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Theorem List for Metamath Proof Explorer - 13001-13100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembrfi1indOLD 13001* Obsolete version of brfi1ind 12995 as of 28-Mar-2021. (Contributed by AV, 7-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel 𝐺    &   𝐹𝑈    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)
 
Theorembrfi1indALTOLD 13002* Obsolete version of brfi1indALT 12996 as of 28-Mar-2021. (Contributed by AV, 7-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel 𝐺    &   𝐹𝑈    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)
 
Theoremopfi1uzindOLD 13003* Obsolete version of opfi1uzind 12997 as of 28-Mar-2021. (Contributed by AV, 22-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐸𝑌    &   𝐹𝑈    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
 
Theoremopfi1indOLD 13004* Obsolete version of opfi1ind 12998 as of 28-Mar-2021. (Contributed by AV, 22-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐸𝑌    &   𝐹𝑈    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin) → 𝜑)
 
5.7  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 is nonempty, finite and not a proper class, these restrictions are not made in the current definition df-word 13013, see, for example, s1cli 13096: ⟨“𝐴”⟩ ∈ Word V. Note that the empty word (i.e. the empty set) is the only word over an empty alphabet, see 0wrd0 13045. Besides the definition of words themselves, several operations on words are defined in this section:

NameReferenceExplanationExample Remarks
Length (or size) of a word df-hash 12848: (#‘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 5697: (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 13014: ( 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 13017: (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 13015: (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 𝑆𝑈 ∈ 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.
Reversal of a word df-reverse 13019: (reverse‘𝑊) Operation which reverses the order of symbols in a word. (𝑊 ∈ Word 𝑉 → (#‘(reverse‘𝑊)) = (#‘𝑊))
Cyclical shift of a word df-csh 13245: (𝑊 cyclShift 𝑁) Operation cyclically shifting the symbols by a number of positions. (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (#‘𝑊)) = 𝑊)
Splicing of a word df-splice 13018: (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) Operation which replaces portions of words. ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
Singleton word df-s1 13016: ⟨“𝑆”⟩ Constructor building a word of length 1 from a symbol. (#‘⟨“𝑆”⟩) = 1
Conventions:
  • Words are usually represented by class variable 𝑊, 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, 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 position to cyclically shift a word is usually represented by class variables 𝑁 or 𝐿.
 
5.7.1  Definitions and basic theorems
 
Syntaxcword 13005 Syntax for the Word operator.
class Word 𝑆
 
Syntaxclsw 13006 Extend class notation with the Last Symbol of a word.
class lastS
 
Syntaxcconcat 13007 Syntax for the concatenation operator.
class ++
 
Syntaxcs1 13008 Syntax for the singleton word constructor.
class ⟨“𝐴”⟩
 
Syntaxcsubstr 13009 Syntax for the subword operator.
class substr
 
Syntaxcsplice 13010 Syntax for the word splicing operator.
class splice
 
Syntaxcreverse 13011 Syntax for the word reverse operator.
class reverse
 
Syntaxcreps 13012 Extend class notation with words consisting of one repeated symbol.
class repeatS
 
Definitiondf-word 13013* Define the class of words over a set. A word (or 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 so that two words with the same symbols in the same order will be the same. This is sometimes denoted with the Kleene star, although properly speaking that is an operator on languages. (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}
 
Definitiondf-lsw 13014 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)))
 
Definitiondf-concat 13015* 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)), (𝑠𝑥), (𝑡‘(𝑥 − (#‘𝑠))))))
 
Definitiondf-s1 13016 Define the canonical injection from symbols to words. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
 
Definitiondf-substr 13017* Define an operation which extracts portions of words. Definition in section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 15-Aug-2015.)
substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))), ∅))
 
Definitiondf-splice 13018* Define an operation which replaces portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩)))
 
Definitiondf-reverse 13019* Define an operation which reverses the order of symbols in a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(#‘𝑠)) ↦ (𝑠‘(((#‘𝑠) − 1) − 𝑥))))
 
Definitiondf-reps 13020* Definition to construct a word consisting of one repeated symbol, often called "repeated symbol word" for short in the following. (Contributed by Alexander van der Vekens, 4-Nov-2018.)
repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
 
Theoremiswrd 13021* Property of being a word over a set with a 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)
 
Theoremwrdval 13022* Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝑆𝑉 → Word 𝑆 = 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)))
 
Theoremiswrdi 13023 A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝑊:(0..^𝐿)⟶𝑆𝑊 ∈ Word 𝑆)
 
Theoremwrdf 13024 A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝑊 ∈ Word 𝑆𝑊:(0..^(#‘𝑊))⟶𝑆)
 
Theoremiswrdb 13025 A word over an alphabet is a function of an open range of nonnegative integers (of length equal to the length of the word) into the alphabet. (Contributed by Alexander van der Vekens, 30-Jul-2018.)
(𝑊 ∈ Word 𝑆𝑊:(0..^(#‘𝑊))⟶𝑆)
 
Theoremwrddm 13026 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)))
 
Theoremsswrd 13027 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.)
(𝑆𝑇 → Word 𝑆 ⊆ Word 𝑇)
 
Theoremsnopiswrd 13028 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.)
(𝑆𝑉 → {⟨0, 𝑆⟩} ∈ Word 𝑉)
 
Theoremwrdexg 13029 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝑆𝑉 → Word 𝑆 ∈ V)
 
Theoremwrdexb 13030 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)
 
Theoremwrdexi 13031 The set of words over a set is a set, inference form. (Contributed by AV, 23-May-2021.)
𝑆 ∈ V       Word 𝑆 ∈ V
 
Theoremwrdsymbcl 13032 A symbol within a word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)
((𝑊 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝑊𝐼) ∈ 𝑉)
 
Theoremwrdfn 13033 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 𝑆𝑊 Fn (0..^(#‘𝑊)))
 
Theoremwrdv 13034 A word over an alphabet is a word over the universal class. (Contributed by AV, 8-Feb-2021.)
(𝑊 ∈ Word 𝑉𝑊 ∈ Word V)
 
Theoremwrdlndm 13035 The length of a word is not in the domain of the word (regarded as function). (Contributed by AV, 3-Mar-2021.)
(𝑊 ∈ Word 𝑉 → (#‘𝑊) ∉ dom 𝑊)
 
Theoremiswrdsymb 13036* 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))(𝑊𝑖) ∈ 𝑉) → 𝑊 ∈ Word 𝑉)
 
Theoremwrdfin 13037 A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.) (Proof shortened by AV, 18-Nov-2018.)
(𝑊 ∈ Word 𝑆𝑊 ∈ Fin)
 
Theoremlencl 13038 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)
 
Theoremlennncl 13039 The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)
((𝑊 ∈ Word 𝑆𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ)
 
Theoremwrdffz 13040 A word is a function from a finite interval of integers. (Contributed by AV, 10-Feb-2021.)
(𝑊 ∈ Word 𝑆𝑊:(0...((#‘𝑊) − 1))⟶𝑆)
 
Theoremwrdeq 13041 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝑆 = 𝑇 → Word 𝑆 = Word 𝑇)
 
Theoremwrdeqi 13042 Equality theorem for the set of words, inference form. (Contributed by AV, 23-May-2021.)
𝑆 = 𝑇       Word 𝑆 = Word 𝑇
 
Theoremiswrddm0 13043 A function with empty domain is a word. (Contributed by AV, 13-Oct-2018.)
(𝑊:∅⟶𝑆𝑊 ∈ Word 𝑆)
 
Theoremwrd0 13044 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
 
Theorem0wrd0 13045 The empty word is the only word over an empty alphabet. (Contributed by AV, 25-Oct-2018.)
(𝑊 ∈ Word ∅ ↔ 𝑊 = ∅)
 
Theoremffz0iswrd 13046 A sequence with zero-based indices is a word. (Contributed by AV, 31-Jan-2018.) (Proof shortened by AV, 13-Oct-2018.)
(𝑊:(0...𝐿)⟶𝑆𝑊 ∈ Word 𝑆)
 
Theoremnfwrd 13047 Hypothesis builder for Word 𝑆. (Contributed by Mario Carneiro, 26-Feb-2016.)
𝑥𝑆       𝑥Word 𝑆
 
Theoremcsbwrdg 13048* Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑆𝑉𝑆 / 𝑥Word 𝑥 = Word 𝑆)
 
Theoremwrdnval 13049* 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.)
((𝑉𝑋𝑁 ∈ ℕ0) → {𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = 𝑁} = (𝑉𝑚 (0..^𝑁)))
 
Theoremwrdmap 13050 Words as a mapping. (Contributed by Thierry Arnoux, 4-Mar-2020.)
((𝑉𝑋𝑁 ∈ ℕ0) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ↔ 𝑊 ∈ (𝑉𝑚 (0..^𝑁))))
 
Theoremhashwrdn 13051* If there is only a finite number of symbols, the number of words of a fixed length over these sysmbols is the number of these symbols raised to the power of the length. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (#‘{𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = 𝑁}) = ((#‘𝑉)↑𝑁))
 
Theoremwrdnfi 13052* If there is only a finite number of symbols, the number of words of a fixed length over these symbols is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = 𝑁} ∈ Fin)
 
Theoremwrdsymb0 13053 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 𝑉𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (#‘𝑊) ≤ 𝐼) → (𝑊𝐼) = ∅))
 
Theoremwrdlenge1n0 13054 A word with length at least 1 is not empty. (Contributed by AV, 14-Oct-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ 1 ≤ (#‘𝑊)))
 
Theoremwrdlenge2n0 13055 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 ≠ ∅)
 
Theoremwrdsymb1 13056 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)
 
Theoremwrdlen1 13057* 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) → ∃𝑣𝑉 (𝑊‘0) = 𝑣)
 
Theoremfstwrdne 13058 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 𝑉𝑊 ≠ ∅) → (𝑊‘0) ∈ 𝑉)
 
Theoremfstwrdne0 13059 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)
 
Theoremeqwrd 13060* Two words are equal iff they have the same length and the same symbol at each position. (Contributed by AV, 13-Apr-2018.)
((𝑈 ∈ Word 𝑉𝑊 ∈ Word 𝑉) → (𝑈 = 𝑊 ↔ ((#‘𝑈) = (#‘𝑊) ∧ ∀𝑖 ∈ (0..^(#‘𝑈))(𝑈𝑖) = (𝑊𝑖))))
 
Theoremelovmpt2wrd 13061* Implications for the value of an operation defined by the maps-to notation with a class abstration 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 𝑣𝜑})       (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))
 
Theoremelovmptnn0wrd 13062* Implications for the value of an operation defined by the maps-to notation with a function of nonnegative integers into 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 and 𝑛. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)
𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑧 ∈ Word 𝑣𝜑}))       (𝑍 ∈ ((𝑉𝑂𝑌)‘𝑁) → ((𝑉 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑁 ∈ ℕ0𝑍 ∈ Word 𝑉)))
 
5.7.2  Last symbol of a word
 
Theoremlsw 13063 Extract the last symbol of a word. May be not meaningful for other sets which are not words. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
(𝑊𝑋 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
 
Theoremlsw0 13064 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) = ∅)
 
Theoremlsw0g 13065 The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 11-Nov-2018.)
( lastS ‘∅) = ∅
 
Theoremlsw1 13066 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))
 
Theoremlswcl 13067 Closure of the last symbol: the last symbol of a not empty word belongs to the alphabet for the word. (Contributed by AV, 2-Aug-2018.) (Proof shortened by AV, 29-Apr-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → ( lastS ‘𝑊) ∈ 𝑉)
 
Theoremlswlgt0cl 13068 The last symbol of a nonempty word is 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)
 
5.7.3  Concatenations of words
 
Theoremccatfn 13069 The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.) (Proof shortened by AV, 29-Apr-2020.)
++ Fn (V × V)
 
Theoremccatfval 13070* Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆𝑉𝑇𝑊) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆𝑥), (𝑇‘(𝑥 − (#‘𝑆))))))
 
Theoremccatcl 13071 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 𝐵𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
 
Theoremccatlen 13072 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑆) + (#‘𝑇)))
 
Theoremccatval1 13073 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.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵𝐼 ∈ (0..^(#‘𝑆))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑆𝐼))
 
Theoremccatval2 13074 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 𝐵𝑇 ∈ Word 𝐵𝐼 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑇‘(𝐼 − (#‘𝑆))))
 
Theoremccatval3 13075 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 𝐵𝑇 ∈ Word 𝐵𝐼 ∈ (0..^(#‘𝑇))) → ((𝑆 ++ 𝑇)‘(𝐼 + (#‘𝑆))) = (𝑇𝐼))
 
Theoremelfzelfzccat 13076 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 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(#‘𝐴)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
 
Theoremccatvalfn 13077 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 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) Fn (0..^((#‘𝐴) + (#‘𝐵))))
 
Theoremccatsymb 13078 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 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (#‘𝐴), (𝐴𝐼), (𝐵‘(𝐼 − (#‘𝐴)))))
 
Theoremccatfv0 13079 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 𝑉𝐵 ∈ Word 𝑉 ∧ 0 < (#‘𝐴)) → ((𝐴 ++ 𝐵)‘0) = (𝐴‘0))
 
Theoremccatval1lsw 13080 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 𝑉𝐵 ∈ Word 𝑉𝐴 ≠ ∅) → ((𝐴 ++ 𝐵)‘((#‘𝐴) − 1)) = ( lastS ‘𝐴))
 
Theoremccatlid 13081 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)
 
Theoremccatrid 13082 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)
 
Theoremccatass 13083 Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) = (𝑆 ++ (𝑇 ++ 𝑈)))
 
Theoremccatrn 13084 The range of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → ran (𝑆 ++ 𝑇) = (ran 𝑆 ∪ ran 𝑇))
 
Theoremlswccatn0lsw 13085 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 𝑉𝐵 ∈ Word 𝑉𝐵 ≠ ∅) → ( lastS ‘(𝐴 ++ 𝐵)) = ( lastS ‘𝐵))
 
Theoremlswccat0lsw 13086 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))
 
Theoremccatalpha 13087 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 𝑆𝐵 ∈ Word 𝑆)))
 
Theoremccatrcl1 13088 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 𝑋𝐵 ∈ Word 𝑌 ∧ (𝑊 = (𝐴 ++ 𝐵) ∧ 𝑊 ∈ Word 𝑆)) → 𝐴 ∈ Word 𝑆)
 
5.7.4  Singleton words
 
Theoremids1 13089 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
 
Theorems1val 13090 Value of a single-symbol word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
 
Theorems1rn 13091 The range of a single-symbol word. (Contributed by Mario Carneiro, 18-Jul-2016.)
(𝐴𝑉 → ran ⟨“𝐴”⟩ = {𝐴})
 
Theorems1eq 13092 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
 
Theorems1eqd 13093 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
 
Theorems1cl 13094 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.)
(𝐴𝐵 → ⟨“𝐴”⟩ ∈ Word 𝐵)
 
Theorems1cld 13095 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝜑𝐴𝐵)       (𝜑 → ⟨“𝐴”⟩ ∈ Word 𝐵)
 
Theorems1cli 13096 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴”⟩ ∈ Word V
 
Theorems1len 13097 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴”⟩) = 1
 
Theorems1nz 13098 A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.) (Proof shortened by Kyle Wyonch, 18-Jul-2021.)
⟨“𝐴”⟩ ≠ ∅
 
Theorems1nzOLD 13099 Obsolete proof of s1nz 13098 as of 18-Jul-2021. A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⟨“𝐴”⟩ ≠ ∅
 
Theorems1dm 13100 The domain of a singleton word is a singleton. (Contributed by AV, 9-Jan-2020.)
dom ⟨“𝐴”⟩ = {0}
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