P. 112 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

resunimafz0 11101

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 11102

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 11103

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 11104

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 11105

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 11106*

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 11107

Version of isorel 5954 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 11108

Lemma for zfz1iso 11111. 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 11109*

Lemma for zfz1iso 11111. 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 11110*

Lemma for zfz1iso 11111. 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 11111*

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 11112*

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 11113

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 11114

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)

Theorem

hashtpgim 11115

The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.) (Revised by Jim Kingdon, 17-Apr-2026.)

⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) → (♯‘{𝐴, 𝐵, 𝐶}) = 3))

Theorem

hashtpglem 11116

Lemma for hashtpg 11117. This is one of the three not-equal conclusions required for the reverse direction. (Contributed by Jim Kingdon, 18-Apr-2026.)

⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → (♯‘{𝐴, 𝐵, 𝐶}) = 3)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 𝐶)

Theorem

hashtpg 11117

The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.)

⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) ↔ (♯‘{𝐴, 𝐵, 𝐶}) = 3))

4.6.10.2  Functions with a domain containing at least two different elements

Theorem

fundm2domnop0 11118

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 13118. (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 11119

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 11120

A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 11121 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 13118. (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 11121

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 11123. Note that the empty word ∅ (i.e., the empty set) is the only word over an empty alphabet, see 0wrd0 11148. 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 11044: (♯‘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 5336: (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 11168: (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 11236: (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 11177: (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 11202: ⟨“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 11122

Syntax for the Word operator.

class Word 𝑆

Definition

df-word 11123*

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 11124*

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 11125*

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 11126

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 11127

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 11128

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 11129

A zero-based sequence is a word. In iswrdinn0 11127 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 11130

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 11131

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 11132

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 11133

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 11134

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 11135

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 11136

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 11137

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 11138

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 11139

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 11140*

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 11141

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 11142

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 11143

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 11144

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

⊢ (𝑆 = 𝑇 → Word 𝑆 = Word 𝑇)

Theorem

wrdeqi 11145

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

⊢ 𝑆 = 𝑇    ⇒   ⊢ Word 𝑆 = Word 𝑇

Theorem

iswrddm0 11146

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

⊢ (𝑊:∅⟶𝑆 → 𝑊 ∈ Word 𝑆)

Theorem

wrd0 11147

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 11148

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

⊢ (𝑊 ∈ Word ∅ ↔ 𝑊 = ∅)

Theorem

ffz0iswrdnn0 11149

A sequence with zero-based indices is a word. (Contributed by AV, 31-Jan-2018.) (Proof shortened by AV, 13-Oct-2018.) (Proof shortened by JJ, 18-Nov-2022.)

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

Theorem

wrdsymb 11150

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 11151

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

⊢ Ⅎ𝑥𝑆    ⇒   ⊢ Ⅎ𝑥Word 𝑆

Theorem

csbwrdg 11152*

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 11153*

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 11154

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 11155

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 11156

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

⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ 1 ≤ (♯‘𝑊)))

Theorem

len0nnbi 11157

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 11158

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 11159

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 11160*

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 11161

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 11162

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 11163*

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 11164*

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 11165

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 11166

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 11167

Extend class notation with the Last Symbol of a word.

class lastS

Definition

df-lsw 11168

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 11169

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 11170

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 11171

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

⊢ (lastS‘∅) = ∅

Theorem

lsw1 11172

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 11173

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 11174

Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11171 or lswcl 11173 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 11175

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 11176

Syntax for the concatenation operator.

class ++

Definition

df-concat 11177*

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 11178*

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 11179

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 11180

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 11181

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 11182

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 11183

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 11184

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 11185

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))

Theorem

elfzelfzccat 11186

An element of a finite set of sequential integers up to the length of a word is an element of an extended finite set of sequential integers up to the length of a concatenation of this word with another word. (Contributed by Alexander van der Vekens, 28-Mar-2018.)

⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘𝐴)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))

Theorem

ccatvalfn 11187

The concatenation of two words is a function over the half-open integer range having the sum of the lengths of the word as length. (Contributed by Alexander van der Vekens, 30-Mar-2018.)

⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) Fn (0..^((♯‘𝐴) + (♯‘𝐵))))

Theorem

ccatsymb 11188

The symbol at a given position in a concatenated word. (Contributed by AV, 26-May-2018.) (Proof shortened by AV, 24-Nov-2018.)

⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))))

Theorem

ccatfv0 11189

The first symbol of a concatenation of two words is the first symbol of the first word if the first word is not empty. (Contributed by Alexander van der Vekens, 22-Sep-2018.)

⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 0 < (♯‘𝐴)) → ((𝐴 ++ 𝐵)‘0) = (𝐴‘0))

Theorem

ccatval1lsw 11190

The last symbol of the left (nonempty) half of a concatenated word. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Proof shortened by AV, 1-May-2020.)

⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐴 ≠ ∅) → ((𝐴 ++ 𝐵)‘((♯‘𝐴) − 1)) = (lastS‘𝐴))

Theorem

ccatval21sw 11191

The first symbol of the right (nonempty) half of a concatenated word. (Contributed by AV, 23-Apr-2022.)

⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐵 ≠ ∅) → ((𝐴 ++ 𝐵)‘(♯‘𝐴)) = (𝐵‘0))

Theorem

ccatlid 11192

Concatenation of a word by the empty word on the left. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 1-May-2020.)

⊢ (𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) = 𝑆)

Theorem

ccatrid 11193

Concatenation of a word by the empty word on the right. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 1-May-2020.)

⊢ (𝑆 ∈ Word 𝐵 → (𝑆 ++ ∅) = 𝑆)

Theorem

ccatass 11194

Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)

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

Theorem

ccatrn 11195

The range of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)

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

Theorem

ccatidid 11196

Concatenation of the empty word by the empty word. (Contributed by AV, 26-Mar-2022.)

⊢ (∅ ++ ∅) = ∅

Theorem

lswccatn0lsw 11197

The last symbol of a word concatenated with a nonempty word is the last symbol of the nonempty word. (Contributed by AV, 22-Oct-2018.) (Proof shortened by AV, 1-May-2020.)

⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐵 ≠ ∅) → (lastS‘(𝐴 ++ 𝐵)) = (lastS‘𝐵))

Theorem

lswccat0lsw 11198

The last symbol of a word concatenated with the empty word is the last symbol of the word. (Contributed by AV, 22-Oct-2018.) (Proof shortened by AV, 1-May-2020.)

⊢ (𝑊 ∈ Word 𝑉 → (lastS‘(𝑊 ++ ∅)) = (lastS‘𝑊))

Theorem

ccatalpha 11199

A concatenation of two arbitrary words is a word over an alphabet iff the symbols of both words belong to the alphabet. (Contributed by AV, 28-Feb-2021.)

⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))

Theorem

ccatrcl1 11200

Reverse closure of a concatenation: If the concatenation of two arbitrary words is a word over an alphabet then the symbols of the first word belong to the alphabet. (Contributed by AV, 3-Mar-2021.)

⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑌 ∧ (𝑊 = (𝐴 ++ 𝐵) ∧ 𝑊 ∈ Word 𝑆)) → 𝐴 ∈ Word 𝑆)