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Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqwrd 13201* Two words are equal iff they have the same length and the same symbol at each position. (Contributed by AV, 13-Apr-2018.)
((𝑈 ∈ Word 𝑉𝑊 ∈ Word 𝑉) → (𝑈 = 𝑊 ↔ ((#‘𝑈) = (#‘𝑊) ∧ ∀𝑖 ∈ (0..^(#‘𝑈))(𝑈𝑖) = (𝑊𝑖))))

Theoremelovmpt2wrd 13202* Implications for the value of an operation defined by the maps-to notation with a class abstration of words as a result having an element. Note that 𝜑 may depend on 𝑧 as well as on 𝑣 and 𝑦. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣𝜑})       (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))

Theoremelovmptnn0wrd 13203* Implications for the value of an operation defined by the maps-to notation with a function of nonnegative integers into a class abstraction of words as a result having an element. Note that 𝜑 may depend on 𝑧 as well as on 𝑣 and 𝑦 and 𝑛. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)
𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑧 ∈ Word 𝑣𝜑}))       (𝑍 ∈ ((𝑉𝑂𝑌)‘𝑁) → ((𝑉 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑁 ∈ ℕ0𝑍 ∈ Word 𝑉)))

5.7.2  Last symbol of a word

Theoremlsw 13204 Extract the last symbol of a word. May be not meaningful for other sets which are not words. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
(𝑊𝑋 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))

Theoremlsw0 13205 The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 19-Mar-2018.) (Proof shortened by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 0) → ( lastS ‘𝑊) = ∅)

Theoremlsw0g 13206 The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 11-Nov-2018.)
( lastS ‘∅) = ∅

Theoremlsw1 13207 The last symbol of a word of length 1 is the first symbol of this word. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1) → ( lastS ‘𝑊) = (𝑊‘0))

Theoremlswcl 13208 Closure of the last symbol: the last symbol of a not empty word belongs to the alphabet for the word. (Contributed by AV, 2-Aug-2018.) (Proof shortened by AV, 29-Apr-2020.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → ( lastS ‘𝑊) ∈ 𝑉)

Theoremlswlgt0cl 13209 The last symbol of a nonempty word is element of the alphabet for the word. (Contributed by Alexander van der Vekens, 1-Oct-2018.) (Proof shortened by AV, 29-Apr-2020.)
((𝑁 ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁)) → ( lastS ‘𝑊) ∈ 𝑉)

5.7.3  Concatenations of words

Theoremccatfn 13210 The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.) (Proof shortened by AV, 29-Apr-2020.)
++ Fn (V × V)

Theoremccatfval 13211* Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆𝑉𝑇𝑊) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆𝑥), (𝑇‘(𝑥 − (#‘𝑆))))))

Theoremccatcl 13212 The concatenation of two words is a word. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 29-Apr-2020.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word 𝐵)

Theoremccatlen 13213 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑆) + (#‘𝑇)))

Theoremccatval1 13214 Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 30-Apr-2020.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵𝐼 ∈ (0..^(#‘𝑆))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑆𝐼))

Theoremccatval2 13215 Value of a symbol in the right half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵𝐼 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑇‘(𝐼 − (#‘𝑆))))

Theoremccatval3 13216 Value of a symbol in the right half of a concatenated word, using an index relative to the subword. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Proof shortened by AV, 30-Apr-2020.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵𝐼 ∈ (0..^(#‘𝑇))) → ((𝑆 ++ 𝑇)‘(𝐼 + (#‘𝑆))) = (𝑇𝐼))

Theoremelfzelfzccat 13217 An element of a finite set of sequential integers up to the length of a word is an element of an extended finite set of sequential integers up to the length of a concatenation of this word with another word. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(#‘𝐴)) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))

Theoremccatvalfn 13218 The concatenation of two words is a function over the half-open integer range having the sum of the lengths of the word as length. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) Fn (0..^((#‘𝐴) + (#‘𝐵))))

Theoremccatsymb 13219 The symbol at a given position in a concatenated word. (Contributed by AV, 26-May-2018.) (Proof shortened by AV, 24-Nov-2018.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (#‘𝐴), (𝐴𝐼), (𝐵‘(𝐼 − (#‘𝐴)))))

Theoremccatfv0 13220 The first symbol of a concatenation of two words is the first symbol of the first word if the first word is not empty. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ 0 < (#‘𝐴)) → ((𝐴 ++ 𝐵)‘0) = (𝐴‘0))

Theoremccatval1lsw 13221 The last symbol of the left (nonempty) half of a concatenated word. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Proof shortened by AV, 1-May-2020.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐴 ≠ ∅) → ((𝐴 ++ 𝐵)‘((#‘𝐴) − 1)) = ( lastS ‘𝐴))

Theoremccatlid 13222 Concatenation of a word by the empty word on the left. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 1-May-2020.)
(𝑆 ∈ Word 𝐵 → (∅ ++ 𝑆) = 𝑆)

Theoremccatrid 13223 Concatenation of a word by the empty word on the right. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 1-May-2020.)
(𝑆 ∈ Word 𝐵 → (𝑆 ++ ∅) = 𝑆)

Theoremccatass 13224 Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) = (𝑆 ++ (𝑇 ++ 𝑈)))

Theoremccatrn 13225 The range of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐵𝑇 ∈ Word 𝐵) → ran (𝑆 ++ 𝑇) = (ran 𝑆 ∪ ran 𝑇))

Theoremlswccatn0lsw 13226 The last symbol of a word concatenated with a nonempty word is the last symbol of the nonempty word. (Contributed by AV, 22-Oct-2018.) (Proof shortened by AV, 1-May-2020.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐵 ≠ ∅) → ( lastS ‘(𝐴 ++ 𝐵)) = ( lastS ‘𝐵))

Theoremlswccat0lsw 13227 The last symbol of a word concatenated with the empty word is the last symbol of the word. (Contributed by AV, 22-Oct-2018.) (Proof shortened by AV, 1-May-2020.)
(𝑊 ∈ Word 𝑉 → ( lastS ‘(𝑊 ++ ∅)) = ( lastS ‘𝑊))

Theoremccatalpha 13228 A concatenation of two arbitrary words is a word over an alphabet iff the symbols of both words belong to the alphabet. (Contributed by AV, 28-Feb-2021.)
((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆𝐵 ∈ Word 𝑆)))

Theoremccatrcl1 13229 Reverse closure of a concatenation: If the concatenation of two arbitrary words is a word over an alphabet then the symbols of the first word belong to the alphabet. (Contributed by AV, 3-Mar-2021.)
((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑌 ∧ (𝑊 = (𝐴 ++ 𝐵) ∧ 𝑊 ∈ Word 𝑆)) → 𝐴 ∈ Word 𝑆)

5.7.4  Singleton words

Theoremids1 13230 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩

Theorems1val 13231 Value of a single-symbol word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})

Theorems1rn 13232 The range of a single-symbol word. (Contributed by Mario Carneiro, 18-Jul-2016.)
(𝐴𝑉 → ran ⟨“𝐴”⟩ = {𝐴})

Theorems1eq 13233 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Theorems1eqd 13234 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Theorems1cl 13235 A singleton word is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 23-Nov-2018.)
(𝐴𝐵 → ⟨“𝐴”⟩ ∈ Word 𝐵)

Theorems1cld 13236 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝜑𝐴𝐵)       (𝜑 → ⟨“𝐴”⟩ ∈ Word 𝐵)

Theorems1cli 13237 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴”⟩ ∈ Word V

Theorems1len 13238 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴”⟩) = 1

Theorems1nz 13239 A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.) (Proof shortened by Kyle Wyonch, 18-Jul-2021.)
⟨“𝐴”⟩ ≠ ∅

Theorems1nzOLD 13240 Obsolete proof of s1nz 13239 as of 18-Jul-2021. A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⟨“𝐴”⟩ ≠ ∅

Theorems1dm 13241 The domain of a singleton word is a singleton. (Contributed by AV, 9-Jan-2020.)
dom ⟨“𝐴”⟩ = {0}

Theorems1dmALT 13242 Alternate version of s1dm 13241, having a shorter proof, but requiring that 𝐴 ia a set. (Contributed by AV, 9-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑆 → dom ⟨“𝐴”⟩ = {0})

Theorems1fv 13243 Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴𝐵 → (⟨“𝐴”⟩‘0) = 𝐴)

Theoremlsws1 13244 The last symbol of a singleton word is its symbol. (Contributed by AV, 22-Oct-2018.)
(𝐴𝑉 → ( lastS ‘⟨“𝐴”⟩) = 𝐴)

Theoremeqs1 13245 A word of length 1 is a singleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 1-May-2020.)
((𝑊 ∈ Word 𝐴 ∧ (#‘𝑊) = 1) → 𝑊 = ⟨“(𝑊‘0)”⟩)

Theoremwrdl1exs1 13246* A word of length 1 is a singleton word. (Contributed by AV, 24-Jan-2021.)
((𝑊 ∈ Word 𝑆 ∧ (#‘𝑊) = 1) → ∃𝑠𝑆 𝑊 = ⟨“𝑠”⟩)

Theoremwrdl1s1 13247 A word of length 1 is a singleton word consisting of the first symbol of the word. (Contributed by AV, 22-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.)
(𝑆𝑉 → (𝑊 = ⟨“𝑆”⟩ ↔ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆)))

Theorems111 13248 The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))

5.7.5  Concatenations with singleton words

Theoremccatws1cl 13249 The concatenation of a word with a singleton word is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → (𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉)

Theoremccat2s1cl 13250 The concatenation of two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word 𝑉)

Theoremccatws1len 13251 The length of the concatenation of a word with a singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → (#‘(𝑊 ++ ⟨“𝑋”⟩)) = ((#‘𝑊) + 1))

Theoremwrdlenccats1lenm1 13252 The length of a word is the length of the word concatenated with a singleton word minus 1. (Contributed by AV, 28-Jun-2018.) (Proof shortened by AV, 1-May-2020.)
((𝑊 ∈ Word 𝑉𝑆𝑉) → (#‘𝑊) = ((#‘(𝑊 ++ ⟨“𝑆”⟩)) − 1))

Theoremccat2s1len 13253 The length of the concatenation of two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → (#‘(⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)) = 2)

Theoremccatw2s1len 13254 The length of the concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → (#‘((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)) = ((#‘𝑊) + 2))

Theoremccats1val1 13255 Value of a symbol in the left half of a word concatenated with a single symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
((𝑊 ∈ Word 𝑉𝑆𝑉𝐼 ∈ (0..^(#‘𝑊))) → ((𝑊 ++ ⟨“𝑆”⟩)‘𝐼) = (𝑊𝐼))

Theoremccats1val2 13256 Value of the symbol concatenated with a word. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Proof shortened by Alexander van der Vekens, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑆𝑉𝐼 = (#‘𝑊)) → ((𝑊 ++ ⟨“𝑆”⟩)‘𝐼) = 𝑆)

Theoremccat2s1p1 13257 Extract the first of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘0) = 𝑋)

Theoremccat2s1p2 13258 Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = 𝑌)

Theoremccatw2s1ass 13259 Associative law for a concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)))

Theoremccatws1lenrev 13260 The length of a word concatenated with a singleton word. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → ((#‘(𝑊 ++ ⟨“𝑋”⟩)) = 𝑁 → (#‘𝑊) = (𝑁 − 1)))

Theoremccatws1n0 13261 The concatenation of a word with a singleton word is not the empty set. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → (𝑊 ++ ⟨“𝑋”⟩) ≠ ∅)

Theoremccatws1ls 13262 The last symbol of the concatenation of a word with a singleton word is the symbol of the singleton word. (Contributed by AV, 29-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → ((𝑊 ++ ⟨“𝑋”⟩)‘(#‘𝑊)) = 𝑋)

Theoremlswccats1 13263 The last symbol of a word concatenated with a singleton word is the symbol of the singleton word. (Contributed by AV, 6-Aug-2018.) (Proof shortened by AV, 22-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑆𝑉) → ( lastS ‘(𝑊 ++ ⟨“𝑆”⟩)) = 𝑆)

Theoremlswccats1fst 13264 The last symbol of a nonempty word concatenated with its first symbol is the first symbol. (Contributed by AV, 28-Jun-2018.) (Proof shortened by AV, 1-May-2020.)
((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ( lastS ‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0))

Theoremccatw2s1p1 13265 Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.)
(((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝑁) = 𝑋)

Theoremccatw2s1p2 13266 Extract the second of two single symbols concatenated with a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.)
(((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑁 + 1)) = 𝑌)

Theoremccat2s1fvw 13267 Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.)
(((𝑊 ∈ Word 𝑉𝐼 ∈ ℕ0𝐼 < (#‘𝑊)) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊𝐼))

Theoremccat2s1fst 13268 The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
(((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘0) = (𝑊‘0))

5.7.6  Subwords

Theoremswrdval 13269* Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))

Theoremswrd00 13270 A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.)
(𝑆 substr ⟨𝑋, 𝑋⟩) = ∅

Theoremswrdcl 13271 Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)

Theoremswrdval2 13272* Value of the subword extractor in its intended domain. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) = (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))))

Theoremswrd0val 13273 Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(#‘𝑆))) → (𝑆 substr ⟨0, 𝐿⟩) = (𝑆 ↾ (0..^𝐿)))

Theoremswrd0len 13274 Length of a left-anchored subword. (Contributed by Stefan O'Rear, 24-Aug-2015.)
((𝑆 ∈ Word 𝐴𝐿 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨0, 𝐿⟩)) = 𝐿)

Theoremswrdlen 13275 Length of an extracted subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
((𝑆 ∈ Word 𝐴𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨𝐹, 𝐿⟩)) = (𝐿𝐹))

Theoremswrdfv 13276 A symbol in an extracted subword, indexed using the subword's indices. (Contributed by Stefan O'Rear, 16-Aug-2015.)
(((𝑆 ∈ Word 𝐴𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) ∧ 𝑋 ∈ (0..^(𝐿𝐹))) → ((𝑆 substr ⟨𝐹, 𝐿⟩)‘𝑋) = (𝑆‘(𝑋 + 𝐹)))

Theoremswrdf 13277 A subword of a word is a function from a half-open range of nonnegative integers of the same length as the subword to the set of symbols for the original word. (Contributed by AV, 13-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) → (𝑊 substr ⟨𝑀, 𝑁⟩):(0..^(𝑁𝑀))⟶𝑉)

Theoremswrdvalfn 13278 Value of the subword extractor as function with domain. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by AV, 2-May-2020.)
((𝑆 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → (𝑆 substr ⟨𝐹, 𝐿⟩) Fn (0..^(𝐿𝐹)))

Theoremswrd0f 13279 A left-anchored subword of a word is a function from a half-open range of nonnegative integers of the same length as the subword to the set of symbols for the original word. (Contributed by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑊))) → (𝑊 substr ⟨0, 𝑁⟩):(0..^𝑁)⟶𝑉)

Theoremswrdid 13280 A word is a subword of itself. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Proof shortened by AV, 2-May-2020.)
(𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨0, (#‘𝑆)⟩) = 𝑆)

Theoremswrdrn 13281 The range of a subword of a word is a subset of the set of symbols for the word. (Contributed by AV, 13-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ 𝑉)

Theoremswrdn0 13282 A prefixing subword consisting of at least one symbol is not empty. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Proof shortened by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ ∧ 𝑁 ≤ (#‘𝑊)) → (𝑊 substr ⟨0, 𝑁⟩) ≠ ∅)

Theoremswrdlend 13283 The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.) (Proof shortened by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝐿𝐹 → (𝑊 substr ⟨𝐹, 𝐿⟩) = ∅))

Theoremswrdnd 13284 The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.) (Proof shortened by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ((𝐹 < 0 ∨ 𝐿𝐹 ∨ (#‘𝑊) < 𝐿) → (𝑊 substr ⟨𝐹, 𝐿⟩) = ∅))

Theoremswrdnd2 13285 Value of the subword extractor outside its intended domain. (Contributed by Alexander van der Vekens, 24-May-2018.)
((𝑊 ∈ Word 𝑉𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐵𝐴 ∨ (#‘𝑊) ≤ 𝐴𝐵 ≤ 0) → (𝑊 substr ⟨𝐴, 𝐵⟩) = ∅))

Theoremswrd0 13286 A subword of an empty set is always the empty set. (Contributed by AV, 31-Mar-2018.) (Revised by AV, 20-Oct-2018.) (Proof shortened by AV, 2-May-2020.)
(∅ substr ⟨𝐹, 𝐿⟩) = ∅

Theoremswrdrlen 13287 Length of a right-anchored subword. (Contributed by Alexander van der Vekens, 5-Apr-2018.)
((𝑊 ∈ Word 𝑉𝐼 ∈ (0...(#‘𝑊))) → (#‘(𝑊 substr ⟨𝐼, (#‘𝑊)⟩)) = ((#‘𝑊) − 𝐼))

Theoremswrd0len0 13288 Length of a prefix of a word reduced by a single symbol, analogous to swrd0len 13274. (Contributed by AV, 4-Aug-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0 ∧ (#‘𝑊) = (𝑁 + 1)) → (#‘(𝑊 substr ⟨0, 𝑁⟩)) = 𝑁)

Theoremaddlenrevswrd 13289 The sum of the lengths of two reversed parts of a word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((#‘(𝑊 substr ⟨𝑀, (#‘𝑊)⟩)) + (#‘(𝑊 substr ⟨0, 𝑀⟩))) = (#‘𝑊))

Theoremaddlenswrd 13290 The sum of the lengths of two parts of a word is the length of the word. (Contributed by AV, 21-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → ((#‘(𝑊 substr ⟨0, 𝑀⟩)) + (#‘(𝑊 substr ⟨𝑀, (#‘𝑊)⟩))) = (#‘𝑊))

Theoremswrd0fv 13291 A symbol in an left-anchored subword, indexed using the subword's indices. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝑊)) ∧ 𝐼 ∈ (0..^𝐿)) → ((𝑊 substr ⟨0, 𝐿⟩)‘𝐼) = (𝑊𝐼))

Theoremswrd0fv0 13292 The first symbol in a left-anchored subword. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
((𝑊 ∈ Word 𝑉𝐼 ∈ (1...(#‘𝑊))) → ((𝑊 substr ⟨0, 𝐼⟩)‘0) = (𝑊‘0))

Theoremswrdtrcfv 13293 A symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅ ∧ 𝐼 ∈ (0..^((#‘𝑊) − 1))) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩)‘𝐼) = (𝑊𝐼))

Theoremswrdtrcfv0 13294 The first symbol in a word truncated by one symbol. (Contributed by Alexander van der Vekens, 16-Jun-2018.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩)‘0) = (𝑊‘0))

Theoremswrd0fvlsw 13295 The last symbol in a left-anchored subword. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (1...(#‘𝑊))) → ( lastS ‘(𝑊 substr ⟨0, 𝐿⟩)) = (𝑊‘(𝐿 − 1)))

Theoremswrdeq 13296* Two subwords of words are equal iff they have the same length and the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨0, 𝑀⟩) = (𝑈 substr ⟨0, 𝑁⟩) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊𝑖) = (𝑈𝑖))))

Theoremswrdlen2 13297 Length of an extracted subword. (Contributed by AV, 5-May-2020.)
((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0𝐿 ∈ (ℤ𝐹)) ∧ 𝐿 ≤ (#‘𝑆)) → (#‘(𝑆 substr ⟨𝐹, 𝐿⟩)) = (𝐿𝐹))

Theoremswrdfv2 13298 A symbol in an extracted subword, indexed using the word's indices. (Contributed by AV, 5-May-2020.)
(((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ ℕ0𝐿 ∈ (ℤ𝐹)) ∧ 𝐿 ≤ (#‘𝑆)) ∧ 𝑋 ∈ (𝐹..^𝐿)) → ((𝑆 substr ⟨𝐹, 𝐿⟩)‘(𝑋𝐹)) = (𝑆𝑋))

Theoremswrdsb0eq 13299 Two subwords with the same bounds are equal if the range is not valid. (Contributed by AV, 4-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ 𝑁𝑀) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))

Theoremswrdsbslen 13300 Two subwords with the same bounds have the same length. (Contributed by AV, 4-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (#‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (#‘(𝑈 substr ⟨𝑀, 𝑁⟩)))

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