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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | difreicc 13101 | The class difference of ℝ and a closed interval. (Contributed by FL, 18-Jun-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℝ ∖ (𝐴[,]𝐵)) = ((-∞(,)𝐴) ∪ (𝐵(,)+∞))) | ||
Theorem | iccsplit 13102 | Split a closed interval into the union of two closed intervals. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝐴[,]𝐶) ∪ (𝐶[,]𝐵))) | ||
Theorem | iccshftr 13103 | Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 + 𝑅) = 𝐶 & ⊢ (𝐵 + 𝑅) = 𝐷 ⇒ ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 + 𝑅) ∈ (𝐶[,]𝐷))) | ||
Theorem | iccshftri 13104 | Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝑅 ∈ ℝ & ⊢ (𝐴 + 𝑅) = 𝐶 & ⊢ (𝐵 + 𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 + 𝑅) ∈ (𝐶[,]𝐷)) | ||
Theorem | iccshftl 13105 | Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 − 𝑅) = 𝐶 & ⊢ (𝐵 − 𝑅) = 𝐷 ⇒ ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 − 𝑅) ∈ (𝐶[,]𝐷))) | ||
Theorem | iccshftli 13106 | Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝑅 ∈ ℝ & ⊢ (𝐴 − 𝑅) = 𝐶 & ⊢ (𝐵 − 𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 − 𝑅) ∈ (𝐶[,]𝐷)) | ||
Theorem | iccdil 13107 | Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 · 𝑅) = 𝐶 & ⊢ (𝐵 · 𝑅) = 𝐷 ⇒ ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 · 𝑅) ∈ (𝐶[,]𝐷))) | ||
Theorem | iccdili 13108 | Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝑅 ∈ ℝ+ & ⊢ (𝐴 · 𝑅) = 𝐶 & ⊢ (𝐵 · 𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 · 𝑅) ∈ (𝐶[,]𝐷)) | ||
Theorem | icccntr 13109 | Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 / 𝑅) = 𝐶 & ⊢ (𝐵 / 𝑅) = 𝐷 ⇒ ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝑋 ∈ ℝ ∧ 𝑅 ∈ ℝ+)) → (𝑋 ∈ (𝐴[,]𝐵) ↔ (𝑋 / 𝑅) ∈ (𝐶[,]𝐷))) | ||
Theorem | icccntri 13110 | Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝑅 ∈ ℝ+ & ⊢ (𝐴 / 𝑅) = 𝐶 & ⊢ (𝐵 / 𝑅) = 𝐷 ⇒ ⊢ (𝑋 ∈ (𝐴[,]𝐵) → (𝑋 / 𝑅) ∈ (𝐶[,]𝐷)) | ||
Theorem | divelunit 13111 | A condition for a ratio to be a member of the closed unit interval. (Contributed by Scott Fenton, 11-Jun-2013.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐴 / 𝐵) ∈ (0[,]1) ↔ 𝐴 ≤ 𝐵)) | ||
Theorem | lincmb01cmp 13112 | A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝐴) + (𝑇 · 𝐵)) ∈ (𝐴[,]𝐵)) | ||
Theorem | iccf1o 13113* | Describe a bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵]. (Contributed by Mario Carneiro, 8-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴))) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (𝐹:(0[,]1)–1-1-onto→(𝐴[,]𝐵) ∧ ◡𝐹 = (𝑦 ∈ (𝐴[,]𝐵) ↦ ((𝑦 − 𝐴) / (𝐵 − 𝐴))))) | ||
Theorem | iccen 13114 | Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 8-Sep-2015.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → (0[,]1) ≈ (𝐴[,]𝐵)) | ||
Theorem | xov1plusxeqvd 13115 | A complex number 𝑋 is positive real iff 𝑋 / (1 + 𝑋) is in (0(,)1). Deduction form. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ≠ -1) ⇒ ⊢ (𝜑 → (𝑋 ∈ ℝ+ ↔ (𝑋 / (1 + 𝑋)) ∈ (0(,)1))) | ||
Theorem | unitssre 13116 | (0[,]1) is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (0[,]1) ⊆ ℝ | ||
Theorem | unitsscn 13117 | The closed unit interval is a subset of the set of the complex numbers. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.) |
⊢ (0[,]1) ⊆ ℂ | ||
Theorem | supicc 13118 | Supremum of a bounded set of real numbers. (Contributed by Thierry Arnoux, 17-May-2019.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶)) & ⊢ (𝜑 → 𝐴 ≠ ∅) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ (𝐵[,]𝐶)) | ||
Theorem | supiccub 13119 | The supremum of a bounded set of real numbers is an upper bound. (Contributed by Thierry Arnoux, 20-May-2019.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶)) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐷 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐷 ≤ sup(𝐴, ℝ, < )) | ||
Theorem | supicclub 13120* | The supremum of a bounded set of real numbers is the least upper bound. (Contributed by Thierry Arnoux, 23-May-2019.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶)) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐷 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐷 < sup(𝐴, ℝ, < ) ↔ ∃𝑧 ∈ 𝐴 𝐷 < 𝑧)) | ||
Theorem | supicclub2 13121* | The supremum of a bounded set of real numbers is the least upper bound. (Contributed by Thierry Arnoux, 23-May-2019.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ⊆ (𝐵[,]𝐶)) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → 𝐷 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ 𝐷) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐷) | ||
Theorem | zltaddlt1le 13122 | The sum of an integer and a real number between 0 and 1 is less than or equal to a second integer iff the sum is less than the second integer. (Contributed by AV, 1-Jul-2021.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ (0(,)1)) → ((𝑀 + 𝐴) < 𝑁 ↔ (𝑀 + 𝐴) ≤ 𝑁)) | ||
Theorem | xnn0xrge0 13123 | An extended nonnegative integer is an extended nonnegative real. (Contributed by AV, 10-Dec-2020.) |
⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ (0[,]+∞)) | ||
Syntax | cfz 13124 |
Extend class notation to include the notation for a contiguous finite set
of integers. Read "𝑀...𝑁 " as "the set of integers
from 𝑀 to
𝑁 inclusive".
This symbol is also used informally in some comments to denote an ellipsis, e.g., 𝐴 + 𝐴↑2 + ... + 𝐴↑(𝑁 − 1). |
class ... | ||
Definition | df-fz 13125* | Define an operation that produces a finite set of sequential integers. Read "𝑀...𝑁 " as "the set of integers from 𝑀 to 𝑁 inclusive". See fzval 13126 for its value and additional comments. (Contributed by NM, 6-Sep-2005.) |
⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) | ||
Theorem | fzval 13126* | The value of a finite set of sequential integers. E.g., 2...5 means the set {2, 3, 4, 5}. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where ℕk means our 1...𝑘; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) | ||
Theorem | fzval2 13127 | An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ)) | ||
Theorem | fzf 13128 | Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.) |
⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ | ||
Theorem | elfz1 13129 | Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | ||
Theorem | elfz 13130 | Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.) |
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | ||
Theorem | elfz2 13131 | Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show 𝑀 ∈ ℤ and 𝑁 ∈ ℤ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | ||
Theorem | elfzd 13132 | Membership in a finite set of sequential integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≤ 𝐾) & ⊢ (𝜑 → 𝐾 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) | ||
Theorem | elfz5 13133 | Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.) |
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (𝑀...𝑁) ↔ 𝐾 ≤ 𝑁)) | ||
Theorem | elfz4 13134 | Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝐾 ∈ (𝑀...𝑁)) | ||
Theorem | elfzuzb 13135 | Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) ↔ (𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾))) | ||
Theorem | eluzfz 13136 | Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → 𝐾 ∈ (𝑀...𝑁)) | ||
Theorem | elfzuz 13137 | A member of a finite set of sequential integers belongs to an upper set of integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀)) | ||
Theorem | elfzuz3 13138 | Membership in a finite set of sequential integers implies membership in an upper set of integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) | ||
Theorem | elfzel2 13139 | Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ) | ||
Theorem | elfzel1 13140 | Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | ||
Theorem | elfzelz 13141 | A member of a finite set of sequential integers is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ) | ||
Theorem | elfzelzd 13142 | A member of a finite set of sequential integers is an integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) ⇒ ⊢ (𝜑 → 𝐾 ∈ ℤ) | ||
Theorem | fzssz 13143 | A finite sequence of integers is a set of integers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝑀...𝑁) ⊆ ℤ | ||
Theorem | elfzle1 13144 | A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐾) | ||
Theorem | elfzle2 13145 | A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁) | ||
Theorem | elfzuz2 13146 | Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑀)) | ||
Theorem | elfzle3 13147 | Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝑁) | ||
Theorem | eluzfz1 13148 | Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | ||
Theorem | eluzfz2 13149 | Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | ||
Theorem | eluzfz2b 13150 | Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑁 ∈ (𝑀...𝑁)) | ||
Theorem | elfz3 13151 | Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.) |
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁)) | ||
Theorem | elfz1eq 13152 | Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.) |
⊢ (𝐾 ∈ (𝑁...𝑁) → 𝐾 = 𝑁) | ||
Theorem | elfzubelfz 13153 | If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (𝑀...𝑁)) | ||
Theorem | peano2fzr 13154 | A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.) |
⊢ ((𝐾 ∈ (ℤ≥‘𝑀) ∧ (𝐾 + 1) ∈ (𝑀...𝑁)) → 𝐾 ∈ (𝑀...𝑁)) | ||
Theorem | fzn0 13155 | Properties of a finite interval of integers which is nonempty. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝑀...𝑁) ≠ ∅ ↔ 𝑁 ∈ (ℤ≥‘𝑀)) | ||
Theorem | fz0 13156 | A finite set of sequential integers is empty if its bounds are not integers. (Contributed by AV, 13-Oct-2018.) |
⊢ ((𝑀 ∉ ℤ ∨ 𝑁 ∉ ℤ) → (𝑀...𝑁) = ∅) | ||
Theorem | fzn 13157 | A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅)) | ||
Theorem | fzen 13158 | A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.) |
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀...𝑁) ≈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))) | ||
Theorem | fz1n 13159 | A 1-based finite set of sequential integers is empty iff it ends at index 0. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝑁 ∈ ℕ0 → ((1...𝑁) = ∅ ↔ 𝑁 = 0)) | ||
Theorem | 0nelfz1 13160 | 0 is not an element of a finite interval of integers starting at 1. (Contributed by AV, 27-Aug-2020.) |
⊢ 0 ∉ (1...𝑁) | ||
Theorem | 0fz1 13161 | Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹 Fn (1...𝑁)) → (𝐹 = ∅ ↔ 𝑁 = 0)) | ||
Theorem | fz10 13162 | There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
⊢ (1...0) = ∅ | ||
Theorem | uzsubsubfz 13163 | Membership of an integer greater than L decreased by ( L - M ) in an M-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.) |
⊢ ((𝐿 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐿)) → (𝑁 − (𝐿 − 𝑀)) ∈ (𝑀...𝑁)) | ||
Theorem | uzsubsubfz1 13164 | Membership of an integer greater than L decreased by ( L - 1 ) in a 1-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.) |
⊢ ((𝐿 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘𝐿)) → (𝑁 − (𝐿 − 1)) ∈ (1...𝑁)) | ||
Theorem | ige3m2fz 13165 | Membership of an integer greater than 2 decreased by 2 in a 1-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.) |
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ (1...𝑁)) | ||
Theorem | fzsplit2 13166 | Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.) |
⊢ (((𝐾 + 1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘𝐾)) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) | ||
Theorem | fzsplit 13167 | Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...𝐾) ∪ ((𝐾 + 1)...𝑁))) | ||
Theorem | fzdisj 13168 | Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝐾 < 𝑀 → ((𝐽...𝐾) ∩ (𝑀...𝑁)) = ∅) | ||
Theorem | fz01en 13169 | 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.) |
⊢ (𝑁 ∈ ℤ → (0...(𝑁 − 1)) ≈ (1...𝑁)) | ||
Theorem | elfznn 13170 | A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.) |
⊢ (𝐾 ∈ (1...𝑁) → 𝐾 ∈ ℕ) | ||
Theorem | elfz1end 13171 | A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.) |
⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴)) | ||
Theorem | fz1ssnn 13172 | A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
⊢ (1...𝐴) ⊆ ℕ | ||
Theorem | fznn0sub 13173 | Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑁 − 𝐾) ∈ ℕ0) | ||
Theorem | fzmmmeqm 13174 | Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.) |
⊢ (𝑀 ∈ (𝐿...𝑁) → ((𝑁 − 𝐿) − (𝑀 − 𝐿)) = (𝑁 − 𝑀)) | ||
Theorem | fzaddel 13175 | Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.) |
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 + 𝐾) ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾)))) | ||
Theorem | fzadd2 13176 | Membership of a sum in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑂 ∈ ℤ ∧ 𝑃 ∈ ℤ)) → ((𝐽 ∈ (𝑀...𝑁) ∧ 𝐾 ∈ (𝑂...𝑃)) → (𝐽 + 𝐾) ∈ ((𝑀 + 𝑂)...(𝑁 + 𝑃)))) | ||
Theorem | fzsubel 13177 | Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.) |
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐽 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (𝐽 ∈ (𝑀...𝑁) ↔ (𝐽 − 𝐾) ∈ ((𝑀 − 𝐾)...(𝑁 − 𝐾)))) | ||
Theorem | fzopth 13178 | A finite set of sequential integers has the ordered pair property (compare opth 5376) under certain conditions. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))) | ||
Theorem | fzass4 13179 | Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ ((𝐵 ∈ (𝐴...𝐷) ∧ 𝐶 ∈ (𝐵...𝐷)) ↔ (𝐵 ∈ (𝐴...𝐶) ∧ 𝐶 ∈ (𝐴...𝐷))) | ||
Theorem | fzss1 13180 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾...𝑁) ⊆ (𝑀...𝑁)) | ||
Theorem | fzss2 13181 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝑀...𝐾) ⊆ (𝑀...𝑁)) | ||
Theorem | fzssuz 13182 | A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.) |
⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀) | ||
Theorem | fzsn 13183 | A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) | ||
Theorem | fzssp1 13184 | Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) | ||
Theorem | fzssnn 13185 | Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.) |
⊢ (𝑀 ∈ ℕ → (𝑀...𝑁) ⊆ ℕ) | ||
Theorem | ssfzunsnext 13186 | A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 13-Nov-2021.) |
⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑆 ∪ {𝐼}) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))) | ||
Theorem | ssfzunsn 13187 | A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 8-Jun-2021.) (Proof shortened by AV, 13-Nov-2021.) |
⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))) | ||
Theorem | fzsuc 13188 | 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 13189 | 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 13190 | 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 13191 | 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 13192 | 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 13193 | 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 13194 | 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 13195 | 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 13196 | 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 13197 | 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 | fz12pr 13198 | An integer range between 1 and 2 is a pair. (Contributed by AV, 11-Jan-2023.) |
⊢ (1...2) = {1, 2} | ||
Theorem | fzsuc2 13199 | 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 13200 | (𝑀...(𝑁 + 1)) is the disjoint union of (𝑀...𝑁) with {(𝑁 + 1)}. (Contributed by Mario Carneiro, 7-Mar-2014.) |
⊢ ((𝑀...𝑁) ∩ {(𝑁 + 1)}) = ∅ |
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