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Theorem List for Metamath Proof Explorer - 11401-11500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhashxplem 11401 Lemma for hashxp 11402. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  B  e.  Fin   =>    |-  ( A  e.  Fin  ->  ( # `  ( A  X.  B ) )  =  ( ( # `  A )  x.  ( # `
  B ) ) )
 
Theoremhashxp 11402 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  X.  B ) )  =  ( ( # `  A )  x.  ( # `
  B ) ) )
 
Theoremhashmap 11403 The size of the set exponential of two finite sets is the exponential of their sizes. (This is the original motivation behind the notation for set exponentiation.) (Contributed by Mario Carneiro, 5-Aug-2014.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  ( A  ^m  B ) )  =  ( ( # `  A ) ^ ( # `
  B ) ) )
 
Theoremhashpw 11404 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.)
 |-  ( A  e.  Fin  ->  ( # `  ~P A )  =  ( 2 ^ ( # `  A ) ) )
 
Theoremhashfun 11405 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.)
 |-  ( F  e.  Fin  ->  ( Fun  F  <->  ( # `  F )  =  ( # `  dom  F ) ) )
 
Theoremhashbclem 11406* Lemma for hashbc 11407: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  -.  z  e.  A )   &    |-  ( ph  ->  A. j  e. 
 ZZ  ( ( # `  A )  _C  j
 )  =  ( # ` 
 { x  e.  ~P A  |  ( # `  x )  =  j }
 ) )   &    |-  ( ph  ->  K  e.  ZZ )   =>    |-  ( ph  ->  ( ( # `  ( A  u.  { z }
 ) )  _C  K )  =  ( # `  { x  e.  ~P ( A  u.  { z } )  |  ( # `  x )  =  K }
 ) )
 
Theoremhashbc 11407* The binomial coefficient counts the number of subsets of a finite set of a given size. (Contributed by Mario Carneiro, 13-Jul-2014.)
 |-  ( ( A  e.  Fin  /\  K  e.  ZZ )  ->  ( ( # `  A )  _C  K )  =  ( # `  { x  e.  ~P A  |  ( # `  x )  =  K } ) )
 
Theoremhashfacen 11408* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  { f  |  f : A -1-1-onto-> C }  ~~  { f  |  f : B -1-1-onto-> D } )
 
Theoremhashf1lem1 11409* Lemma for hashf1 11411. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  -.  z  e.  A )   &    |-  ( ph  ->  ( ( # `  A )  +  1 )  <_  ( # `  B ) )   &    |-  ( ph  ->  F : A -1-1-> B )   =>    |-  ( ph  ->  { f  |  ( ( f  |`  A )  =  F  /\  f : ( A  u.  { z }
 ) -1-1-> B ) }  ~~  ( B  \  ran  F ) )
 
Theoremhashf1lem2 11410* Lemma for hashf1 11411. (Contributed by Mario Carneiro, 17-Apr-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  -.  z  e.  A )   &    |-  ( ph  ->  ( ( # `  A )  +  1 )  <_  ( # `  B ) )   =>    |-  ( ph  ->  ( # `
  { f  |  f : ( A  u.  { z }
 ) -1-1-> B } )  =  ( ( ( # `  B )  -  ( # `
  A ) )  x.  ( # `  { f  |  f : A -1-1-> B } ) ) )
 
Theoremhashf1 11411* The permutation number  |  A  |  !  x.  (  |  B  |  _C  |  A  | 
)  =  |  B  |  !  /  (  |  B  |  -  |  A  | 
) ! counts the number of injections from  A to  B. (Contributed by Mario Carneiro, 21-Jan-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( # `  { f  |  f : A -1-1-> B } )  =  (
 ( ! `  ( # `
  A ) )  x.  ( ( # `  B )  _C  ( # `
  A ) ) ) )
 
Theoremhashfac 11412* 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.)
 |-  ( A  e.  Fin  ->  ( # `  { f  |  f : A -1-1-onto-> A } )  =  ( ! `  ( # `
  A ) ) )
 
Theoremleiso 11413 Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( A  C_  RR*  /\  B  C_  RR* )  ->  ( F  Isom  <  ,  <  ( A ,  B ) 
 <->  F  Isom  <_  ,  <_  ( A ,  B ) ) )
 
Theoremleisorel 11414 Version of isorel 5839 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A 
 C_  RR*  /\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C  <_  D  <->  ( F `  C )  <_  ( F `
  D ) ) )
 
Theoremfz1isolem 11415* Lemma for fz1iso 11416. (Contributed by Mario Carneiro, 2-Apr-2014.)
 |-  G  =  ( rec ( ( n  e. 
 _V  |->  ( n  +  1 ) ) ,  1 )  |`  om )   &    |-  B  =  ( NN  i^i  ( `'  <  " { ( ( # `  A )  +  1 ) } )
 )   &    |-  C  =  ( om  i^i  ( `' G `  ( ( # `  A )  +  1 )
 ) )   &    |-  O  = OrdIso ( R ,  A )   =>    |-  (
 ( R  Or  A  /\  A  e.  Fin )  ->  E. f  f  Isom  <  ,  R  ( (
 1 ... ( # `  A ) ) ,  A ) )
 
Theoremfz1iso 11416* Any finite ordered set has an order isometry to a one-based finite sequence. (Contributed by Mario Carneiro, 2-Apr-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  E. f  f  Isom  <  ,  R  ( (
 1 ... ( # `  A ) ) ,  A ) )
 
Theoremseqcoll 11417* The function  F contains a sparse set of non-zero values to be summed. The function  G is an order isomorphism from the set of non-zero values of  F to a 1-based finite sequence, and  H collects these non-zero values together. Under these conditions, the sum over the values in  H yields the same result as the sum over the original set  F. (Contributed by Mario Carneiro, 2-Apr-2014.)
 |-  ( ( ph  /\  k  e.  S )  ->  ( Z  .+  k )  =  k )   &    |-  ( ( ph  /\  k  e.  S ) 
 ->  ( k  .+  Z )  =  k )   &    |-  (
 ( ph  /\  ( k  e.  S  /\  n  e.  S ) )  ->  ( k  .+  n )  e.  S )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  G  Isom  <  ,  <  (
 ( 1 ... ( # `
  A ) ) ,  A ) )   &    |-  ( ph  ->  N  e.  ( 1 ... ( # `
  A ) ) )   &    |-  ( ph  ->  A 
 C_  ( ZZ>= `  M ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... ( G `  ( # `  A ) ) ) )  ->  ( F `  k )  e.  S )   &    |-  (
 ( ph  /\  k  e.  ( ( M ... ( G `  ( # `  A ) ) ) 
 \  A ) ) 
 ->  ( F `  k
 )  =  Z )   &    |-  ( ( ph  /\  n  e.  ( 1 ... ( # `
  A ) ) )  ->  ( H `  n )  =  ( F `  ( G `
  n ) ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  ( G `  N ) )  =  (  seq  1
 (  .+  ,  H ) `  N ) )
 
Theoremseqcoll2 11418* The function  F contains a sparse set of non-zero values to be summed. The function  G is an order isomorphism from the set of non-zero values of  F to a 1-based finite sequence, and  H collects these non-zero values together. Under these conditions, the sum over the values in  H yields the same result as the sum over the original set  F. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  ( ( ph  /\  k  e.  S )  ->  ( Z  .+  k )  =  k )   &    |-  ( ( ph  /\  k  e.  S ) 
 ->  ( k  .+  Z )  =  k )   &    |-  (
 ( ph  /\  ( k  e.  S  /\  n  e.  S ) )  ->  ( k  .+  n )  e.  S )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  G  Isom  <  ,  <  (
 ( 1 ... ( # `
  A ) ) ,  A ) )   &    |-  ( ph  ->  A  =/=  (/) )   &    |-  ( ph  ->  A 
 C_  ( M ... N ) )   &    |-  ( ( ph  /\  k  e.  ( M
 ... N ) ) 
 ->  ( F `  k
 )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ( M ... N )  \  A ) )  ->  ( F `  k )  =  Z )   &    |-  ( ( ph  /\  n  e.  ( 1 ... ( # `
  A ) ) )  ->  ( H `  n )  =  ( F `  ( G `
  n ) ) )   =>    |-  ( ph  ->  (  seq  M (  .+  ,  F ) `  N )  =  (  seq  1 (  .+  ,  H ) `  ( # `  A ) ) )
 
5.6.9  Words over a set
 
Syntaxcword 11419 Syntax for the Word operator.
 class Word  S
 
Syntaxcconcat 11420 Syntax for the concatenation operator.
 class concat
 
Syntaxcs1 11421 Syntax for the singleton word constructor.
 class  <" A ">
 
Syntaxcsubstr 11422 Syntax for the word slicing operator.
 class substr
 
Syntaxcsplice 11423 Syntax for the word splicing operator.
 class splice
 
Syntaxcreverse 11424 Syntax for the word reverse operator.
 class reverse
 
Definitiondf-word 11425* Define the class of words over a set. A word is a finite sequence of symbols from a set. The domain is forced so that two words with the same symbols in the same order will be the same. This is sometimes denoted with the Kleene star, although properly speaking that is an operator on languages. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |- Word  S  =  { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> S }
 
Definitiondf-concat 11426* Define the concatenation operator which combines two words. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 15-Aug-2015.)
 |- concat  =  ( s  e.  _V ,  t  e.  _V  |->  ( x  e.  (
 0..^ ( ( # `  s )  +  ( # `
  t ) ) )  |->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x ) ,  ( t `  ( x  -  ( # `
  s ) ) ) ) ) )
 
Definitiondf-s1 11427 Define the canonical injection from symbols to words. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A ">  =  { <. 0 ,  (  _I  `  A ) >. }
 
Definitiondf-substr 11428* Define an operation which extracts portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |- substr  =  ( s  e.  _V ,  b  e.  ( ZZ  X.  ZZ )  |->  if ( ( ( 1st `  b )..^ ( 2nd `  b ) )  C_  dom  s ,  ( x  e.  ( 0..^ ( ( 2nd `  b
 )  -  ( 1st `  b ) ) ) 
 |->  ( s `  ( x  +  ( 1st `  b ) ) ) ) ,  (/) ) )
 
Definitiondf-splice 11429* Define an operation which replaces portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |- splice  =  ( s  e.  _V ,  b  e.  _V  |->  ( ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
 ) ) >. ) concat  ( 2nd `  b ) ) concat 
 ( s substr  <. ( 2nd `  ( 1st `  b
 ) ) ,  ( # `
  s ) >. ) ) )
 
Definitiondf-reverse 11430* Define an operation which reverses the order of symbols in a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |- reverse  =  ( s  e.  _V  |->  ( x  e.  (
 0..^ ( # `  s
 ) )  |->  ( s `
  ( ( ( # `  s )  -  1 )  -  x ) ) ) )
 
Theoremiswrd 11431* Property of being a word over a set with a quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( W  e. Word  S  <->  E. l  e.  NN0  W : ( 0..^ l ) --> S )
 
Theoremwrdval 11432* Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  e.  V  -> Word 
 S  =  U_ l  e.  NN0  ( S  ^m  ( 0..^ l ) ) )
 
Theoremiswrdi 11433 A one-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( W : ( 0..^ L ) --> S  ->  W  e. Word  S )
 
Theoremwrd0 11434 The empty set is a word (frequently denoted ε in this context). (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  (/)  e. Word  S
 
Theoremwrdf 11435 A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( W  e. Word  S  ->  W : ( 0..^ ( # `  W ) ) --> S )
 
Theoremwrdfin 11436 A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.)
 |-  ( W  e. Word  S  ->  W  e.  Fin )
 
Theoremlencl 11437 The length of a word is a nonnegative integer. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( W  e. Word  S  ->  ( # `  W )  e.  NN0 )
 
Theoremlennncl 11438 The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( W  e. Word  S 
 /\  W  =/=  (/) )  ->  ( # `  W )  e.  NN )
 
Theoremsswrd 11439 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.)
 |-  ( S  C_  T  -> Word 
 S  C_ Word  T )
 
Theoremwrdeq 11440 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  =  T  -> Word 
 S  = Word  T )
 
Theoremwrdexg 11441 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  e.  V  -> Word 
 S  e.  _V )
 
Theoremnfwrd 11442 Hypothesis builder for Word  S. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  F/_ x S   =>    |-  F/_ xWord  S
 
Theoremccatfn 11443 The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |- concat  Fn  ( _V  X.  _V )
 
Theoremccatfval 11444* Value of the concatenation operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S concat  T )  =  ( x  e.  ( 0..^ ( ( # `  S )  +  ( # `  T ) ) )  |->  if ( x  e.  ( 0..^ ( # `  S ) ) ,  ( S `
  x ) ,  ( T `  ( x  -  ( # `  S ) ) ) ) ) )
 
Theoremccatcl 11445 The concatenation of two words is a word. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( S  e. Word  B 
 /\  T  e. Word  B )  ->  ( S concat  T )  e. Word  B )
 
Theoremccatlen 11446 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( S  e. Word  B 
 /\  T  e. Word  B )  ->  ( # `  ( S concat  T ) )  =  ( ( # `  S )  +  ( # `  T ) ) )
 
Theoremccatval1 11447 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.)
 |-  ( ( S  e. Word  B 
 /\  T  e. Word  B  /\  I  e.  (
 0..^ ( # `  S ) ) )  ->  ( ( S concat  T ) `  I )  =  ( S `  I
 ) )
 
Theoremccatval2 11448 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  e. Word  B 
 /\  T  e. Word  B  /\  I  e.  (
 ( # `  S )..^ ( ( # `  S )  +  ( # `  T ) ) ) ) 
 ->  ( ( S concat  T ) `  I )  =  ( T `  ( I  -  ( # `  S ) ) ) )
 
Theoremccatval3 11449 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.)
 |-  ( ( S  e. Word  B 
 /\  T  e. Word  B  /\  I  e.  (
 0..^ ( # `  T ) ) )  ->  ( ( S concat  T ) `  ( I  +  ( # `  S ) ) )  =  ( T `  I ) )
 
Theoremccatlid 11450 Concatenation of a word by the empty word on the left. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( S  e. Word  B  ->  ( (/) concat  S )  =  S )
 
Theoremccatrid 11451 Concatenation of a word by the empty word on the right. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( S  e. Word  B  ->  ( S concat  (/) )  =  S )
 
Theoremccatass 11452 Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( S  e. Word  B 
 /\  T  e. Word  B  /\  U  e. Word  B )  ->  ( ( S concat  T ) concat  U )  =  ( S concat  ( T concat  U ) ) )
 
Theoremids1 11453 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A ">  = 
 <" (  _I  `  A ) ">
 
Theorems1val 11454 Value of a single-symbol word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( A  e.  V  -> 
 <" A ">  =  { <. 0 ,  A >. } )
 
Theorems1eq 11455 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( A  =  B  -> 
 <" A ">  = 
 <" B "> )
 
Theorems1eqd 11456 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <" A ">  =  <" B "> )
 
Theorems1cl 11457 A singleton word is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( A  e.  B  -> 
 <" A ">  e. Word  B )
 
Theorems1cld 11458 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  <" A ">  e. Word  B )
 
Theorems1cli 11459 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A ">  e. Word  _V
 
Theorems1len 11460 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A "> )  =  1
 
Theorems1nz 11461 A singleton is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |- 
 <" A ">  =/=  (/)
 
Theorems1fv 11462 Sole symbol of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( A  e.  B  ->  ( <" A "> `  0 )  =  A )
 
Theoremeqs1 11463 A word of length 1 is a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( ( W  e. Word  A 
 /\  ( # `  W )  =  1 )  ->  W  =  <" ( W `  0 ) "> )
 
Theorems111 11464 The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( ( S  e.  A  /\  T  e.  A )  ->  ( <" S ">  =  <" T "> 
 <->  S  =  T ) )
 
Theoremwrdexb 11465 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  e.  _V  <-> Word  S  e.  _V )
 
Theoremswrdval 11466* Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( S substr  <. F ,  L >. )  =  if ( ( F..^ L )  C_  dom  S ,  ( x  e.  (
 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F ) ) ) ,  (/) ) )
 
Theoremswrd00 11467 A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( S substr  <. X ,  X >. )  =  (/)
 
Theoremswrdcl 11468 Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  e. Word  A  ->  ( S substr  <. F ,  L >. )  e. Word  A )
 
Theoremswrdval2 11469* Value of the subword extractor in its intended domain. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( S  e. Word  A 
 /\  F  e.  (
 0 ... L )  /\  L  e.  ( 0 ... ( # `  S ) ) )  ->  ( S substr  <. F ,  L >. )  =  ( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `
  ( x  +  F ) ) ) )
 
Theoremswrd0val 11470 Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  ( ( S  e. Word  A 
 /\  L  e.  (
 0 ... ( # `  S ) ) )  ->  ( S substr  <. 0 ,  L >. )  =  ( S  |`  ( 0..^ L ) ) )
 
Theoremswrd0len 11471 Length of a left-anchored subword. (Contributed by Stefan O'Rear, 24-Aug-2015.)
 |-  ( ( S  e. Word  A 
 /\  L  e.  (
 0 ... ( # `  S ) ) )  ->  ( # `  ( S substr  <. 0 ,  L >. ) )  =  L )
 
Theoremswrdlen 11472 Length of an extracted subword. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  ( ( S  e. Word  A 
 /\  F  e.  (
 0 ... L )  /\  L  e.  ( 0 ... ( # `  S ) ) )  ->  ( # `  ( S substr  <. F ,  L >. ) )  =  ( L  -  F ) )
 
Theoremswrdfv 11473 A symbol in an extracted subword, indexed using the subword's indices. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  ( ( ( S  e. Word  A  /\  F  e.  ( 0 ... L )  /\  L  e.  (
 0 ... ( # `  S ) ) )  /\  X  e.  ( 0..^ ( L  -  F ) ) )  ->  ( ( S substr  <. F ,  L >. ) `  X )  =  ( S `  ( X  +  F ) ) )
 
Theoremswrdid 11474 A word is a subword of itself. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  ( # `  S ) >. )  =  S )
 
Theoremccatswrd 11475 Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
 |-  ( ( S  e. Word  A 
 /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  (
 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) ) 
 ->  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  =  ( S substr  <. X ,  Z >. ) )
 
Theoremswrdccat1 11476 Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( ( S  e. Word  B 
 /\  T  e. Word  B )  ->  ( ( S concat  T ) substr  <. 0 ,  ( # `  S ) >. )  =  S )
 
Theoremswrdccat2 11477 Recover the right half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( ( S  e. Word  B 
 /\  T  e. Word  B )  ->  ( ( S concat  T ) substr  <. ( # `  S ) ,  (
 ( # `  S )  +  ( # `  T ) ) >. )  =  T )
 
Theoremccatopth 11478 An opth 4261-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `  A )  =  ( # `  C ) ) 
 ->  ( ( A concat  B )  =  ( C concat  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremccatopth2 11479 An opth 4261-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( ( A  e. Word  X  /\  B  e. Word  X )  /\  ( C  e. Word  X  /\  D  e. Word  X )  /\  ( # `  B )  =  ( # `  D ) ) 
 ->  ( ( A concat  B )  =  ( C concat  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremccatlcan 11480 Concatenation of words is left-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ( A  e. Word  X 
 /\  B  e. Word  X  /\  C  e. Word  X )  ->  ( ( C concat  A )  =  ( C concat  B )  <->  A  =  B ) )
 
Theoremccatrcan 11481 Concatenation of words is right-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ( ( A  e. Word  X 
 /\  B  e. Word  X  /\  C  e. Word  X )  ->  ( ( A concat  C )  =  ( B concat  C )  <->  A  =  B ) )
 
Theoremsplval 11482 Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y ) )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  ( # `
  S ) >. ) ) )
 
Theoremsplcl 11483 Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  ( ( S  e. Word  A 
 /\  R  e. Word  A )  ->  ( S splice  <. F ,  T ,  R >. )  e. Word  A )
 
Theoremsplid 11484 Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.)
 |-  ( ( S  e. Word  A 
 /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  (
 0 ... ( # `  S ) ) ) ) 
 ->  ( S splice  <. X ,  Y ,  ( S substr  <. X ,  Y >. )
 >. )  =  S )
 
Theoremspllen 11485 The length of a splice. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( ph  ->  S  e. Word  A )   &    |-  ( ph  ->  F  e.  ( 0 ...
 T ) )   &    |-  ( ph  ->  T  e.  (
 0 ... ( # `  S ) ) )   &    |-  ( ph  ->  R  e. Word  A )   =>    |-  ( ph  ->  ( # `
  ( S splice  <. F ,  T ,  R >. ) )  =  ( ( # `  S )  +  ( ( # `  R )  -  ( T  -  F ) ) ) )
 
Theoremsplfv1 11486 Symbols to the left of a splice are unaffected. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( ph  ->  S  e. Word  A )   &    |-  ( ph  ->  F  e.  ( 0 ...
 T ) )   &    |-  ( ph  ->  T  e.  (
 0 ... ( # `  S ) ) )   &    |-  ( ph  ->  R  e. Word  A )   &    |-  ( ph  ->  X  e.  ( 0..^ F ) )   =>    |-  ( ph  ->  (
 ( S splice  <. F ,  T ,  R >. ) `
  X )  =  ( S `  X ) )
 
Theoremsplfv2a 11487 Symbols within the replacement region of a splice, expressed using the coordinates of the replacement region. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( ph  ->  S  e. Word  A )   &    |-  ( ph  ->  F  e.  ( 0 ...
 T ) )   &    |-  ( ph  ->  T  e.  (
 0 ... ( # `  S ) ) )   &    |-  ( ph  ->  R  e. Word  A )   &    |-  ( ph  ->  X  e.  ( 0..^ ( # `  R ) ) )   =>    |-  ( ph  ->  ( ( S splice 
 <. F ,  T ,  R >. ) `  ( F  +  X )
 )  =  ( R `
  X ) )
 
Theoremsplval2 11488 Value of a splice, assuming the input word  S has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( ph  ->  A  e. Word  X )   &    |-  ( ph  ->  B  e. Word  X )   &    |-  ( ph  ->  C  e. Word  X )   &    |-  ( ph  ->  R  e. Word  X )   &    |-  ( ph  ->  S  =  ( ( A concat  B ) concat  C ) )   &    |-  ( ph  ->  F  =  ( # `  A ) )   &    |-  ( ph  ->  T  =  ( F  +  ( # `  B ) ) )   =>    |-  ( ph  ->  ( S splice 
 <. F ,  T ,  R >. )  =  ( ( A concat  R ) concat  C ) )
 
Theoremswrds1 11489 Extract a single symbol from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( ( W  e. Word  A 
 /\  I  e.  (
 0..^ ( # `  W ) ) )  ->  ( W substr  <. I ,  ( I  +  1
 ) >. )  =  <" ( W `  I
 ) "> )
 
Theoremwrdeqcats1 11490 Decompose a non-empty word by separating off the last symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( W  e. Word  A 
 /\  W  =/=  (/) )  ->  W  =  ( ( W substr 
 <. 0 ,  ( ( # `  W )  -  1 ) >. ) concat  <" ( W `  ( ( # `  W )  -  1
 ) ) "> ) )
 
Theoremwrdeqs1cat 11491 Decompose a non-empty word by separating off the first symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( W  e. Word  A 
 /\  W  =/=  (/) )  ->  W  =  ( <" ( W `  0
 ) "> concat  ( W substr  <. 1 ,  ( # `  W ) >. ) ) )
 
Theoremcats1un 11492 Express a word with an extra symbol as the union of the word and the new value. (Contributed by Mario Carneiro, 28-Feb-2016.)
 |-  ( ( A  e. Word  X 
 /\  B  e.  X )  ->  ( A concat  <" B "> )  =  ( A  u.  { <. ( # `  A ) ,  B >. } ) )
 
Theoremwrdind 11493* Perform induction over the structure of a word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y concat  <" z "> )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 ( y  e. Word  B  /\  z  e.  B )  ->  ( ch  ->  th ) )   =>    |-  ( A  e. Word  B  ->  ta )
 
Theoremrevval 11494* Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  ( W  e.  V  ->  (reverse `  W )  =  ( x  e.  (
 0..^ ( # `  W ) )  |->  ( W `
  ( ( ( # `  W )  -  1 )  -  x ) ) ) )
 
Theoremrevcl 11495 The reverse of a word is a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  ( W  e. Word  A  ->  (reverse `  W )  e. Word  A )
 
Theoremrevlen 11496 The reverse of a word has the same length as the original. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  ( W  e. Word  A  ->  ( # `  (reverse `  W ) )  =  ( # `  W ) )
 
Theoremrevfv 11497 Reverse of a word at a point. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  ( ( W  e. Word  A 
 /\  X  e.  (
 0..^ ( # `  W ) ) )  ->  ( (reverse `  W ) `  X )  =  ( W `  ( ( ( # `  W )  -  1 )  -  X ) ) )
 
Theoremrev0 11498 The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  (reverse `  (/) )  =  (/)
 
Theoremrevs1 11499 Singleton words are their own reverses. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  (reverse `  <" S "> )  =  <" S ">
 
Theoremrevccat 11500 Antiautomorphic property of the reversal operation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( ( S  e. Word  A 
 /\  T  e. Word  A )  ->  (reverse `  ( S concat  T ) )  =  ( (reverse `  T ) concat  (reverse `  S ) ) )
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