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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-s5 11501 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 11502 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 11503 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 11504 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 11505 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 11506 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 11507 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 11508 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 11509 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 11510 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 11511 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 11512 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 11513 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 11514 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 11515 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 11516 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 11517 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 11518 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 11519 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 11520 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 11521 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 11522 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 11523 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 11524 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 11525 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 11526 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 11527 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 11528 A doubleton is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B ">  e. Word  _V
 
Theorems3cli 11529 A length 3 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C ">  e. Word  _V
 
Theorems4cli 11530 A length 4 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  e. Word  _V
 
Theorems5cli 11531 A length 5 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E ">  e. Word  _V
 
Theorems6cli 11532 A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E F ">  e. Word  _V
 
Theorems7cli 11533 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 11534 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 11535 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 11536 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 11537 The length of a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B "> )  =  2
 
Theorems3fv0 11538 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 11539 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 11540 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 11541 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C "> )  =  3
 
Theorems4len 11542 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D "> )  =  4
 
Theorems5len 11543 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E "> )  =  5
 
Theorems6len 11544 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F "> )  =  6
 
Theorems7len 11545 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F G "> )  =  7
 
Theorems8len 11546 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F G H "> )  =  8
 
Theorems2co 11547 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 11548 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 11549 Concatenation of fixed length strings. (This special case of ccatlid 11434 is provided to complete the pattern s0s1 11549, df-s2 11498, df-s3 11499, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
 |- 
 <" A ">  =  ( (/) concat  <" A "> )
 
Theorems1s2 11550 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C ">  =  ( <" A "> concat  <" B C "> )
 
Theorems1s3 11551 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A "> concat  <" B C D "> )
 
Theorems1s4 11552 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E ">  =  ( <" A "> concat 
 <" B C D E "> )
 
Theorems1s5 11553 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 11554 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 11555 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 11556 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A B "> concat 
 <" C D "> )
 
Theorems4s2 11557 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 11558 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 11559 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 11560 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 11561 Extend class notation with function shifter.
 class  shift
 
Definitiondf-shft 11562* 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 11569 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 11563* 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 11564* 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 11565* 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 11566* 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 11567 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 11568* 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 11569 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 11570 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 ) ) )
 
Theoremshftval3 11571 Value of a sequence shifted by  A  -  B. (Contributed by NM, 20-Jul-2005.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  ( A  -  B ) ) `
  A )  =  ( F `  B ) )
 
Theoremshftval4 11572 Value of a sequence shifted by  -u A. (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  -u A ) `  B )  =  ( F `  ( A  +  B )
 ) )
 
Theoremshftval5 11573 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A ) `
  ( B  +  A ) )  =  ( F `  B ) )
 
Theoremshftf 11574* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( F : B --> C  /\  A  e.  CC )  ->  ( F  shift  A ) : { x  e. 
 CC  |  ( x  -  A )  e.  B } --> C )
 
Theorem2shfti 11575 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( F  shift  A ) 
 shift  B )  =  ( F  shift  ( A  +  B ) ) )
 
Theoremshftidt2 11576 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( F  shift  0 )  =  ( F  |`  CC )
 
Theoremshftidt 11577 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( A  e.  CC  ->  ( ( F 
 shift  0 ) `  A )  =  ( F `  A ) )
 
Theoremshftcan1 11578 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( ( F  shift  A )  shift  -u A ) `  B )  =  ( F `  B ) )
 
Theoremshftcan2 11579 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  (
 ( ( F  shift  -u A )  shift  A ) `
  B )  =  ( F `  B ) )
 
Theoremseqshft 11580 Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-Feb-2014.)
 |-  F  e.  _V   =>    |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  seq  M (  .+  ,  ( F 
 shift  N ) )  =  (  seq  ( M  -  N ) ( 
 .+  ,  F )  shift  N ) )
 
5.7.2  Real and imaginary parts; conjugate
 
Syntaxccj 11581 Extend class notation to include complex conjugate function.
 class  *
 
Syntaxcre 11582 Extend class notation to include real part of a complex number.
 class  Re
 
Syntaxcim 11583 Extend class notation to include imaginary part of a complex number.
 class  Im
 
Definitiondf-cj 11584* Define the complex conjugate function. See cjcli 11654 for its closure and cjval 11587 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  *  =  ( x  e.  CC  |->  ( iota_ y  e.  CC ( ( x  +  y )  e.  RR  /\  ( _i  x.  ( x  -  y ) )  e. 
 RR ) ) )
 
Definitiondf-re 11585 Define a function whose value is the real part of a complex number. See reval 11591 for its value, recli 11652 for its closure, and replim 11601 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Re  =  ( x  e.  CC  |->  ( ( x  +  ( * `
  x ) ) 
 /  2 ) )
 
Definitiondf-im 11586 Define a function whose value is the imaginary part of a complex number. See imval 11592 for its value, imcli 11653 for its closure, and replim 11601 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Im  =  ( x  e.  CC  |->  ( Re
 `  ( x  /  _i ) ) )
 
Theoremcjval 11587* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( * `  A )  =  ( iota_ x  e. 
 CC ( ( A  +  x )  e. 
 RR  /\  ( _i  x.  ( A  -  x ) )  e.  RR ) ) )
 
Theoremcjth 11588 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( ( A  +  ( * `  A ) )  e.  RR  /\  ( _i  x.  ( A  -  ( * `  A ) ) )  e.  RR ) )
 
Theoremcjf 11589 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  * : CC --> CC
 
Theoremcjcl 11590 The conjugate of a complex number is a complex number (closure law). (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( * `  A )  e.  CC )
 
Theoremreval 11591 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Re `  A )  =  ( ( A  +  ( * `  A ) )  / 
 2 ) )
 
Theoremimval 11592 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Im `  A )  =  ( Re `  ( A  /  _i ) ) )
 
Theoremimre 11593 The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Im `  A )  =  ( Re `  ( -u _i  x.  A ) ) )
 
Theoremreim 11594 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)
 |-  ( A  e.  CC  ->  ( Re `  A )  =  ( Im `  ( _i  x.  A ) ) )
 
Theoremrecl 11595 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Re `  A )  e.  RR )
 
Theoremimcl 11596 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  ( A  e.  CC  ->  ( Im `  A )  e.  RR )
 
Theoremref 11597 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  Re : CC --> RR
 
Theoremimf 11598 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)
 |-  Im : CC --> RR
 
Theoremcrre 11599 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Re `  ( A  +  ( _i  x.  B ) ) )  =  A )
 
Theoremcrim 11600 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Im `  ( A  +  ( _i  x.  B ) ) )  =  B )
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