P. 149 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	repswpfx 14801	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 14802	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 14803	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 14805. The main theorems in this section are about counting the number of different necklaces resulting from cyclically shifting a given word, see cshwrepswhash1 17120 for words consisting of identical symbols and cshwshash 17122 for words having lengths which are prime numbers.
Syntax	ccsh 14804	Extend class notation with Cyclical Shifts.
class cyclShift
Definition	df-csh 14805*	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 14806*	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 14807	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 14808	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 14809	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 14810	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 14811	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 14812	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 14813	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 14814	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 14815	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 14816	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 14817	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 14818	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 14819	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 14820	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 14821	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 14822	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 14823	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 14824	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 14825	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 14826	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 14827	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 14828	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 14829	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 14830	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 14831	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 14832	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 14833	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 14834	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 14835	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 14836	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 14837*	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 14838*	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 14839	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 14840*	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 14841*	Obsolete version of cshwsexa 14840 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 14842*	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 14843*	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 14844*	A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 29948 and erclwwlknsym 29997. (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 14845*	A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 29949 and erclwwlkntr 29998. (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 14846	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 14847	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 14848	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 14849	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 14850	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 14851	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 14852	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 14853	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 14854	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 14855	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 14856	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 14857	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 14858	Syntax for the length 2 word constructor.
class ⟨“𝐴𝐵”⟩
Syntax	cs3 14859	Syntax for the length 3 word constructor.
class ⟨“𝐴𝐵𝐶”⟩
Syntax	cs4 14860	Syntax for the length 4 word constructor.
class ⟨“𝐴𝐵𝐶𝐷”⟩
Syntax	cs5 14861	Syntax for the length 5 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩
Syntax	cs6 14862	Syntax for the length 6 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩
Syntax	cs7 14863	Syntax for the length 7 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩
Syntax	cs8 14864	Syntax for the length 8 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩
Definition	df-s2 14865	Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)
Definition	df-s3 14866	Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)
Definition	df-s4 14867	Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)
Definition	df-s5 14868	Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)
Definition	df-s6 14869	Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)
Definition	df-s7 14870	Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
Definition	df-s8 14871	Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
Theorem	cats1cld 14872	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 14873	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 14874	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 14875	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 14876	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 14877	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 14878	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 14879	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 14880	Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)
Theorem	s3eqd 14881	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 14882	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”⟩)
Theorem	s5eqd 14883	Equality theorem for a length 5 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    &   ⊢ (𝜑 → 𝐷 = 𝑄)    &   ⊢ (𝜑 → 𝐸 = 𝑅)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅”⟩)
Theorem	s6eqd 14884	Equality theorem for a length 6 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    &   ⊢ (𝜑 → 𝐷 = 𝑄)    &   ⊢ (𝜑 → 𝐸 = 𝑅)    &   ⊢ (𝜑 → 𝐹 = 𝑆)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩)
Theorem	s7eqd 14885	Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    &   ⊢ (𝜑 → 𝐷 = 𝑄)    &   ⊢ (𝜑 → 𝐸 = 𝑅)    &   ⊢ (𝜑 → 𝐹 = 𝑆)    &   ⊢ (𝜑 → 𝐺 = 𝑇)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)
Theorem	s8eqd 14886	Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 = 𝑁)    &   ⊢ (𝜑 → 𝐵 = 𝑂)    &   ⊢ (𝜑 → 𝐶 = 𝑃)    &   ⊢ (𝜑 → 𝐷 = 𝑄)    &   ⊢ (𝜑 → 𝐸 = 𝑅)    &   ⊢ (𝜑 → 𝐹 = 𝑆)    &   ⊢ (𝜑 → 𝐺 = 𝑇)    &   ⊢ (𝜑 → 𝐻 = 𝑈)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩)
Theorem	s3eq2 14887	Equality theorem for a length 3 word for the second symbol. (Contributed by AV, 4-Jan-2022.)
⊢ (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)
Theorem	s2cld 14888	A doubleton word is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word 𝑋)
Theorem	s3cld 14889	A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)
Theorem	s4cld 14890	A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑋)
Theorem	s5cld 14891	A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word 𝑋)
Theorem	s6cld 14892	A length 6 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ∈ Word 𝑋)
Theorem	s7cld 14893	A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ∈ Word 𝑋)
Theorem	s8cld 14894	A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ ∈ Word 𝑋)
Theorem	s2cl 14895	A doubleton word is a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨“𝐴𝐵”⟩ ∈ Word 𝑋)
Theorem	s3cl 14896	A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)
Theorem	s2cli 14897	A doubleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵”⟩ ∈ Word V
Theorem	s3cli 14898	A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶”⟩ ∈ Word V
Theorem	s4cli 14899	A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word V
Theorem	s5cli 14900	A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⊢ ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word V