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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremccatfv0 13401 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 13402 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))
 
Theoremccatval21sw 13403 The first symbol of the right (nonempty) half of a concatenated word. (Contributed by AV, 23-Apr-2022.)
((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝐵 ≠ ∅) → ((𝐴 ++ 𝐵)‘(#‘𝐴)) = (𝐵‘0))
 
Theoremccatlid 13404 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 13405 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 13406 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 13407 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))
 
Theoremccatidid 13408 Concatenation of the empty word by the empty word. (Contributed by AV, 26-Mar-2022.)
(∅ ++ ∅) = ∅
 
Theoremlswccatn0lsw 13409 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 13410 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 13411 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 13412 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 13413 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴”⟩ = ⟨“( I ‘𝐴)”⟩
 
Theorems1val 13414 Value of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴𝑉 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
 
Theorems1rn 13415 The range of a singleton word. (Contributed by Mario Carneiro, 18-Jul-2016.)
(𝐴𝑉 → ran ⟨“𝐴”⟩ = {𝐴})
 
Theorems1eq 13416 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
 
Theorems1eqd 13417 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
 
Theorems1cl 13418 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 13419 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝜑𝐴𝐵)       (𝜑 → ⟨“𝐴”⟩ ∈ Word 𝐵)
 
Theorems1prc 13420 Value of a singleton word if the symbol is a proper class. (Contributed by AV, 26-Mar-2022.)
𝐴 ∈ V → ⟨“𝐴”⟩ = ⟨“∅”⟩)
 
Theorems1cli 13421 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴”⟩ ∈ Word V
 
Theorems1len 13422 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(#‘⟨“𝐴”⟩) = 1
 
Theorems1nz 13423 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 13424 Obsolete proof of s1nz 13423 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 13425 The domain of a singleton word is a singleton. (Contributed by AV, 9-Jan-2020.)
dom ⟨“𝐴”⟩ = {0}
 
Theorems1dmALT 13426 Alternate version of s1dm 13425, having a shorter proof, but requiring that 𝐴 is a set. (Contributed by AV, 9-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑆 → dom ⟨“𝐴”⟩ = {0})
 
Theorems1fv 13427 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 13428 The last symbol of a singleton word is its symbol. (Contributed by AV, 22-Oct-2018.)
(𝐴𝑉 → ( lastS ‘⟨“𝐴”⟩) = 𝐴)
 
Theoremeqs1 13429 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 13430* A word of length 1 is a singleton word. (Contributed by AV, 24-Jan-2021.)
((𝑊 ∈ Word 𝑆 ∧ (#‘𝑊) = 1) → ∃𝑠𝑆 𝑊 = ⟨“𝑠”⟩)
 
Theoremwrdl1s1 13431 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 13432 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 13433 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)
 
Theoremccatws1clv 13434 The concatenation of a word with a singleton word (which can be over a different alphabet) is a word. (Contributed by AV, 5-Mar-2022.)
(𝑊 ∈ Word 𝑉 → (𝑊 ++ ⟨“𝑋”⟩) ∈ Word V)
 
Theoremccat2s1cl 13435 The concatenation of two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → (⟨“𝑋”⟩ ++ ⟨“𝑌”⟩) ∈ Word 𝑉)
 
Theoremccats1alpha 13436 A concatenation of a word with a singleton word is a word over an alphabet 𝑆 iff the symbols of both words belong to the alphabet 𝑆. (Contributed by AV, 27-Mar-2022.)
((𝐴 ∈ Word 𝑉𝑋𝑈) → ((𝐴 ++ ⟨“𝑋”⟩) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆𝑋𝑆)))
 
Theoremccatws1len 13437 The length of the concatenation of a word with a singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 4-Mar-2022.)
(𝑊 ∈ Word 𝑉 → (#‘(𝑊 ++ ⟨“𝑋”⟩)) = ((#‘𝑊) + 1))
 
Theoremccatws1lenOLD 13438 Obsolete version of ccatws1len 13437 as of 4-Mar-2022. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → (#‘(𝑊 ++ ⟨“𝑋”⟩)) = ((#‘𝑊) + 1))
 
Theoremccatws1lenp1b 13439 The length of a word is 𝑁 iff the length of the concatenation of the word with a singleton word is 𝑁 + 1. (Contributed by AV, 4-Mar-2022.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) → ((#‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ↔ (#‘𝑊) = 𝑁))
 
Theoremwrdlenccats1lenm1 13440 The length of a word is the length of the word concatenated with a singleton word minus 1. (Contributed by AV, 28-Jun-2018.) (Revised by AV, 5-Mar-2022.)
(𝑊 ∈ Word 𝑉 → ((#‘(𝑊 ++ ⟨“𝑆”⟩)) − 1) = (#‘𝑊))
 
Theoremwrdlenccats1lenm1OLD 13441 Obsolete version of wrdlenccats1lenm1 13440 as of 5-Mar-2022. (Contributed by AV, 28-Jun-2018.) (Proof shortened by AV, 1-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑊 ∈ Word 𝑉𝑆𝑉) → (#‘𝑊) = ((#‘(𝑊 ++ ⟨“𝑆”⟩)) − 1))
 
Theoremccat2s1len 13442 The length of the concatenation of two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → (#‘(⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)) = 2)
 
Theoremccatw2s1cl 13443 The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ Word 𝑉)
 
Theoremccatw2s1len 13444 The length of the concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 5-Mar-2022.)
(𝑊 ∈ Word 𝑉 → (#‘((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)) = ((#‘𝑊) + 2))
 
Theoremccatw2s1lenOLD 13445 Obsolete version of ccatw2s1len 13444 as of 5-Mar-2022. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → (#‘((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)) = ((#‘𝑊) + 2))
 
Theoremccats1val1 13446 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 13447 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)
 
Theoremccat1st1st 13448 The first symbol of a word concatenated with its first symbol is the first symbol of the word. This theorem holds even if 𝑊 is the empty word. (Contributed by AV, 26-Mar-2022.)
(𝑊 ∈ Word 𝑉 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0))
 
Theoremccat2s1p1 13449 Extract the first of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘0) = 𝑋)
 
Theoremccat2s1p2 13450 Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋𝑉𝑌𝑉) → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = 𝑌)
 
Theoremccatw2s1ass 13451 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”⟩)))
 
Theoremccatws1lenrevOLD 13452 Obsolete theorem as of 24-Jan-2022. Use wrdlenccats1lenm1 13440 instead. The length of a word concatenated with a singleton word. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → ((#‘(𝑊 ++ ⟨“𝑋”⟩)) = 𝑁 → (#‘𝑊) = (𝑁 − 1)))
 
Theoremccatws1n0 13453 The concatenation of a word with a singleton word is not the empty set. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 5-Mar-2022.)
(𝑊 ∈ Word 𝑉 → (𝑊 ++ ⟨“𝑋”⟩) ≠ ∅)
 
Theoremccatws1n0OLD 13454 Obsolete version of ccatws1n0 13453 as of 5-Mar-2022. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑊 ∈ Word 𝑉𝑋𝑉) → (𝑊 ++ ⟨“𝑋”⟩) ≠ ∅)
 
Theoremccatws1ls 13455 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 13456 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 13457 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 13458 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 13459 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 13460 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 13461 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 13462* 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 13463 A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.)
(𝑆 substr ⟨𝑋, 𝑋⟩) = ∅
 
Theoremswrdcl 13464 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 13465* 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 13466 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 13467 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 13468 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 13469 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)))
 
Theoremswrdfv0 13470 The first symbol in an extracted subword. (Contributed by AV, 27-Apr-2022.)
((𝑆 ∈ Word 𝐴𝐹 ∈ (0..^𝐿) ∧ 𝐿 ∈ (0...(#‘𝑆))) → ((𝑆 substr ⟨𝐹, 𝐿⟩)‘0) = (𝑆𝐹))
 
Theoremswrdf 13471 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 13472 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 13473 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 13474 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 13475 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 13476 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 13477 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 13478 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 13479 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 13480 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 13481 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 13482 Length of a prefix of a word reduced by a single symbol, analogous to swrd0len 13467. (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 13483 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 13484 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 13485 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 13486 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 13487 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 13488 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 13489 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 13490* 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 13491 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 13492 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 13493 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 13494 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⟩)))
 
Theoremswrdspsleq 13495* Two words have a common subword (starting at the same position with the same length) iff they have the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Proof shortened by AV, 7-May-2020.)
(((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
 
Theoremswrdtrcfvl 13496 The last symbol in a word truncated by one symbol. (Contributed by AV, 16-Jun-2018.) (Proof shortened by Mario Carneiro/AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑊)) → ( lastS ‘(𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩)) = (𝑊‘((#‘𝑊) − 2)))
 
Theoremswrds1 13497 Extract a single symbol from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.)
((𝑊 ∈ Word 𝐴𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 substr ⟨𝐼, (𝐼 + 1)⟩) = ⟨“(𝑊𝐼)”⟩)
 
Theoremswrdlsw 13498 Extract the last single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑊)”⟩)
 
Theorem2swrdeqwrdeq 13499 Two words are equal if and only if they have the same prefix and the same suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 substr ⟨0, 𝐼⟩) = (𝑆 substr ⟨0, 𝐼⟩) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))
 
Theorem2swrd1eqwrdeq 13500 Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))
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