Home Metamath Proof ExplorerTheorem List (p. 131 of 425) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26947) Hilbert Space Explorer (26948-28472) Users' Mathboxes (28473-42426)

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}

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
 Copyright terms: Public domain < Previous  Next >