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Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwrdeq 11301 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( S  =  T  -> Word 
 S  = Word  T )
 
Theoremwrdexg 11302 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 11303 Hypothesis builder for Word  S. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  F/_ x S   =>    |-  F/_ xWord  S
 
Theoremccatfn 11304 The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |- concat  Fn  ( _V  X.  _V )
 
Theoremccatfval 11305* 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 11306 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 11307 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 11308 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 11309 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 11310 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 11311 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 11312 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 11313 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 11314 Identity function protection for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A ">  = 
 <" (  _I  `  A ) ">
 
Theorems1val 11315 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 11316 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( A  =  B  -> 
 <" A ">  = 
 <" B "> )
 
Theorems1eqd 11317 Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  <" A ">  =  <" B "> )
 
Theorems1cl 11318 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 11319 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  <" A ">  e. Word  B )
 
Theorems1cli 11320 A singleton word is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A ">  e. Word  _V
 
Theorems1len 11321 Length of a singleton word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A "> )  =  1
 
Theorems1nz 11322 A singleton is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |- 
 <" A ">  =/=  (/)
 
Theorems1fv 11323 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 11324 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 11325 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 11326 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 11327* 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 11328 A zero length substring. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  ( S substr  <. X ,  X >. )  =  (/)
 
Theoremswrdcl 11329 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 11330* 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 11331 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 11332 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 11333 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 11334 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 11335 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 11336 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 11337 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 11338 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 11339 An opth 4138-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 11340 An opth 4138-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 11341 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 11342 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 11343 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 11344 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 11345 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 11346 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 11347 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 11348 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 11349 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 11350 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 11351 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 11352 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 11353 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 11354* 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 11355* 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 11356 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 11357 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 11358 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 11359 The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  (reverse `  (/) )  =  (/)
 
Theoremrevs1 11360 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 11361 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 11362 Reversion is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W ) )  =  W )
 
Theoremwrdco 11363 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 11364 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 11365 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 11366 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 11367 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 11368 Syntax for the length 2 word constructor.
 class  <" A B ">
 
Syntaxcs3 11369 Syntax for the length 3 word constructor.
 class  <" A B C ">
 
Syntaxcs4 11370 Syntax for the length 4 word constructor.
 class  <" A B C D ">
 
Syntaxcs5 11371 Syntax for the length 5 word constructor.
 class  <" A B C D E ">
 
Syntaxcs6 11372 Syntax for the length 6 word constructor.
 class  <" A B C D E F ">
 
Syntaxcs7 11373 Syntax for the length 7 word constructor.
 class  <" A B C D E F G ">
 
Syntaxcs8 11374 Syntax for the length 8 word constructor.
 class  <" A B C D E F G H ">
 
Definitiondf-s2 11375 Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B ">  =  ( <" A "> concat  <" B "> )
 
Definitiondf-s3 11376 Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C ">  =  ( <" A B "> concat  <" C "> )
 
Definitiondf-s4 11377 Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A B C "> concat  <" D "> )
 
Definitiondf-s5 11378 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 11379 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 11380 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 11381 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 "> )
 
Theoremcats1cld 11382 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  T  =  ( S concat  <" X "> )   &    |-  ( ph  ->  S  e. Word  A )   &    |-  ( ph  ->  X  e.  A )   =>    |-  ( ph  ->  T  e. Word  A )
 
Theoremcats1co 11383 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  T  =  ( S concat  <" X "> )   &    |-  ( ph  ->  S  e. Word  A )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( F  o.  S )  =  U )   &    |-  V  =  ( U concat  <" ( F `  X ) "> )   =>    |-  ( ph  ->  ( F  o.  T )  =  V )
 
Theoremcats1cli 11384 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  T  =  ( S concat  <" X "> )   &    |-  S  e. Word  _V   =>    |-  T  e. Word  _V
 
Theoremcats1fvn 11385 The last symbol of a concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  T  =  ( S concat  <" X "> )   &    |-  S  e. Word  _V   &    |-  ( # `  S )  =  M   =>    |-  ( X  e.  V  ->  ( T `  M )  =  X )
 
Theoremcats1fv 11386 A symbol other than the last in a concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  T  =  ( S concat  <" X "> )   &    |-  S  e. Word  _V   &    |-  ( # `  S )  =  M   &    |-  ( Y  e.  V  ->  ( S `  N )  =  Y )   &    |-  N  e.  NN0   &    |-  N  <  M   =>    |-  ( Y  e.  V  ->  ( T `  N )  =  Y )
 
Theoremcats1len 11387 The length of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  T  =  ( S concat  <" X "> )   &    |-  S  e. Word  _V   &    |-  ( # `  S )  =  M   &    |-  ( M  +  1 )  =  N   =>    |-  ( # `
  T )  =  N
 
Theoremcats1cat 11388 Closure of concatenation with a singleton. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  T  =  ( S concat  <" X "> )   &    |-  A  e. Word  _V   &    |-  S  e. Word  _V   &    |-  C  =  ( B concat  <" X "> )   &    |-  B  =  ( A concat  S )   =>    |-  C  =  ( A concat  T )
 
Theorems2eqd 11389 Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  =  N )   &    |-  ( ph  ->  B  =  O )   =>    |-  ( ph  ->  <" A B ">  = 
 <" N O "> )
 
Theorems3eqd 11390 Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  =  N )   &    |-  ( ph  ->  B  =  O )   &    |-  ( ph  ->  C  =  P )   =>    |-  ( ph  ->  <" A B C ">  =  <" N O P "> )
 
Theorems4eqd 11391 Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  =  N )   &    |-  ( ph  ->  B  =  O )   &    |-  ( ph  ->  C  =  P )   &    |-  ( ph  ->  D  =  Q )   =>    |-  ( ph  ->  <" A B C D ">  = 
 <" N O P Q "> )
 
Theorems5eqd 11392 Equality theorem for a length 5 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  =  N )   &    |-  ( ph  ->  B  =  O )   &    |-  ( ph  ->  C  =  P )   &    |-  ( ph  ->  D  =  Q )   &    |-  ( ph  ->  E  =  R )   =>    |-  ( ph  ->  <" A B C D E ">  =  <" N O P Q R "> )
 
Theorems6eqd 11393 Equality theorem for a length 6 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  =  N )   &    |-  ( ph  ->  B  =  O )   &    |-  ( ph  ->  C  =  P )   &    |-  ( ph  ->  D  =  Q )   &    |-  ( ph  ->  E  =  R )   &    |-  ( ph  ->  F  =  S )   =>    |-  ( ph  ->  <" A B C D E F ">  =  <" N O P Q R S "> )
 
Theorems7eqd 11394 Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  =  N )   &    |-  ( ph  ->  B  =  O )   &    |-  ( ph  ->  C  =  P )   &    |-  ( ph  ->  D  =  Q )   &    |-  ( ph  ->  E  =  R )   &    |-  ( ph  ->  F  =  S )   &    |-  ( ph  ->  G  =  T )   =>    |-  ( ph  ->  <" A B C D E F G ">  =  <" N O P Q R S T "> )
 
Theorems8eqd 11395 Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  =  N )   &    |-  ( ph  ->  B  =  O )   &    |-  ( ph  ->  C  =  P )   &    |-  ( ph  ->  D  =  Q )   &    |-  ( ph  ->  E  =  R )   &    |-  ( ph  ->  F  =  S )   &    |-  ( ph  ->  G  =  T )   &    |-  ( ph  ->  H  =  U )   =>    |-  ( ph  ->  <" A B C D E F G H ">  =  <" N O P Q R S T U "> )
 
Theorems2cld 11396 A doubleton is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   =>    |-  ( ph  ->  <" A B ">  e. Word  X )
 
Theorems3cld 11397 A length 3 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   =>    |-  ( ph  ->  <" A B C ">  e. Word  X )
 
Theorems4cld 11398 A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  X )   =>    |-  ( ph  ->  <" A B C D ">  e. Word  X )
 
Theorems5cld 11399 A length 5 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  E  e.  X )   =>    |-  ( ph  ->  <" A B C D E ">  e. Word  X )
 
Theorems6cld 11400 A length 6 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  X )   &    |-  ( ph  ->  E  e.  X )   &    |-  ( ph  ->  F  e.  X )   =>    |-  ( ph  ->  <" A B C D E F ">  e. Word  X )
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