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Theorem List for Metamath Proof Explorer - 12801-12900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorems8cli 12801 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 12802 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 12803 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 12804 The length of a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B "> )  =  2
 
Theorems3fv0 12805 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 12806 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 12807 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 12808 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C "> )  =  3
 
Theorems4len 12809 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D "> )  =  4
 
Theorems5len 12810 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E "> )  =  5
 
Theorems6len 12811 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F "> )  =  6
 
Theorems7len 12812 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F G "> )  =  7
 
Theorems8len 12813 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |-  ( # `  <" A B C D E F G H "> )  =  8
 
Theorems2prop 12814 A length 2 word is an unordered pair of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  ( ( A  e.  S  /\  B  e.  S )  ->  <" A B ">  =  { <. 0 ,  A >. ,  <. 1 ,  B >. } )
 
Theorems4prop 12815 A length 4 word is a union of two unordered pairs of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  <" A B C D ">  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
 <. 3 ,  D >. } ) )
 
Theorems2f1o 12816 A length 2 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  ( ( A  e.  S  /\  B  e.  S  /\  A  =/=  B ) 
 ->  ( E  =  <" A B ">  ->  E : { 0 ,  1 } -1-1-onto-> { A ,  B } ) )
 
Theoremf1oun2prg 12817 A union of unordered pairs of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
 )  ->  ( (
 ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D ) )  ->  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  D >. } ) : ( { 0 ,  1 }  u.  { 2 ,  3 } ) -1-1-onto-> ( { A ,  B }  u.  { C ,  D } ) ) )
 
Theorems4f1o 12818 A length 4 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
 |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( (
 ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D ) )  ->  ( E  =  <" A B C D ">  ->  E : dom  E -1-1-onto-> ( { A ,  B }  u.  { C ,  D } ) ) ) )
 
Theorems4dom 12819 The domain of a length 4 word is the union of two (disjunct) pairs. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
 |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( E  =  <" A B C D ">  ->  dom 
 E  =  ( {
 0 ,  1 }  u.  { 2 ,  3 } ) ) )
 
Theorems2co 12820 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 12821 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 12822 Concatenation of fixed length strings. (This special case of ccatlid 12557 is provided to complete the pattern s0s1 12822, df-s2 12765, df-s3 12766, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
 |- 
 <" A ">  =  ( (/) concat  <" A "> )
 
Theorems1s2 12823 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C ">  =  ( <" A "> concat  <" B C "> )
 
Theorems1s3 12824 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A "> concat  <" B C D "> )
 
Theorems1s4 12825 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D E ">  =  ( <" A "> concat 
 <" B C D E "> )
 
Theorems1s5 12826 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 12827 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 12828 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 12829 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
 |- 
 <" A B C D ">  =  ( <" A B "> concat 
 <" C D "> )
 
Theorems4s2 12830 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 12831 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 12832 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 "> )
 
Theorems2eq2s1eq 12833 Two length 2 words are equal iff the corresponding singleton words consisting of their symbols are equal. (Contributed by Alexander van der Vekens, 24-Sep-2018.)
 |-  ( ( ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
 )  ->  ( <" A B ">  = 
 <" C D ">  <-> 
 ( <" A ">  =  <" C ">  /\  <" B ">  =  <" D "> ) ) )
 
Theorems2eq2seq 12834 Two length 2 words are equal iff the corresponding symbols are equal. (Contributed by AV, 20-Oct-2018.)
 |-  ( ( ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
 )  ->  ( <" A B ">  = 
 <" C D ">  <-> 
 ( A  =  C  /\  B  =  D ) ) )
 
Theoremswrds2 12835 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
 ) ) "> )
 
Theoremwrdlen2i 12836 Implications of a word of length 2. (Contributed by AV, 27-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.)
 |-  ( ( S  e.  V  /\  T  e.  V )  ->  ( W  =  { <. 0 ,  S >. ,  <. 1 ,  T >. }  ->  ( ( W  e. Word  V  /\  ( # `
  W )  =  2 )  /\  (
 ( W `  0
 )  =  S  /\  ( W `  1 )  =  T ) ) ) )
 
Theoremwrd2pr2op 12837 A word of length 2 represented as unordered pair of ordered pairs. (Contributed by AV, 20-Oct-2018.)
 |-  ( ( W  e. Word  V 
 /\  ( # `  W )  =  2 )  ->  W  =  { <. 0 ,  ( W `  0 ) >. ,  <. 1 ,  ( W `  1 ) >. } )
 
Theoremwrdlen2 12838 A word of length 2. (Contributed by AV, 20-Oct-2018.)
 |-  ( ( S  e.  V  /\  T  e.  V )  ->  ( W  =  { <. 0 ,  S >. ,  <. 1 ,  T >. }  <->  ( ( W  e. Word  V  /\  ( # `  W )  =  2 )  /\  ( ( W `  0 )  =  S  /\  ( W `  1 )  =  T ) ) ) )
 
Theoremwrdlen2s2 12839 A word of length 2 as doubleton. (Contributed by AV, 20-Oct-2018.)
 |-  ( ( W  e. Word  V 
 /\  ( # `  W )  =  2 )  ->  W  =  <" ( W `  0 ) ( W `  1 ) "> )
 
Theoremrepsw2 12840 The "repeated symbol word" of length 2. (Contributed by AV, 6-Nov-2018.)
 |-  ( S  e.  V  ->  ( S repeatS  2 )  =  <" S S "> )
 
Theoremrepsw3 12841 The "repeated symbol word" of length 3. (Contributed by AV, 6-Nov-2018.)
 |-  ( S  e.  V  ->  ( S repeatS  3 )  =  <" S S S "> )
 
Theoremswrd2lsw 12842 Extract the last two single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
 |-  ( ( W  e. Word  V 
 /\  1  <  ( # `
  W ) ) 
 ->  ( W substr  <. ( ( # `  W )  -  2 ) ,  ( # `
  W ) >. )  =  <" ( W `
  ( ( # `  W )  -  2
 ) ) ( lastS  `  W ) "> )
 
Theorem2swrd2eqwrdeq 12843 Two words of length at least 2 are equal if and only if they have the same prefix and the same two single symbols suffix. (Contributed by AV, 24-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)
 |-  ( ( W  e. Word  V 
 /\  U  e. Word  V  /\  1  <  ( # `  W ) )  ->  ( W  =  U  <->  ( ( # `  W )  =  ( # `  U )  /\  ( ( W substr  <. 0 ,  ( ( # `  W )  -  2 ) >. )  =  ( U substr  <. 0 ,  ( ( # `  W )  -  2 ) >. ) 
 /\  ( W `  ( ( # `  W )  -  2 ) )  =  ( U `  ( ( # `  W )  -  2 ) ) 
 /\  ( lastS  `  W )  =  ( lastS  `  U ) ) ) ) )
 
Theoremccatw2s1ccatws2 12844 The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.)
 |-  ( ( W  e. Word  V 
 /\  X  e.  V  /\  Y  e.  V ) 
 ->  ( ( W concat  <" X "> ) concat  <" Y "> )  =  ( W concat  <" X Y "> ) )
 
Theoremccat2s1fvwALT 12845 Alternate proof of ccat2s1fvw 12594 using words of length 2, see df-s2 12765. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (New usage is discouraged.)
 |-  ( ( ( W  e. Word  V  /\  I  e. 
 NN0  /\  I  <  ( # `
  W ) ) 
 /\  ( X  e.  V  /\  Y  e.  V ) )  ->  ( ( ( W concat  <" X "> ) concat  <" Y "> ) `  I
 )  =  ( W `
  I ) )
 
Theoremwwlktovf 12846* Lemma 1 for wrd2f1tovbij 12850. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
 |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w `
  0 )  =  P  /\  { ( w `  0 ) ,  ( w `  1
 ) }  e.  X ) }   &    |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }   &    |-  F  =  ( t  e.  D  |->  ( t `  1 ) )   =>    |-  F : D --> R
 
Theoremwwlktovf1 12847* Lemma 2 for wrd2f1tovbij 12850. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
 |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w `
  0 )  =  P  /\  { ( w `  0 ) ,  ( w `  1
 ) }  e.  X ) }   &    |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }   &    |-  F  =  ( t  e.  D  |->  ( t `  1 ) )   =>    |-  F : D -1-1-> R
 
Theoremwwlktovfo 12848* Lemma 3 for wrd2f1tovbij 12850. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
 |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w `
  0 )  =  P  /\  { ( w `  0 ) ,  ( w `  1
 ) }  e.  X ) }   &    |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }   &    |-  F  =  ( t  e.  D  |->  ( t `  1 ) )   =>    |-  ( P  e.  V  ->  F : D -onto-> R )
 
Theoremwwlktovf1o 12849* Lemma 4 for wrd2f1tovbij 12850. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
 |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w `
  0 )  =  P  /\  { ( w `  0 ) ,  ( w `  1
 ) }  e.  X ) }   &    |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }   &    |-  F  =  ( t  e.  D  |->  ( t `  1 ) )   =>    |-  ( P  e.  V  ->  F : D -1-1-onto-> R )
 
Theoremwrd2f1tovbij 12850* There is a bijection between words of length two with a fixed first symbol contained in a pair and the symbols contained in a pair together with the fixed symbol. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
 |-  ( ( V  e.  Y  /\  P  e.  V )  ->  E. f  f : { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w `  0 )  =  P  /\  {
 ( w `  0
 ) ,  ( w `
  1 ) }  e.  X ) } -1-1-onto-> { n  e.  V  |  { P ,  n }  e.  X }
 )
 
5.8  Elementary real and complex functions
 
5.8.1  The "shift" operation
 
Syntaxcshi 12851 Extend class notation with function shifter.
 class  shift
 
Definitiondf-shft 12852* 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 12859 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 12853* 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 12854* 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 12855* 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 12856* 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 12857 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 12858* 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 12859 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 12860 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 12861 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 12862 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 12863 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 12864* 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 12865 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 12866 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( F  shift  0 )  =  ( F  |`  CC )
 
Theoremshftidt 12867 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 12868 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 12869 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 12870 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 ) )
 
5.8.2  Signum (sgn or sign) function
 
Syntaxcsgn 12871 Extend class notation to include the Signum function.
 class sgn
 
Definitiondf-sgn 12872 Signum function. Pronounced "signum" , otherwise it might be confused with "sine". Defined as "sgn" in ISO 80000-2:2009(E) operation 2-9.13. It is named "sign" (with the same definition) in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. We define this over  RR* (df-xr 9623) instead of  RR so that it can accept +oo and -oo. Note that df-psgn 16307 defines the sign of a permutation, which is different. Value shown in sgnval 12873. (Contributed by David A. Wheeler, 15-May-2015.)
 |- sgn 
 =  ( x  e.  RR*  |->  if ( x  =  0 ,  0 ,  if ( x  < 
 0 ,  -u 1 ,  1 ) ) )
 
Theoremsgnval 12873 Value of Signum function. Pronounced "signum" . See df-sgn 12872. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgn0 12874 Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (sgn `  0 )  =  0
 
Theoremsgnp 12875 Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( ( A  e.  RR*  /\  0  <  A ) 
 ->  (sgn `  A )  =  1 )
 
Theoremsgnrrp 12876 Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
 |-  ( A  e.  RR+  ->  (sgn `  A )  =  1 )
 
Theoremsgn1 12877 Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn `  1 )  =  1
 
Theoremsgnpnf 12878 Proof that the signum of +oo is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` +oo )  =  1
 
Theoremsgnn 12879 Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( ( A  e.  RR*  /\  A  <  0 ) 
 ->  (sgn `  A )  =  -u 1 )
 
Theoremsgnmnf 12880 Proof that the signum of -oo is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
 |-  (sgn ` -oo )  =  -u 1
 
5.8.3  Real and imaginary parts; conjugate
 
Syntaxccj 12881 Extend class notation to include complex conjugate function.
 class  *
 
Syntaxcre 12882 Extend class notation to include real part of a complex number.
 class  Re
 
Syntaxcim 12883 Extend class notation to include imaginary part of a complex number.
 class  Im
 
Definitiondf-cj 12884* Define the complex conjugate function. See cjcli 12954 for its closure and cjval 12887 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 12885 Define a function whose value is the real part of a complex number. See reval 12891 for its value, recli 12952 for its closure, and replim 12901 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Re  =  ( x  e.  CC  |->  ( ( x  +  ( * `
  x ) ) 
 /  2 ) )
 
Definitiondf-im 12886 Define a function whose value is the imaginary part of a complex number. See imval 12892 for its value, imcli 12953 for its closure, and replim 12901 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Im  =  ( x  e.  CC  |->  ( Re
 `  ( x  /  _i ) ) )
 
Theoremcjval 12887* 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 12888 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 12889 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  * : CC --> CC
 
Theoremcjcl 12890 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 12891 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 12892 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 12893 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 12894 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 12895 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 12896 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 12897 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 12898 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 12899 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 12900 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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