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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcshwidxmodr 13401 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 13402 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 13403 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 13404 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 13405 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 13406 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 13407 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 13408 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 13409 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 13410 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 13411 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 13412 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 13413 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 13414 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 13415 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 13416 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 13417 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 13418* 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 13419* 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 13420 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 13421* 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 13422* 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 13423* 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 13424* A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlktr 26343 and erclwwlkntr 26355. (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 13425* A cyclically shifted word can be reconstructed by cyclically shifting it again twice. Lemma for erclwwlktr 26343 and erclwwlkntr 26355. (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 13426 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 13427 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 13428 Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝐹𝑊) ∈ Word 𝐵)

Theoremlenco 13429 Length of a mapped word is unchanged. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (#‘(𝐹𝑊)) = (#‘𝑊))

Theorems1co 13430 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 13431 Mapping of words commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝐹 ∘ (reverse‘𝑊)) = (reverse‘(𝐹𝑊)))

Theoremccatco 13432 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 13433 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 13434 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 13435 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 13436 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 13437 Syntax for the length 2 word constructor.
class ⟨“𝐴𝐵”⟩

Syntaxcs3 13438 Syntax for the length 3 word constructor.
class ⟨“𝐴𝐵𝐶”⟩

Syntaxcs4 13439 Syntax for the length 4 word constructor.
class ⟨“𝐴𝐵𝐶𝐷”⟩

Syntaxcs5 13440 Syntax for the length 5 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸”⟩

Syntaxcs6 13441 Syntax for the length 6 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩

Syntaxcs7 13442 Syntax for the length 7 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩

Syntaxcs8 13443 Syntax for the length 8 word constructor.
class ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩

Definitiondf-s2 13444 Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵”⟩ = (⟨“𝐴”⟩ ++ ⟨“𝐵”⟩)

Definitiondf-s3 13445 Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ = (⟨“𝐴𝐵”⟩ ++ ⟨“𝐶”⟩)

Definitiondf-s4 13446 Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ = (⟨“𝐴𝐵𝐶”⟩ ++ ⟨“𝐷”⟩)

Definitiondf-s5 13447 Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = (⟨“𝐴𝐵𝐶𝐷”⟩ ++ ⟨“𝐸”⟩)

Definitiondf-s6 13448 Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ++ ⟨“𝐹”⟩)

Definitiondf-s7 13449 Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)

Definitiondf-s8 13450 Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)

Theoremcats1cld 13451 Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝑋𝐴)       (𝜑𝑇 ∈ Word 𝐴)

Theoremcats1co 13452 Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   (𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝑋𝐴)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹𝑆) = 𝑈)    &   𝑉 = (𝑈 ++ ⟨“(𝐹𝑋)”⟩)       (𝜑 → (𝐹𝑇) = 𝑉)

Theoremcats1cli 13453 Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
𝑇 = (𝑆 ++ ⟨“𝑋”⟩)    &   𝑆 ∈ Word V       𝑇 ∈ Word V

Theoremcats1fvn 13454 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       (𝑋𝑉 → (𝑇𝑀) = 𝑋)

Theoremcats1fv 13455 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    &   (𝑌𝑉 → (𝑆𝑁) = 𝑌)    &   𝑁 ∈ ℕ0    &   𝑁 < 𝑀       (𝑌𝑉 → (𝑇𝑁) = 𝑌)

Theoremcats1len 13456 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

Theoremcats1cat 13457 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)

Theoremcats2cat 13458 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)

Theorems2eqd 13459 Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)       (𝜑 → ⟨“𝐴𝐵”⟩ = ⟨“𝑁𝑂”⟩)

Theorems3eqd 13460 Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝑁𝑂𝑃”⟩)

Theorems4eqd 13461 Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)    &   (𝜑𝐷 = 𝑄)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ = ⟨“𝑁𝑂𝑃𝑄”⟩)

Theorems5eqd 13462 Equality theorem for a length 5 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)    &   (𝜑𝐷 = 𝑄)    &   (𝜑𝐸 = 𝑅)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅”⟩)

Theorems6eqd 13463 Equality theorem for a length 6 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)    &   (𝜑𝐷 = 𝑄)    &   (𝜑𝐸 = 𝑅)    &   (𝜑𝐹 = 𝑆)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩)

Theorems7eqd 13464 Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)    &   (𝜑𝐷 = 𝑄)    &   (𝜑𝐸 = 𝑅)    &   (𝜑𝐹 = 𝑆)    &   (𝜑𝐺 = 𝑇)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)

Theorems8eqd 13465 Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴 = 𝑁)    &   (𝜑𝐵 = 𝑂)    &   (𝜑𝐶 = 𝑃)    &   (𝜑𝐷 = 𝑄)    &   (𝜑𝐸 = 𝑅)    &   (𝜑𝐹 = 𝑆)    &   (𝜑𝐺 = 𝑇)    &   (𝜑𝐻 = 𝑈)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩)

Theorems2cld 13466 A doubleton word is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)       (𝜑 → ⟨“𝐴𝐵”⟩ ∈ Word 𝑋)

Theorems3cld 13467 A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)

Theorems4cld 13468 A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word 𝑋)

Theorems5cld 13469 A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑𝐸𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word 𝑋)

Theorems6cld 13470 A length 6 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑𝐸𝑋)    &   (𝜑𝐹𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ∈ Word 𝑋)

Theorems7cld 13471 A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑𝐸𝑋)    &   (𝜑𝐹𝑋)    &   (𝜑𝐺𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ∈ Word 𝑋)

Theorems8cld 13472 A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑋)    &   (𝜑𝐸𝑋)    &   (𝜑𝐹𝑋)    &   (𝜑𝐺𝑋)    &   (𝜑𝐻𝑋)       (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ ∈ Word 𝑋)

Theorems2cl 13473 A doubleton word is a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝐴𝑋𝐵𝑋) → ⟨“𝐴𝐵”⟩ ∈ Word 𝑋)

Theorems3cl 13474 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
((𝐴𝑋𝐵𝑋𝐶𝑋) → ⟨“𝐴𝐵𝐶”⟩ ∈ Word 𝑋)

Theorems2cli 13475 A doubleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵”⟩ ∈ Word V

Theorems3cli 13476 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶”⟩ ∈ Word V

Theorems4cli 13477 A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷”⟩ ∈ Word V

Theorems5cli 13478 A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸”⟩ ∈ Word V

Theorems6cli 13479 A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ∈ Word V

Theorems7cli 13480 A length 7 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ∈ Word V

Theorems8cli 13481 A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ ∈ Word V

Theorems2fv0 13482 Extract the first symbol from a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴𝑉 → (⟨“𝐴𝐵”⟩‘0) = 𝐴)

Theorems2fv1 13483 Extract the second symbol from a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐵𝑉 → (⟨“𝐴𝐵”⟩‘1) = 𝐵)

Theorems2len 13484 The length of a doubleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵”⟩) = 2

Theorems2dm 13485 The domain of a doubleton word is an unordered pair. (Contributed by AV, 9-Jan-2020.)
dom ⟨“𝐴𝐵”⟩ = {0, 1}

Theorems3fv0 13486 Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐴𝑉 → (⟨“𝐴𝐵𝐶”⟩‘0) = 𝐴)

Theorems3fv1 13487 Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐵𝑉 → (⟨“𝐴𝐵𝐶”⟩‘1) = 𝐵)

Theorems3fv2 13488 Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
(𝐶𝑉 → (⟨“𝐴𝐵𝐶”⟩‘2) = 𝐶)

Theorems3len 13489 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶”⟩) = 3

Theorems4fv0 13490 Extract the first symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐴𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴)

Theorems4fv1 13491 Extract the second symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐵𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵)

Theorems4fv2 13492 Extract the third symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐶𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶)

Theorems4fv3 13493 Extract the fourth symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020.)
(𝐷𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷)

Theorems4len 13494 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶𝐷”⟩) = 4

Theorems5len 13495 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶𝐷𝐸”⟩) = 5

Theorems6len 13496 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩) = 6

Theorems7len 13497 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩) = 7

Theorems8len 13498 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩) = 8

Theoremlsws2 13499 The last symbol of a doubleton word is its second symbol. (Contributed by AV, 8-Feb-2021.)
(𝐵𝑉 → ( lastS ‘⟨“𝐴𝐵”⟩) = 𝐵)

Theoremlsws3 13500 The last symbol of a 3 letter word is its third symbol. (Contributed by AV, 8-Feb-2021.)
(𝐶𝑉 → ( lastS ‘⟨“𝐴𝐵𝐶”⟩) = 𝐶)

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