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Theorem List for Metamath Proof Explorer - 11401-11500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsplfv2a 11401 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 11402 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 11403 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 11404 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 11405 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 11406 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 11407* 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 11408* 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 11409 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 11410 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 11411 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 11412 The empty word is its own reverse. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  (reverse `  (/) )  =  (/)
 
Theoremrevs1 11413 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 11414 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 11415 Reversion is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( W  e. Word  A  ->  (reverse `  (reverse `  W ) )  =  W )
 
Theoremwrdco 11416 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 11417 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 11418 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 11419 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 11420 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 11421 Syntax for the length 2 word constructor.
 class  <" A B ">
 
Syntaxcs3 11422 Syntax for the length 3 word constructor.
 class  <" A B C ">
 
Syntaxcs4 11423 Syntax for the length 4 word constructor.
 class  <" A B C D ">
 
Syntaxcs5 11424 Syntax for the length 5 word constructor.
 class  <" A B C D E ">
 
Syntaxcs6 11425 Syntax for the length 6 word constructor.
 class  <" A B C D E F ">
 
Syntaxcs7 11426 Syntax for the length 7 word constructor.
 class  <" A B C D E F G ">
 
Syntaxcs8 11427 Syntax for the length 8 word constructor.
 class  <" A B C D E F G H ">
 
Definitiondf-s2 11428 Define the length 2 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B ">  =  ( <" A "> concat  <" B "> )
 
Definitiondf-s3 11429 Define the length 3 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C ">  =  ( <" A B "> concat  <" C "> )
 
Definitiondf-s4 11430 Define the length 4 word constructor. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A B C "> concat  <" D "> )
 
Definitiondf-s5 11431 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 11432 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 11433 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 11434 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 11435 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 11436 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 11437 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 11438 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 11439 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 11440 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 11441 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 11442 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 11443 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 11444 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 11445 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 11446 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 11447 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 11448 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 11449 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 11450 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 11451 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 11452 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 11453 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 )
 
Theorems7cld 11454 A length 7 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  ->  G  e.  X )   =>    |-  ( ph  ->  <" A B C D E F G ">  e. Word  X )
 
Theorems8cld 11455 A length 7 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  ->  G  e.  X )   &    |-  ( ph  ->  H  e.  X )   =>    |-  ( ph  ->  <" A B C D E F G H ">  e. Word  X )
 
Theorems2cl 11456 A doubleton is a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( ( A  e.  X  /\  B  e.  X )  ->  <" A B ">  e. Word  X )
 
Theorems3cl 11457 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) 
 ->  <" A B C ">  e. Word  X )
 
Theorems2cli 11458 A doubleton is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B ">  e. Word  _V
 
Theorems3cli 11459 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C ">  e. Word  _V
 
Theorems4cli 11460 A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  e. Word  _V
 
Theorems5cli 11461 A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E ">  e. Word  _V
 
Theorems6cli 11462 A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F ">  e. Word  _V
 
Theorems7cli 11463 A length 7 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G ">  e. Word  _V
 
Theorems8cli 11464 A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G H ">  e. Word  _V
 
Theorems2fv0 11465 Extract the first symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( A  e.  V  ->  ( <" A B "> `  0 )  =  A )
 
Theorems2fv1 11466 Extract the second symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( B  e.  V  ->  ( <" A B "> `  1 )  =  B )
 
Theorems2len 11467 The length of a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B "> )  =  2
 
Theorems3fv0 11468 Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  ( A  e.  V  ->  ( <" A B C "> `  0
 )  =  A )
 
Theorems3fv1 11469 Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  ( B  e.  V  ->  ( <" A B C "> `  1
 )  =  B )
 
Theorems3fv2 11470 Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)
 |-  ( C  e.  V  ->  ( <" A B C "> `  2
 )  =  C )
 
Theorems3len 11471 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C "> )  =  3
 
Theorems4len 11472 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D "> )  =  4
 
Theorems5len 11473 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E "> )  =  5
 
Theorems6len 11474 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F "> )  =  6
 
Theorems7len 11475 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F G "> )  =  7
 
Theorems8len 11476 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F G H "> )  =  8
 
Theorems2co 11477 Mapping a doubleton by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   =>    |-  ( ph  ->  ( F  o.  <" A B "> )  =  <" ( F `  A ) ( F `  B ) "> )
 
Theorems3co 11478 Mapping a length 3 string by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   =>    |-  ( ph  ->  ( F  o.  <" A B C "> )  =  <" ( F `
  A ) ( F `  B ) ( F `  C ) "> )
 
Theorems0s1 11479 Concatenation of fixed length strings. (This special case of ccatlid 11364 is provided to complete the pattern s0s1 11479, df-s2 11428, df-s3 11429, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
 |- 
 <" A ">  =  ( (/) concat  <" A "> )
 
Theorems1s2 11480 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C ">  =  ( <" A "> concat  <" B C "> )
 
Theorems1s3 11481 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A "> concat  <" B C D "> )
 
Theorems1s4 11482 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E ">  =  ( <" A "> concat 
 <" B C D E "> )
 
Theorems1s5 11483 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F ">  =  ( <" A "> concat 
 <" B C D E F "> )
 
Theorems1s6 11484 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G ">  =  ( <" A "> concat  <" B C D E F G "> )
 
Theorems1s7 11485 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G H ">  =  ( <" A "> concat  <" B C D E F G H "> )
 
Theorems2s2 11486 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A B "> concat 
 <" C D "> )
 
Theorems4s2 11487 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F ">  =  ( <" A B C D "> concat  <" E F "> )
 
Theorems4s3 11488 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G ">  =  ( <" A B C D "> concat  <" E F G "> )
 
Theorems4s4 11489 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F G H ">  =  ( <" A B C D "> concat  <" E F G H "> )
 
Theoremswrds2 11490 Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( ( W  e. Word  A 
 /\  I  e.  NN0  /\  ( I  +  1 )  e.  ( 0..^ ( # `  W ) ) )  ->  ( W substr  <. I ,  ( I  +  2
 ) >. )  =  <" ( W `  I
 ) ( W `  ( I  +  1
 ) ) "> )
 
5.7  Elementary real and complex functions
 
5.7.1  The "shift" operation
 
Syntaxcshi 11491 Extend class notation with function shifter.
 class  shift
 
Definitiondf-shft 11492* Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of  CC) and produces a new function on  CC. See shftval 11499 for its value. (Contributed by NM, 20-Jul-2005.)
 |- 
 shift  =  ( f  e.  _V ,  x  e. 
 CC  |->  { <. y ,  z >.  |  ( y  e. 
 CC  /\  ( y  -  x ) f z ) } )
 
Theoremshftlem 11493* Two ways to write a shifted set  ( B  +  A
). (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  CC  /\  B  C_  CC )  ->  { x  e. 
 CC  |  ( x  -  A )  e.  B }  =  { x  |  E. y  e.  B  x  =  ( y  +  A ) } )
 
Theoremshftuz 11494* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  { x  e. 
 CC  |  ( x  -  A )  e.  ( ZZ>= `  B ) }  =  ( ZZ>= `  ( B  +  A ) ) )
 
Theoremshftfval 11495* The value of the sequence shifter operation is a function on  CC.  A is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( A  e.  CC  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }
 )
 
Theoremshftdm 11496* Domain of a relation shifted by  A. The set on the right is more commonly notated as  ( dom  F  +  A ) (meaning add  A to every element of  dom  F). (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( A  e.  CC  ->  dom  (  F  shift  A )  =  { x  e.  CC  |  ( x  -  A )  e.  dom  F }
 )
 
Theoremshftfib 11497 Value of a fiber of the relation  F. (Contributed by Mario Carneiro, 4-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A )
 " { B }
 )  =  ( F
 " { ( B  -  A ) }
 ) )
 
Theoremshftfn 11498* Functionality and domain of a sequence shifted by  A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( F  Fn  B  /\  A  e.  CC )  ->  ( F  shift  A )  Fn 
 { x  e.  CC  |  ( x  -  A )  e.  B }
 )
 
Theoremshftval 11499 Value of a sequence shifted by  A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A ) `
  B )  =  ( F `  ( B  -  A ) ) )
 
Theoremshftval2 11500 Value of a sequence shifted by  A  -  B. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( F  shift  ( A  -  B ) ) `  ( A  +  C ) )  =  ( F `  ( B  +  C ) ) )
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