HomeHome Metamath Proof Explorer
Theorem List (p. 133 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  Metamath Proof Explorer
(1-26941)
  Hilbert Space Explorer  Hilbert Space Explorer
(26942-28466)
  Users' Mathboxes  Users' Mathboxes
(28467-42420)
 

Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremswrdccatin12lem2c 13201 Lemma for swrdccatin12lem2 13202 and swrdccatin12lem3 13203. (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 13202 Lemma 2 for swrdccatin12 13204. (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 13203 Lemma 3 for swrdccatin12 13204. (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)))
 
Theoremswrdccatin12 13204 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 Alexander van der Vekens, 27-May-2018.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩))))
 
Theoremswrdccat3 13205 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 Alexander van der Vekens, 28-May-2018.)
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 substr ⟨0, (𝑁𝐿)⟩))))))
 
Theoremswrdccat 13206 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)⟩))))
 
Theoremswrdccat3a 13207 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 Alexander van der Vekens, 29-May-2018.)
𝐿 = (#‘𝐴)       ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁𝐿, (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩)))))
 
Theoremswrdccat3blem 13208 Lemma for swrdccat3b 13209. (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 13209 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.)
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))))
 
Theoremswrdccatid 13210 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.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 = (#‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = 𝐴)
 
Theoremccats1swrdeqbi 13211 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.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) ↔ 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))
 
Theoremswrdccatin1d 13212 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 13213 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)⟩))
 
Theoremswrdccatin12d 13214 The subword of a concatenation of two words within both 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...𝐿))    &   (𝜑𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))       (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩)))
 
5.7.10  Splicing words (substring replacement)
 
Theoremsplval 13215 Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))
 
Theoremsplcl 13216 Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)
((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
 
Theoremsplid 13217 Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.)
((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = 𝑆)
 
Theoremspllen 13218 The length of a splice. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(#‘𝑆)))    &   (𝜑𝑅 ∈ Word 𝐴)       (𝜑 → (#‘(𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)) = ((#‘𝑆) + ((#‘𝑅) − (𝑇𝐹))))
 
Theoremsplfv1 13219 Symbols to the left of a splice are unaffected. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(#‘𝑆)))    &   (𝜑𝑅 ∈ Word 𝐴)    &   (𝜑𝑋 ∈ (0..^𝐹))       (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘𝑋) = (𝑆𝑋))
 
Theoremsplfv2a 13220 Symbols within the replacement region of a splice, expressed using the coordinates of the replacement region. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(#‘𝑆)))    &   (𝜑𝑅 ∈ Word 𝐴)    &   (𝜑𝑋 ∈ (0..^(#‘𝑅)))       (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘(𝐹 + 𝑋)) = (𝑅𝑋))
 
Theoremsplval2 13221 Value of a splice, assuming the input word 𝑆 has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.)
(𝜑𝐴 ∈ Word 𝑋)    &   (𝜑𝐵 ∈ Word 𝑋)    &   (𝜑𝐶 ∈ Word 𝑋)    &   (𝜑𝑅 ∈ Word 𝑋)    &   (𝜑𝑆 = ((𝐴 ++ 𝐵) ++ 𝐶))    &   (𝜑𝐹 = (#‘𝐴))    &   (𝜑𝑇 = (𝐹 + (#‘𝐵)))       (𝜑 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = ((𝐴 ++ 𝑅) ++ 𝐶))
 
5.7.11  Reversing words
 
Theoremrevval 13222* Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(𝑊𝑉 → (reverse‘𝑊) = (𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥))))
 
Theoremrevcl 13223 The reverse of a word is a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(𝑊 ∈ Word 𝐴 → (reverse‘𝑊) ∈ Word 𝐴)
 
Theoremrevlen 13224 The reverse of a word has the same length as the original. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(𝑊 ∈ Word 𝐴 → (#‘(reverse‘𝑊)) = (#‘𝑊))
 
Theoremrevfv 13225 Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.)
((𝑊 ∈ Word 𝐴𝑋 ∈ (0..^(#‘𝑊))) → ((reverse‘𝑊)‘𝑋) = (𝑊‘(((#‘𝑊) − 1) − 𝑋)))
 
Theoremrev0 13226 The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
(reverse‘∅) = ∅
 
Theoremrevs1 13227 Singleton words are their own reverses. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(reverse‘⟨“𝑆”⟩) = ⟨“𝑆”⟩
 
Theoremrevccat 13228 Antiautomorphic property of the reversal operation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑆 ∈ Word 𝐴𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) = ((reverse‘𝑇) ++ (reverse‘𝑆)))
 
Theoremrevrev 13229 Reversion is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
(𝑊 ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) = 𝑊)
 
5.7.12  Repeated symbol words
 
Theoremreps 13230* Construct a function mapping a half-open range of nonnegative integers to a constant. (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
 
Theoremrepsundef 13231 A function mapping a half-open range of nonnegative integers with an upper bound not being a nonnegative integer to a constant is the empty set (in the meaning of "undefined"). (Contributed by AV, 5-Nov-2018.)
(𝑁 ∉ ℕ0 → (𝑆 repeatS 𝑁) = ∅)
 
Theoremrepsconst 13232 Construct a function mapping a half-open range of nonnegative integers to a constant, see also fconstmpt 4979. (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = ((0..^𝑁) × {𝑆}))
 
Theoremrepsf 13233 The constructed function mapping a half-open range of nonnegative integers to a constant is a function. (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉)
 
Theoremrepswsymb 13234 The symbols of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0𝐼 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝐼) = 𝑆)
 
Theoremrepsw 13235 A function mapping a half-open range of nonnegative integers to a constant is a word consisting of one symbol repeated several times ("repeated symbol word"). (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) ∈ Word 𝑉)
 
Theoremrepswlen 13236 The length of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (#‘(𝑆 repeatS 𝑁)) = 𝑁)
 
Theoremrepsw0 13237 The "repeated symbol word" of length 0. (Contributed by AV, 4-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 0) = ∅)
 
Theoremrepsdf2 13238* Alternative definition of a "repeated symbol word". (Contributed by AV, 7-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (𝑊 = (𝑆 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊𝑖) = 𝑆)))
 
Theoremrepswsymball 13239* All the symbols of a "repeated symbol word" are the same. (Contributed by AV, 10-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑆𝑉) → (𝑊 = (𝑆 repeatS (#‘𝑊)) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = 𝑆))
 
Theoremrepswsymballbi 13240* A word is a "repeated symbol word" iff each of its symbols equals the first symbol of the word. (Contributed by AV, 10-Nov-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 = ((𝑊‘0) repeatS (#‘𝑊)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑊‘0)))
 
Theoremrepswfsts 13241 The first symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ) → ((𝑆 repeatS 𝑁)‘0) = 𝑆)
 
Theoremrepswlsw 13242 The last symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ) → ( lastS ‘(𝑆 repeatS 𝑁)) = 𝑆)
 
Theoremrepsw1 13243 The "repeated symbol word" of length 1. (Contributed by AV, 4-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 1) = ⟨“𝑆”⟩)
 
Theoremrepswswrd 13244 A subword of a "repeated symbol word" is again a "repeated symbol word". The assumption N <_ L is required, because otherwise ( L < N ): ((𝑆 repeatS 𝐿) substr ⟨𝑀, 𝑁⟩) = ∅, but for M < N (𝑆 repeatS (𝑁𝑀))) ≠ ∅! The proof is relatively long because the border cases (𝑀 = 𝑁, ¬ (𝑀..^𝑁) ⊆ (0..^𝐿) must have been considered. (Contributed by AV, 6-Nov-2018.)
(((𝑆𝑉𝐿 ∈ ℕ0) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ 𝑁𝐿) → ((𝑆 repeatS 𝐿) substr ⟨𝑀, 𝑁⟩) = (𝑆 repeatS (𝑁𝑀)))
 
Theoremrepswccat 13245 The concatenation of two "repeated symbol words" with the same symbol is again a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → ((𝑆 repeatS 𝑁) ++ (𝑆 repeatS 𝑀)) = (𝑆 repeatS (𝑁 + 𝑀)))
 
Theoremrepswrevw 13246 The reverse of a "repeated symbol word". (Contributed by AV, 6-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0) → (reverse‘(𝑆 repeatS 𝑁)) = (𝑆 repeatS 𝑁))
 
5.7.13  Cyclical shifts of words

A word/string can be regarded as "necklace" by connecting the two ends of the word/string together (see Wikipedia "Necklace (combinatorics)", https://en.wikipedia.org/wiki/Necklace_(combinatorics)).

Two strings are regarded as the same necklace if one string can be rotated/circularly shifted/cyclically shifted to obtain the second string. To cope with words in the sense of necklaces, the rotation/cyclic shift cyclShift is defined as the basic operation, see df-csh 13248. The main theorems in this section are about counting the number of different necklaces resulting from cyclically shifting a given word, see cshwrepswhash1 15534 for words consisting of identical symbols and cshwshash 15536 for words having lengths which are prime numbers.

 
Syntaxccsh 13247 Extend class notation with Cyclical Shifts.
class cyclShift
 
Definitiondf-csh 13248* Perform a cyclical shift for an arbitrary class. Meaningful only for words 𝑤 ∈ Word 𝑆 or at least functions over half-open ranges of nonnegative integers. (Contributed by Alexander van der Vekens, 20-May-2018.) (Revised by Mario Carneiro/Alexander van der Vekens/ Gerard Lang, 17-Nov-2018.)
cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (#‘𝑤)), (#‘𝑤)⟩) ++ (𝑤 substr ⟨0, (𝑛 mod (#‘𝑤))⟩))))
 
Theoremcshfn 13249* Perform a cyclical shift for a function over a half-open range of nonnegative integers. (Contributed by AV, 20-May-2018.) (Revised by AV, 17-Nov-2018.)
((𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩))))
 
Theoremcshword 13250 Perform a cyclical shift for a word. (Contributed by Alexander van der Vekens, 20-May-2018.) (Revised by AV, 17-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 substr ⟨0, (𝑁 mod (#‘𝑊))⟩)))
 
Theoremcshnz 13251 A cyclical shift is the empty set if the number of shifts is not an integer. (Contributed by Alexander van der Vekens, 21-May-2018.) (Revised by AV, 17-Nov-2018.)
𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅)
 
Theorem0csh0 13252 Cyclically shifting an empty set/word always results in the empty word/set. (Contributed by AV, 25-Oct-2018.) (Revised by AV, 17-Nov-2018.)
(∅ cyclShift 𝑁) = ∅
 
Theoremcshw0 13253 A word cyclically shifted by 0 is the word itself. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 26-Oct-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 0) = 𝑊)
 
Theoremcshwmodn 13254 Cyclically shifting a word is invariant regarding modulo the word's length. (Contributed by AV, 26-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (#‘𝑊))))
 
Theoremcshwsublen 13255 Cyclically shifting a word is invariant regarding subtraction of the word's length. (Contributed by AV, 3-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 − (#‘𝑊))))
 
Theoremcshwn 13256 A word cyclically shifted by its length is the word itself. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 26-Oct-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (#‘𝑊)) = 𝑊)
 
Theoremcshwcl 13257 A cyclically shifted word is a word over the same set as for the original word. (Contributed by AV, 16-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 27-Oct-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑁) ∈ Word 𝑉)
 
Theoremcshwlen 13258 The length of a cyclically shifted word is the same as the length of the original word. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 27-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑁)) = (#‘𝑊))
 
Theoremcshwf 13259 A cyclically shifted word is a function from a half-open range of integers of the same length as the word as domain to the set of symbols for the word. (Contributed by AV, 12-Nov-2018.)
((𝑊 ∈ Word 𝐴𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁):(0..^(#‘𝑊))⟶𝐴)
 
Theoremcshwfn 13260 A cyclically shifted word is a function with a half-open range of integers of the same length as the word as domain. (Contributed by AV, 12-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) Fn (0..^(#‘𝑊)))
 
Theoremcshwrn 13261 The range of a cyclically shifted word is a subset of the set of symbols for the word. (Contributed by AV, 12-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → ran (𝑊 cyclShift 𝑁) ⊆ 𝑉)
 
Theoremcshwidxmod 13262 The symbol at a given index of a cyclically shifted nonempty word is the symbol at the shifted index of the original word. (Contributed by AV, 13-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑁)‘𝐼) = (𝑊‘((𝐼 + 𝑁) mod (#‘𝑊))))
 
Theoremcshwidxmodr 13263 The symbol at a given index of a cyclically shifted nonempty word is the symbol at the shifted index of the original word. (Contributed by AV, 17-Mar-2021.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((𝐼𝑁) mod (#‘𝑊))) = (𝑊𝐼))
 
Theoremcshwidx0mod 13264 The symbol at index 0 of a cyclically shifted nonempty word is the symbol at index N (modulo the length of the word) of the original word. (Contributed by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑁)‘0) = (𝑊‘(𝑁 mod (#‘𝑊))))
 
Theoremcshwidx0 13265 The symbol at index 0 of a cyclically shifted nonempty word is the symbol at index N of the original word. (Contributed by AV, 15-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑁)‘0) = (𝑊𝑁))
 
Theoremcshwidxm1 13266 The symbol at index ((n-N)-1) of a word of length n (not 0) cyclically shifted by N positions is the symbol at index (n-1) of the original word. (Contributed by AV, 23-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0..^(#‘𝑊))) → ((𝑊 cyclShift 𝑁)‘(((#‘𝑊) − 𝑁) − 1)) = (𝑊‘((#‘𝑊) − 1)))
 
Theoremcshwidxm 13267 The symbol at index (n-N) of a word of length n (not 0) cyclically shifted by N positions (not 0) is the symbol at index 0 of the original word. (Contributed by AV, 18-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (1...(#‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((#‘𝑊) − 𝑁)) = (𝑊‘0))
 
Theoremcshwidxn 13268 The symbol at index (n-1) of a word of length n (not 0) cyclically shifted by N positions (not 0) is the symbol at index (N-1) of the original word. (Contributed by AV, 18-May-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (1...(#‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((#‘𝑊) − 1)) = (𝑊‘(𝑁 − 1)))
 
Theoremcshf1 13269 Cyclically shifting a word which contains a symbol at most once results in a word which contains a symbol at most once. (Contributed by AV, 14-Mar-2021.)
((𝐹:(0..^(#‘𝐹))–1-1𝐴𝑆 ∈ ℤ ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(#‘𝐹))–1-1𝐴)
 
Theoremcshinj 13270 If a word is injectiv (regarded as function), the cyclically shifted word is also injective. (Contributed by AV, 14-Mar-2021.)
((𝐹 ∈ Word 𝐴 ∧ Fun 𝐹𝑆 ∈ ℤ) → (𝐺 = (𝐹 cyclShift 𝑆) → Fun 𝐺))
 
Theoremrepswcshw 13271 A cyclically shifted "repeated symbol word". (Contributed by Alexander van der Vekens, 7-Nov-2018.)
((𝑆𝑉𝑁 ∈ ℕ0𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = (𝑆 repeatS 𝑁))
 
Theorem2cshw 13272 Cyclically shifting a word two times. (Contributed by AV, 7-Apr-2018.) (Revised by AV, 4-Jun-2018.) (Revised by AV, 31-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) = (𝑊 cyclShift (𝑀 + 𝑁)))
 
Theorem2cshwid 13273 Cyclically shifting a word two times resulting in the word itself. (Contributed by AV, 7-Apr-2018.) (Revised by AV, 5-Jun-2018.) (Revised by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑁) cyclShift ((#‘𝑊) − 𝑁)) = 𝑊)
 
Theoremlswcshw 13274 The last symbol of a word cyclically shifted by N positions is the symbol at index (N-1) of the original word. (Contributed by AV, 21-Mar-2018.) (Revised by AV, 5-Jun-2018.) (Revised by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (1...(#‘𝑊))) → ( lastS ‘(𝑊 cyclShift 𝑁)) = (𝑊‘(𝑁 − 1)))
 
Theorem2cshwcom 13275 Cyclically shifting a word two times is commutative. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 5-Jun-2018.) (Revised by Mario Carneiro/AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑊 cyclShift 𝑁) cyclShift 𝑀) = ((𝑊 cyclShift 𝑀) cyclShift 𝑁))
 
Theoremcshwleneq 13276 If the results of cyclically shifting two words are equal, the length of the two words was equal. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 5-Jun-2018.) (Revised by AV, 1-Nov-2018.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (#‘𝑊) = (#‘𝑈))
 
Theorem3cshw 13277 Cyclically shifting a word three times results in a once cyclically shifted word under certain circumstances. (Contributed by AV, 6-Jun-2018.) (Revised by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (((𝑊 cyclShift 𝑀) cyclShift 𝑁) cyclShift ((#‘𝑊) − 𝑀)))
 
Theoremcshweqdif2 13278 If cyclically shifting two words (of the same length) results in the same word, cyclically shifting one of the words by the difference of the numbers of shifts results in the other word. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 6-Jun-2018.) (Revised by AV, 1-Nov-2018.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀) → (𝑈 cyclShift (𝑀𝑁)) = 𝑊))
 
Theoremcshweqdifid 13279 If cyclically shifting a word by two positions results in the same word, cyclically shifting the word by the difference of these two positions results in the original word itself. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 7-Jun-2018.) (Revised by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑊 cyclShift 𝑁) = (𝑊 cyclShift 𝑀) → (𝑊 cyclShift (𝑀𝑁)) = 𝑊))
 
Theoremcshweqrep 13280* If cyclically shifting a word by L position results in the word itself, the symbol at any position is repeated at multiples of L (modulo the length of the word) positions in the word. (Contributed by AV, 13-May-2018.) (Revised by AV, 7-Jun-2018.) (Revised by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉𝐿 ∈ ℤ) → (((𝑊 cyclShift 𝐿) = 𝑊𝐼 ∈ (0..^(#‘𝑊))) → ∀𝑗 ∈ ℕ0 (𝑊𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (#‘𝑊)))))
 
Theoremcshw1 13281* If cyclically shifting a word by 1 position results in the word itself, the word is build of identical symbols. Remark: also "valid" for an empty word! (Contributed by AV, 13-May-2018.) (Revised by AV, 7-Jun-2018.) (Proof shortened by AV, 1-Nov-2018.)
((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑊‘0))
 
Theoremcshw1repsw 13282 If cyclically shifting a word by 1 position results in the word itself, the word is a "repeated symbol word". Remark: also "valid" for an empty word! (Contributed by AV, 8-Nov-2018.) (Proof shortened by AV, 10-Nov-2018.)
((𝑊 ∈ Word 𝑉 ∧ (𝑊 cyclShift 1) = 𝑊) → 𝑊 = ((𝑊‘0) repeatS (#‘𝑊)))
 
Theoremcshwsexa 13283* The class of (different!) words resulting by cyclically shifting something (not necessarily a word) is a set. (Contributed by AV, 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.)
{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
 
Theorem2cshwcshw 13284* If a word is a cyclically shifted word, and a second word is the result of cyclically shifting the same word, then the second word is the result of cyclically shifting the first word. (Contributed by AV, 11-May-2018.) (Revised by AV, 12-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
((𝑌 ∈ Word 𝑉 ∧ (#‘𝑌) = 𝑁) → ((𝐾 ∈ (0...𝑁) ∧ 𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))
 
Theoremscshwfzeqfzo 13285* For a nonempty word the sets of shifted words, expressd by a finite interval of integers or by a half-open integer range are identical. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
((𝑋 ∈ Word 𝑉𝑋 ≠ ∅ ∧ 𝑁 = (#‘𝑋)) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)})
 
Theoremcshwcshid 13286* A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlktr 26075 and erclwwlkntr 26087. (Contributed by AV, 8-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
(𝜑𝑦 ∈ Word 𝑉)    &   (𝜑 → (#‘𝑥) = (#‘𝑦))       (𝜑 → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
 
Theoremcshwcsh2id 13287* A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 26075 and erclwwlkntr 26087. (Contributed by AV, 9-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
(𝜑𝑧 ∈ Word 𝑉)    &   (𝜑 → ((#‘𝑦) = (#‘𝑧) ∧ (#‘𝑥) = (#‘𝑦)))       (𝜑 → (((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) ∧ (𝑘 ∈ (0...(#‘𝑧)) ∧ 𝑦 = (𝑧 cyclShift 𝑘))) → ∃𝑛 ∈ (0...(#‘𝑧))𝑥 = (𝑧 cyclShift 𝑛)))
 
Theoremcshimadifsn 13288 The image of a cyclically shifted word under its domain without its left bound is the image of a cyclically shifted word under its domain without the number of shifted symbols. (Contributed by AV, 19-Mar-2021.)
((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))
 
Theoremcshimadifsn0 13289 The image of a cyclically shifted word under its domain without its upper bound is the image of a cyclically shifted word under its domain without the number of shifted symbols. (Contributed by AV, 19-Mar-2021.)
((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))
 
5.7.14  Mapping words by a function
 
Theoremwrdco 13290 Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝐹𝑊) ∈ Word 𝐵)
 
Theoremlenco 13291 Length of a mapped word is unchanged. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (#‘(𝐹𝑊)) = (#‘𝑊))
 
Theorems1co 13292 Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = ⟨“(𝐹𝑆)”⟩)
 
Theoremrevco 13293 Mapping of words commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝐹 ∘ (reverse‘𝑊)) = (reverse‘(𝐹𝑊)))
 
Theoremccatco 13294 Mapping of words commutes with concatenation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑆 ∈ Word 𝐴𝑇 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 ++ 𝑇)) = ((𝐹𝑆) ++ (𝐹𝑇)))
 
Theoremcshco 13295 Mapping of words commutes with the "cyclical shift" operation. (Contributed by AV, 12-Nov-2018.)
((𝑊 ∈ Word 𝐴𝑁 ∈ ℤ ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 cyclShift 𝑁)) = ((𝐹𝑊) cyclShift 𝑁))
 
Theoremswrdco 13296 Mapping of words commutes with the substring operation. (Contributed by AV, 11-Nov-2018.)
((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩)) = ((𝐹𝑊) substr ⟨𝑀, 𝑁⟩))
 
Theoremlswco 13297 Mapping of (nonempty) words commutes with the "last symbol" operation. This theorem would not hold if 𝑊 = ∅, (𝐹‘∅) ≠ ∅ and ∅ ∈ 𝐴, because then ( lastS ‘(𝐹𝑊)) = ( lastS ‘∅) = ∅ ≠ (𝐹‘∅) = (𝐹( lastS ‘𝑊)). (Contributed by AV, 11-Nov-2018.)
((𝑊 ∈ Word 𝐴𝑊 ≠ ∅ ∧ 𝐹:𝐴𝐵) → ( lastS ‘(𝐹𝑊)) = (𝐹‘( lastS ‘𝑊)))
 
Theoremrepsco 13298 Mapping of words commutes with the "repeated symbol" operation. (Contributed by AV, 11-Nov-2018.)
((𝑆𝐴𝑁 ∈ ℕ0𝐹:𝐴𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹𝑆) repeatS 𝑁))
 
5.7.15  Longer string literals
 
Syntaxcs2 13299 Syntax for the length 2 word constructor.
class ⟨“𝐴𝐵”⟩
 
Syntaxcs3 13300 Syntax for the length 3 word constructor.
class ⟨“𝐴𝐵𝐶”⟩
    < Previous  Next >

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 42401-42420
  Copyright terms: Public domain < Previous  Next >