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	revccat 14701	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 14702	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 14703	Extend class notation with words consisting of one repeated symbol.
class repeatS
Definition	df-reps 14704*	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 14705*	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 14706	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 14707	Construct a function mapping a half-open range of nonnegative integers to a constant, see also fconstmpt 5694. (Contributed by AV, 4-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑆 repeatS 𝑁) = ((0..^𝑁) × {𝑆}))
Theorem	repsf 14708	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 14709	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 14710	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 14711	The length of a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)
⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (♯‘(𝑆 repeatS 𝑁)) = 𝑁)
Theorem	repsw0 14712	The "repeated symbol word" of length 0. (Contributed by AV, 4-Nov-2018.)
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 0) = ∅)
Theorem	repsdf2 14713*	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 14714*	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 14715*	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 14716	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 14717	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 14718	The "repeated symbol word" of length 1. (Contributed by AV, 4-Nov-2018.)
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 1) = ⟨“𝑆”⟩)
Theorem	repswswrd 14719	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 14720	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 14721	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 14722	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 14724. The main theorems in this section are about counting the number of different necklaces resulting from cyclically shifting a given word, see cshwrepswhash1 17042 for words consisting of identical symbols and cshwshash 17044 for words having lengths which are prime numbers.
Syntax	ccsh 14723	Extend class notation with Cyclical Shifts.
class cyclShift
Definition	df-csh 14724*	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 14725*	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 14726	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 14727	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 14728	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 14729	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 14730	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 14731	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 14732	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 14733	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 14734	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 14735	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 14736	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 14737	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 14738	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 14739	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 14740	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 14741	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 14742	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 14743	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 14744	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 14745	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 14746	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 14747	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 14748	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 14749	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 14750	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 14751	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 14752	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 14753	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 14754	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 14755	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 14756*	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 14757*	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 14758	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 14759*	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	2cshwcshw 14760*	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 14761*	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 14762*	A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 30108 and erclwwlknsym 30157. (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)))
Theorem	cshwcsh2id 14763*	A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 30109 and erclwwlkntr 30158. (Contributed by AV, 9-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
⊢ (𝜑 → 𝑧 ∈ 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)))
Theorem	cshimadifsn 14764	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 𝑆 ∧ 𝑁 = (♯‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))
Theorem	cshimadifsn0 14765	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 𝑆 ∧ 𝑁 = (♯‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))
5.7.15  Mapping words by a function
Theorem	wrdco 14766	Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑊) ∈ Word 𝐵)
Theorem	lenco 14767	Length of a mapped word is unchanged. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑊)) = (♯‘𝑊))
Theorem	s1co 14768	Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = ⟨“(𝐹‘𝑆)”⟩)
Theorem	revco 14769	Mapping of words (i.e., a letterwise mapping) commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (reverse‘𝑊)) = (reverse‘(𝐹 ∘ 𝑊)))
Theorem	ccatco 14770	Mapping of words commutes with concatenation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 ++ 𝑇)) = ((𝐹 ∘ 𝑆) ++ (𝐹 ∘ 𝑇)))
Theorem	cshco 14771	Mapping of words commutes with the "cyclical shift" operation. (Contributed by AV, 12-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑁 ∈ ℤ ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑊 cyclShift 𝑁)) = ((𝐹 ∘ 𝑊) cyclShift 𝑁))
Theorem	swrdco 14772	Mapping of words commutes with the substring operation. (Contributed by AV, 11-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩)) = ((𝐹 ∘ 𝑊) substr ⟨𝑀, 𝑁⟩))
Theorem	pfxco 14773	Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑁 ∈ (0...(♯‘𝑊)) ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑊 prefix 𝑁)) = ((𝐹 ∘ 𝑊) prefix 𝑁))
Theorem	lswco 14774	Mapping of (nonempty) words commutes with the "last symbol" operation. This theorem would not hold if 𝑊 = ∅, (𝐹‘∅) ≠ ∅ and ∅ ∈ 𝐴, because then (lastS‘(𝐹 ∘ 𝑊)) = (lastS‘∅) = ∅ ≠ (𝐹‘∅) = (𝐹(lastS‘𝑊)). (Contributed by AV, 11-Nov-2018.)
⊢ ((𝑊 ∈ Word 𝐴 ∧ 𝑊 ≠ ∅ ∧ 𝐹:𝐴⟶𝐵) → (lastS‘(𝐹 ∘ 𝑊)) = (𝐹‘(lastS‘𝑊)))
Theorem	repsco 14775	Mapping of words commutes with the "repeated symbol" operation. (Contributed by AV, 11-Nov-2018.)
⊢ ((𝑆 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 repeatS 𝑁)) = ((𝐹‘𝑆) repeatS 𝑁))
5.7.16  Longer string literals
Syntax	cs2 14776	Syntax for the length 2 word constructor.
class ⟨“𝐴𝐵”⟩
Syntax	cs3 14777	Syntax for the length 3 word constructor.
class ⟨“𝐴𝐵𝐶”⟩
Syntax	cs4 14778	Syntax for the length 4 word constructor.
class ⟨“𝐴𝐵𝐶𝐷”⟩
Syntax	cs5 14779	Syntax for the length 5 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩
Syntax	cs6 14780	Syntax for the length 6 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩
Syntax	cs7 14781	Syntax for the length 7 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩
Syntax	cs8 14782	Syntax for the length 8 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩
Definition	df-s2 14783	Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
Definition	df-s3 14784	Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
Definition	df-s4 14785	Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
Definition	df-s5 14786	Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
Definition	df-s6 14787	Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
Definition	df-s7 14788	Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
Definition	df-s8 14789	Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
Theorem	cats1cld 14790	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ (𝜑 → 𝑆 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝑇 ∈ Word 𝐴)
Theorem	cats1co 14791	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ (𝜑 → 𝑆 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 ∘ 𝑆) = 𝑈)    &   ⊢ 𝑉 = (𝑈 ++ ⟨“(𝐹‘𝑋)”⟩)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝑇) = 𝑉)
Theorem	cats1cli 14792	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ 𝑆 ∈ Word V    ⇒   ⊢ 𝑇 ∈ Word V
Theorem	cats1fvn 14793	The last symbol of a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ 𝑆 ∈ Word V    &   ⊢ (♯‘𝑆) = 𝑀    ⇒   ⊢ (𝑋 ∈ 𝑉 → (𝑇‘𝑀) = 𝑋)
Theorem	cats1fv 14794	A symbol other than the last in a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ 𝑆 ∈ Word V    &   ⊢ (♯‘𝑆) = 𝑀    &   ⊢ (𝑌 ∈ 𝑉 → (𝑆‘𝑁) = 𝑌)    &   ⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑁 < 𝑀    ⇒   ⊢ (𝑌 ∈ 𝑉 → (𝑇‘𝑁) = 𝑌)
Theorem	cats1len 14795	The length of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ 𝑆 ∈ Word V    &   ⊢ (♯‘𝑆) = 𝑀    &   ⊢ (𝑀 + 1) = 𝑁    ⇒   ⊢ (♯‘𝑇) = 𝑁
Theorem	cats1cat 14796	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ 𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   ⊢ 𝐴 ∈ Word V    &   ⊢ 𝑆 ∈ Word V    &   ⊢ 𝐶 = (𝐵 ++ ⟨“𝑋”⟩)    &   ⊢ 𝐵 = (𝐴 ++ 𝑆)    ⇒   ⊢ 𝐶 = (𝐴 ++ 𝑇)
Theorem	cats2cat 14797	Closure of concatenation of concatenations with singleton words. (Contributed by AV, 1-Mar-2021.)
⊢ 𝐵 ∈ Word V    &   ⊢ 𝐷 ∈ Word V    &   ⊢ 𝐴 = (𝐵 ++ ⟨“𝑋”⟩)    &   ⊢ 𝐶 = (⟨“𝑌”⟩ ++ 𝐷)    ⇒   ⊢ (𝐴 ++ 𝐶) = ((𝐵 ++ ⟨“𝑋𝑌”⟩) ++ 𝐷)
Theorem	s2eqd 14798	Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
Theorem	s3eqd 14799	Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)
Theorem	s4eqd 14800	Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    &   ⊢ (𝜑 → 𝐷 = 𝑄)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ = ⟨“𝑁𝑂𝑃𝑄”⟩)