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Theorem List for Metamath Proof Explorer - 14001-14100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremswrdfv0 14001 The first symbol in an extracted subword. (Contributed by AV, 27-Apr-2022.)
((𝑆 ∈ Word 𝐴𝐹 ∈ (0..^𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → ((𝑆 substr ⟨𝐹, 𝐿⟩)‘0) = (𝑆𝐹))
 
Theoremswrdf 14002 A subword of a word is a function from a half-open range of nonnegative integers of the same length as the subword to the set of symbols for the original word. (Contributed by AV, 13-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝑀, 𝑁⟩):(0..^(𝑁𝑀))⟶𝑉)
 
Theoremswrdvalfn 14003 Value of the subword extractor as function with domain. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by AV, 2-May-2020.)
((𝑆 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) Fn (0..^(𝐿𝐹)))
 
Theoremswrdrn 14004 The range of a subword of a word is a subset of the set of symbols for the word. (Contributed by AV, 13-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ 𝑉)
 
Theoremswrdlend 14005 The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.) (Proof shortened by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐿𝐹 → (𝑊 substr ⟨𝐹, 𝐿⟩) = ∅))
 
Theoremswrdnd 14006 The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.) (Proof shortened by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐹 < 0 ∨ 𝐿𝐹 ∨ (♯‘𝑊) < 𝐿) → (𝑊 substr ⟨𝐹, 𝐿⟩) = ∅))
 
Theoremswrdnd2 14007 Value of the subword extractor outside its intended domain. (Contributed by Alexander van der Vekens, 24-May-2018.)
((𝑊 ∈ Word 𝑉𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐵𝐴 ∨ (♯‘𝑊) ≤ 𝐴𝐵 ≤ 0) → (𝑊 substr ⟨𝐴, 𝐵⟩) = ∅))
 
Theoremswrdnnn0nd 14008 The value of a subword operation for arguments not being nonnegative integers is the empty set. (Contributed by AV, 2-Dec-2022.)
((𝑆 ∈ Word 𝑉 ∧ ¬ (𝐹 ∈ ℕ0𝐿 ∈ ℕ0)) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅)
 
Theoremswrdnd0 14009 The value of a subword operation for inproper arguments is the empty set. (Contributed by AV, 2-Dec-2022.)
(𝑆 ∈ Word 𝑉 → (¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) = ∅))
 
Theoremswrd0 14010 A subword of an empty set is always the empty set. (Contributed by AV, 31-Mar-2018.) (Revised by AV, 20-Oct-2018.) (Proof shortened by AV, 2-May-2020.)
(∅ substr ⟨𝐹, 𝐿⟩) = ∅
 
Theoremswrdrlen 14011 Length of a right-anchored subword. (Contributed by Alexander van der Vekens, 5-Apr-2018.)
((𝑊 ∈ Word 𝑉𝐼 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 substr ⟨𝐼, (♯‘𝑊)⟩)) = ((♯‘𝑊) − 𝐼))
 
Theoremswrdlen2 14012 Length of an extracted subword. (Contributed by AV, 5-May-2020.)
((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0𝐿 ∈ (ℤ𝐹)) ∧ 𝐿 ≤ (♯‘𝑆)) → (♯‘(𝑆 substr ⟨𝐹, 𝐿⟩)) = (𝐿𝐹))
 
Theoremswrdfv2 14013 A symbol in an extracted subword, indexed using the word's indices. (Contributed by AV, 5-May-2020.)
(((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0𝐿 ∈ (ℤ𝐹)) ∧ 𝐿 ≤ (♯‘𝑆)) ∧ 𝑋 ∈ (𝐹..^𝐿)) → ((𝑆 substr ⟨𝐹, 𝐿⟩)‘(𝑋𝐹)) = (𝑆𝑋))
 
Theoremswrdwrdsymb 14014 A subword is a word over the symbols it consists of. (Contributed by AV, 2-Dec-2022.)
(𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑀, 𝑁⟩) ∈ Word (𝑆 “ (𝑀..^𝑁)))
 
Theoremswrdsb0eq 14015 Two subwords with the same bounds are equal if the range is not valid. (Contributed by AV, 4-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ 𝑁𝑀) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
 
Theoremswrdsbslen 14016 Two subwords with the same bounds have the same length. (Contributed by AV, 4-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (♯‘(𝑈 substr ⟨𝑀, 𝑁⟩)))
 
Theoremswrdspsleq 14017* Two words have a common subword (starting at the same position with the same length) iff they have the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Proof shortened by AV, 7-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
 
Theoremswrds1 14018 Extract a single symbol from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐼 ∈ (0..^(♯‘𝑊))) → (𝑊 substr ⟨𝐼, (𝐼 + 1)⟩) = ⟨“(𝑊𝐼)”⟩)
 
Theoremswrdlsw 14019 Extract the last single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑊 substr ⟨((♯‘𝑊) − 1), (♯‘𝑊)⟩) = ⟨“(lastS‘𝑊)”⟩)
 
Theoremccatswrd 14020 Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 substr ⟨𝑋, 𝑍⟩))
 
Theoremswrdccat2 14021 Recover the right half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) substr ⟨(♯‘𝑆), ((♯‘𝑆) + (♯‘𝑇))⟩) = 𝑇)
 
5.7.7  Prefixes of a word
 
Syntaxcpfx 14022 Syntax for the prefix operator.
class prefix
 
Definitiondf-pfx 14023* Define an operation which extracts prefixes of words, i.e. subwords (or substrings) starting at the beginning of a word (or string). In other words, (𝑆 prefix 𝐿) is the prefix of the word 𝑆 of length 𝐿. Definition in Section 9.1 of [AhoHopUll] p. 318. See also Wikipedia "Substring" https://en.wikipedia.org/wiki/Substring#Prefix. (Contributed by AV, 2-May-2020.)
prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr ⟨0, 𝑙⟩))
 
Theorempfxnndmnd 14024 The value of a prefix operation for out-of-domain arguments. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6694). (Contributed by AV, 3-Dec-2022.) (New usage is discouraged.)
(¬ (𝑆 ∈ V ∧ 𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = ∅)
 
Theorempfxval 14025 Value of a prefix operation. (Contributed by AV, 2-May-2020.)
((𝑆𝑉𝐿 ∈ ℕ0) → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
 
Theorempfx00 14026 The zero length prefix is the empty set. (Contributed by AV, 2-May-2020.)
(𝑆 prefix 0) = ∅
 
Theorempfx0 14027 A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
(∅ prefix 𝐿) = ∅
 
Theorempfxval0 14028 Value of a prefix operation. This theorem should only be used in proofs if 𝐿 ∈ ℕ0 is not available. Otherwise (and usually), pfxval 14025 should be used. (Contributed by AV, 3-Dec-2022.) (New usage is discouraged.)
(𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
 
Theorempfxcl 14029 Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
(𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) ∈ Word 𝐴)
 
Theorempfxmpt 14030* Value of the prefix extractor as a mapping. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝐿) = (𝑥 ∈ (0..^𝐿) ↦ (𝑆𝑥)))
 
Theorempfxres 14031 Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝐿) = (𝑆 ↾ (0..^𝐿)))
 
Theorempfxf 14032 A prefix of a word is a function from a half-open range of nonnegative integers of the same length as the prefix to the set of symbols for the original word. (Contributed by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐿):(0..^𝐿)⟶𝑉)
 
Theorempfxfn 14033 Value of the prefix extractor as function with domain. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝐿) Fn (0..^𝐿))
 
Theorempfxfv 14034 A symbol in a prefix of a word, indexed using the prefix' indices. (Contributed by Alexander van der Vekens, 16-Jun-2018.) (Revised by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 prefix 𝐿)‘𝐼) = (𝑊𝐼))
 
Theorempfxlen 14035 Length of a prefix. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 prefix 𝐿)) = 𝐿)
 
Theorempfxid 14036 A word is a prefix of itself. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by AV, 2-May-2020.)
(𝑆 ∈ Word 𝐴 → (𝑆 prefix (♯‘𝑆)) = 𝑆)
 
Theorempfxrn 14037 The range of a prefix of a word is a subset of the set of symbols for the word. (Contributed by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ 𝑉)
 
Theorempfxn0 14038 A prefix consisting of at least one symbol is not empty. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ ℕ ∧ 𝐿 ≤ (♯‘𝑊)) → (𝑊 prefix 𝐿) ≠ ∅)
 
Theorempfxnd 14039 The value of a prefix operation for a length argument larger than the word length is the empty set. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6694). (Contributed by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ ℕ0 ∧ (♯‘𝑊) < 𝐿) → (𝑊 prefix 𝐿) = ∅)
 
Theorempfxnd0 14040 The value of a prefix operation for a length argument not in the range of the word length is the empty set. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6694). (Contributed by AV, 3-Dec-2022.)
((𝑊 ∈ Word 𝑉𝐿 ∉ (0...(♯‘𝑊))) → (𝑊 prefix 𝐿) = ∅)
 
Theorempfxwrdsymb 14041 A prefix of a word is a word over the symbols it consists of. (Contributed by AV, 3-Dec-2022.)
(𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝐿) ∈ Word (𝑆 “ (0..^𝐿)))
 
Theoremaddlenrevpfx 14042 The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.) (Revised by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(♯‘𝑊))) → ((♯‘(𝑊 substr ⟨𝑀, (♯‘𝑊)⟩)) + (♯‘(𝑊 prefix 𝑀))) = (♯‘𝑊))
 
Theoremaddlenpfx 14043 The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 21-Oct-2018.) (Revised by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(♯‘𝑊))) → ((♯‘(𝑊 prefix 𝑀)) + (♯‘(𝑊 substr ⟨𝑀, (♯‘𝑊)⟩))) = (♯‘𝑊))
 
Theorempfxfv0 14044 The first symbol of a prefix is the first symbol of the word. (Contributed by Alexander van der Vekens, 16-Jun-2018.) (Revised by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (1...(♯‘𝑊))) → ((𝑊 prefix 𝐿)‘0) = (𝑊‘0))
 
Theorempfxtrcfv 14045 A symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.) (Revised by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅ ∧ 𝐼 ∈ (0..^((♯‘𝑊) − 1))) → ((𝑊 prefix ((♯‘𝑊) − 1))‘𝐼) = (𝑊𝐼))
 
Theorempfxtrcfv0 14046 The first symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.) (Revised by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → ((𝑊 prefix ((♯‘𝑊) − 1))‘0) = (𝑊‘0))
 
Theorempfxfvlsw 14047 The last symbol in a nonempty prefix of a word. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 3-May-2020.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 prefix 𝐿)) = (𝑊‘(𝐿 − 1)))
 
Theorempfxeq 14048* The prefixes of two words are equal iff they have the same length and the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 4-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑊 prefix 𝑀) = (𝑈 prefix 𝑁) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊𝑖) = (𝑈𝑖))))
 
Theorempfxtrcfvl 14049 The last symbol in a word truncated by one symbol. (Contributed by AV, 16-Jun-2018.) (Revised by AV, 5-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (lastS‘(𝑊 prefix ((♯‘𝑊) − 1))) = (𝑊‘((♯‘𝑊) − 2)))
 
Theorempfxsuffeqwrdeq 14050 Two words are equal if and only if they have the same prefix and the same suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by AV, 5-May-2020.)
((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(♯‘𝑊))) → (𝑊 = 𝑆 ↔ ((♯‘𝑊) = (♯‘𝑆) ∧ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (♯‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (♯‘𝑊)⟩)))))
 
Theorempfxsuff1eqwrdeq 14051 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by AV, 6-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (♯‘𝑊)) → (𝑊 = 𝑈 ↔ ((♯‘𝑊) = (♯‘𝑈) ∧ ((𝑊 prefix ((♯‘𝑊) − 1)) = (𝑈 prefix ((♯‘𝑊) − 1)) ∧ (lastS‘𝑊) = (lastS‘𝑈)))))
 
Theoremdisjwrdpfx 14052* Sets of words are disjoint if each set contains exactly the extensions of distinct words of a fixed length. Remark: A word 𝑊 is called an "extension" of a word 𝑃 if 𝑃 is a prefix of 𝑊. (Contributed by AV, 29-Jul-2018.) (Revised by AV, 6-May-2020.)
Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}
 
Theoremccatpfx 14053 Concatenating a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
((𝑆 ∈ Word 𝐴𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 prefix 𝑍))
 
Theorempfxccat1 14054 Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by AV, 6-May-2020.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) prefix (♯‘𝑆)) = 𝑆)
 
Theorempfx1 14055 The prefix of length one of a nonempty word expressed as a singleton word. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑊 prefix 1) = ⟨“(𝑊‘0)”⟩)
 
5.7.8  Subwords of subwords
 
Theoremswrdswrdlem 14056 Lemma for swrdswrd 14057. (Contributed by Alexander van der Vekens, 4-Apr-2018.)
(((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) ∧ (𝐾 ∈ (0...(𝑁𝑀)) ∧ 𝐿 ∈ (𝐾...(𝑁𝑀)))) → (𝑊 ∈ Word 𝑉 ∧ (𝑀 + 𝐾) ∈ (0...(𝑀 + 𝐿)) ∧ (𝑀 + 𝐿) ∈ (0...(♯‘𝑊))))
 
Theoremswrdswrd 14057 A subword of a subword is a subword. (Contributed by Alexander van der Vekens, 4-Apr-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → ((𝐾 ∈ (0...(𝑁𝑀)) ∧ 𝐿 ∈ (𝐾...(𝑁𝑀))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) substr ⟨𝐾, 𝐿⟩) = (𝑊 substr ⟨(𝑀 + 𝐾), (𝑀 + 𝐿)⟩)))
 
Theorempfxswrd 14058 A prefix of a subword is a subword. (Contributed by AV, 2-Apr-2018.) (Revised by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝐿 ∈ (0...(𝑁𝑀)) → ((𝑊 substr ⟨𝑀, 𝑁⟩) prefix 𝐿) = (𝑊 substr ⟨𝑀, (𝑀 + 𝐿)⟩)))
 
Theoremswrdpfx 14059 A subword of a prefix is a subword. (Contributed by Alexander van der Vekens, 6-Apr-2018.) (Revised by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(♯‘𝑊))) → ((𝐾 ∈ (0...𝑁) ∧ 𝐿 ∈ (𝐾...𝑁)) → ((𝑊 prefix 𝑁) substr ⟨𝐾, 𝐿⟩) = (𝑊 substr ⟨𝐾, 𝐿⟩)))
 
Theorempfxpfx 14060 A prefix of a prefix is a prefix. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝐿 ∈ (0...𝑁)) → ((𝑊 prefix 𝑁) prefix 𝐿) = (𝑊 prefix 𝐿))
 
Theorempfxpfxid 14061 A prefix of a prefix with the same length is the original prefix. In other words, the operation "prefix of length 𝑁 " is idempotent. (Contributed by AV, 5-Apr-2018.) (Revised by AV, 8-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 𝑁) prefix 𝑁) = (𝑊 prefix 𝑁))
 
5.7.9  Subwords and concatenations
 
Theorempfxcctswrd 14062 The concatenation of the prefix of a word and the rest of the word yields the word itself. (Contributed by AV, 21-Oct-2018.) (Revised by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 𝑀) ++ (𝑊 substr ⟨𝑀, (♯‘𝑊)⟩)) = 𝑊)
 
Theoremlenpfxcctswrd 14063 The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. (Contributed by AV, 21-Oct-2018.) (Revised by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(♯‘𝑊))) → (♯‘((𝑊 prefix 𝑀) ++ (𝑊 substr ⟨𝑀, (♯‘𝑊)⟩))) = (♯‘𝑊))
 
Theoremlenrevpfxcctswrd 14064 The length of the concatenation of the rest of a word and the prefix of the word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.) (Revised by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(♯‘𝑊))) → (♯‘((𝑊 substr ⟨𝑀, (♯‘𝑊)⟩) ++ (𝑊 prefix 𝑀))) = (♯‘𝑊))
 
Theorempfxlswccat 14065 Reconstruct a nonempty word from its prefix and last symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → ((𝑊 prefix ((♯‘𝑊) − 1)) ++ ⟨“(lastS‘𝑊)”⟩) = 𝑊)
 
Theoremccats1pfxeq 14066 The last symbol of a word concatenated with the word with the last symbol removed results in the word itself. (Contributed by Alexander van der Vekens, 24-Oct-2018.) (Revised by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (♯‘𝑈) = ((♯‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (♯‘𝑊)) → 𝑈 = (𝑊 ++ ⟨“(lastS‘𝑈)”⟩)))
 
Theoremccats1pfxeqrex 14067* There exists a symbol such that its concatenation after the prefix obtained by deleting the last symbol of a nonempty word results in the word itself. (Contributed by AV, 5-Oct-2018.) (Revised by AV, 9-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (♯‘𝑈) = ((♯‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (♯‘𝑊)) → ∃𝑠𝑉 𝑈 = (𝑊 ++ ⟨“𝑠”⟩)))
 
Theoremccatopth 14068 An opth 5360-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 12-Oct-2022.)
(((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theoremccatopth2 14069 An opth 5360-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
(((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
 
Theoremccatlcan 14070 Concatenation of words is left-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋𝐶 ∈ Word 𝑋) → ((𝐶 ++ 𝐴) = (𝐶 ++ 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremccatrcan 14071 Concatenation of words is right-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋𝐶 ∈ Word 𝑋) → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremwrdeqs1cat 14072 Decompose a nonempty word by separating off the first symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 12-Oct-2022.)
((𝑊 ∈ Word 𝐴𝑊 ≠ ∅) → 𝑊 = (⟨“(𝑊‘0)”⟩ ++ (𝑊 substr ⟨1, (♯‘𝑊)⟩)))
 
Theoremcats1un 14073 Express a word with an extra symbol as the union of the word and the new value. (Contributed by Mario Carneiro, 28-Feb-2016.)
((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}))
 
Theoremwrdind 14074* Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 12-Oct-2022.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   ((𝑦 ∈ Word 𝐵𝑧𝐵) → (𝜒𝜃))       (𝐴 ∈ Word 𝐵𝜏)
 
Theoremwrd2ind 14075* Perform induction over the structure of two words of the same length. (Contributed by AV, 23-Jan-2019.) (Proof shortened by AV, 12-Oct-2022.)
((𝑥 = ∅ ∧ 𝑤 = ∅) → (𝜑𝜓))    &   ((𝑥 = 𝑦𝑤 = 𝑢) → (𝜑𝜒))    &   ((𝑥 = (𝑦 ++ ⟨“𝑧”⟩) ∧ 𝑤 = (𝑢 ++ ⟨“𝑠”⟩)) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜌𝜏))    &   (𝑤 = 𝐵 → (𝜑𝜌))    &   𝜓    &   (((𝑦 ∈ Word 𝑋𝑧𝑋) ∧ (𝑢 ∈ Word 𝑌𝑠𝑌) ∧ (♯‘𝑦) = (♯‘𝑢)) → (𝜒𝜃))       ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑌 ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝜏)
 
5.7.10  Subwords of concatenations
 
Theoremswrdccatfn 14076 The subword of a concatenation as function. (Contributed by Alexander van der Vekens, 27-May-2018.)
(((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
 
Theoremswrdccatin1 14077 The subword of a concatenation of two words within the first of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
 
Theorempfxccatin12lem4 14078 Lemma 4 for pfxccatin12 14085. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by Alexander van der Vekens, 23-May-2018.)
((𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → 𝐾 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿)))))
 
Theorempfxccatin12lem2a 14079 Lemma for pfxccatin12lem2 14083. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → (𝐾 + 𝑀) ∈ (𝐿..^𝑋)))
 
Theorempfxccatin12lem1 14080 Lemma 1 for pfxccatin12 14085. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 9-May-2020.)
((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...𝑋)) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → (𝐾 − (𝐿𝑀)) ∈ (0..^(𝑁𝐿))))
 
Theoremswrdccatin2 14081 The subword of a concatenation of two words within the second of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)
𝐿 = (♯‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩)))
 
Theorempfxccatin12lem2c 14082 Lemma for pfxccatin12lem2 14083 and pfxccatin12lem3 14084. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
𝐿 = (♯‘𝐴)       (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
 
Theorempfxccatin12lem2 14083 Lemma 2 for pfxccatin12 14085. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 9-May-2020.)
𝐿 = (♯‘𝐴)       (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝐾) = ((𝐵 prefix (𝑁𝐿))‘(𝐾 − (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))))))
 
Theorempfxccatin12lem3 14084 Lemma 3 for pfxccatin12 14085. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
𝐿 = (♯‘𝐴)       (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐾 ∈ (0..^(𝑁𝑀)) ∧ 𝐾 ∈ (0..^(𝐿𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝐾) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝐾)))
 
Theorempfxccatin12 14085 The subword of a concatenation of two words within both of the concatenated words. (Contributed by Alexander van der Vekens, 5-Apr-2018.) (Revised by AV, 9-May-2020.)
𝐿 = (♯‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))
 
Theorempfxccat3 14086 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by AV, 10-May-2020.)
𝐿 = (♯‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))))
 
Theoremswrdccat 14087 The subword of a concatenation of two words as concatenation of subwords of the two concatenated words. (Contributed by Alexander van der Vekens, 29-May-2018.)
𝐿 = (♯‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, if(𝑁𝐿, 𝑁, 𝐿)⟩) ++ (𝐵 substr ⟨if(0 ≤ (𝑀𝐿), (𝑀𝐿), 0), (𝑁𝐿)⟩))))
 
Theorempfxccatpfx1 14088 A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
𝐿 = (♯‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ (0...𝐿)) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))
 
Theorempfxccatpfx2 14089 A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)
𝐿 = (♯‘𝐴)    &   𝑀 = (♯‘𝐵)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))
 
Theorempfxccat3a 14090 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by AV, 10-May-2020.)
𝐿 = (♯‘𝐴)    &   𝑀 = (♯‘𝐵)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿))))))
 
Theoremswrdccat3blem 14091 Lemma for swrdccat3b 14092. (Contributed by AV, 30-May-2018.)
𝐿 = (♯‘𝐴)       ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (♯‘𝐵)))) ∧ (𝐿 + (♯‘𝐵)) ≤ 𝐿) → if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩))
 
Theoremswrdccat3b 14092 A suffix of a concatenation is either a suffix of the second concatenated word or a concatenation of a suffix of the first word with the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by Alexander van der Vekens, 30-May-2018.) (Proof shortened by AV, 14-Oct-2022.)
𝐿 = (♯‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩) = if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))))
 
Theorempfxccatid 14093 A prefix of a concatenation of length of the first concatenated word is the first word itself. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 10-May-2020.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (♯‘𝐴)) → ((𝐴 ++ 𝐵) prefix 𝑁) = 𝐴)
 
Theoremccats1pfxeqbi 14094 A word is a prefix of a word with length greater by 1 than the first word iff the second word is the first word concatenated with the last symbol of the second word. (Contributed by AV, 24-Oct-2018.) (Revised by AV, 10-May-2020.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (♯‘𝑈) = ((♯‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (♯‘𝑊)) ↔ 𝑈 = (𝑊 ++ ⟨“(lastS‘𝑈)”⟩)))
 
Theoremswrdccatin1d 14095 The subword of a concatenation of two words within the first of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)
(𝜑 → (♯‘𝐴) = 𝐿)    &   (𝜑 → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))    &   (𝜑𝑀 ∈ (0...𝑁))    &   (𝜑𝑁 ∈ (0...𝐿))       (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
 
Theoremswrdccatin2d 14096 The subword of a concatenation of two words within the second of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by Mario Carneiro/AV, 21-Oct-2018.)
(𝜑 → (♯‘𝐴) = 𝐿)    &   (𝜑 → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))    &   (𝜑𝑀 ∈ (𝐿...𝑁))    &   (𝜑𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))       (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩))
 
Theorempfxccatin12d 14097 The subword of a concatenation of two words within both of the concatenated words. (Contributed by AV, 31-May-2018.) (Revised by AV, 10-May-2020.)
(𝜑 → (♯‘𝐴) = 𝐿)    &   (𝜑 → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))    &   (𝜑𝑀 ∈ (0...𝐿))    &   (𝜑𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))       (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))))
 
Theoremreuccatpfxs1lem 14098* Lemma for reuccatpfxs1 14099. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised by AV, 9-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈𝑋) ∧ ∀𝑠𝑉 ((𝑊 ++ ⟨“𝑠”⟩) ∈ 𝑋𝑆 = 𝑠) ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (𝑊 = (𝑈 prefix (♯‘𝑊)) → 𝑈 = (𝑊 ++ ⟨“𝑆”⟩)))
 
Theoremreuccatpfxs1 14099* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 13-Oct-2022.)
𝑣𝑋       ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))
 
Theoremreuccatpfxs1v 14100* There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 10-May-2022.) (Proof shortened by AV, 13-Oct-2022.)
((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))
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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 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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 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