P. 111 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

hash2 11001

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

⊢ (♯‘2_o) = 2

Theorem

hash3 11002

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

⊢ (♯‘3_o) = 3

Theorem

hash4 11003

Size of a natural number ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)

⊢ (♯‘4_o) = 4

Theorem

pr0hash2ex 11004

There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.)

⊢ (♯‘{∅, {∅}}) = 2

Theorem

fihashss 11005

The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (♯‘𝐵) ≤ (♯‘𝐴))

Theorem

fiprsshashgt1 11006

The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)

⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 ∈ Fin) → ({𝐴, 𝐵} ⊆ 𝐶 → 2 ≤ (♯‘𝐶)))

Theorem

fihashssdif 11007

The size of the difference of a finite set and a finite subset is the set's size minus the subset's. (Contributed by Jim Kingdon, 31-May-2022.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (♯‘(𝐴 ∖ 𝐵)) = ((♯‘𝐴) − (♯‘𝐵)))

Theorem

hashdifsn 11008

The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → (♯‘(𝐴 ∖ {𝐵})) = ((♯‘𝐴) − 1))

Theorem

hashdifpr 11009

The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)

⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2))

Theorem

hashfz 11010

Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)

⊢ (𝐵 ∈ (ℤ_≥‘𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1))

Theorem

hashfzo 11011

Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)

⊢ (𝐵 ∈ (ℤ_≥‘𝐴) → (♯‘(𝐴..^𝐵)) = (𝐵 − 𝐴))

Theorem

hashfzo0 11012

Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)

⊢ (𝐵 ∈ ℕ₀ → (♯‘(0..^𝐵)) = 𝐵)

Theorem

hashfzp1 11013

Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)

⊢ (𝐵 ∈ (ℤ_≥‘𝐴) → (♯‘((𝐴 + 1)...𝐵)) = (𝐵 − 𝐴))

Theorem

hashfz0 11014

Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)

⊢ (𝐵 ∈ ℕ₀ → (♯‘(0...𝐵)) = (𝐵 + 1))

Theorem

hashxp 11015

The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)

⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 × 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))

Theorem

fimaxq 11016*

A finite set of rational numbers has a maximum. (Contributed by Jim Kingdon, 6-Sep-2022.)

⊢ ((𝐴 ⊆ ℚ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

Theorem

fiubm 11017*

Lemma for fiubz 11018 and fiubnn 11019. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.)

⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ ℚ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

Theorem

fiubz 11018*

A finite set of integers has an upper bound which is an integer. (Contributed by Jim Kingdon, 29-Oct-2024.)

⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

Theorem

fiubnn 11019*

A finite set of natural numbers has an upper bound which is a a natural number. (Contributed by Jim Kingdon, 29-Oct-2024.)

⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)

Theorem

resunimafz0 11020

The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)

⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    ⇒   ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))

Theorem

fnfz0hash 11021

The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (0...𝑁)) → (♯‘𝐹) = (𝑁 + 1))

Theorem

ffz0hash 11022

The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(0...𝑁)⟶𝐵) → (♯‘𝐹) = (𝑁 + 1))

Theorem

ffzo0hash 11023

The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹 Fn (0..^𝑁)) → (♯‘𝐹) = 𝑁)

Theorem

fnfzo0hash 11024

The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)

⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐹:(0..^𝑁)⟶𝐵) → (♯‘𝐹) = 𝑁)

Theorem

hashfacen 11025*

The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)

⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ≈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷})

Theorem

leisorel 11026

Version of isorel 5905 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)

⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷)))

Theorem

zfz1isolemsplit 11027

Lemma for zfz1iso 11030. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.)

⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))

Theorem

zfz1isolemiso 11028*

Lemma for zfz1iso 11030. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)

⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ⊆ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑋)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)    &   ⊢ (𝜑 → 𝐺 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))    &   ⊢ (𝜑 → 𝐴 ∈ (1...(♯‘𝑋)))    &   ⊢ (𝜑 → 𝐵 ∈ (1...(♯‘𝑋)))    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐴) < ((𝐺 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝐵)))

Theorem

zfz1isolem1 11029*

Lemma for zfz1iso 11030. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)

⊢ (𝜑 → 𝐾 ∈ ω)    &   ⊢ (𝜑 → ∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦 ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)))    &   ⊢ (𝜑 → 𝑋 ⊆ ℤ)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≈ suc 𝐾)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑋)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)    ⇒   ⊢ (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋))

Theorem

zfz1iso 11030*

A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)

⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))

Theorem

seq3coll 11031*

The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 2-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)

⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑆 ∧ 𝑛 ∈ 𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 Isom < , < ((1...(♯‘𝐴)), 𝐴))    &   ⊢ (𝜑 → 𝑁 ∈ (1...(♯‘𝐴)))    &   ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐻‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀...(𝐺‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑘) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑛) = (𝐹‘(𝐺‘𝑛)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺‘𝑁)) = (seq1( + , 𝐻)‘𝑁))

4.6.10.1  Proper unordered pairs and triples (sets of size 2 and 3)

Theorem

hash2en 11032

Two equivalent ways to say a set has two elements. (Contributed by Jim Kingdon, 4-Dec-2025.)

⊢ (𝑉 ≈ 2_o ↔ (𝑉 ∈ Fin ∧ (♯‘𝑉) = 2))

Theorem

hashdmprop2dom 11033

A class which contains two ordered pairs with different first components has at least two elements. (Contributed by AV, 12-Nov-2021.)

⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ 𝐹)    ⇒   ⊢ (𝜑 → 2_o ≼ dom 𝐹)

4.6.10.2  Functions with a domain containing at least two different elements

Theorem

fundm2domnop0 11034

A function with a domain containing (at least) two different elements is not an ordered pair. This theorem (which requires that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 13011. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.)

⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2_o ≼ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Theorem

fundm2domnop 11035

A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.) (Proof shortened by AV, 9-Jun-2021.)

⊢ ((Fun 𝐺 ∧ 2_o ≼ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Theorem

fun2dmnop0 11036

A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 11037 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 13011. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Theorem

fun2dmnop 11037

A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 9-Jun-2021.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun 𝐺 ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

4.7  Words over a set

This section is about words (or strings) over a set (alphabet) defined as finite sequences of symbols (or characters) being elements of the alphabet. Although it is often required that the underlying set/alphabet be nonempty, finite and not a proper class, these restrictions are not made in the current definition df-word 11039. Note that the empty word ∅ (i.e., the empty set) is the only word over an empty alphabet, see 0wrd0 11064. The set Word 𝑆 of words over 𝑆 is the free monoid over 𝑆, where the monoid law is concatenation and the monoid unit is the empty word. Besides the definition of words themselves, several operations on words are defined in this section:

Name	Reference	Explanation	Example	Remarks
Length (or size) of a word	df-ihash 10965: (♯‘𝑊)	Operation which determines the number of the symbols within the word.	𝑊:(0..^𝑁)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 𝑁	This is not a special definition for words, but for arbitrary sets.
First symbol of a word	df-fv 5302: (𝑊‘0)	Operation which extracts the first symbol of a word. The result is the symbol at the first place in the sequence representing the word.	𝑊:(0..^1)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (𝑊‘0) ∈ 𝑆	This is not a special definition for words, but for arbitrary functions with a half-open range of nonnegative integers as domain.
Last symbol of a word	df-lsw 11083: (lastS‘𝑊)	Operation which extracts the last symbol of a word. The result is the symbol at the last place in the sequence representing the word.	𝑊:(0..^3)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (lastS‘𝑊) = (𝑊‘2)	Note that the index of the last symbol is less by 1 than the length of the word.
Subword (or substring) of a word	df-substr 11144: (𝑊 substr ⟨𝐼, 𝐽⟩)	Operation which extracts a portion of a word. The result is a consecutive, reindexed part of the sequence representing the word.	𝑊:(0..^3)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (𝑊 substr ⟨1, 2⟩) ∈ Word 𝑆 ∧ (♯‘(𝑊 substr ⟨1, 2⟩)) = 1	Note that the last index of the range defining the subword is greater by 1 than the index of the last symbol of the subword in the sequence of the original word.
Concatenation of two words	df-concat 11092: (𝑊 ++ 𝑈)	Operation combining two words to one new word. The result is a combined, reindexed sequence build from the sequences representing the two words.	(𝑊 ∈ Word 𝑆 ∧ 𝑈 ∈ Word 𝑆) → (♯‘(𝑊 ++ 𝑈)) = ((♯‘𝑊) + (♯‘𝑈))	Note that the index of the first symbol of the second concatenated word is the length of the first word in the concatenation.
Singleton word	df-s1 11115: ⟨“𝑆”⟩	Constructor building a word of length 1 from a symbol.	(♯‘⟨“𝑆”⟩) = 1

Conventions:

• Words are usually represented by class variable 𝑊, or if two words are involved, by 𝑊 and 𝑈 or by 𝐴 and 𝐵.
• The alphabets are usually represented by class variable 𝑉 (because any arbitrary set can be an alphabet), sometimes also by 𝑆 (especially as codomain of the function representing a word, because the alphabet is the set of symbols).
• The symbols are usually represented by class variables 𝑆 or 𝐴, or if two symbols are involved, by 𝑆 and 𝑇 or by 𝐴 and 𝐵.
• The indices of the sequence representing a word are usually represented by class variable 𝐼, if two indices are involved (especially for subwords) by 𝐼 and 𝐽, or by 𝑀 and 𝑁.
• The length of a word is usually represented by class variables 𝑁 or 𝐿.
• The number of positions by which to cyclically shift a word is usually represented by class variables 𝑁 or 𝐿.

4.7.1  Definitions and basic theorems

Syntax

cword 11038

Syntax for the Word operator.

class Word 𝑆

Definition

df-word 11039*

Define the class of words over a set. A word (sometimes also called a string) is a finite sequence of symbols from a set (alphabet) 𝑆. Definition in Section 9.1 of [AhoHopUll] p. 318. The domain is forced to be an initial segment of ℕ₀ so that two words with the same symbols in the same order be equal. The set Word 𝑆 is sometimes denoted by S*, using the Kleene star, although the Kleene star, or Kleene closure, is sometimes reserved to denote an operation on languages. The set Word 𝑆 equipped with concatenation is the free monoid over 𝑆, and the monoid unit is the empty word. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ₀ 𝑤:(0..^𝑙)⟶𝑆}

Theorem

iswrd 11040*

Property of being a word over a set with an existential quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)

⊢ (𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ₀ 𝑊:(0..^𝑙)⟶𝑆)

Theorem

wrdval 11041*

Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_𝑚 (0..^𝑙)))

Theorem

lencl 11042

The length of a word is a nonnegative integer. This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 27-Aug-2015.)

⊢ (𝑊 ∈ Word 𝑆 → (♯‘𝑊) ∈ ℕ₀)

Theorem

iswrdinn0 11043

A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 16-Aug-2025.)

⊢ ((𝑊:(0..^𝐿)⟶𝑆 ∧ 𝐿 ∈ ℕ₀) → 𝑊 ∈ Word 𝑆)

Theorem

wrdf 11044

A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆)

Theorem

iswrdiz 11045

A zero-based sequence is a word. In iswrdinn0 11043 we can specify a length as an nonnegative integer. However, it will occasionally be helpful to allow a negative length, as well as zero, to specify an empty sequence. (Contributed by Jim Kingdon, 18-Aug-2025.)

⊢ ((𝑊:(0..^𝐿)⟶𝑆 ∧ 𝐿 ∈ ℤ) → 𝑊 ∈ Word 𝑆)

Theorem

wrddm 11046

The indices of a word (i.e. its domain regarded as function) are elements of an open range of nonnegative integers (of length equal to the length of the word). (Contributed by AV, 2-May-2020.)

⊢ (𝑊 ∈ Word 𝑆 → dom 𝑊 = (0..^(♯‘𝑊)))

Theorem

sswrd 11047

The set of words respects ordering on the base set. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)

⊢ (𝑆 ⊆ 𝑇 → Word 𝑆 ⊆ Word 𝑇)

Theorem

snopiswrd 11048

A singleton of an ordered pair (with 0 as first component) is a word. (Contributed by AV, 23-Nov-2018.) (Proof shortened by AV, 18-Apr-2021.)

⊢ (𝑆 ∈ 𝑉 → {⟨0, 𝑆⟩} ∈ Word 𝑉)

Theorem

wrdexg 11049

The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by JJ, 18-Nov-2022.)

⊢ (𝑆 ∈ 𝑉 → Word 𝑆 ∈ V)

Theorem

wrdexb 11050

The set of words over a set is a set, bidirectional version. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 23-Nov-2018.)

⊢ (𝑆 ∈ V ↔ Word 𝑆 ∈ V)

Theorem

wrdexi 11051

The set of words over a set is a set, inference form. (Contributed by AV, 23-May-2021.)

⊢ 𝑆 ∈ V    ⇒   ⊢ Word 𝑆 ∈ V

Theorem

wrdsymbcl 11052

A symbol within a word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝐼) ∈ 𝑉)

Theorem

wrdfn 11053

A word is a function with a zero-based sequence of integers as domain. (Contributed by Alexander van der Vekens, 13-Apr-2018.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊 Fn (0..^(♯‘𝑊)))

Theorem

wrdv 11054

A word over an alphabet is a word over the universal class. (Contributed by AV, 8-Feb-2021.) (Proof shortened by JJ, 18-Nov-2022.)

⊢ (𝑊 ∈ Word 𝑉 → 𝑊 ∈ Word V)

Theorem

wrdlndm 11055

The length of a word is not in the domain of the word (regarded as a function). (Contributed by AV, 3-Mar-2021.) (Proof shortened by JJ, 18-Nov-2022.)

⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∉ dom 𝑊)

Theorem

iswrdsymb 11056*

An arbitrary word is a word over an alphabet if all of its symbols belong to the alphabet. (Contributed by AV, 23-Jan-2021.)

⊢ ((𝑊 ∈ Word V ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ∈ 𝑉) → 𝑊 ∈ Word 𝑉)

Theorem

wrdfin 11057

A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.) (Proof shortened by AV, 18-Nov-2018.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊 ∈ Fin)

Theorem

lennncl 11058

The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)

⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ)

Theorem

wrdffz 11059

A word is a function from a finite interval of integers. (Contributed by AV, 10-Feb-2021.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0...((♯‘𝑊) − 1))⟶𝑆)

Theorem

wrdeq 11060

Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)

⊢ (𝑆 = 𝑇 → Word 𝑆 = Word 𝑇)

Theorem

wrdeqi 11061

Equality theorem for the set of words, inference form. (Contributed by AV, 23-May-2021.)

⊢ 𝑆 = 𝑇    ⇒   ⊢ Word 𝑆 = Word 𝑇

Theorem

iswrddm0 11062

A function with empty domain is a word. (Contributed by AV, 13-Oct-2018.)

⊢ (𝑊:∅⟶𝑆 → 𝑊 ∈ Word 𝑆)

Theorem

wrd0 11063

The empty set is a word (the empty word, frequently denoted ε in this context). This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 13-May-2020.)

⊢ ∅ ∈ Word 𝑆

Theorem

0wrd0 11064

The empty word is the only word over an empty alphabet. (Contributed by AV, 25-Oct-2018.)

⊢ (𝑊 ∈ Word ∅ ↔ 𝑊 = ∅)

Theorem

wrdsymb 11065

A word is a word over the symbols it consists of. (Contributed by AV, 1-Dec-2022.)

⊢ (𝑆 ∈ Word 𝐴 → 𝑆 ∈ Word (𝑆 “ (0..^(♯‘𝑆))))

Theorem

nfwrd 11066

Hypothesis builder for Word 𝑆. (Contributed by Mario Carneiro, 26-Feb-2016.)

⊢ Ⅎ𝑥𝑆    ⇒   ⊢ Ⅎ𝑥Word 𝑆

Theorem

csbwrdg 11067*

Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌Word 𝑥 = Word 𝑆)

Theorem

wrdnval 11068*

Words of a fixed length are mappings from a fixed half-open integer interval. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Proof shortened by AV, 13-May-2020.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} = (𝑉 ↑_𝑚 (0..^𝑁)))

Theorem

wrdmap 11069

Words as a mapping. (Contributed by Thierry Arnoux, 4-Mar-2020.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ↔ 𝑊 ∈ (𝑉 ↑_𝑚 (0..^𝑁))))

Theorem

wrdsymb0 11070

A symbol at a position "outside" of a word. (Contributed by Alexander van der Vekens, 26-May-2018.) (Proof shortened by AV, 2-May-2020.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘𝑊) ≤ 𝐼) → (𝑊‘𝐼) = ∅))

Theorem

wrdlenge1n0 11071

A word with length at least 1 is not empty. (Contributed by AV, 14-Oct-2018.)

⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ 1 ≤ (♯‘𝑊)))

Theorem

len0nnbi 11072

The length of a word is a positive integer iff the word is not empty. (Contributed by AV, 22-Mar-2022.)

⊢ (𝑊 ∈ Word 𝑆 → (𝑊 ≠ ∅ ↔ (♯‘𝑊) ∈ ℕ))

Theorem

wrdlenge2n0 11073

A word with length at least 2 is not empty. (Contributed by AV, 18-Jun-2018.) (Proof shortened by AV, 14-Oct-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → 𝑊 ≠ ∅)

Theorem

wrdsymb1 11074

The first symbol of a nonempty word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑊)) → (𝑊‘0) ∈ 𝑉)

Theorem

wrdlen1 11075*

A word of length 1 starts with a symbol. (Contributed by AV, 20-Jul-2018.) (Proof shortened by AV, 19-Oct-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → ∃𝑣 ∈ 𝑉 (𝑊‘0) = 𝑣)

Theorem

fstwrdne 11076

The first symbol of a nonempty word is element of the alphabet for the word. (Contributed by AV, 28-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊‘0) ∈ 𝑉)

Theorem

fstwrdne0 11077

The first symbol of a nonempty word is element of the alphabet for the word. (Contributed by AV, 29-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.)

⊢ ((𝑁 ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)) → (𝑊‘0) ∈ 𝑉)

Theorem

eqwrd 11078*

Two words are equal iff they have the same length and the same symbol at each position. (Contributed by AV, 13-Apr-2018.) (Revised by JJ, 30-Dec-2023.)

⊢ ((𝑈 ∈ Word 𝑆 ∧ 𝑊 ∈ Word 𝑇) → (𝑈 = 𝑊 ↔ ((♯‘𝑈) = (♯‘𝑊) ∧ ∀𝑖 ∈ (0..^(♯‘𝑈))(𝑈‘𝑖) = (𝑊‘𝑖))))

Theorem

elovmpowrd 11079*

Implications for the value of an operation defined by the maps-to notation with a class abstraction of words as a result having an element. Note that 𝜑 may depend on 𝑧 as well as on 𝑣 and 𝑦. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

⊢ 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣 ∣ 𝜑})    ⇒   ⊢ (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))

Theorem

wrdred1 11080

A word truncated by a symbol is a word. (Contributed by AV, 29-Jan-2021.)

⊢ (𝐹 ∈ Word 𝑆 → (𝐹 ↾ (0..^((♯‘𝐹) − 1))) ∈ Word 𝑆)

Theorem

wrdred1hash 11081

The length of a word truncated by a symbol. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)

⊢ ((𝐹 ∈ Word 𝑆 ∧ 1 ≤ (♯‘𝐹)) → (♯‘(𝐹 ↾ (0..^((♯‘𝐹) − 1)))) = ((♯‘𝐹) − 1))

4.7.2  Last symbol of a word

Syntax

clsw 11082

Extend class notation with the Last Symbol of a word.

class lastS

Definition

df-lsw 11083

Extract the last symbol of a word. May be not meaningful for other sets which are not words. The name lastS (as abbreviation of "lastSymbol") is a compromise between usually used names for corresponding functions in computer programs (as last() or lastChar()), the terminology used for words in set.mm ("symbol" instead of "character") and brevity ("lastS" is shorter than "lastChar" and "lastSymbol"). Labels of theorems about last symbols of a word will contain the abbreviation "lsw" (Last Symbol of a Word). (Contributed by Alexander van der Vekens, 18-Mar-2018.)

⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1)))

Theorem

lswwrd 11084

Extract the last symbol of a word. (Contributed by Alexander van der Vekens, 18-Mar-2018.) (Revised by Jim Kingdon, 18-Dec-2025.)

⊢ (𝑊 ∈ Word 𝑉 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))

Theorem

lsw0 11085

The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 19-Mar-2018.) (Proof shortened by AV, 2-May-2020.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (lastS‘𝑊) = ∅)

Theorem

lsw0g 11086

The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 11-Nov-2018.)

⊢ (lastS‘∅) = ∅

Theorem

lsw1 11087

The last symbol of a word of length 1 is the first symbol of this word. (Contributed by Alexander van der Vekens, 19-Mar-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → (lastS‘𝑊) = (𝑊‘0))

Theorem

lswcl 11088

Closure of the last symbol: the last symbol of a nonempty word belongs to the alphabet for the word. (Contributed by AV, 2-Aug-2018.) (Proof shortened by AV, 29-Apr-2020.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (lastS‘𝑊) ∈ 𝑉)

Theorem

lswex 11089

Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11086 or lswcl 11088 if you want more specific results for empty or nonempty words, respectively. (Contributed by Jim Kingdon, 27-Dec-2025.)

⊢ (𝑊 ∈ Word 𝑉 → (lastS‘𝑊) ∈ V)

Theorem

lswlgt0cl 11090

The last symbol of a nonempty word is an element of the alphabet for the word. (Contributed by Alexander van der Vekens, 1-Oct-2018.) (Proof shortened by AV, 29-Apr-2020.)

⊢ ((𝑁 ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)) → (lastS‘𝑊) ∈ 𝑉)

4.7.3  Concatenations of words

Syntax

cconcat 11091

Syntax for the concatenation operator.

class ++

Definition

df-concat 11092*

Define the concatenation operator which combines two words. Definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 15-Aug-2015.)

⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((♯‘𝑠) + (♯‘𝑡))) ↦ if(𝑥 ∈ (0..^(♯‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (♯‘𝑠))))))

Theorem

ccatfvalfi 11093*

Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)

⊢ ((𝑆 ∈ Fin ∧ 𝑇 ∈ Fin) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))))

Theorem

ccatcl 11094

The concatenation of two words is a word. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 29-Apr-2020.)

⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word 𝐵)

Theorem

ccatclab 11095

The concatenation of words over two sets is a word over the union of those sets. (Contributed by Jim Kingdon, 19-Dec-2025.)

⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word (𝐴 ∪ 𝐵))

Theorem

ccatlen 11096

The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by JJ, 1-Jan-2024.)

⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))

Theorem

ccat0 11097

The concatenation of two words is empty iff the two words are empty. (Contributed by AV, 4-Mar-2022.) (Revised by JJ, 18-Jan-2024.)

⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) = ∅ ↔ (𝑆 = ∅ ∧ 𝑇 = ∅)))

Theorem

ccatval1 11098

Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 30-Apr-2020.) (Revised by JJ, 18-Jan-2024.)

⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑆‘𝐼))

Theorem

ccatval2 11099

Value of a symbol in the right half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)

⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑇‘(𝐼 − (♯‘𝑆))))

Theorem

ccatval3 11100

Value of a symbol in the right half of a concatenated word, using an index relative to the subword. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Proof shortened by AV, 30-Apr-2020.)

⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘(𝐼 + (♯‘𝑆))) = (𝑇‘𝐼))