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Theorem | hashpw 14401 | The size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.) (Proof shortened by Mario Carneiro, 5-Aug-2014.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π΄ β Fin β (β―βπ« π΄) = (2β(β―βπ΄))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashfun 14402 | A finite set is a function iff it is equinumerous to its domain. (Contributed by Mario Carneiro, 26-Sep-2013.) (Revised by Mario Carneiro, 12-Mar-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (πΉ β Fin β (Fun πΉ β (β―βπΉ) = (β―βdom πΉ))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashres 14403 | The number of elements of a finite function restricted to a subset of its domain is equal to the number of elements of that subset. (Contributed by AV, 15-Dec-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((Fun π΄ β§ π΄ β Fin β§ π΅ β dom π΄) β (β―β(π΄ βΎ π΅)) = (β―βπ΅)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashreshashfun 14404 | The number of elements of a finite function expressed by a restriction. (Contributed by AV, 15-Dec-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((Fun π΄ β§ π΄ β Fin β§ π΅ β dom π΄) β (β―βπ΄) = ((β―β(π΄ βΎ π΅)) + (β―β(dom π΄ β π΅)))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashimarn 14405 | The size of the image of a one-to-one function πΈ under the range of a function πΉ which is a one-to-one function into the domain of πΈ equals the size of the function πΉ. (Contributed by Alexander van der Vekens, 4-Feb-2018.) (Proof shortened by AV, 4-May-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π) β (πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β (β―β(πΈ β ran πΉ)) = (β―βπΉ))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashimarni 14406 | If the size of the image of a one-to-one function πΈ under the range of a function πΉ which is a one-to-one function into the domain of πΈ is a nonnegative integer, the size of the function πΉ is the same nonnegative integer. (Contributed by Alexander van der Vekens, 4-Feb-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((πΈ:dom πΈβ1-1βran πΈ β§ πΈ β π) β ((πΉ:(0..^(β―βπΉ))β1-1βdom πΈ β§ π = (πΈ β ran πΉ) β§ (β―βπ) = π) β (β―βπΉ) = π)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashfundm 14407 | The size of a set function is equal to the size of its domain. (Contributed by BTernaryTau, 30-Sep-2023.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((πΉ β π β§ Fun πΉ) β (β―βπΉ) = (β―βdom πΉ)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashf1dmrn 14408 | The size of the domain of a one-to-one set function is equal to the size of its range. (Contributed by BTernaryTau, 1-Oct-2023.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((πΉ β π β§ πΉ:π΄β1-1βπ΅) β (β―βπ΄) = (β―βran πΉ)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | resunimafz0 14409 | TODO-AV: Revise using πΉ β Word dom πΌ? Formerly part of proof of eupth2lem3 29757: The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Fun πΌ) & β’ (π β πΉ:(0..^(β―βπΉ))βΆdom πΌ) & β’ (π β π β (0..^(β―βπΉ))) β β’ (π β (πΌ βΎ (πΉ β (0...π))) = ((πΌ βΎ (πΉ β (0..^π))) βͺ {β¨(πΉβπ), (πΌβ(πΉβπ))β©})) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fnfz0hash 14410 | The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β β0 β§ πΉ Fn (0...π)) β (β―βπΉ) = (π + 1)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | ffz0hash 14411 | The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β β0 β§ πΉ:(0...π)βΆπ΅) β (β―βπΉ) = (π + 1)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fnfz0hashnn0 14412 | The size of a function on a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (πΉ Fn (0...π) β (β―βπΉ) β β0) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | ffzo0hash 14413 | The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β β0 β§ πΉ Fn (0..^π)) β (β―βπΉ) = π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fnfzo0hash 14414 | The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β β0 β§ πΉ:(0..^π)βΆπ΅) β (β―βπΉ) = π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fnfzo0hashnn0 14415 | The value of the size function on a half-open range of nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (πΉ Fn (0..^π) β (β―βπΉ) β β0) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashbclem 14416* | Lemma for hashbc 14417: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π΄ β Fin) & β’ (π β Β¬ π§ β π΄) & β’ (π β βπ β β€ ((β―βπ΄)Cπ) = (β―β{π₯ β π« π΄ β£ (β―βπ₯) = π})) & β’ (π β πΎ β β€) β β’ (π β ((β―β(π΄ βͺ {π§}))CπΎ) = (β―β{π₯ β π« (π΄ βͺ {π§}) β£ (β―βπ₯) = πΎ})) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashbc 14417* | The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). (Contributed by Mario Carneiro, 13-Jul-2014.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π΄ β Fin β§ πΎ β β€) β ((β―βπ΄)CπΎ) = (β―β{π₯ β π« π΄ β£ (β―βπ₯) = πΎ})) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashfacen 14418* | The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by AV, 7-Aug-2024.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π΄ β π΅ β§ πΆ β π·) β {π β£ π:π΄β1-1-ontoβπΆ} β {π β£ π:π΅β1-1-ontoβπ·}) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashfacenOLD 14419* | Obsolete version of hashfacen 14418 as of 7-Aug-2024. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π΄ β π΅ β§ πΆ β π·) β {π β£ π:π΄β1-1-ontoβπΆ} β {π β£ π:π΅β1-1-ontoβπ·}) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashf1lem1 14420* | Lemma for hashf1 14423. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 14-Aug-2024.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π΄ β Fin) & β’ (π β π΅ β Fin) & β’ (π β Β¬ π§ β π΄) & β’ (π β ((β―βπ΄) + 1) β€ (β―βπ΅)) & β’ (π β πΉ:π΄β1-1βπ΅) β β’ (π β {π β£ ((π βΎ π΄) = πΉ β§ π:(π΄ βͺ {π§})β1-1βπ΅)} β (π΅ β ran πΉ)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashf1lem1OLD 14421* | Obsolete version of hashf1lem1 14420 as of 7-Aug-2024. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π΄ β Fin) & β’ (π β π΅ β Fin) & β’ (π β Β¬ π§ β π΄) & β’ (π β ((β―βπ΄) + 1) β€ (β―βπ΅)) & β’ (π β πΉ:π΄β1-1βπ΅) β β’ (π β {π β£ ((π βΎ π΄) = πΉ β§ π:(π΄ βͺ {π§})β1-1βπ΅)} β (π΅ β ran πΉ)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashf1lem2 14422* | Lemma for hashf1 14423. (Contributed by Mario Carneiro, 17-Apr-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π΄ β Fin) & β’ (π β π΅ β Fin) & β’ (π β Β¬ π§ β π΄) & β’ (π β ((β―βπ΄) + 1) β€ (β―βπ΅)) β β’ (π β (β―β{π β£ π:(π΄ βͺ {π§})β1-1βπ΅}) = (((β―βπ΅) β (β―βπ΄)) Β· (β―β{π β£ π:π΄β1-1βπ΅}))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashf1 14423* | The permutation number β£ π΄ β£ ! Β· ( β£ π΅ β£ C β£ π΄ β£ ) = β£ π΅ β£ ! / ( β£ π΅ β£ β β£ π΄ β£ )! counts the number of injections from π΄ to π΅. (Contributed by Mario Carneiro, 21-Jan-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π΄ β Fin β§ π΅ β Fin) β (β―β{π β£ π:π΄β1-1βπ΅}) = ((!β(β―βπ΄)) Β· ((β―βπ΅)C(β―βπ΄)))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashfac 14424* | A factorial counts the number of bijections on a finite set. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by Mario Carneiro, 17-Apr-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π΄ β Fin β (β―β{π β£ π:π΄β1-1-ontoβπ΄}) = (!β(β―βπ΄))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | leiso 14425 | Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π΄ β β* β§ π΅ β β*) β (πΉ Isom < , < (π΄, π΅) β πΉ Isom β€ , β€ (π΄, π΅))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | leisorel 14426 | Version of isorel 7326 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((πΉ Isom < , < (π΄, π΅) β§ (π΄ β β* β§ π΅ β β*) β§ (πΆ β π΄ β§ π· β π΄)) β (πΆ β€ π· β (πΉβπΆ) β€ (πΉβπ·))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fz1isolem 14427* | Lemma for fz1iso 14428. (Contributed by Mario Carneiro, 2-Apr-2014.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ πΊ = (rec((π β V β¦ (π + 1)), 1) βΎ Ο) & β’ π΅ = (β β© (β‘ < β {((β―βπ΄) + 1)})) & β’ πΆ = (Ο β© (β‘πΊβ((β―βπ΄) + 1))) & β’ π = OrdIso(π , π΄) β β’ ((π Or π΄ β§ π΄ β Fin) β βπ π Isom < , π ((1...(β―βπ΄)), π΄)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fz1iso 14428* | Any finite ordered set has an order isomorphism to a one-based finite sequence. (Contributed by Mario Carneiro, 2-Apr-2014.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π Or π΄ β§ π΄ β Fin) β βπ π Isom < , π ((1...(β―βπ΄)), π΄)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | ishashinf 14429* | Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 9264. (Contributed by Thierry Arnoux, 5-Jul-2017.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (Β¬ π΄ β Fin β βπ β β βπ₯ β π« π΄(β―βπ₯) = π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | seqcoll 14430* | The function πΉ contains a sparse set of nonzero values to be summed. The function πΊ is an order isomorphism from the set of nonzero values of πΉ to a 1-based finite sequence, and π» collects these nonzero values together. Under these conditions, the sum over the values in π» yields the same result as the sum over the original set πΉ. (Contributed by Mario Carneiro, 2-Apr-2014.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β§ π β π) β (π + π) = π) & β’ ((π β§ π β π) β (π + π) = π) & β’ ((π β§ (π β π β§ π β π)) β (π + π) β π) & β’ (π β π β π) & β’ (π β πΊ Isom < , < ((1...(β―βπ΄)), π΄)) & β’ (π β π β (1...(β―βπ΄))) & β’ (π β π΄ β (β€β₯βπ)) & β’ ((π β§ π β (π...(πΊβ(β―βπ΄)))) β (πΉβπ) β π) & β’ ((π β§ π β ((π...(πΊβ(β―βπ΄))) β π΄)) β (πΉβπ) = π) & β’ ((π β§ π β (1...(β―βπ΄))) β (π»βπ) = (πΉβ(πΊβπ))) β β’ (π β (seqπ( + , πΉ)β(πΊβπ)) = (seq1( + , π»)βπ)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | seqcoll2 14431* | The function πΉ contains a sparse set of nonzero values to be summed. The function πΊ is an order isomorphism from the set of nonzero values of πΉ to a 1-based finite sequence, and π» collects these nonzero values together. Under these conditions, the sum over the values in π» yields the same result as the sum over the original set πΉ. (Contributed by Mario Carneiro, 13-Dec-2014.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β§ π β π) β (π + π) = π) & β’ ((π β§ π β π) β (π + π) = π) & β’ ((π β§ (π β π β§ π β π)) β (π + π) β π) & β’ (π β π β π) & β’ (π β πΊ Isom < , < ((1...(β―βπ΄)), π΄)) & β’ (π β π΄ β β ) & β’ (π β π΄ β (π...π)) & β’ ((π β§ π β (π...π)) β (πΉβπ) β π) & β’ ((π β§ π β ((π...π) β π΄)) β (πΉβπ) = π) & β’ ((π β§ π β (1...(β―βπ΄))) β (π»βπ) = (πΉβ(πΊβπ))) β β’ (π β (seqπ( + , πΉ)βπ) = (seq1( + , π»)β(β―βπ΄))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | phphashd 14432 | Corollary of the Pigeonhole Principle using equality. Equivalent of phpeqd 9219 expressed using the hash function. (Contributed by Rohan Ridenour, 3-Aug-2023.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π΄ β Fin) & β’ (π β π΅ β π΄) & β’ (π β (β―βπ΄) = (β―βπ΅)) β β’ (π β π΄ = π΅) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | phphashrd 14433 | Corollary of the Pigeonhole Principle using equality. Equivalent of phphashd 14432 with reversed arguments. (Contributed by Rohan Ridenour, 3-Aug-2023.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π΅ β Fin) & β’ (π β π΄ β π΅) & β’ (π β (β―βπ΄) = (β―βπ΅)) β β’ (π β π΄ = π΅) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashprlei 14434 | An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ({π΄, π΅} β Fin β§ (β―β{π΄, π΅}) β€ 2) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hash2pr 14435* | A set of size two is an unordered pair. (Contributed by Alexander van der Vekens, 8-Dec-2017.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β π β§ (β―βπ) = 2) β βπβπ π = {π, π}) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hash2prde 14436* | A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β π β§ (β―βπ) = 2) β βπβπ(π β π β§ π = {π, π})) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hash2exprb 14437* | A set of size two is an unordered pair if and only if it contains two different elements. (Contributed by Alexander van der Vekens, 14-Jan-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π β ((β―βπ) = 2 β βπβπ(π β π β§ π = {π, π}))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hash2prb 14438* | A set of size two is a proper unordered pair. (Contributed by AV, 1-Nov-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π β ((β―βπ) = 2 β βπ β π βπ β π (π β π β§ π = {π, π}))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | prprrab 14439 | The set of proper pairs of elements of a given set expressed in two ways. (Contributed by AV, 24-Nov-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ {π₯ β (π« π΄ β {β }) β£ (β―βπ₯) = 2} = {π₯ β π« π΄ β£ (β―βπ₯) = 2} | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | nehash2 14440 | The cardinality of a set with two distinct elements. (Contributed by Thierry Arnoux, 27-Aug-2019.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π β π) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β π΄ β π΅) β β’ (π β 2 β€ (β―βπ)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hash2prd 14441 | A set of size two is an unordered pair if it contains two different elements. (Contributed by Alexander van der Vekens, 9-Dec-2018.) (Proof shortened by AV, 16-Jun-2022.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β π β§ (β―βπ) = 2) β ((π β π β§ π β π β§ π β π) β π = {π, π})) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hash2pwpr 14442 | If the size of a subset of an unordered pair is 2, the subset is the pair itself. (Contributed by Alexander van der Vekens, 9-Dec-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (((β―βπ) = 2 β§ π β π« {π, π}) β π = {π, π}) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashle2pr 14443* | A nonempty set of size less than or equal to two is an unordered pair of sets. (Contributed by AV, 24-Nov-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β π β§ π β β ) β ((β―βπ) β€ 2 β βπβπ π = {π, π})) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashle2prv 14444* | A nonempty subset of a powerset of a class π has size less than or equal to two iff it is an unordered pair of elements of π. (Contributed by AV, 24-Nov-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β (π« π β {β }) β ((β―βπ) β€ 2 β βπ β π βπ β π π = {π, π})) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | pr2pwpr 14445* | The set of subsets of a pair having length 2 is the set of the pair as singleton. (Contributed by AV, 9-Dec-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π΄ β π β§ π΅ β π β§ π΄ β π΅) β {π β π« {π΄, π΅} β£ π β 2o} = {{π΄, π΅}}) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashge2el2dif 14446* | A set with size at least 2 has at least 2 different elements. (Contributed by AV, 18-Mar-2019.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π· β π β§ 2 β€ (β―βπ·)) β βπ₯ β π· βπ¦ β π· π₯ β π¦) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashge2el2difr 14447* | A set with at least 2 different elements has size at least 2. (Contributed by AV, 14-Oct-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π· β π β§ βπ₯ β π· βπ¦ β π· π₯ β π¦) β 2 β€ (β―βπ·)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashge2el2difb 14448* | A set has size at least 2 iff it has at least 2 different elements. (Contributed by AV, 14-Oct-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π· β π β (2 β€ (β―βπ·) β βπ₯ β π· βπ¦ β π· π₯ β π¦)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashdmpropge2 14449 | The size of the domain of a class which contains two ordered pairs with different first components is greater than or equal to 2. (Contributed by AV, 12-Nov-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) & β’ (π β πΉ β π) & β’ (π β π΄ β π΅) & β’ (π β {β¨π΄, πΆβ©, β¨π΅, π·β©} β πΉ) β β’ (π β 2 β€ (β―βdom πΉ)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashtplei 14450 | An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ({π΄, π΅, πΆ} β Fin β§ (β―β{π΄, π΅, πΆ}) β€ 3) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashtpg 14451 | The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π΄ β π β§ π΅ β π β§ πΆ β π) β ((π΄ β π΅ β§ π΅ β πΆ β§ πΆ β π΄) β (β―β{π΄, π΅, πΆ}) = 3)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashge3el3dif 14452* | A set with size at least 3 has at least 3 different elements. In contrast to hashge2el2dif 14446, which has an elementary proof, the dominance relation and 1-1 functions from a set with three elements which are known to be different are used to prove this theorem. Although there is also an elementary proof for this theorem, it might be much longer. After all, this proof should be kept because it can be used as template for proofs for higher cardinalities. (Contributed by AV, 20-Mar-2019.) (Proof modification is discouraged.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π· β π β§ 3 β€ (β―βπ·)) β βπ₯ β π· βπ¦ β π· βπ§ β π· (π₯ β π¦ β§ π₯ β π§ β§ π¦ β π§)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | elss2prb 14453* | An element of the set of subsets with two elements is a proper unordered pair. (Contributed by AV, 1-Nov-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β {π§ β π« π β£ (β―βπ§) = 2} β βπ₯ β π βπ¦ β π (π₯ β π¦ β§ π = {π₯, π¦})) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hash2sspr 14454* | A subset of size two is an unordered pair of elements of its superset. (Contributed by Alexander van der Vekens, 12-Jul-2017.) (Proof shortened by AV, 4-Nov-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β π« π β§ (β―βπ) = 2) β βπ β π βπ β π π = {π, π}) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | exprelprel 14455* | If there is an element of the set of subsets with two elements in a set, an unordered pair of sets is in the set. (Contributed by Alexander van der Vekens, 12-Jul-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (βπ β {π β π« π β£ (β―βπ) = 2}π β π β βπ£ β π βπ€ β π {π£, π€} β π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hash3tr 14456* | A set of size three is an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β π β§ (β―βπ) = 3) β βπβπβπ π = {π, π, π}) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hash1to3 14457* | If the size of a set is between 1 and 3 (inclusively), the set is a singleton or an unordered pair or an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β Fin β§ 1 β€ (β―βπ) β§ (β―βπ) β€ 3) β βπβπβπ(π = {π} β¨ π = {π, π} β¨ π = {π, π, π})) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fundmge2nop0 14458 | A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fundmge2nop 14459 (with the less restrictive requirement that (πΊ β {β }) needs to be a function instead of πΊ) is useful for proofs for extensible structures, see structn0fun 17089. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((Fun (πΊ β {β }) β§ 2 β€ (β―βdom πΊ)) β Β¬ πΊ β (V Γ V)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fundmge2nop 14459 | A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.) (Proof shortened by AV, 9-Jun-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((Fun πΊ β§ 2 β€ (β―βdom πΊ)) β Β¬ πΊ β (V Γ V)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fun2dmnop0 14460 | A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 14461 (with the less restrictive requirement that (πΊ β {β }) needs to be a function instead of πΊ) is useful for proofs for extensible structures, see structn0fun 17089. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ π΄ β V & β’ π΅ β V β β’ ((Fun (πΊ β {β }) β§ π΄ β π΅ β§ {π΄, π΅} β dom πΊ) β Β¬ πΊ β (V Γ V)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fun2dmnop 14461 | A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 9-Jun-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ π΄ β V & β’ π΅ β V β β’ ((Fun πΊ β§ π΄ β π΅ β§ {π΄, π΅} β dom πΊ) β Β¬ πΊ β (V Γ V)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | hashdifsnp1 14462 | If the size of a set is a nonnegative integer increased by 1, the size of the set with one of its elements removed is this nonnegative integer. (Contributed by Alexander van der Vekens, 7-Jan-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β π β§ π β π β§ π β β0) β ((β―βπ) = (π + 1) β (β―β(π β {π})) = π)) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | fi1uzind 14463* | Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, usually with πΏ = 0 (see opfi1ind 14468) or πΏ = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ πΉ β V & β’ πΏ β β0 & β’ ((π£ = π β§ π = πΈ) β (π β π)) & β’ ((π£ = π€ β§ π = π) β (π β π)) & β’ (([π£ / π][π / π]π β§ π β π£) β [(π£ β {π}) / π][πΉ / π]π) & β’ ((π€ = (π£ β {π}) β§ π = πΉ) β (π β π)) & β’ (([π£ / π][π / π]π β§ (β―βπ£) = πΏ) β π) & β’ ((((π¦ + 1) β β0 β§ ([π£ / π][π / π]π β§ (β―βπ£) = (π¦ + 1) β§ π β π£)) β§ π) β π) β β’ (([π / π][πΈ / π]π β§ π β Fin β§ πΏ β€ (β―βπ)) β π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | brfi1uzind 14464* | Properties of a binary relation with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, usually with πΏ = 0 (see brfi1ind 14465) or πΏ = 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (Proof shortened by AV, 23-Oct-2020.) (Revised by AV, 28-Mar-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ Rel πΊ & β’ πΉ β V & β’ πΏ β β0 & β’ ((π£ = π β§ π = πΈ) β (π β π)) & β’ ((π£ = π€ β§ π = π) β (π β π)) & β’ ((π£πΊπ β§ π β π£) β (π£ β {π})πΊπΉ) & β’ ((π€ = (π£ β {π}) β§ π = πΉ) β (π β π)) & β’ ((π£πΊπ β§ (β―βπ£) = πΏ) β π) & β’ ((((π¦ + 1) β β0 β§ (π£πΊπ β§ (β―βπ£) = (π¦ + 1) β§ π β π£)) β§ π) β π) β β’ ((ππΊπΈ β§ π β Fin β§ πΏ β€ (β―βπ)) β π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | brfi1ind 14465* | Properties of a binary relation with a finite first component, proven by finite induction on the size of the first component. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (Revised by AV, 28-Mar-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ Rel πΊ & β’ πΉ β V & β’ ((π£ = π β§ π = πΈ) β (π β π)) & β’ ((π£ = π€ β§ π = π) β (π β π)) & β’ ((π£πΊπ β§ π β π£) β (π£ β {π})πΊπΉ) & β’ ((π€ = (π£ β {π}) β§ π = πΉ) β (π β π)) & β’ ((π£πΊπ β§ (β―βπ£) = 0) β π) & β’ ((((π¦ + 1) β β0 β§ (π£πΊπ β§ (β―βπ£) = (π¦ + 1) β§ π β π£)) β§ π) β π) β β’ ((ππΊπΈ β§ π β Fin) β π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | brfi1indALT 14466* | Alternate proof of brfi1ind 14465, which does not use brfi1uzind 14464. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ Rel πΊ & β’ πΉ β V & β’ ((π£ = π β§ π = πΈ) β (π β π)) & β’ ((π£ = π€ β§ π = π) β (π β π)) & β’ ((π£πΊπ β§ π β π£) β (π£ β {π})πΊπΉ) & β’ ((π€ = (π£ β {π}) β§ π = πΉ) β (π β π)) & β’ ((π£πΊπ β§ (β―βπ£) = 0) β π) & β’ ((((π¦ + 1) β β0 β§ (π£πΊπ β§ (β―βπ£) = (π¦ + 1) β§ π β π£)) β§ π) β π) β β’ ((ππΊπΈ β§ π β Fin) β π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | opfi1uzind 14467* | Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, usually with πΏ = 0 (see opfi1ind 14468) or πΏ = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ πΈ β V & β’ πΉ β V & β’ πΏ β β0 & β’ ((π£ = π β§ π = πΈ) β (π β π)) & β’ ((π£ = π€ β§ π = π) β (π β π)) & β’ ((β¨π£, πβ© β πΊ β§ π β π£) β β¨(π£ β {π}), πΉβ© β πΊ) & β’ ((π€ = (π£ β {π}) β§ π = πΉ) β (π β π)) & β’ ((β¨π£, πβ© β πΊ β§ (β―βπ£) = πΏ) β π) & β’ ((((π¦ + 1) β β0 β§ (β¨π£, πβ© β πΊ β§ (β―βπ£) = (π¦ + 1) β§ π β π£)) β§ π) β π) β β’ ((β¨π, πΈβ© β πΊ β§ π β Fin β§ πΏ β€ (β―βπ)) β π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | opfi1ind 14468* | Properties of an ordered pair with a finite first component, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, e.g., fusgrfis 28855. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ πΈ β V & β’ πΉ β V & β’ ((π£ = π β§ π = πΈ) β (π β π)) & β’ ((π£ = π€ β§ π = π) β (π β π)) & β’ ((β¨π£, πβ© β πΊ β§ π β π£) β β¨(π£ β {π}), πΉβ© β πΊ) & β’ ((π€ = (π£ β {π}) β§ π = πΉ) β (π β π)) & β’ ((β¨π£, πβ© β πΊ β§ (β―βπ£) = 0) β π) & β’ ((((π¦ + 1) β β0 β§ (β¨π£, πβ© β πΊ β§ (β―βπ£) = (π¦ + 1) β§ π β π£)) β§ π) β π) β β’ ((β¨π, πΈβ© β πΊ β§ π β Fin) β π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
This section is about words (or strings) over a set (alphabet) defined as finite sequences of symbols (or characters) being elements of the alphabet. Although it is often required that the underlying set/alphabet be nonempty, finite and not a proper class, these restrictions are not made in the current definition df-word 14470, see, for example, s1cli 14560: β¨βπ΄ββ© β Word V. Note that the empty word β (i.e., the empty set) is the only word over an empty alphabet, see 0wrd0 14495. The set Word π of words over π is the free monoid over π, where the monoid law is concatenation and the monoid unit is the empty word. Besides the definition of words themselves, several operations on words are defined in this section:
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Syntax | cword 14469 | Syntax for the Word operator. | ||||||||||||||||||||||||||||||||||||||||||||||||||
class Word π | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Definition | df-word 14470* | Define the class of words over a set. A word (sometimes also called a string) is a finite sequence of symbols from a set (alphabet) π. Definition in Section 9.1 of [AhoHopUll] p. 318. The domain is forced to be an initial segment of β0 so that two words with the same symbols in the same order be equal. The set Word π is sometimes denoted by S*, using the Kleene star, although the Kleene star, or Kleene closure, is sometimes reserved to denote an operation on languages. The set Word π equipped with concatenation is the free monoid over π, and the monoid unit is the empty word (see frmdval 18769). (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ Word π = {π€ β£ βπ β β0 π€:(0..^π)βΆπ} | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | iswrd 14471* | Property of being a word over a set with an existential quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π β βπ β β0 π:(0..^π)βΆπ) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdval 14472* | Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π β Word π = βͺ π β β0 (π βm (0..^π))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | iswrdi 14473 | A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π:(0..^πΏ)βΆπ β π β Word π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdf 14474 | A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π β π:(0..^(β―βπ))βΆπ) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | iswrdb 14475 | A word over an alphabet is a function from an open range of nonnegative integers (of length equal to the length of the word) into the alphabet. (Contributed by Alexander van der Vekens, 30-Jul-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π β π:(0..^(β―βπ))βΆπ) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrddm 14476 | The indices of a word (i.e. its domain regarded as function) are elements of an open range of nonnegative integers (of length equal to the length of the word). (Contributed by AV, 2-May-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π β dom π = (0..^(β―βπ))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | sswrd 14477 | The set of words respects ordering on the base set. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π β Word π β Word π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | snopiswrd 14478 | A singleton of an ordered pair (with 0 as first component) is a word. (Contributed by AV, 23-Nov-2018.) (Proof shortened by AV, 18-Apr-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π β {β¨0, πβ©} β Word π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdexg 14479 | The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by JJ, 18-Nov-2022.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π β Word π β V) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdexb 14480 | The set of words over a set is a set, bidirectional version. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 23-Nov-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β V β Word π β V) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdexi 14481 | The set of words over a set is a set, inference form. (Contributed by AV, 23-May-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ π β V β β’ Word π β V | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdsymbcl 14482 | A symbol within a word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β Word π β§ πΌ β (0..^(β―βπ))) β (πβπΌ) β π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdfn 14483 | A word is a function with a zero-based sequence of integers as domain. (Contributed by Alexander van der Vekens, 13-Apr-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π β π Fn (0..^(β―βπ))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdv 14484 | A word over an alphabet is a word over the universal class. (Contributed by AV, 8-Feb-2021.) (Proof shortened by JJ, 18-Nov-2022.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π β π β Word V) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdlndm 14485 | The length of a word is not in the domain of the word (regarded as a function). (Contributed by AV, 3-Mar-2021.) (Proof shortened by JJ, 18-Nov-2022.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π β (β―βπ) β dom π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | iswrdsymb 14486* | An arbitrary word is a word over an alphabet if all of its symbols belong to the alphabet. (Contributed by AV, 23-Jan-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β Word V β§ βπ β (0..^(β―βπ))(πβπ) β π) β π β Word π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdfin 14487 | A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.) (Proof shortened by AV, 18-Nov-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π β π β Fin) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | lencl 14488 | The length of a word is a nonnegative integer. This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 27-Aug-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π β (β―βπ) β β0) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | lennncl 14489 | The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β Word π β§ π β β ) β (β―βπ) β β) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdffz 14490 | A word is a function from a finite interval of integers. (Contributed by AV, 10-Feb-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π β π:(0...((β―βπ) β 1))βΆπ) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdeq 14491 | Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π = π β Word π = Word π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdeqi 14492 | Equality theorem for the set of words, inference form. (Contributed by AV, 23-May-2021.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ π = π β β’ Word π = Word π | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | iswrddm0 14493 | A function with empty domain is a word. (Contributed by AV, 13-Oct-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π:β βΆπ β π β Word π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrd0 14494 | The empty set is a word (the empty word, frequently denoted Ξ΅ in this context). This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 13-May-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ β β Word π | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | 0wrd0 14495 | The empty word is the only word over an empty alphabet. (Contributed by AV, 25-Oct-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word β β π = β ) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | ffz0iswrd 14496 | A sequence with zero-based indices is a word. (Contributed by AV, 31-Jan-2018.) (Proof shortened by AV, 13-Oct-2018.) (Proof shortened by JJ, 18-Nov-2022.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π:(0...πΏ)βΆπ β π β Word π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdsymb 14497 | A word is a word over the symbols it consists of. (Contributed by AV, 1-Dec-2022.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β Word π΄ β π β Word (π β (0..^(β―βπ)))) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | nfwrd 14498 | Hypothesis builder for Word π. (Contributed by Mario Carneiro, 26-Feb-2016.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ β²π₯π β β’ β²π₯Word π | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | csbwrdg 14499* | Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ (π β π β β¦π / π₯β¦Word π₯ = Word π) | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Theorem | wrdnval 14500* | Words of a fixed length are mappings from a fixed half-open integer interval. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Proof shortened by AV, 13-May-2020.) | ||||||||||||||||||||||||||||||||||||||||||||||||||
β’ ((π β π β§ π β β0) β {π€ β Word π β£ (β―βπ€) = π} = (π βm (0..^π))) |
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