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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremccatval21sw 14401 The first symbol of the right (nonempty) half of a concatenated word. (Contributed by AV, 23-Apr-2022.)
((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ ((𝐴 ++ 𝐡)β€˜(β™―β€˜π΄)) = (π΅β€˜0))
 
Theoremccatlid 14402 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 𝐡 β†’ (βˆ… ++ 𝑆) = 𝑆)
 
Theoremccatrid 14403 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 𝐡 β†’ (𝑆 ++ βˆ…) = 𝑆)
 
Theoremccatass 14404 Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡 ∧ π‘ˆ ∈ Word 𝐡) β†’ ((𝑆 ++ 𝑇) ++ π‘ˆ) = (𝑆 ++ (𝑇 ++ π‘ˆ)))
 
Theoremccatrn 14405 The range of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ran (𝑆 ++ 𝑇) = (ran 𝑆 βˆͺ ran 𝑇))
 
Theoremccatidid 14406 Concatenation of the empty word by the empty word. (Contributed by AV, 26-Mar-2022.)
(βˆ… ++ βˆ…) = βˆ…
 
Theoremlswccatn0lsw 14407 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 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (lastSβ€˜(𝐴 ++ 𝐡)) = (lastSβ€˜π΅))
 
Theoremlswccat0lsw 14408 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.)
(π‘Š ∈ Word 𝑉 β†’ (lastSβ€˜(π‘Š ++ βˆ…)) = (lastSβ€˜π‘Š))
 
Theoremccatalpha 14409 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 ∧ 𝐡 ∈ Word V) β†’ ((𝐴 ++ 𝐡) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝐡 ∈ Word 𝑆)))
 
Theoremccatrcl1 14410 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 𝑋 ∧ 𝐡 ∈ Word π‘Œ ∧ (π‘Š = (𝐴 ++ 𝐡) ∧ π‘Š ∈ Word 𝑆)) β†’ 𝐴 ∈ Word 𝑆)
 
5.7.4  Singleton words
 
Syntaxcs1 14411 Syntax for the singleton word constructor.
class βŸ¨β€œπ΄β€βŸ©
 
Definitiondf-s1 14412 Define the canonical injection from symbols to words. Although not required, 𝐴 should usually be a set. Otherwise, the singleton word βŸ¨β€œπ΄β€βŸ© would be the singleton word consisting of the empty set, see s1prc 14420, and not, as maybe expected, the empty word, see also s1nz 14423. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
βŸ¨β€œπ΄β€βŸ© = {⟨0, ( I β€˜π΄)⟩}
 
Theoremids1 14413 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
βŸ¨β€œπ΄β€βŸ© = βŸ¨β€œ( I β€˜π΄)β€βŸ©
 
Theorems1val 14414 Value of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴 ∈ 𝑉 β†’ βŸ¨β€œπ΄β€βŸ© = {⟨0, 𝐴⟩})
 
Theorems1rn 14415 The range of a singleton word. (Contributed by Mario Carneiro, 18-Jul-2016.)
(𝐴 ∈ 𝑉 β†’ ran βŸ¨β€œπ΄β€βŸ© = {𝐴})
 
Theorems1eq 14416 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝐴 = 𝐡 β†’ βŸ¨β€œπ΄β€βŸ© = βŸ¨β€œπ΅β€βŸ©)
 
Theorems1eqd 14417 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄β€βŸ© = βŸ¨β€œπ΅β€βŸ©)
 
Theorems1cl 14418 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.)
(𝐴 ∈ 𝐡 β†’ βŸ¨β€œπ΄β€βŸ© ∈ Word 𝐡)
 
Theorems1cld 14419 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
(πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ βŸ¨β€œπ΄β€βŸ© ∈ Word 𝐡)
 
Theorems1prc 14420 Value of a singleton word if the symbol is a proper class. (Contributed by AV, 26-Mar-2022.)
(Β¬ 𝐴 ∈ V β†’ βŸ¨β€œπ΄β€βŸ© = βŸ¨β€œβˆ…β€βŸ©)
 
Theorems1cli 14421 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
βŸ¨β€œπ΄β€βŸ© ∈ Word V
 
Theorems1len 14422 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(β™―β€˜βŸ¨β€œπ΄β€βŸ©) = 1
 
Theorems1nz 14423 A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.) (Proof shortened by Kyle Wyonch, 18-Jul-2021.)
βŸ¨β€œπ΄β€βŸ© β‰  βˆ…
 
Theorems1dm 14424 The domain of a singleton word is a singleton. (Contributed by AV, 9-Jan-2020.)
dom βŸ¨β€œπ΄β€βŸ© = {0}
 
Theorems1dmALT 14425 Alternate version of s1dm 14424, 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.)
(𝐴 ∈ 𝑆 β†’ dom βŸ¨β€œπ΄β€βŸ© = {0})
 
Theorems1fv 14426 Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝐴 ∈ 𝐡 β†’ (βŸ¨β€œπ΄β€βŸ©β€˜0) = 𝐴)
 
Theoremlsws1 14427 The last symbol of a singleton word is its symbol. (Contributed by AV, 22-Oct-2018.)
(𝐴 ∈ 𝑉 β†’ (lastSβ€˜βŸ¨β€œπ΄β€βŸ©) = 𝐴)
 
Theoremeqs1 14428 A word of length 1 is a singleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 1-May-2020.)
((π‘Š ∈ Word 𝐴 ∧ (β™―β€˜π‘Š) = 1) β†’ π‘Š = βŸ¨β€œ(π‘Šβ€˜0)β€βŸ©)
 
Theoremwrdl1exs1 14429* A word of length 1 is a singleton word. (Contributed by AV, 24-Jan-2021.)
((π‘Š ∈ Word 𝑆 ∧ (β™―β€˜π‘Š) = 1) β†’ βˆƒπ‘  ∈ 𝑆 π‘Š = βŸ¨β€œπ‘ β€βŸ©)
 
Theoremwrdl1s1 14430 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.)
(𝑆 ∈ 𝑉 β†’ (π‘Š = βŸ¨β€œπ‘†β€βŸ© ↔ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 1 ∧ (π‘Šβ€˜0) = 𝑆)))
 
Theorems111 14431 The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) β†’ (βŸ¨β€œπ‘†β€βŸ© = βŸ¨β€œπ‘‡β€βŸ© ↔ 𝑆 = 𝑇))
 
5.7.5  Concatenations with singleton words
 
Theoremccatws1cl 14432 The concatenation of a word with a singleton word is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((π‘Š ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉) β†’ (π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ∈ Word 𝑉)
 
Theoremccatws1clv 14433 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.)
(π‘Š ∈ Word 𝑉 β†’ (π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ∈ Word V)
 
Theoremccat2s1cl 14434 The concatenation of two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ Word 𝑉)
 
Theoremccats1alpha 14435 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 𝑉 ∧ 𝑋 ∈ π‘ˆ) β†’ ((𝐴 ++ βŸ¨β€œπ‘‹β€βŸ©) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝑋 ∈ 𝑆)))
 
Theoremccatws1len 14436 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.)
(π‘Š ∈ Word 𝑉 β†’ (β™―β€˜(π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)) = ((β™―β€˜π‘Š) + 1))
 
Theoremccatws1lenp1b 14437 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.)
((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ β„•0) β†’ ((β™―β€˜(π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)) = (𝑁 + 1) ↔ (β™―β€˜π‘Š) = 𝑁))
 
Theoremwrdlenccats1lenm1 14438 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.)
(π‘Š ∈ Word 𝑉 β†’ ((β™―β€˜(π‘Š ++ βŸ¨β€œπ‘†β€βŸ©)) βˆ’ 1) = (β™―β€˜π‘Š))
 
Theoremccat2s1len 14439 The length of the concatenation of two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 14-Jan-2024.)
(β™―β€˜(βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘Œβ€βŸ©)) = 2
 
Theoremccatw2s1cl 14440 The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((π‘Š ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ Word 𝑉)
 
Theoremccatw2s1len 14441 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.)
(π‘Š ∈ Word 𝑉 β†’ (β™―β€˜((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)) = ((β™―β€˜π‘Š) + 2))
 
Theoremccats1val1 14442 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.) (Revised by JJ, 20-Jan-2024.)
((π‘Š ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(β™―β€˜π‘Š))) β†’ ((π‘Š ++ βŸ¨β€œπ‘†β€βŸ©)β€˜πΌ) = (π‘Šβ€˜πΌ))
 
Theoremccats1val2 14443 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.)
((π‘Š ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ∧ 𝐼 = (β™―β€˜π‘Š)) β†’ ((π‘Š ++ βŸ¨β€œπ‘†β€βŸ©)β€˜πΌ) = 𝑆)
 
Theoremccat1st1st 14444 The first symbol of a word concatenated with its first symbol is the first symbol of the word. This theorem holds even if π‘Š is the empty word. (Contributed by AV, 26-Mar-2022.)
(π‘Š ∈ Word 𝑉 β†’ ((π‘Š ++ βŸ¨β€œ(π‘Šβ€˜0)β€βŸ©)β€˜0) = (π‘Šβ€˜0))
 
Theoremccat2s1p1 14445 Extract the first of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 20-Jan-2024.)
(𝑋 ∈ 𝑉 β†’ ((βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜0) = 𝑋)
 
Theoremccat2s1p2 14446 Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 20-Jan-2024.)
(π‘Œ ∈ 𝑉 β†’ ((βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜1) = π‘Œ)
 
Theoremccatw2s1ass 14447 Associative law for a concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
(π‘Š ∈ Word 𝑉 β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) = (π‘Š ++ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘Œβ€βŸ©)))
 
Theoremccatw2s1assOLD 14448 Obsolete version of ccatw2s1ass 14447 as of 29-Jan-2024. Associative law for a concatenation of a word with two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
((π‘Š ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) = (π‘Š ++ (βŸ¨β€œπ‘‹β€βŸ© ++ βŸ¨β€œπ‘Œβ€βŸ©)))
 
Theoremccatws1n0 14449 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.)
(π‘Š ∈ Word 𝑉 β†’ (π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) β‰  βˆ…)
 
Theoremccatws1ls 14450 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.)
((π‘Š ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜(β™―β€˜π‘Š)) = 𝑋)
 
Theoremlswccats1 14451 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.)
((π‘Š ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉) β†’ (lastSβ€˜(π‘Š ++ βŸ¨β€œπ‘†β€βŸ©)) = 𝑆)
 
Theoremlswccats1fst 14452 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 ≀ (β™―β€˜π‘ƒ)) β†’ (lastSβ€˜(𝑃 ++ βŸ¨β€œ(π‘ƒβ€˜0)β€βŸ©)) = ((𝑃 ++ βŸ¨β€œ(π‘ƒβ€˜0)β€βŸ©)β€˜0))
 
Theoremccatw2s1p1 14453 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.) (Revised by AV, 1-May-2020.) (Revised by AV, 29-Jan-2024.)
((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 𝑁 ∧ 𝑋 ∈ 𝑉) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜π‘) = 𝑋)
 
Theoremccatw2s1p1OLD 14454 Obsolete version of ccatw2s1p1 14453 as of 29-Jan-2024. 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.) (Proof modification is discouraged.) (New usage is discouraged.)
(((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 𝑁) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜π‘) = 𝑋)
 
Theoremccatw2s1p2 14455 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.)
(((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = 𝑁) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 + 1)) = π‘Œ)
 
Theoremccat2s1fvw 14456 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.) (Revised by AV, 28-Jan-2024.)
((π‘Š ∈ Word 𝑉 ∧ 𝐼 ∈ β„•0 ∧ 𝐼 < (β™―β€˜π‘Š)) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜πΌ) = (π‘Šβ€˜πΌ))
 
Theoremccat2s1fvwOLD 14457 Obsolete version of ccat2s1fvw 14456 as of 28-Jan-2024. 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.) (New usage is discouraged.) (Proof modification is discouraged.)
(((π‘Š ∈ Word 𝑉 ∧ 𝐼 ∈ β„•0 ∧ 𝐼 < (β™―β€˜π‘Š)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜πΌ) = (π‘Šβ€˜πΌ))
 
Theoremccat2s1fst 14458 The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 28-Jan-2024.)
((π‘Š ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜0) = (π‘Šβ€˜0))
 
Theoremccat2s1fstOLD 14459 Obsolete version of ccat2s1fst 14458 as of 28-Jan-2024. The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(((π‘Š ∈ Word 𝑉 ∧ 0 < (β™―β€˜π‘Š)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜0) = (π‘Šβ€˜0))
 
5.7.6  Subwords/substrings
 
Syntaxcsubstr 14460 Syntax for the subword operator.
class substr
 
Definitiondf-substr 14461* Define an operation which extracts portions (called subwords or substrings) of words. Definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 15-Aug-2015.)
substr = (𝑠 ∈ V, 𝑏 ∈ (β„€ Γ— β„€) ↦ if(((1st β€˜π‘)..^(2nd β€˜π‘)) βŠ† dom 𝑠, (π‘₯ ∈ (0..^((2nd β€˜π‘) βˆ’ (1st β€˜π‘))) ↦ (π‘ β€˜(π‘₯ + (1st β€˜π‘)))), βˆ…))
 
Theoremswrdnznd 14462 The value of a subword operation for noninteger arguments is the empty set. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6873). (Contributed by AV, 2-Dec-2022.) (New usage is discouraged.)
(Β¬ (𝐹 ∈ β„€ ∧ 𝐿 ∈ β„€) β†’ (𝑆 substr ⟨𝐹, 𝐿⟩) = βˆ…)
 
Theoremswrdval 14463* Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝑆 ∈ 𝑉 ∧ 𝐹 ∈ β„€ ∧ 𝐿 ∈ β„€) β†’ (𝑆 substr ⟨𝐹, 𝐿⟩) = if((𝐹..^𝐿) βŠ† dom 𝑆, (π‘₯ ∈ (0..^(𝐿 βˆ’ 𝐹)) ↦ (π‘†β€˜(π‘₯ + 𝐹))), βˆ…))
 
Theoremswrd00 14464 A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.)
(𝑆 substr βŸ¨π‘‹, π‘‹βŸ©) = βˆ…
 
Theoremswrdcl 14465 Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝑆 ∈ Word 𝐴 β†’ (𝑆 substr ⟨𝐹, 𝐿⟩) ∈ Word 𝐴)
 
Theoremswrdval2 14466* 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 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘†))) β†’ (𝑆 substr ⟨𝐹, 𝐿⟩) = (π‘₯ ∈ (0..^(𝐿 βˆ’ 𝐹)) ↦ (π‘†β€˜(π‘₯ + 𝐹))))
 
Theoremswrdlen 14467 Length of an extracted subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘†))) β†’ (β™―β€˜(𝑆 substr ⟨𝐹, 𝐿⟩)) = (𝐿 βˆ’ 𝐹))
 
Theoremswrdfv 14468 A symbol in an extracted subword, indexed using the subword's indices. (Contributed by Stefan O'Rear, 16-Aug-2015.)
(((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘†))) ∧ 𝑋 ∈ (0..^(𝐿 βˆ’ 𝐹))) β†’ ((𝑆 substr ⟨𝐹, 𝐿⟩)β€˜π‘‹) = (π‘†β€˜(𝑋 + 𝐹)))
 
Theoremswrdfv0 14469 The first symbol in an extracted subword. (Contributed by AV, 27-Apr-2022.)
((𝑆 ∈ Word 𝐴 ∧ 𝐹 ∈ (0..^𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘†))) β†’ ((𝑆 substr ⟨𝐹, 𝐿⟩)β€˜0) = (π‘†β€˜πΉ))
 
Theoremswrdf 14470 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.)
((π‘Š ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š))) β†’ (π‘Š substr βŸ¨π‘€, π‘βŸ©):(0..^(𝑁 βˆ’ 𝑀))βŸΆπ‘‰)
 
Theoremswrdvalfn 14471 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 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘†))) β†’ (𝑆 substr ⟨𝐹, 𝐿⟩) Fn (0..^(𝐿 βˆ’ 𝐹)))
 
Theoremswrdrn 14472 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.)
((π‘Š ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(β™―β€˜π‘Š))) β†’ ran (π‘Š substr βŸ¨π‘€, π‘βŸ©) βŠ† 𝑉)
 
Theoremswrdlend 14473 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.)
((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ β„€ ∧ 𝐿 ∈ β„€) β†’ (𝐿 ≀ 𝐹 β†’ (π‘Š substr ⟨𝐹, 𝐿⟩) = βˆ…))
 
Theoremswrdnd 14474 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.)
((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ β„€ ∧ 𝐿 ∈ β„€) β†’ ((𝐹 < 0 ∨ 𝐿 ≀ 𝐹 ∨ (β™―β€˜π‘Š) < 𝐿) β†’ (π‘Š substr ⟨𝐹, 𝐿⟩) = βˆ…))
 
Theoremswrdnd2 14475 Value of the subword extractor outside its intended domain. (Contributed by Alexander van der Vekens, 24-May-2018.)
((π‘Š ∈ Word 𝑉 ∧ 𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ ((𝐡 ≀ 𝐴 ∨ (β™―β€˜π‘Š) ≀ 𝐴 ∨ 𝐡 ≀ 0) β†’ (π‘Š substr ⟨𝐴, 𝐡⟩) = βˆ…))
 
Theoremswrdnnn0nd 14476 The value of a subword operation for arguments not being nonnegative integers is the empty set. (Contributed by AV, 2-Dec-2022.)
((𝑆 ∈ Word 𝑉 ∧ Β¬ (𝐹 ∈ β„•0 ∧ 𝐿 ∈ β„•0)) β†’ (𝑆 substr ⟨𝐹, 𝐿⟩) = βˆ…)
 
Theoremswrdnd0 14477 The value of a subword operation for inproper arguments is the empty set. (Contributed by AV, 2-Dec-2022.)
(𝑆 ∈ Word 𝑉 β†’ (Β¬ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘†))) β†’ (𝑆 substr ⟨𝐹, 𝐿⟩) = βˆ…))
 
Theoremswrd0 14478 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.)
(βˆ… substr ⟨𝐹, 𝐿⟩) = βˆ…
 
Theoremswrdrlen 14479 Length of a right-anchored subword. (Contributed by Alexander van der Vekens, 5-Apr-2018.)
((π‘Š ∈ Word 𝑉 ∧ 𝐼 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜(π‘Š substr ⟨𝐼, (β™―β€˜π‘Š)⟩)) = ((β™―β€˜π‘Š) βˆ’ 𝐼))
 
Theoremswrdlen2 14480 Length of an extracted subword. (Contributed by AV, 5-May-2020.)
((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ β„•0 ∧ 𝐿 ∈ (β„€β‰₯β€˜πΉ)) ∧ 𝐿 ≀ (β™―β€˜π‘†)) β†’ (β™―β€˜(𝑆 substr ⟨𝐹, 𝐿⟩)) = (𝐿 βˆ’ 𝐹))
 
Theoremswrdfv2 14481 A symbol in an extracted subword, indexed using the word's indices. (Contributed by AV, 5-May-2020.)
(((𝑆 ∈ Word 𝑉 ∧ (𝐹 ∈ β„•0 ∧ 𝐿 ∈ (β„€β‰₯β€˜πΉ)) ∧ 𝐿 ≀ (β™―β€˜π‘†)) ∧ 𝑋 ∈ (𝐹..^𝐿)) β†’ ((𝑆 substr ⟨𝐹, 𝐿⟩)β€˜(𝑋 βˆ’ 𝐹)) = (π‘†β€˜π‘‹))
 
Theoremswrdwrdsymb 14482 A subword is a word over the symbols it consists of. (Contributed by AV, 2-Dec-2022.)
(𝑆 ∈ Word 𝐴 β†’ (𝑆 substr βŸ¨π‘€, π‘βŸ©) ∈ Word (𝑆 β€œ (𝑀..^𝑁)))
 
Theoremswrdsb0eq 14483 Two subwords with the same bounds are equal if the range is not valid. (Contributed by AV, 4-May-2020.)
(((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉) ∧ (𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) ∧ 𝑁 ≀ 𝑀) β†’ (π‘Š substr βŸ¨π‘€, π‘βŸ©) = (π‘ˆ substr βŸ¨π‘€, π‘βŸ©))
 
Theoremswrdsbslen 14484 Two subwords with the same bounds have the same length. (Contributed by AV, 4-May-2020.)
(((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉) ∧ (𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) ∧ (𝑁 ≀ (β™―β€˜π‘Š) ∧ 𝑁 ≀ (β™―β€˜π‘ˆ))) β†’ (β™―β€˜(π‘Š substr βŸ¨π‘€, π‘βŸ©)) = (β™―β€˜(π‘ˆ substr βŸ¨π‘€, π‘βŸ©)))
 
Theoremswrdspsleq 14485* 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.)
(((π‘Š ∈ Word 𝑉 ∧ π‘ˆ ∈ Word 𝑉) ∧ (𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) ∧ (𝑁 ≀ (β™―β€˜π‘Š) ∧ 𝑁 ≀ (β™―β€˜π‘ˆ))) β†’ ((π‘Š substr βŸ¨π‘€, π‘βŸ©) = (π‘ˆ substr βŸ¨π‘€, π‘βŸ©) ↔ βˆ€π‘– ∈ (𝑀..^𝑁)(π‘Šβ€˜π‘–) = (π‘ˆβ€˜π‘–)))
 
Theoremswrds1 14486 Extract a single symbol from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.)
((π‘Š ∈ Word 𝐴 ∧ 𝐼 ∈ (0..^(β™―β€˜π‘Š))) β†’ (π‘Š substr ⟨𝐼, (𝐼 + 1)⟩) = βŸ¨β€œ(π‘Šβ€˜πΌ)β€βŸ©)
 
Theoremswrdlsw 14487 Extract the last single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((π‘Š ∈ Word 𝑉 ∧ π‘Š β‰  βˆ…) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 1), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(lastSβ€˜π‘Š)β€βŸ©)
 
Theoremccatswrd 14488 Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...π‘Œ) ∧ π‘Œ ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(β™―β€˜π‘†)))) β†’ ((𝑆 substr βŸ¨π‘‹, π‘ŒβŸ©) ++ (𝑆 substr βŸ¨π‘Œ, π‘βŸ©)) = (𝑆 substr βŸ¨π‘‹, π‘βŸ©))
 
Theoremswrdccat2 14489 Recover the right half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
((𝑆 ∈ Word 𝐡 ∧ 𝑇 ∈ Word 𝐡) β†’ ((𝑆 ++ 𝑇) substr ⟨(β™―β€˜π‘†), ((β™―β€˜π‘†) + (β™―β€˜π‘‡))⟩) = 𝑇)
 
5.7.7  Prefixes of a word
 
Syntaxcpfx 14490 Syntax for the prefix operator.
class prefix
 
Definitiondf-pfx 14491* Define an operation which extracts prefixes of words, i.e. subwords (or substrings) starting at the beginning of a word (or string). In other words, (𝑆 prefix 𝐿) is the prefix of the word 𝑆 of length 𝐿. Definition in Section 9.1 of [AhoHopUll] p. 318. See also Wikipedia "Substring" https://en.wikipedia.org/wiki/Substring#Prefix. (Contributed by AV, 2-May-2020.)
prefix = (𝑠 ∈ V, 𝑙 ∈ β„•0 ↦ (𝑠 substr ⟨0, π‘™βŸ©))
 
Theorempfxnndmnd 14492 The value of a prefix operation for out-of-domain arguments. (This is due to our definition of function values for out-of-domain arguments, see ndmfv 6873). (Contributed by AV, 3-Dec-2022.) (New usage is discouraged.)
(Β¬ (𝑆 ∈ V ∧ 𝐿 ∈ β„•0) β†’ (𝑆 prefix 𝐿) = βˆ…)
 
Theorempfxval 14493 Value of a prefix operation. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ 𝑉 ∧ 𝐿 ∈ β„•0) β†’ (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
 
Theorempfx00 14494 The zero length prefix is the empty set. (Contributed by AV, 2-May-2020.)
(𝑆 prefix 0) = βˆ…
 
Theorempfx0 14495 A prefix of an empty set is always the empty set. (Contributed by AV, 3-May-2020.)
(βˆ… prefix 𝐿) = βˆ…
 
Theorempfxval0 14496 Value of a prefix operation. This theorem should only be used in proofs if 𝐿 ∈ β„•0 is not available. Otherwise (and usually), pfxval 14493 should be used. (Contributed by AV, 3-Dec-2022.) (New usage is discouraged.)
(𝑆 ∈ Word 𝐴 β†’ (𝑆 prefix 𝐿) = (𝑆 substr ⟨0, 𝐿⟩))
 
Theorempfxcl 14497 Closure of the prefix extractor. (Contributed by AV, 2-May-2020.)
(𝑆 ∈ Word 𝐴 β†’ (𝑆 prefix 𝐿) ∈ Word 𝐴)
 
Theorempfxmpt 14498* Value of the prefix extractor as a mapping. (Contributed by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(β™―β€˜π‘†))) β†’ (𝑆 prefix 𝐿) = (π‘₯ ∈ (0..^𝐿) ↦ (π‘†β€˜π‘₯)))
 
Theorempfxres 14499 Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by AV, 2-May-2020.)
((𝑆 ∈ Word 𝐴 ∧ 𝐿 ∈ (0...(β™―β€˜π‘†))) β†’ (𝑆 prefix 𝐿) = (𝑆 β†Ύ (0..^𝐿)))
 
Theorempfxf 14500 A prefix of a word is a function from a half-open range of nonnegative integers of the same length as the prefix to the set of symbols for the original word. (Contributed by AV, 2-May-2020.)
((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (π‘Š prefix 𝐿):(0..^𝐿)βŸΆπ‘‰)
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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