P. 110 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

isfinite4im 10901

A finite set is equinumerous to the range of integers from one up to the hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.)

|- ( A e. Fin -> ( 1 ... (♯ ` A ) ) ~~ A )

Theorem

fihasheq0 10902

Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)

|- ( A e. Fin -> ( (♯ ` A ) = 0 <-> A = (/) ) )

Theorem

fihashneq0 10903

Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 6955. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)

|- ( A e. Fin -> ( 0 < (♯ ` A ) <-> A =/= (/) ) )

Theorem

hashnncl 10904

Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)

|- ( A e. Fin -> ( (♯ ` A ) e. NN <-> A =/= (/) ) )

Theorem

hash0 10905

The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)

|- (♯ ` (/) ) = 0

Theorem

fihashelne0d 10906

A finite set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.)

|- ( ph -> B e. A )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> -. (♯ ` A ) = 0 )

Theorem

hashsng 10907

The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)

|- ( A e. V -> (♯ ` { A } ) = 1 )

Theorem

fihashen1 10908

A finite set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)

|- ( A e. Fin -> ( (♯ ` A ) = 1 <-> A ~~ 1o ) )

Theorem

fihashfn 10909

A function on a finite set is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) (Intuitionized by Jim Kingdon, 24-Feb-2022.)

|- ( ( F Fn A /\ A e. Fin ) -> (♯ ` F ) = (♯ ` A ) )

Theorem

fseq1hash 10910

The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)

|- ( ( N e. NN0 /\ F Fn ( 1 ... N ) ) -> (♯ ` F ) = N )

Theorem

omgadd 10911

Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)

|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( ( A e. om /\ B e. om ) -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) )

Theorem

fihashdom 10912

Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)

|- ( ( A e. Fin /\ B e. Fin ) -> ( (♯ ` A ) <_ (♯ ` B ) <-> A ~<_ B ) )

Theorem

hashunlem 10913

Lemma for hashun 10914. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)

|- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> N e. om )   &    |- ( ph -> M e. om )   &    |- ( ph -> A ~~ N )   &    |- ( ph -> B ~~ M )   =>    |- ( ph -> ( A u. B ) ~~ ( N +o M ) )

Theorem

hashun 10914

The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)

|- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> (♯ ` ( A u. B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )

Theorem

1elfz0hash 10915

1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)

|- ( ( A e. Fin /\ A =/= (/) ) -> 1 e. ( 0 ... (♯ ` A ) ) )

Theorem

hashunsng 10916

The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)

|- ( B e. V -> ( ( A e. Fin /\ -. B e. A ) -> (♯ ` ( A u. { B } ) ) = ( (♯ ` A ) + 1 ) ) )

Theorem

hashprg 10917

The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)

|- ( ( A e. V /\ B e. W ) -> ( A =/= B <-> (♯ ` { A , B } ) = 2 ) )

Theorem

prhash2ex 10918

There is (at least) one set with two different elements: the unordered pair containing 0 and 1. In contrast to pr0hash2ex 10924, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)

|- (♯ ` { 0 , 1 } ) = 2

Theorem

hashp1i 10919

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

|- A e. om   &    |- B = suc A   &    |- (♯ ` A ) = M   &    |- ( M + 1 ) = N   =>    |- (♯ ` B ) = N

Theorem

hash1 10920

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

|- (♯ ` 1o ) = 1

Theorem

hash2 10921

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

|- (♯ ` 2o ) = 2

Theorem

hash3 10922

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

|- (♯ ` 3o ) = 3

Theorem

hash4 10923

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

|- (♯ ` 4o ) = 4

Theorem

pr0hash2ex 10924

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 10925

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 e. Fin /\ B e. Fin /\ B C_ A ) -> (♯ ` B ) <_ (♯ ` A ) )

Theorem

fiprsshashgt1 10926

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

|- ( ( ( A e. V /\ B e. W /\ A =/= B ) /\ C e. Fin ) -> ( { A , B } C_ C -> 2 <_ (♯ ` C ) ) )

Theorem

fihashssdif 10927

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 e. Fin /\ B e. Fin /\ B C_ A ) -> (♯ ` ( A \ B ) ) = ( (♯ ` A ) - (♯ ` B ) ) )

Theorem

hashdifsn 10928

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 e. Fin /\ B e. A ) -> (♯ ` ( A \ { B } ) ) = ( (♯ ` A ) - 1 ) )

Theorem

hashdifpr 10929

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 e. Fin /\ ( B e. A /\ C e. A /\ B =/= C ) ) -> (♯ ` ( A \ { B , C } ) ) = ( (♯ ` A ) - 2 ) )

Theorem

hashfz 10930

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 e. ( ZZ>= ` A ) -> (♯ ` ( A ... B ) ) = ( ( B - A ) + 1 ) )

Theorem

hashfzo 10931

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

|- ( B e. ( ZZ>= ` A ) -> (♯ ` ( A..^ B ) ) = ( B - A ) )

Theorem

hashfzo0 10932

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

|- ( B e. NN0 -> (♯ ` ( 0..^ B ) ) = B )

Theorem

hashfzp1 10933

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

|- ( B e. ( ZZ>= ` A ) -> (♯ ` ( ( A + 1 ) ... B ) ) = ( B - A ) )

Theorem

hashfz0 10934

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

|- ( B e. NN0 -> (♯ ` ( 0 ... B ) ) = ( B + 1 ) )

Theorem

hashxp 10935

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

|- ( ( A e. Fin /\ B e. Fin ) -> (♯ ` ( A X. B ) ) = ( (♯ ` A ) x. (♯ ` B ) ) )

Theorem

fimaxq 10936*

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

|- ( ( A C_ QQ /\ A e. Fin /\ A =/= (/) ) -> E. x e. A A. y e. A y <_ x )

Theorem

fiubm 10937*

Lemma for fiubz 10938 and fiubnn 10939. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.)

|- ( ph -> A C_ B )   &    |- ( ph -> B C_ QQ )   &    |- ( ph -> C e. B )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> E. x e. B A. y e. A y <_ x )

Theorem

fiubz 10938*

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

|- ( ( A C_ ZZ /\ A e. Fin ) -> E. x e. ZZ A. y e. A y <_ x )

Theorem

fiubnn 10939*

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

|- ( ( A C_ NN /\ A e. Fin ) -> E. x e. NN A. y e. A y <_ x )

Theorem

resunimafz0 10940

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.)

|- ( ph -> Fun I )   &    |- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )   &    |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )   =>    |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )

Theorem

fnfz0hash 10941

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

|- ( ( N e. NN0 /\ F Fn ( 0 ... N ) ) -> (♯ ` F ) = ( N + 1 ) )

Theorem

ffz0hash 10942

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 e. NN0 /\ F : ( 0 ... N ) --> B ) -> (♯ ` F ) = ( N + 1 ) )

Theorem

ffzo0hash 10943

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

|- ( ( N e. NN0 /\ F Fn ( 0..^ N ) ) -> (♯ ` F ) = N )

Theorem

fnfzo0hash 10944

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 e. NN0 /\ F : ( 0..^ N ) --> B ) -> (♯ ` F ) = N )

Theorem

hashfacen 10945*

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 10946

Version of isorel 5858 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 C_ RR* /\ B C_ RR* ) /\ ( C e. A /\ D e. A ) ) -> ( C <_ D <-> ( F ` C ) <_ ( F ` D ) ) )

Theorem

zfz1isolemsplit 10947

Lemma for zfz1iso 10950. 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 10948*

Lemma for zfz1iso 10950. 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 10949*

Lemma for zfz1iso 10950. 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 10950*

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 10951*

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 ) )

4.7  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 10953. Note that the empty word (/) (i.e., the empty set) is the only word over an empty alphabet, see 0wrd0 10978. 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 10885: (♯ ` 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 5267: ( 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.

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 10952

Syntax for the Word operator.

class Word S

Definition

df-word 10953*

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 10954*

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 10955*

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 10956

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 10957

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 10958

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 10959

A zero-based sequence is a word. In iswrdinn0 10957 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 10960

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 10961

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 10962

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 10963

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 10964

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 10965

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 10966

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 10967

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 10968

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 10969

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 10970*

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 10971

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 10972

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 10973

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 10974

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

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

Theorem

wrdeqi 10975

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

|- S = T   =>    |- Word S = Word T

Theorem

iswrddm0 10976

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

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

Theorem

wrd0 10977

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 10978

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

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

Theorem

wrdsymb 10979

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 10980

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

|- F/_ x S   =>    |- F/_ xWord S

Theorem

csbwrdg 10981*

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 10982*

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 10983

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 10984

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 10985

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 10986

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 10987

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 10988

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 10989*

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 10990

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 10991

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 10992*

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 10993*

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 10994

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 10995

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 ) )

4.8  Elementary real and complex functions

4.8.1  The "shift" operation

Syntax

cshi 10996

Extend class notation with function shifter.

class shift

Definition

df-shft 10997*

Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of CC) and produces a new function on CC. See shftval 11007 for its value. (Contributed by NM, 20-Jul-2005.)

|- shift = ( f e. _V , x e. CC |-> { <. y , z >. | ( y e. CC /\ ( y - x ) f z ) } )

Theorem

shftlem 10998*

Two ways to write a shifted set ( B + A ). (Contributed by Mario Carneiro, 3-Nov-2013.)

|- ( ( A e. CC /\ B C_ CC ) -> { x e. CC | ( x - A ) e. B } = { x | E. y e. B x = ( y + A ) } )

Theorem

shftuz 10999*

A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)

|- ( ( A e. ZZ /\ B e. ZZ ) -> { x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = ( ZZ>= ` ( B + A ) ) )

Theorem

shftfvalg 11000*

The value of the sequence shifter operation is a function on CC. A is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

|- ( ( A e. CC /\ F e. V ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )