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Theorem List for Metamath Proof Explorer - 13301-13400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-reps 13301* 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 13302* 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 13303* 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 13304 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 13305 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 13306 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 13307 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 13308 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 13309 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 13310 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝑆𝑉 → Word 𝑆 ∈ V)

Theoremwrdexb 13311 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 13312 The set of words over a set is a set, inference form. (Contributed by AV, 23-May-2021.)
𝑆 ∈ V       Word 𝑆 ∈ V

Theoremwrdsymbcl 13313 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 13314 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 13315 A word over an alphabet is a word over the universal class. (Contributed by AV, 8-Feb-2021.)
(𝑊 ∈ Word 𝑉𝑊 ∈ Word V)

Theoremwrdlndm 13316 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 13317* 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 13318 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 13319 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 13320 The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)
((𝑊 ∈ Word 𝑆𝑊 ≠ ∅) → (#‘𝑊) ∈ ℕ)

Theoremwrdffz 13321 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 13322 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝑆 = 𝑇 → Word 𝑆 = Word 𝑇)

Theoremwrdeqi 13323 Equality theorem for the set of words, inference form. (Contributed by AV, 23-May-2021.)
𝑆 = 𝑇       Word 𝑆 = Word 𝑇

Theoremiswrddm0 13324 A function with empty domain is a word. (Contributed by AV, 13-Oct-2018.)
(𝑊:∅⟶𝑆𝑊 ∈ Word 𝑆)

Theoremwrd0 13325 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 13326 The empty word is the only word over an empty alphabet. (Contributed by AV, 25-Oct-2018.)
(𝑊 ∈ Word ∅ ↔ 𝑊 = ∅)

Theoremffz0iswrd 13327 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 13328 Hypothesis builder for Word 𝑆. (Contributed by Mario Carneiro, 26-Feb-2016.)
𝑥𝑆       𝑥Word 𝑆

Theoremcsbwrdg 13329* Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑆𝑉𝑆 / 𝑥Word 𝑥 = Word 𝑆)

Theoremwrdnval 13330* 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 13331 Words as a mapping. (Contributed by Thierry Arnoux, 4-Mar-2020.)
((𝑉𝑋𝑁 ∈ ℕ0) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ↔ 𝑊 ∈ (𝑉𝑚 (0..^𝑁))))

Theoremhashwrdn 13332* 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 13333* 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 13334 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 13335 A word with length at least 1 is not empty. (Contributed by AV, 14-Oct-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ 1 ≤ (#‘𝑊)))

Theoremwrdlenge2n0 13336 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 13337 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 13338* 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 13339 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 13340 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 13341* 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 13342* 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 13343* 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)))

Theoremwrdred1 13344 A word truncated by a symbol is a word. (Contributed by AV, 29-Jan-2021.)
(𝐹 ∈ Word 𝑆 → (𝐹 ↾ (0..^((#‘𝐹) − 1))) ∈ Word 𝑆)

Theoremwrdred1hash 13345 The length of a word truncated by a symbol. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)
((𝐹 ∈ Word 𝑆 ∧ 1 ≤ (#‘𝐹)) → (#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))) = ((#‘𝐹) − 1))

5.7.2  Last symbol of a word

Theoremlsw 13346 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 13347 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 13348 The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 11-Nov-2018.)
( lastS ‘∅) = ∅

Theoremlsw1 13349 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 13350 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 13351 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 13352 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 13353* 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 13354 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 13355 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑆) + (#‘𝑇)))

Theoremccatval1 13356 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 13357 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 13358 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 13359 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 13360 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 13361 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 13362 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 13363 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 13364 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 13365 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 13366 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 13367 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 13368 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 13369 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 13370 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 13371 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 13372 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩

Theorems1val 13373 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 13374 The range of a single-symbol word. (Contributed by Mario Carneiro, 18-Jul-2016.)
(𝐴𝑉 → ran ⟨“𝐴”⟩ = {𝐴})

Theorems1eq 13375 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Theorems1eqd 13376 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Theorems1cl 13377 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 13378 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝜑𝐴𝐵)       (𝜑 → ⟨“𝐴”⟩ ∈ Word 𝐵)

Theorems1cli 13379 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴”⟩ ∈ Word V

Theorems1len 13380 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴”⟩) = 1

Theorems1nz 13381 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 13382 Obsolete proof of s1nz 13381 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 13383 The domain of a singleton word is a singleton. (Contributed by AV, 9-Jan-2020.)
dom ⟨“𝐴”⟩ = {0}

Theorems1dmALT 13384 Alternate version of s1dm 13383, having a shorter proof, but requiring that 𝐴 ia a set. (Contributed by AV, 9-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑆 → dom ⟨“𝐴”⟩ = {0})

Theorems1fv 13385 Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴𝐵 → (⟨“𝐴”⟩‘0) = 𝐴)

Theoremlsws1 13386 The last symbol of a singleton word is its symbol. (Contributed by AV, 22-Oct-2018.)
(𝐴𝑉 → ( lastS ‘⟨“𝐴”⟩) = 𝐴)

Theoremeqs1 13387 A word of length 1 is a singleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 1-May-2020.)
((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → 𝑊 = ⟨“(𝑊‘0)”⟩)

Theoremwrdl1exs1 13388* A word of length 1 is a singleton word. (Contributed by AV, 24-Jan-2021.)
((𝑊 ∈ Word 𝑆 ∧ (#‘𝑊) = 1) → ∃𝑠𝑆 𝑊 = ⟨“𝑠”⟩)

Theoremwrdl1s1 13389 A word of length 1 is a singleton word consisting of the first symbol of the word. (Contributed by AV, 22-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.)
(𝑆𝑉 → (𝑊 = ⟨“𝑆”⟩ ↔ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆)))

Theorems111 13390 The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))

5.7.5  Concatenations with singleton words

Theoremccatws1cl 13391 The concatenation of a word with a singleton word is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → (𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉)

Theoremccat2s1cl 13392 The concatenation of two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word 𝑉)

Theoremccatws1len 13393 The length of the concatenation of a word with a singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → (#‘(𝑊 ++ ⟨“𝑋”⟩)) = ((#‘𝑊) + 1))

Theoremwrdlenccats1lenm1 13394 The length of a word is the length of the word concatenated with a singleton word minus 1. (Contributed by AV, 28-Jun-2018.) (Proof shortened by AV, 1-May-2020.)
((𝑊 ∈ Word 𝑉𝑆𝑉) → (#‘𝑊) = ((#‘(𝑊 ++ ⟨“𝑆”⟩)) − 1))

Theoremccat2s1len 13395 The length of the concatenation of two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → (#‘(⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)) = 2)

Theoremccatw2s1len 13396 The length of the concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → (#‘((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)) = ((#‘𝑊) + 2))

Theoremccats1val1 13397 Value of a symbol in the left half of a word concatenated with a single symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
((𝑊 ∈ Word 𝑉𝑆𝑉𝐼 ∈ (0..^(#‘𝑊))) → ((𝑊 ++ ⟨“𝑆”⟩)‘𝐼) = (𝑊𝐼))

Theoremccats1val2 13398 Value of the symbol concatenated with a word. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Proof shortened by Alexander van der Vekens, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑆𝑉𝐼 = (#‘𝑊)) → ((𝑊 ++ ⟨“𝑆”⟩)‘𝐼) = 𝑆)

Theoremccat2s1p1 13399 Extract the first of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘0) = 𝑋)

Theoremccat2s1p2 13400 Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = 𝑌)

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