8788 | 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 8789 | 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 8790* | 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 8791 | 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 8792 | 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 8793 | 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 8794 | 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 8795 | 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 8796 | 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 8797 | 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 8798 | 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 8799 | Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.) |
⊢ (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁)) | ||
Theorem | fzo0ss1 8800 | 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) |
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