P. 112 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

zfz1isolemsplit 11101

Lemma for zfz1iso 11104. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.)

|- ( ph -> X e. Fin )   &    |- ( ph -> M e. X )   =>    |- ( ph -> ( 1 ... (♯ ` X ) ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )

Theorem

zfz1isolemiso 11102*

Lemma for zfz1iso 11104. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)

|- ( ph -> X e. Fin )   &    |- ( ph -> X C_ ZZ )   &    |- ( ph -> M e. X )   &    |- ( ph -> A. z e. X z <_ M )   &    |- ( ph -> G Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )   &    |- ( ph -> A e. ( 1 ... (♯ ` X ) ) )   &    |- ( ph -> B e. ( 1 ... (♯ ` X ) ) )   =>    |- ( ph -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )

Theorem

zfz1isolem1 11103*

Lemma for zfz1iso 11104. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)

|- ( ph -> K e. om )   &    |- ( ph -> A. y ( ( ( y C_ ZZ /\ y e. Fin ) /\ y ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) ) )   &    |- ( ph -> X C_ ZZ )   &    |- ( ph -> X e. Fin )   &    |- ( ph -> X ~~ suc K )   &    |- ( ph -> M e. X )   &    |- ( ph -> A. z e. X z <_ M )   =>    |- ( ph -> E. f f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) )

Theorem

zfz1iso 11104*

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

|- ( ( A C_ ZZ /\ A e. Fin ) -> E. f f Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )

Theorem

seq3coll 11105*

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.)

|- ( ( ph /\ k e. S ) -> ( Z .+ k ) = k )   &    |- ( ( ph /\ k e. S ) -> ( k .+ Z ) = k )   &    |- ( ( ph /\ ( k e. S /\ n e. S ) ) -> ( k .+ n ) e. S )   &    |- ( ph -> Z e. S )   &    |- ( ph -> G Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )   &    |- ( ph -> N e. ( 1 ... (♯ ` A ) ) )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> ( H ` k ) e. S )   &    |- ( ( ph /\ k e. ( ( M ... ( G ` (♯ ` A ) ) ) \ A ) ) -> ( F ` k ) = Z )   &    |- ( ( ph /\ n e. ( 1 ... (♯ ` A ) ) ) -> ( H ` n ) = ( F ` ( G ` n ) ) )   =>    |- ( ph -> ( seq M ( .+ , F ) ` ( G ` N ) ) = ( seq 1 ( .+ , H ) ` N ) )

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

Theorem

hash2en 11106

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

|- ( V ~~ 2o <-> ( V e. Fin /\ (♯ ` V ) = 2 ) )

Theorem

hashdmprop2dom 11107

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

|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. Y )   &    |- ( ph -> F e. Z )   &    |- ( ph -> A =/= B )   &    |- ( ph -> { <. A , C >. , <. B , D >. } C_ F )   =>    |- ( ph -> 2o ~<_ dom F )

4.6.10.2  Functions with a domain containing at least two different elements

Theorem

fundm2domnop0 11108

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 13094. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.)

|- ( ( Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> -. G e. ( _V X. _V ) )

Theorem

fundm2domnop 11109

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 /\ 2o ~<_ dom G ) -> -. G e. ( _V X. _V ) )

Theorem

fun2dmnop0 11110

A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 11111 (with the less restrictive requirement that ( G \ { (/) } ) needs to be a function instead of G) is useful for proofs for extensible structures, see structn0fun 13094. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.)

|- A e. _V   &    |- B e. _V   =>    |- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> -. G e. ( _V X. _V ) )

Theorem

fun2dmnop 11111

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 e. _V   &    |- B e. _V   =>    |- ( ( Fun G /\ A =/= B /\ { A , B } C_ dom G ) -> -. G e. ( _V X. _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 11113. Note that the empty word (/) (i.e., the empty set) is the only word over an empty alphabet, see 0wrd0 11138. 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 11037: (♯ ` W )	Operation which determines the number of the symbols within the word.	W : ( 0..^ N ) --> S -> ( W e. Word S /\ (♯ ` W ) = N	This is not a special definition for words, but for arbitrary sets.
First symbol of a word	df-fv 5334: ( 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 e. Word S /\ ( W ` 0 ) e. 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 11158: (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 e. 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 11226: ( 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 e. Word S /\ ( W substr <. 1 , 2 >. ) e. 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 11167: ( 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 e. Word S /\ U e. 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 11192: <" 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 11112

Syntax for the Word operator.

class Word S

Definition

df-word 11113*

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 NN0 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 | E. l e. NN0 w : ( 0..^ l ) --> S }

Theorem

iswrd 11114*

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 e. Word S <-> E. l e. NN0 W : ( 0..^ l ) --> S )

Theorem

wrdval 11115*

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 e. V -> Word S = U_ l e. NN0 ( S ^m ( 0..^ l ) ) )

Theorem

lencl 11116

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 e. Word S -> (♯ ` W ) e. NN0 )

Theorem

iswrdinn0 11117

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 e. NN0 ) -> W e. Word S )

Theorem

wrdf 11118

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

|- ( W e. Word S -> W : ( 0..^ (♯ ` W ) ) --> S )

Theorem

iswrdiz 11119

A zero-based sequence is a word. In iswrdinn0 11117 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 e. ZZ ) -> W e. Word S )

Theorem

wrddm 11120

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 e. Word S -> dom W = ( 0..^ (♯ ` W ) ) )

Theorem

sswrd 11121

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 C_ T -> Word S C_ Word T )

Theorem

snopiswrd 11122

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 e. V -> { <. 0 , S >. } e. Word V )

Theorem

wrdexg 11123

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 e. V -> Word S e. _V )

Theorem

wrdexb 11124

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 e. _V <-> Word S e. _V )

Theorem

wrdexi 11125

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

|- S e. _V   =>    |- Word S e. _V

Theorem

wrdsymbcl 11126

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

|- ( ( W e. Word V /\ I e. ( 0..^ (♯ ` W ) ) ) -> ( W ` I ) e. V )

Theorem

wrdfn 11127

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

|- ( W e. Word S -> W Fn ( 0..^ (♯ ` W ) ) )

Theorem

wrdv 11128

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 e. Word V -> W e. Word _V )

Theorem

wrdlndm 11129

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 e. Word V -> (♯ ` W ) e/ dom W )

Theorem

iswrdsymb 11130*

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 e. Word _V /\ A. i e. ( 0..^ (♯ ` W ) ) ( W ` i ) e. V ) -> W e. Word V )

Theorem

wrdfin 11131

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

|- ( W e. Word S -> W e. Fin )

Theorem

lennncl 11132

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

|- ( ( W e. Word S /\ W =/= (/) ) -> (♯ ` W ) e. NN )

Theorem

wrdffz 11133

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

|- ( W e. Word S -> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> S )

Theorem

wrdeq 11134

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

|- ( S = T -> Word S = Word T )

Theorem

wrdeqi 11135

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

|- S = T   =>    |- Word S = Word T

Theorem

iswrddm0 11136

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

|- ( W : (/) --> S -> W e. Word S )

Theorem

wrd0 11137

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.)

|- (/) e. Word S

Theorem

0wrd0 11138

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

|- ( W e. Word (/) <-> W = (/) )

Theorem

ffz0iswrdnn0 11139

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 e. NN0 ) -> W e. Word S )

Theorem

wrdsymb 11140

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

|- ( S e. Word A -> S e. Word ( S " ( 0..^ (♯ ` S ) ) ) )

Theorem

nfwrd 11141

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

|- F/_ x S   =>    |- F/_ xWord S

Theorem

csbwrdg 11142*

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

|- ( S e. V -> [_ S / x ]_Word x = Word S )

Theorem

wrdnval 11143*

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 e. X /\ N e. NN0 ) -> { w e. Word V | (♯ ` w ) = N } = ( V ^m ( 0..^ N ) ) )

Theorem

wrdmap 11144

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

|- ( ( V e. X /\ N e. NN0 ) -> ( ( W e. Word V /\ (♯ ` W ) = N ) <-> W e. ( V ^m ( 0..^ N ) ) ) )

Theorem

wrdsymb0 11145

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 e. Word V /\ I e. ZZ ) -> ( ( I < 0 \/ (♯ ` W ) <_ I ) -> ( W ` I ) = (/) ) )

Theorem

wrdlenge1n0 11146

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

|- ( W e. Word V -> ( W =/= (/) <-> 1 <_ (♯ ` W ) ) )

Theorem

len0nnbi 11147

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

|- ( W e. Word S -> ( W =/= (/) <-> (♯ ` W ) e. NN ) )

Theorem

wrdlenge2n0 11148

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

|- ( ( W e. Word V /\ 2 <_ (♯ ` W ) ) -> W =/= (/) )

Theorem

wrdsymb1 11149

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

|- ( ( W e. Word V /\ 1 <_ (♯ ` W ) ) -> ( W ` 0 ) e. V )

Theorem

wrdlen1 11150*

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

|- ( ( W e. Word V /\ (♯ ` W ) = 1 ) -> E. v e. V ( W ` 0 ) = v )

Theorem

fstwrdne 11151

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 e. Word V /\ W =/= (/) ) -> ( W ` 0 ) e. V )

Theorem

fstwrdne0 11152

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 e. NN /\ ( W e. Word V /\ (♯ ` W ) = N ) ) -> ( W ` 0 ) e. V )

Theorem

eqwrd 11153*

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 e. Word S /\ W e. Word T ) -> ( U = W <-> ( (♯ ` U ) = (♯ ` W ) /\ A. i e. ( 0..^ (♯ ` U ) ) ( U ` i ) = ( W ` i ) ) ) )

Theorem

elovmpowrd 11154*

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 ph may depend on z as well as on v and y. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

|- O = ( v e. _V , y e. _V |-> { z e. Word v | ph } )   =>    |- ( Z e. ( V O Y ) -> ( V e. _V /\ Y e. _V /\ Z e. Word V ) )

Theorem

wrdred1 11155

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

|- ( F e. Word S -> ( F |` ( 0..^ ( (♯ ` F ) - 1 ) ) ) e. Word S )

Theorem

wrdred1hash 11156

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 e. Word S /\ 1 <_ (♯ ` F ) ) -> (♯ ` ( F |` ( 0..^ ( (♯ ` F ) - 1 ) ) ) ) = ( (♯ ` F ) - 1 ) )

4.7.2  Last symbol of a word

Syntax

clsw 11157

Extend class notation with the Last Symbol of a word.

class lastS

Definition

df-lsw 11158

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 e. _V |-> ( w ` ( (♯ ` w ) - 1 ) ) )

Theorem

lswwrd 11159

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

|- ( W e. Word V -> (lastS ` W ) = ( W ` ( (♯ ` W ) - 1 ) ) )

Theorem

lsw0 11160

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 e. Word V /\ (♯ ` W ) = 0 ) -> (lastS ` W ) = (/) )

Theorem

lsw0g 11161

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

|- (lastS ` (/) ) = (/)

Theorem

lsw1 11162

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 e. Word V /\ (♯ ` W ) = 1 ) -> (lastS ` W ) = ( W ` 0 ) )

Theorem

lswcl 11163

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 e. Word V /\ W =/= (/) ) -> (lastS ` W ) e. V )

Theorem

lswex 11164

Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11161 or lswcl 11163 if you want more specific results for empty or nonempty words, respectively. (Contributed by Jim Kingdon, 27-Dec-2025.)

|- ( W e. Word V -> (lastS ` W ) e. _V )

Theorem

lswlgt0cl 11165

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 e. NN /\ ( W e. Word V /\ (♯ ` W ) = N ) ) -> (lastS ` W ) e. V )

4.7.3  Concatenations of words

Syntax

cconcat 11166

Syntax for the concatenation operator.

class ++

Definition

df-concat 11167*

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 e. _V , t e. _V |-> ( x e. ( 0..^ ( (♯ ` s ) + (♯ ` t ) ) ) |-> if ( x e. ( 0..^ (♯ ` s ) ) , ( s ` x ) , ( t ` ( x - (♯ ` s ) ) ) ) ) )

Theorem

ccatfvalfi 11168*

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

|- ( ( S e. Fin /\ T e. Fin ) -> ( S ++ T ) = ( x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) |-> if ( x e. ( 0..^ (♯ ` S ) ) , ( S ` x ) , ( T ` ( x - (♯ ` S ) ) ) ) ) )

Theorem

ccatcl 11169

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 e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )

Theorem

ccatclab 11170

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

|- ( ( S e. Word A /\ T e. Word B ) -> ( S ++ T ) e. Word ( A u. B ) )

Theorem

ccatlen 11171

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

|- ( ( S e. Word A /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )

Theorem

ccat0 11172

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 e. Word A /\ T e. Word B ) -> ( ( S ++ T ) = (/) <-> ( S = (/) /\ T = (/) ) ) )

Theorem

ccatval1 11173

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 e. Word A /\ T e. Word B /\ I e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` I ) = ( S ` I ) )

Theorem

ccatval2 11174

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 e. Word B /\ T e. Word B /\ I e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` I ) = ( T ` ( I - (♯ ` S ) ) ) )

Theorem

ccatval3 11175

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 e. Word B /\ T e. Word B /\ I e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( I + (♯ ` S ) ) ) = ( T ` I ) )

Theorem

elfzelfzccat 11176

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 e. Word V /\ B e. Word V ) -> ( N e. ( 0 ... (♯ ` A ) ) -> N e. ( 0 ... (♯ ` ( A ++ B ) ) ) ) )

Theorem

ccatvalfn 11177

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 e. Word V /\ B e. Word V ) -> ( A ++ B ) Fn ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) )

Theorem

ccatsymb 11178

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

|- ( ( A e. Word V /\ B e. Word V /\ I e. ZZ ) -> ( ( A ++ B ) ` I ) = if ( I < (♯ ` A ) , ( A ` I ) , ( B ` ( I - (♯ ` A ) ) ) ) )

Theorem

ccatfv0 11179

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 e. Word V /\ B e. Word V /\ 0 < (♯ ` A ) ) -> ( ( A ++ B ) ` 0 ) = ( A ` 0 ) )

Theorem

ccatval1lsw 11180

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 e. Word V /\ B e. Word V /\ A =/= (/) ) -> ( ( A ++ B ) ` ( (♯ ` A ) - 1 ) ) = (lastS ` A ) )

Theorem

ccatval21sw 11181

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

|- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( ( A ++ B ) ` (♯ ` A ) ) = ( B ` 0 ) )

Theorem

ccatlid 11182

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 e. Word B -> ( (/) ++ S ) = S )

Theorem

ccatrid 11183

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 e. Word B -> ( S ++ (/) ) = S )

Theorem

ccatass 11184

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

|- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( S ++ T ) ++ U ) = ( S ++ ( T ++ U ) ) )

Theorem

ccatrn 11185

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

|- ( ( S e. Word B /\ T e. Word B ) -> ran ( S ++ T ) = ( ran S u. ran T ) )

Theorem

ccatidid 11186

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

|- ( (/) ++ (/) ) = (/)

Theorem

lswccatn0lsw 11187

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 e. Word V /\ B e. Word V /\ B =/= (/) ) -> (lastS ` ( A ++ B ) ) = (lastS ` B ) )

Theorem

lswccat0lsw 11188

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 e. Word V -> (lastS ` ( W ++ (/) ) ) = (lastS ` W ) )

Theorem

ccatalpha 11189

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 e. Word _V /\ B e. Word _V ) -> ( ( A ++ B ) e. Word S <-> ( A e. Word S /\ B e. Word S ) ) )

Theorem

ccatrcl1 11190

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 e. Word X /\ B e. Word Y /\ ( W = ( A ++ B ) /\ W e. Word S ) ) -> A e. Word S )

4.7.4  Singleton words

Syntax

cs1 11191

Syntax for the singleton word constructor.

class <" A ">

Definition

df-s1 11192

Define the canonical injection from symbols to words. Although not required, A should usually be a set. Otherwise, the singleton word <" A "> would be the singleton word consisting of the empty set, see s1prc 11199, and not, as maybe expected, the empty word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

|- <" A "> = { <. 0 , ( _I ` A ) >. }

Theorem

s1val 11193

Value of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

|- ( A e. V -> <" A "> = { <. 0 , A >. } )

Theorem

s1rn 11194

The range of a singleton word. (Contributed by Mario Carneiro, 18-Jul-2016.)

|- ( A e. V -> ran <" A "> = { A } )

Theorem

s1eq 11195

Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

|- ( A = B -> <" A "> = <" B "> )

Theorem

s1eqd 11196

Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

|- ( ph -> A = B )   =>    |- ( ph -> <" A "> = <" B "> )

Theorem

s1cl 11197

A singleton word is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 23-Nov-2018.)

|- ( A e. B -> <" A "> e. Word B )

Theorem

s1cld 11198

A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)

|- ( ph -> A e. B )   =>    |- ( ph -> <" A "> e. Word B )

Theorem

s1prc 11199

Value of a singleton word if the symbol is a proper class. (Contributed by AV, 26-Mar-2022.)

|- ( -. A e. _V -> <" A "> = <" (/) "> )

Theorem

s1leng 11200

Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

|- ( A e. V -> (♯ ` <" A "> ) = 1 )