P. 148 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14701-14800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	swrdccatin2 14701	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 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩)))
Theorem	pfxccatin12lem2c 14702	Lemma for pfxccatin12lem2 14703 and pfxccatin12lem3 14704. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
⊢ 𝐿 = (♯‘𝐴)    ⇒   ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
Theorem	pfxccatin12lem2 14703	Lemma 2 for pfxccatin12 14705. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 9-May-2020.)
⊢ 𝐿 = (♯‘𝐴)    ⇒   ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐾 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿 − 𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝐾) = ((𝐵 prefix (𝑁 − 𝐿))‘(𝐾 − (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))))))
Theorem	pfxccatin12lem3 14704	Lemma 3 for pfxccatin12 14705. (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.)
⊢ 𝐿 = (♯‘𝐴)    ⇒   ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐾 ∈ (0..^(𝑁 − 𝑀)) ∧ 𝐾 ∈ (0..^(𝐿 − 𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝐾) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝐾)))
Theorem	pfxccatin12 14705	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 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁 − 𝐿)))))
Theorem	pfxccat3 14706	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 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁 ≤ 𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁 − 𝐿)))))))
Theorem	swrdccat 14707	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 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, if(𝑁 ≤ 𝐿, 𝑁, 𝐿)⟩) ++ (𝐵 substr ⟨if(0 ≤ (𝑀 − 𝐿), (𝑀 − 𝐿), 0), (𝑁 − 𝐿)⟩))))
Theorem	pfxccatpfx1 14708	A prefix of a concatenation being a prefix of the first concatenated word. (Contributed by AV, 10-May-2020.)
⊢ 𝐿 = (♯‘𝐴)    ⇒   ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...𝐿)) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))
Theorem	pfxccatpfx2 14709	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 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))
Theorem	pfxccat3a 14710	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 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁 ≤ 𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁 − 𝐿))))))
Theorem	swrdccat3blem 14711	Lemma for swrdccat3b 14712. (Contributed by AV, 30-May-2018.)
⊢ 𝐿 = (♯‘𝐴)    ⇒   ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...(𝐿 + (♯‘𝐵)))) ∧ (𝐿 + (♯‘𝐵)) ≤ 𝐿) → if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵)) = (𝐴 substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩))
Theorem	swrdccat3b 14712	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 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑀 ∈ (0...(𝐿 + (♯‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, (𝐿 + (♯‘𝐵))⟩) = if(𝐿 ≤ 𝑀, (𝐵 substr ⟨(𝑀 − 𝐿), (♯‘𝐵)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ 𝐵))))
Theorem	pfxccatid 14713	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 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → ((𝐴 ++ 𝐵) prefix 𝑁) = 𝐴)
Theorem	ccats1pfxeqbi 14714	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 𝑉 ∧ 𝑈 ∈ Word 𝑉 ∧ (♯‘𝑈) = ((♯‘𝑊) + 1)) → (𝑊 = (𝑈 prefix (♯‘𝑊)) ↔ 𝑈 = (𝑊 ++ ⟨“(lastS‘𝑈)”⟩)))
Theorem	swrdccatin1d 14715	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 𝑉 ∧ 𝐵 ∈ Word 𝑉))    &   ⊢ (𝜑 → 𝑀 ∈ (0...𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ (0...𝐿))    ⇒   ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
Theorem	swrdccatin2d 14716	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 𝑉 ∧ 𝐵 ∈ Word 𝑉))    &   ⊢ (𝜑 → 𝑀 ∈ (𝐿...𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))    ⇒   ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀 − 𝐿), (𝑁 − 𝐿)⟩))
Theorem	pfxccatin12d 14717	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 𝑉 ∧ 𝐵 ∈ Word 𝑉))    &   ⊢ (𝜑 → 𝑀 ∈ (0...𝐿))    &   ⊢ (𝜑 → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))    ⇒   ⊢ (𝜑 → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁 − 𝐿))))
Theorem	reuccatpfxs1lem 14718*	Lemma for reuccatpfxs1 14719. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised by AV, 9-May-2020.)
⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ 𝑋) ∧ ∀𝑠 ∈ 𝑉 ((𝑊 ++ ⟨“𝑠”⟩) ∈ 𝑋 → 𝑆 = 𝑠) ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (𝑊 = (𝑈 prefix (♯‘𝑊)) → 𝑈 = (𝑊 ++ ⟨“𝑆”⟩)))
Theorem	reuccatpfxs1 14719*	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.)
⊢ Ⅎ𝑣𝑋    ⇒   ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))
Theorem	reuccatpfxs1v 14720*	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 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))
5.7.11  Splicing words (substring replacement)
Syntax	csplice 14721	Syntax for the word splicing operator.
class splice
Definition	df-splice 14722*	Define an operation which replaces portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by AV, 14-Oct-2022.)
⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st ‘𝑏))) ++ (2nd ‘𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st ‘𝑏)), (♯‘𝑠)⟩)))
Theorem	splval 14723	Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by AV, 11-May-2020.) (Revised by AV, 15-Oct-2022.)
⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ 𝑊 ∧ 𝑇 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))
Theorem	splcl 14724	Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
Theorem	splid 14725	Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.) (Proof shortened by AV, 14-Oct-2022.)
⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆)))) → (𝑆 splice ⟨𝑋, 𝑌, (𝑆 substr ⟨𝑋, 𝑌⟩)⟩) = 𝑆)
Theorem	spllen 14726	The length of a splice. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
⊢ (𝜑 → 𝑆 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ (0...𝑇))    &   ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆)))    &   ⊢ (𝜑 → 𝑅 ∈ Word 𝐴)    ⇒   ⊢ (𝜑 → (♯‘(𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)) = ((♯‘𝑆) + ((♯‘𝑅) − (𝑇 − 𝐹))))
Theorem	splfv1 14727	Symbols to the left of a splice are unaffected. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
⊢ (𝜑 → 𝑆 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ (0...𝑇))    &   ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆)))    &   ⊢ (𝜑 → 𝑅 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ (0..^𝐹))    ⇒   ⊢ (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘𝑋) = (𝑆‘𝑋))
Theorem	splfv2a 14728	Symbols within the replacement region of a splice, expressed using the coordinates of the replacement region. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 15-Oct-2022.)
⊢ (𝜑 → 𝑆 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ (0...𝑇))    &   ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆)))    &   ⊢ (𝜑 → 𝑅 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ (0..^(♯‘𝑅)))    ⇒   ⊢ (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘(𝐹 + 𝑋)) = (𝑅‘𝑋))
Theorem	splval2 14729	Value of a splice, assuming the input word 𝑆 has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.)
⊢ (𝜑 → 𝐴 ∈ Word 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ Word 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ Word 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ Word 𝑋)    &   ⊢ (𝜑 → 𝑆 = ((𝐴 ++ 𝐵) ++ 𝐶))    &   ⊢ (𝜑 → 𝐹 = (♯‘𝐴))    &   ⊢ (𝜑 → 𝑇 = (𝐹 + (♯‘𝐵)))    ⇒   ⊢ (𝜑 → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = ((𝐴 ++ 𝑅) ++ 𝐶))
5.7.12  Reversing words
Syntax	creverse 14730	Syntax for the word reverse operator.
class reverse
Definition	df-reverse 14731*	Define an operation which reverses the order of symbols in a word. This operation is also known as "word reversal" and "word mirroring". (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(♯‘𝑠)) ↦ (𝑠‘(((♯‘𝑠) − 1) − 𝑥))))
Theorem	revval 14732*	Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ (𝑊 ∈ 𝑉 → (reverse‘𝑊) = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ (𝑊‘(((♯‘𝑊) − 1) − 𝑥))))
Theorem	revcl 14733	The reverse of a word is a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ (𝑊 ∈ Word 𝐴 → (reverse‘𝑊) ∈ Word 𝐴)
Theorem	revlen 14734	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))
Theorem	revfv 14735	Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑋 ∈ (0..^(♯‘𝑊))) → ((reverse‘𝑊)‘𝑋) = (𝑊‘(((♯‘𝑊) − 1) − 𝑋)))
Theorem	rev0 14736	The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ (reverse‘∅) = ∅
Theorem	revs1 14737	Singleton words are their own reverses. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⊢ (reverse‘⟨“𝑆”⟩) = ⟨“𝑆”⟩
Theorem	revccat 14738	Antiautomorphic property of the reversal operation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) = ((reverse‘𝑇) ++ (reverse‘𝑆)))
Theorem	revrev 14739	Reversal is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ (𝑊 ∈ Word 𝐴 → (reverse‘(reverse‘𝑊)) = 𝑊)
5.7.13  Repeated symbol words
Syntax	creps 14740	Extend class notation with words consisting of one repeated symbol.
class repeatS
Definition	df-reps 14741*	Definition to construct a word consisting of one repeated symbol, often called "repeated symbol word" for short in the following. (Contributed by Alexander van der Vekens, 4-Nov-2018.)
⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))
Theorem	reps 14742*	Construct a function mapping a half-open range of nonnegative integers to a constant. (Contributed by AV, 4-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = (𝑥 ∈ (0..^𝑁) ↦ 𝑆))
Theorem	repsundef 14743	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) = ∅)
Theorem	repsconst 14744	Construct a function mapping a half-open range of nonnegative integers to a constant, see also fconstmpt 5703. (Contributed by AV, 4-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = ((0..^𝑁) × {𝑆}))
Theorem	repsf 14745	The constructed function mapping a half-open range of nonnegative integers to a constant is a function. (Contributed by AV, 4-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁):(0..^𝑁)⟶𝑉)
Theorem	repswsymb 14746	The symbols of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ (0..^𝑁)) → ((𝑆 repeatS 𝑁)‘𝐼) = 𝑆)
Theorem	repsw 14747	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.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) ∈ Word 𝑉)
Theorem	repswlen 14748	The length of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (♯‘(𝑆 repeatS 𝑁)) = 𝑁)
Theorem	repsw0 14749	The "repeated symbol word" of length 0. (Contributed by AV, 4-Nov-2018.)
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 0) = ∅)
Theorem	repsdf2 14750*	Alternative definition of a "repeated symbol word". (Contributed by AV, 7-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑊 = (𝑆 repeatS 𝑁) ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) = 𝑆)))
Theorem	repswsymball 14751*	All the symbols of a "repeated symbol word" are the same. (Contributed by AV, 10-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉) → (𝑊 = (𝑆 repeatS (♯‘𝑊)) → ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = 𝑆))
Theorem	repswsymballbi 14752*	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))(𝑊‘𝑖) = (𝑊‘0)))
Theorem	repswfsts 14753	The first symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑆 repeatS 𝑁)‘0) = 𝑆)
Theorem	repswlsw 14754	The last symbol of a nonempty "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (lastS‘(𝑆 repeatS 𝑁)) = 𝑆)
Theorem	repsw1 14755	The "repeated symbol word" of length 1. (Contributed by AV, 4-Nov-2018.)
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 1) = ⟨“𝑆”⟩)
Theorem	repswswrd 14756	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 (𝑁 − 𝑀))) ≠ ∅! 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.)
⊢ (((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ ℕ0) ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 𝑁 ≤ 𝐿) → ((𝑆 repeatS 𝐿) substr ⟨𝑀, 𝑁⟩) = (𝑆 repeatS (𝑁 − 𝑀)))
Theorem	repswpfx 14757	A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ (0...𝑁)) → ((𝑆 repeatS 𝑁) prefix 𝐿) = (𝑆 repeatS 𝐿))
Theorem	repswccat 14758	The concatenation of two "repeated symbol words" with the same symbol is again a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → ((𝑆 repeatS 𝑁) ++ (𝑆 repeatS 𝑀)) = (𝑆 repeatS (𝑁 + 𝑀)))
Theorem	repswrevw 14759	The reverse of a "repeated symbol word". (Contributed by AV, 6-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (reverse‘(𝑆 repeatS 𝑁)) = (𝑆 repeatS 𝑁))
5.7.14  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 14761. The main theorems in this section are about counting the number of different necklaces resulting from cyclically shifting a given word, see cshwrepswhash1 17080 for words consisting of identical symbols and cshwshash 17082 for words having lengths which are prime numbers.
Syntax	ccsh 14760	Extend class notation with Cyclical Shifts.
class cyclShift
Definition	df-csh 14761*	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.) (Revised by AV, 4-Nov-2022.)
⊢ cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr ⟨(𝑛 mod (♯‘𝑤)), (♯‘𝑤)⟩) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))))
Theorem	cshfn 14762*	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.) (Revised by AV, 4-Nov-2022.)
⊢ ((𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = if(𝑊 = ∅, ∅, ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊))))))
Theorem	cshword 14763	Perform a cyclical shift for a word. (Contributed by Alexander van der Vekens, 20-May-2018.) (Revised by AV, 12-Oct-2022.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (♯‘𝑊)), (♯‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (♯‘𝑊)))))
Theorem	cshnz 14764	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) = ∅)
Theorem	0csh0 14765	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) = ∅
Theorem	cshw0 14766	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)
Theorem	cshwmodn 14767	Cyclically shifting a word is invariant regarding modulo the word's length. (Contributed by AV, 26-Oct-2018.) (Proof shortened by AV, 16-Oct-2022.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 mod (♯‘𝑊))))
Theorem	cshwsublen 14768	Cyclically shifting a word is invariant regarding subtraction of the word's length. (Contributed by AV, 3-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (𝑊 cyclShift (𝑁 − (♯‘𝑊))))
Theorem	cshwn 14769	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)
Theorem	cshwcl 14770	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)
Theorem	cshwlen 14771	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.) (Proof shortened by AV, 16-Oct-2022.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → (♯‘(𝑊 cyclShift 𝑁)) = (♯‘𝑊))
Theorem	cshwf 14772	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 𝐴 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁):(0..^(♯‘𝑊))⟶𝐴)
Theorem	cshwfn 14773	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 𝑉 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) Fn (0..^(♯‘𝑊)))
Theorem	cshwrn 14774	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 𝑉 ∧ 𝑁 ∈ ℤ) → ran (𝑊 cyclShift 𝑁) ⊆ 𝑉)
Theorem	cshwidxmod 14775	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.) (Proof shortened by AV, 12-Oct-2022.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘𝐼) = (𝑊‘((𝐼 + 𝑁) mod (♯‘𝑊))))
Theorem	cshwidxmodr 14776	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 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((𝐼 − 𝑁) mod (♯‘𝑊))) = (𝑊‘𝐼))
Theorem	cshwidx0mod 14777	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 𝑉 ∧ 𝑊 ≠ ∅ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑁)‘0) = (𝑊‘(𝑁 mod (♯‘𝑊))))
Theorem	cshwidx0 14778	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 𝑉 ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘0) = (𝑊‘𝑁))
Theorem	cshwidxm1 14779	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-Mar-2018.) (Revised by AV, 21-May-2018.) (Revised by AV, 30-Oct-2018.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘(((♯‘𝑊) − 𝑁) − 1)) = (𝑊‘((♯‘𝑊) − 1)))
Theorem	cshwidxm 14780	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 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((♯‘𝑊) − 𝑁)) = (𝑊‘0))
Theorem	cshwidxn 14781	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 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → ((𝑊 cyclShift 𝑁)‘((♯‘𝑊) − 1)) = (𝑊‘(𝑁 − 1)))
Theorem	cshf1 14782	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))–1-1→𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(♯‘𝐹))–1-1→𝐴)
Theorem	cshinj 14783	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 ◡𝐹 ∧ 𝑆 ∈ ℤ) → (𝐺 = (𝐹 cyclShift 𝑆) → Fun ◡𝐺))
Theorem	repswcshw 14784	A cyclically shifted "repeated symbol word". (Contributed by Alexander van der Vekens, 7-Nov-2018.) (Proof shortened by AV, 16-Oct-2022.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ∧ 𝐼 ∈ ℤ) → ((𝑆 repeatS 𝑁) cyclShift 𝐼) = (𝑆 repeatS 𝑁))
Theorem	2cshw 14785	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 𝑉 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑀) cyclShift 𝑁) = (𝑊 cyclShift (𝑀 + 𝑁)))
Theorem	2cshwid 14786	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 𝑉 ∧ 𝑁 ∈ ℤ) → ((𝑊 cyclShift 𝑁) cyclShift ((♯‘𝑊) − 𝑁)) = 𝑊)
Theorem	lswcshw 14787	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 𝑉 ∧ 𝑁 ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 cyclShift 𝑁)) = (𝑊‘(𝑁 − 1)))
Theorem	2cshwcom 14788	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 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑊 cyclShift 𝑁) cyclShift 𝑀) = ((𝑊 cyclShift 𝑀) cyclShift 𝑁))
Theorem	cshwleneq 14789	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 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀)) → (♯‘𝑊) = (♯‘𝑈))
Theorem	3cshw 14790	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 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑊 cyclShift 𝑁) = (((𝑊 cyclShift 𝑀) cyclShift 𝑁) cyclShift ((♯‘𝑊) − 𝑀)))
Theorem	cshweqdif2 14791	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 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑊 cyclShift 𝑁) = (𝑈 cyclShift 𝑀) → (𝑈 cyclShift (𝑀 − 𝑁)) = 𝑊))
Theorem	cshweqdifid 14792	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 𝑉 ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑊 cyclShift 𝑁) = (𝑊 cyclShift 𝑀) → (𝑊 cyclShift (𝑀 − 𝑁)) = 𝑊))
Theorem	cshweqrep 14793*	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 𝑉 ∧ 𝐿 ∈ ℤ) → (((𝑊 cyclShift 𝐿) = 𝑊 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ∀𝑗 ∈ ℕ0 (𝑊‘𝐼) = (𝑊‘((𝐼 + (𝑗 · 𝐿)) mod (♯‘𝑊)))))
Theorem	cshw1 14794*	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))(𝑊‘𝑖) = (𝑊‘0))
Theorem	cshw1repsw 14795	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)))
Theorem	cshwsexa 14796*	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.) (Proof shortened by SN, 15-Jan-2025.)
⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
Theorem	cshwsexaOLD 14797*	Obsolete version of cshwsexa 14796 as of 15-Jan-2025. (Contributed by AV, 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
Theorem	2cshwcshw 14798*	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)))
Theorem	scshwfzeqfzo 14799*	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 𝑉 ∧ 𝑋 ≠ ∅ ∧ 𝑁 = (♯‘𝑋)) → {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑋 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^𝑁)𝑦 = (𝑋 cyclShift 𝑛)})
Theorem	cshwcshid 14800*	A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 29957 and erclwwlknsym 30006. (Contributed by AV, 8-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
⊢ (𝜑 → 𝑦 ∈ Word 𝑉)    &   ⊢ (𝜑 → (♯‘𝑥) = (♯‘𝑦))    ⇒   ⊢ (𝜑 → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))