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Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorems1dmALT 13101 Alternate version of s1dm 13100, 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 13102 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 13103 The last symbol of a singleton word is its symbol. (Contributed by AV, 22-Oct-2018.)
(𝐴𝑉 → ( lastS ‘⟨“𝐴”⟩) = 𝐴)

Theoremeqs1 13104 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 13105* A word of length 1 is a singleton word. (Contributed by AV, 24-Jan-2021.)
((𝑊 ∈ Word 𝑆 ∧ (#‘𝑊) = 1) → ∃𝑠𝑆 𝑊 = ⟨“𝑠”⟩)

Theoremwrdl1s1 13106 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 13107 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 13108 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 13109 The concatenation of two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word 𝑉)

Theoremccatws1len 13110 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 13111 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 13112 The length of the concatenation of two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → (#‘(⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)) = 2)

Theoremccatw2s1len 13113 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 13114 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 13115 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 13116 Extract the first of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘0) = 𝑋)

Theoremccat2s1p2 13117 Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = 𝑌)

Theoremccatw2s1ass 13118 Associative law for a concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)))

Theoremccatws1lenrev 13119 The length of a word concatenated with a singleton word. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → ((#‘(𝑊 ++ ⟨“𝑋”⟩)) = 𝑁 → (#‘𝑊) = (𝑁 − 1)))

Theoremccatws1n0 13120 The concatenation of a word with a singleton word is not the empty set. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → (𝑊 ++ ⟨“𝑋”⟩) ≠ ∅)

Theoremccatws1ls 13121 The last symbol of the concatenation of a word with a singleton word is the symbol of the singleton word. (Contributed by AV, 29-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → ((𝑊 ++ ⟨“𝑋”⟩)‘(#‘𝑊)) = 𝑋)

Theoremlswccats1 13122 The last symbol of a word concatenated with a singleton word is the symbol of the singleton word. (Contributed by AV, 6-Aug-2018.) (Proof shortened by AV, 22-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑆𝑉) → ( lastS ‘(𝑊 ++ ⟨“𝑆”⟩)) = 𝑆)

Theoremlswccats1fst 13123 The last symbol of a nonempty word concatenated with its first symbol is the first symbol. (Contributed by AV, 28-Jun-2018.) (Proof shortened by AV, 1-May-2020.)
((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ( lastS ‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0))

Theoremccatw2s1p1 13124 Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.)
(((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝑁) = 𝑋)

Theoremccatw2s1p2 13125 Extract the second of two single symbols concatenated with a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.)
(((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑁 + 1)) = 𝑌)

Theoremccat2s1fvw 13126 Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.)
(((𝑊 ∈ Word 𝑉𝐼 ∈ ℕ0𝐼 < (#‘𝑊)) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊𝐼))

Theoremccat2s1fst 13127 The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
(((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘0) = (𝑊‘0))

5.7.6  Subwords

Theoremswrdval 13128* Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))

Theoremswrd00 13129 A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.)
(𝑆 substr ⟨𝑋, 𝑋⟩) = ∅

Theoremswrdcl 13130 Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)

Theoremswrdval2 13131* Value of the subword extractor in its intended domain. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) = (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))))

Theoremswrd0val 13132 Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(#‘𝑆))) → (𝑆 substr ⟨0, 𝐿⟩) = (𝑆 ↾ (0..^𝐿)))

Theoremswrd0len 13133 Length of a left-anchored subword. (Contributed by Stefan O'Rear, 24-Aug-2015.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨0, 𝐿⟩)) = 𝐿)

Theoremswrdlen 13134 Length of an extracted subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
((𝑆 ∈ Word 𝐴𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨𝐹, 𝐿⟩)) = (𝐿𝐹))

Theoremswrdfv 13135 A symbol in an extracted subword, indexed using the subword's indices. (Contributed by Stefan O'Rear, 16-Aug-2015.)
(((𝑆 ∈ Word 𝐴𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) ∧ 𝑋 ∈ (0..^(𝐿𝐹))) → ((𝑆 substr ⟨𝐹, 𝐿⟩)‘𝑋) = (𝑆‘(𝑋 + 𝐹)))

Theoremswrdf 13136 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 13137 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)))

Theoremswrd0f 13138 A left-anchored 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, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊))) → (𝑊 substr ⟨0, 𝑁⟩):(0..^𝑁)⟶𝑉)

Theoremswrdid 13139 A word is a subword of itself. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Proof shortened by AV, 2-May-2020.)
(𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨0, (#‘𝑆)⟩) = 𝑆)

Theoremswrdrn 13140 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)

Theoremswrdn0 13141 A prefixing subword consisting of at least one symbol is not empty. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Proof shortened by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ ∧ 𝑁 ≤ (#‘𝑊)) → (𝑊 substr ⟨0, 𝑁⟩) ≠ ∅)

Theoremswrdlend 13142 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 13143 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 13144 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⟩) = ∅))

Theoremswrd0 13145 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 13146 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))

Theoremswrd0len0 13147 Length of a prefix of a word reduced by a single symbol, analogous to swrd0len 13133. (Contributed by AV, 4-Aug-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0 ∧ (#‘𝑊) = (𝑁 + 1)) → (#‘(𝑊 substr ⟨0, 𝑁⟩)) = 𝑁)

Theoremaddlenrevswrd 13148 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.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((#‘(𝑊 substr ⟨𝑀, (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, 𝑀⟩))) = (#‘𝑊))

Theoremaddlenswrd 13149 The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 21-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((#‘(𝑊 substr ⟨0, 𝑀⟩)) + (#‘(𝑊 substr ⟨𝑀, (#‘𝑊)⟩))) = (#‘𝑊))

Theoremswrd0fv 13150 A symbol in an left-anchored subword, indexed using the subword's indices. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 substr ⟨0, 𝐿⟩)‘𝐼) = (𝑊𝐼))

Theoremswrd0fv0 13151 The first symbol in a left-anchored subword. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
((𝑊 ∈ Word 𝑉𝐼 ∈ (1...(#‘𝑊))) → ((𝑊 substr ⟨0, 𝐼⟩)‘0) = (𝑊‘0))

Theoremswrdtrcfv 13152 A symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅ ∧ 𝐼 ∈ (0..^((#‘𝑊) − 1))) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩)‘𝐼) = (𝑊𝐼))

Theoremswrdtrcfv0 13153 The first symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩)‘0) = (𝑊‘0))

Theoremswrd0fvlsw 13154 The last symbol in a left-anchored subword. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (1...(#‘𝑊))) → ( lastS ‘(𝑊 substr ⟨0, 𝐿⟩)) = (𝑊‘(𝐿 − 1)))

Theoremswrdeq 13155* Two subwords of 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.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨0, 𝑀⟩) = (𝑈 substr ⟨0, 𝑁⟩) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊𝑖) = (𝑈𝑖))))

Theoremswrdlen2 13156 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 13157 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))

Theoremswrdsb0eq 13158 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 13159 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 13160* 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)))

Theoremswrdtrcfvl 13161 The last symbol in a word truncated by one symbol. (Contributed by AV, 16-Jun-2018.) (Proof shortened by Mario Carneiro/AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → ( lastS ‘(𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩)) = (𝑊‘((#‘𝑊) − 2)))

Theoremswrds1 13162 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 13163 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)”⟩)

Theorem2swrdeqwrdeq 13164 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.)
((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 substr ⟨0, 𝐼⟩) = (𝑆 substr ⟨0, 𝐼⟩) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))

Theorem2swrd1eqwrdeq 13165 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 Mario Carneiro/AV, 23-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))

Theoremdisjxwrd 13166* Sets of words are disjoint if each set contains extensions of distinct words of a fixed length. (Contributed by AV, 29-Jul-2018.) (Proof shortened by AV, 7-May-2020.)
Disj 𝑦𝑊 {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦}

Theoremccatswrd 13167 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⟩))

Theoremswrdccat1 13168 Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) substr ⟨0, (#‘𝑆)⟩) = 𝑆)

Theoremswrdccat2 13169 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  Subwords of subwords

Theoremswrdswrdlem 13170 Lemma for swrdswrd 13171. (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 13171 A subword of 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)⟩)))

Theoremswrd0swrd 13172 A prefix of a subword. (Contributed by AV, 2-Apr-2018.) (Proof shortened by AV, 21-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊)) ∧ 𝑀 ∈ (0...𝑁)) → (𝐿 ∈ (0...(𝑁𝑀)) → ((𝑊 substr ⟨𝑀, 𝑁⟩) substr ⟨0, 𝐿⟩) = (𝑊 substr ⟨𝑀, (𝑀 + 𝐿)⟩)))

Theoremswrdswrd0 13173 A subword of a prefix. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊))) → ((𝐾 ∈ (0...𝑁) ∧ 𝐿 ∈ (𝐾...𝑁)) → ((𝑊 substr ⟨0, 𝑁⟩) substr ⟨𝐾, 𝐿⟩) = (𝑊 substr ⟨𝐾, 𝐿⟩)))

Theoremswrd0swrd0 13174 A prefix of a prefix. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊)) ∧ 𝐿 ∈ (0...𝑁)) → ((𝑊 substr ⟨0, 𝑁⟩) substr ⟨0, 𝐿⟩) = (𝑊 substr ⟨0, 𝐿⟩))

Theoremswrd0swrdid 13175 A prefix of a prefix with the same length is the prefix. (Contributed by AV, 5-Apr-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊))) → ((𝑊 substr ⟨0, 𝑁⟩) substr ⟨0, 𝑁⟩) = (𝑊 substr ⟨0, 𝑁⟩))

5.7.8  Subwords and concatenations

Theoremwrdcctswrd 13176 The concatenation of two parts of a word yields the word itself. (Contributed by AV, 21-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((𝑊 substr ⟨0, 𝑀⟩) ++ (𝑊 substr ⟨𝑀, (#‘𝑊)⟩)) = 𝑊)

Theoremlencctswrd 13177 The length of two concatenated parts of a word is the length of the word. (Contributed by AV, 21-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → (#‘((𝑊 substr ⟨0, 𝑀⟩) ++ (𝑊 substr ⟨𝑀, (#‘𝑊)⟩))) = (#‘𝑊))

Theoremlenrevcctswrd 13178 The length of two reversely concatenated parts of a word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → (#‘((𝑊 substr ⟨𝑀, (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, 𝑀⟩))) = (#‘𝑊))

Theoremswrdccatwrd 13179 Reconstruct a nonempty word from its prefix and last symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩) = 𝑊)

Theoremccats1swrdeq 13180 The last symbol of a word concatenated with the subword of the word having length less by 1 than the word results in the word itself. (Contributed by Alexander van der Vekens, 24-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) → 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))

Theoremccatopth 13181 An opth 4769-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) ∧ (#‘𝐴) = (#‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremccatopth2 13182 An opth 4769-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 13183 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 13184 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 13185 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, 9-May-2020.)
((𝑊 ∈ Word 𝐴𝑊 ≠ ∅) → 𝑊 = (⟨“(𝑊‘0)”⟩ ++ (𝑊 substr ⟨1, (#‘𝑊)⟩)))

Theoremcats1un 13186 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 13187* Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   ((𝑦 ∈ Word 𝐵𝑧𝐵) → (𝜒𝜃))       (𝐴 ∈ Word 𝐵𝜏)

Theoremwrd2ind 13188* Perform induction over the structure of two words of the same length. (Contributed by AV, 23-Jan-2019.)
((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)

Theoremccats1swrdeqrex 13189* There exists a symbol such that its concatenation with the subword obtained by deleting the last symbol of a nonempty word results in the word itself. (Contributed by AV, 5-Oct-2018.) (Proof shortened by AV, 24-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) → ∃𝑠𝑉 𝑈 = (𝑊 ++ ⟨“𝑠”⟩)))

Theoremreuccats1lem 13190* Lemma for reuccats1 13191. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Proof shortened by AV, 15-Jan-2020.)
(((𝑊 ∈ Word 𝑉𝑈𝑋 ∧ (𝑊 ++ ⟨“𝑆”⟩) ∈ 𝑋) ∧ (∀𝑠𝑉 ((𝑊 ++ ⟨“𝑠”⟩) ∈ 𝑋𝑆 = 𝑠) ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)))) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) → 𝑈 = (𝑊 ++ ⟨“𝑆”⟩)))

Theoremreuccats1 13191* A set of words having the length of a given word increased by 1 contains a unique word with the given word as prefix if there is a unique symbol which extends the given word to be a word of the set. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩)))

5.7.9  Subwords of concatenations

Theoremswrdccatfn 13192 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 13193 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⟩)))

Theoremswrdccatin12lem1 13194 Lemma 1 for swrdccatin12 13201. (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)))))

Theoremswrdccatin12lem2a 13195 Lemma 1 for swrdccatin12lem2 13199. (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)))

Theoremswrdccatin12lem2b 13196 Lemma 2 for swrdccatin12lem2 13199. (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 − (𝐿𝑀)) ∈ (0..^((𝑁𝐿) − 0))))

Theoremswrdccatin2 13197 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)⟩)))

Theoremswrdccatin12lem2c 13198 Lemma for swrdccatin12lem2 13199 and swrdccatin12lem3 13200. (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)))))

Theoremswrdccatin12lem2 13199 Lemma 2 for swrdccatin12 13201. (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) = ((𝐵 substr ⟨0, (𝑁𝐿)⟩)‘(𝐾 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩))))))

Theoremswrdccatin12lem3 13200 Lemma 3 for swrdccatin12 13201. (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)))

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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 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