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Theorem List for Metamath Proof Explorer - 11401-11500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcs1 11401 Syntax for the singleton word constructor.
 class  <" A ">
 
Syntaxcsubstr 11402 Syntax for the word slicing operator.
 class substr
 
Syntaxcsplice 11403 Syntax for the word splicing operator.
 class splice
 
Syntaxcreverse 11404 Syntax for the word reverse operator.
 class reverse
 
Definitiondf-word 11405* Define the class of words over a set. A word is an 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 11406* 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 11407 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 11408* 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 11409* 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 11410* 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 11411* 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 11412* 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 11413 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 11414 The empty set is a word (frequently denoted ε in this context). (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  (/)  e. Word  S
 
Theoremwrdf 11415 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 11416 A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.)
 |-  ( W  e. Word  S  ->  W  e.  Fin )
 
Theoremlencl 11417 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 11418 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 11419 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 11420 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  =  T  -> Word 
 S  = Word  T )
 
Theoremwrdexg 11421 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 11422 Hypothesis builder for Word  S. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  F/_ x S   =>    |-  F/_ xWord  S
 
Theoremccatfn 11423 The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |- concat  Fn  ( _V  X.  _V )
 
Theoremccatfval 11424* 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 11425 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 11426 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 11427 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 11428 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 11429 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 11430 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 11431 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 11432 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 11433 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A ">  = 
 <" (  _I  `  A ) ">
 
Theorems1val 11434 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 11435 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( A  =  B  -> 
 <" A ">  = 
 <" B "> )
 
Theorems1eqd 11436 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <" A ">  =  <" B "> )
 
Theorems1cl 11437 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 11438 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  <" A ">  e. Word  B )
 
Theorems1cli 11439 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A ">  e. Word  _V
 
Theorems1len 11440 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A "> )  =  1
 
Theorems1nz 11441 A singleton is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |- 
 <" A ">  =/=  (/)
 
Theorems1fv 11442 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 11443 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 11444 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 11445 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 11446* 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 11447 A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( S substr  <. X ,  X >. )  =  (/)
 
Theoremswrdcl 11448 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 11449* 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 11450 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 11451 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 11452 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 11453 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 11454 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 11455 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 11456 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 11457 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 11458 An opth 4244-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 11459 An opth 4244-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 11460 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 11461 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 11462 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 11463 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 11464 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 11465 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 11466 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 11467 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 11468 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 11469 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 11470 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 11471 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 11472 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 11473* 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 11474* 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 11475 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 11476 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 11477 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 11478 The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  (reverse `  (/) )  =  (/)
 
Theoremrevs1 11479 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 11480 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 ) ) )
 
Theoremrevrev 11481 Reversion is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W ) )  =  W )
 
Theoremwrdco 11482 Mapping a word by a function. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( ( W  e. Word  A 
 /\  F : A --> B )  ->  ( F  o.  W )  e. Word  B )
 
Theoremlenco 11483 Length of a mapped word is unchanged. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( ( W  e. Word  A 
 /\  F : A --> B )  ->  ( # `  ( F  o.  W ) )  =  ( # `
  W ) )
 
Theorems1co 11484 Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( ( S  e.  A  /\  F : A --> B )  ->  ( F  o.  <" S "> )  =  <" ( F `  S ) "> )
 
Theoremrevco 11485 Mapping of words commutes with reversal. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( ( W  e. Word  A 
 /\  F : A --> B )  ->  ( F  o.  (reverse `  W ) )  =  (reverse `  ( F  o.  W ) ) )
 
Theoremccatco 11486 Mapping of words commutes with concatenation. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( ( S  e. Word  A 
 /\  T  e. Word  A  /\  F : A --> B ) 
 ->  ( F  o.  ( S concat  T ) )  =  ( ( F  o.  S ) concat  ( F  o.  T ) ) )
 
5.6.10  Longer string literals
 
Syntaxcs2 11487 Syntax for the length 2 word constructor.
 class  <" A B ">
 
Syntaxcs3 11488 Syntax for the length 3 word constructor.
 class  <" A B C ">
 
Syntaxcs4 11489 Syntax for the length 4 word constructor.
 class  <" A B C D ">
 
Syntaxcs5 11490 Syntax for the length 5 word constructor.
 class  <" A B C D E ">
 
Syntaxcs6 11491 Syntax for the length 6 word constructor.
 class  <" A B C D E F ">
 
Syntaxcs7 11492 Syntax for the length 7 word constructor.
 class  <" A B C D E F G ">
 
Syntaxcs8 11493 Syntax for the length 8 word constructor.
 class  <" A B C D E F G H ">
 
Definitiondf-s2 11494 Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B ">  =  ( <" A "> concat  <" B "> )
 
Definitiondf-s3 11495 Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C ">  =  ( <" A B "> concat  <" C "> )
 
Definitiondf-s4 11496 Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A B C "> concat  <" D "> )
 
Definitiondf-s5 11497 Define the length 5 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E ">  =  ( <" A B C D "> concat  <" E "> )
 
Definitiondf-s6 11498 Define the length 6 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F ">  =  ( <" A B C D E "> concat  <" F "> )
 
Definitiondf-s7 11499 Define the length 7 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G ">  =  ( <" A B C D E F "> concat  <" G "> )
 
Definitiondf-s8 11500 Define the length 8 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G H ">  =  ( <" A B C D E F G "> concat  <" H "> )
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