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Theorem List for Intuitionistic Logic Explorer - 10501-10600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfocdmex 10501 The codomain of an onto function is a set if its domain is a set. (Contributed by AV, 4-May-2021.)
 |-  ( ( A  e.  V  /\  F : A -onto-> B )  ->  B  e.  _V )
 
Theoremfihasheqf1oi 10502 The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  F : A -1-1-onto-> B )  ->  ( `  A )  =  ( `  B ) )
 
Theoremfihashf1rn 10503 The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  F : A -1-1-> B )  ->  ( `  F )  =  ( `  ran  F ) )
 
Theoremfihasheqf1od 10504 The size of two finite sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  F : A -1-1-onto-> B )   =>    |-  ( ph  ->  ( `  A )  =  ( `  B ) )
 
Theoremfz1eqb 10505 Two possibly-empty 1-based finite sets of sequential integers are equal iff their endpoints are equal. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 29-Mar-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( ( 1 ...
 M )  =  ( 1 ... N )  <->  M  =  N )
 )
 
Theoremfiltinf 10506 The size of an infinite set is greater than the size of a finite set. (Contributed by Jim Kingdon, 21-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  om  ~<_  B )  ->  ( `  A )  < 
 ( `  B ) )
 
Theoremisfinite4im 10507 A finite set is equinumerous to the range of integers from one up to the hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( 1 ... ( `  A ) )  ~~  A )
 
Theoremfihasheq0 10508 Two ways of saying a finite set is empty. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 27-Jul-2014.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( ( `  A )  =  0  <->  A  =  (/) ) )
 
Theoremfihashneq0 10509 Two ways of saying a finite set is not empty. Also, "A is inhabited" would be equivalent by fin0 6747. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( 0  <  ( `  A )  <->  A  =/=  (/) ) )
 
Theoremhashnncl 10510 Positive natural closure of the hash function. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( A  e.  Fin  ->  ( ( `  A )  e.  NN  <->  A  =/=  (/) ) )
 
Theoremhash0 10511 The empty set has size zero. (Contributed by Mario Carneiro, 8-Jul-2014.)
 |-  ( `  (/) )  =  0
 
Theoremhashsng 10512 The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
 |-  ( A  e.  V  ->  ( `  { A }
 )  =  1 )
 
Theoremfihashen1 10513 A finite set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) (Intuitionized by Jim Kingdon, 23-Feb-2022.)
 |-  ( A  e.  Fin  ->  ( ( `  A )  =  1  <->  A  ~~  1o )
 )
 
Theoremfihashfn 10514 A function on a finite set is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) (Intuitionized by Jim Kingdon, 24-Feb-2022.)
 |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  ( `  F )  =  ( `  A )
 )
 
Theoremfseq1hash 10515 The value of the size function on a finite 1-based sequence. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( N  e.  NN0  /\  F  Fn  ( 1
 ... N ) ) 
 ->  ( `  F )  =  N )
 
Theoremomgadd 10516 Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)
 |-  G  = frec ( ( x  e.  ZZ  |->  ( x  +  1 ) ) ,  0 )   =>    |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( G `  ( A  +o  B ) )  =  ( ( G `  A )  +  ( G `  B ) ) )
 
Theoremfihashdom 10517 Dominance relation for the size function. (Contributed by Jim Kingdon, 24-Feb-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( ( `  A )  <_  ( `  B )  <->  A  ~<_  B ) )
 
Theoremhashunlem 10518 Lemma for hashun 10519. Ordinal size of the union. (Contributed by Jim Kingdon, 25-Feb-2022.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  M  e.  om )   &    |-  ( ph  ->  A 
 ~~  N )   &    |-  ( ph  ->  B  ~~  M )   =>    |-  ( ph  ->  ( A  u.  B )  ~~  ( N  +o  M ) )
 
Theoremhashun 10519 The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  ( A  i^i  B )  =  (/) )  ->  ( `  ( A  u.  B ) )  =  (
 ( `  A )  +  ( `  B ) ) )
 
Theorem1elfz0hash 10520 1 is an element of the finite set of sequential nonnegative integers bounded by the size of a nonempty finite set. (Contributed by AV, 9-May-2020.)
 |-  ( ( A  e.  Fin  /\  A  =/=  (/) )  -> 
 1  e.  ( 0
 ... ( `  A )
 ) )
 
Theoremhashunsng 10521 The size of the union of a finite set with a disjoint singleton is one more than the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( B  e.  V  ->  ( ( A  e.  Fin  /\  -.  B  e.  A )  ->  ( `  ( A  u.  { B } )
 )  =  ( ( `  A )  +  1 ) ) )
 
Theoremhashprg 10522 The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  =/=  B  <-> 
 ( `  { A ,  B } )  =  2 ) )
 
Theoremprhash2ex 10523 There is (at least) one set with two different elements: the unordered pair containing  0 and  1. In contrast to pr0hash2ex 10529, numbers are used instead of sets because their representation is shorter (and more comprehensive). (Contributed by AV, 29-Jan-2020.)
 |-  ( `  { 0 ,  1 } )  =  2
 
Theoremhashp1i 10524 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  A  e.  om   &    |-  B  =  suc  A   &    |-  ( `  A )  =  M   &    |-  ( M  +  1 )  =  N   =>    |-  ( `  B )  =  N
 
Theoremhash1 10525 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( `  1o )  =  1
 
Theoremhash2 10526 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( `  2o )  =  2
 
Theoremhash3 10527 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( `  3o )  =  3
 
Theoremhash4 10528 Size of a finite ordinal. (Contributed by Mario Carneiro, 5-Jan-2016.)
 |-  ( `  4o )  =  4
 
Theorempr0hash2ex 10529 There is (at least) one set with two different elements: the unordered pair containing the empty set and the singleton containing the empty set. (Contributed by AV, 29-Jan-2020.)
 |-  ( `  { (/) ,  { (/)
 } } )  =  2
 
Theoremfihashss 10530 The size of a subset is less than or equal to the size of its superset. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  B  C_  A )  ->  ( `  B )  <_  ( `  A ) )
 
Theoremfiprsshashgt1 10531 The size of a superset of a proper unordered pair is greater than 1. (Contributed by AV, 6-Feb-2021.)
 |-  ( ( ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  /\  C  e.  Fin )  ->  ( { A ,  B }  C_  C  ->  2  <_  ( `  C ) ) )
 
Theoremfihashssdif 10532 The size of the difference of a finite set and a finite subset is the set's size minus the subset's. (Contributed by Jim Kingdon, 31-May-2022.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin  /\  B  C_  A )  ->  ( `  ( A  \  B ) )  =  ( ( `  A )  -  ( `  B ) ) )
 
Theoremhashdifsn 10533 The size of the difference of a finite set and a singleton subset is the set's size minus 1. (Contributed by Alexander van der Vekens, 6-Jan-2018.)
 |-  ( ( A  e.  Fin  /\  B  e.  A ) 
 ->  ( `  ( A  \  { B } )
 )  =  ( ( `  A )  -  1
 ) )
 
Theoremhashdifpr 10534 The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)
 |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C ) ) 
 ->  ( `  ( A  \  { B ,  C } ) )  =  ( ( `  A )  -  2 ) )
 
Theoremhashfz 10535 Value of the numeric cardinality of a nonempty integer range. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Proof shortened by Mario Carneiro, 15-Apr-2015.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( `  ( A ... B ) )  =  ( ( B  -  A )  +  1 )
 )
 
Theoremhashfzo 10536 Cardinality of a half-open set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( `  ( A..^ B ) )  =  ( B  -  A ) )
 
Theoremhashfzo0 10537 Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( B  e.  NN0  ->  ( `  ( 0..^ B ) )  =  B )
 
Theoremhashfzp1 10538 Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( `  ( ( A  +  1 ) ... B ) )  =  ( B  -  A ) )
 
Theoremhashfz0 10539 Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
 |-  ( B  e.  NN0  ->  ( `  ( 0 ...
 B ) )  =  ( B  +  1 ) )
 
Theoremhashxp 10540 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( `  ( A  X.  B ) )  =  ( ( `  A )  x.  ( `  B ) ) )
 
Theoremfimaxq 10541* A finite set of rational numbers has a maximum. (Contributed by Jim Kingdon, 6-Sep-2022.)
 |-  ( ( A  C_  QQ  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  y 
 <_  x )
 
Theoremresunimafz0 10542 The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
 |-  ( ph  ->  Fun  I
 )   &    |-  ( ph  ->  F : ( 0..^ ( `  F ) ) --> dom  I
 )   &    |-  ( ph  ->  N  e.  ( 0..^ ( `  F ) ) )   =>    |-  ( ph  ->  ( I  |`  ( F " ( 0 ... N ) ) )  =  ( ( I  |`  ( F " ( 0..^ N ) ) )  u.  { <. ( F `
  N ) ,  ( I `  ( F `  N ) )
 >. } ) )
 
Theoremfnfz0hash 10543 The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
 |-  ( ( N  e.  NN0  /\  F  Fn  ( 0
 ... N ) ) 
 ->  ( `  F )  =  ( N  +  1 ) )
 
Theoremffz0hash 10544 The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.)
 |-  ( ( N  e.  NN0  /\  F : ( 0
 ... N ) --> B ) 
 ->  ( `  F )  =  ( N  +  1 ) )
 
Theoremffzo0hash 10545 The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
 |-  ( ( N  e.  NN0  /\  F  Fn  ( 0..^ N ) )  ->  ( `  F )  =  N )
 
Theoremfnfzo0hash 10546 The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)
 |-  ( ( N  e.  NN0  /\  F : ( 0..^ N ) --> B ) 
 ->  ( `  F )  =  N )
 
Theoremhashfacen 10547* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)
 |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  { f  |  f : A -1-1-onto-> C }  ~~  { f  |  f : B -1-1-onto-> D } )
 
Theoremleisorel 10548 Version of isorel 5677 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
 |-  ( ( F  Isom  <  ,  <  ( A ,  B )  /\  ( A 
 C_  RR*  /\  B  C_  RR* )  /\  ( C  e.  A  /\  D  e.  A ) )  ->  ( C  <_  D  <->  ( F `  C )  <_  ( F `
  D ) ) )
 
Theoremzfz1isolemsplit 10549 Lemma for zfz1iso 10552. Removing one element from an integer range. (Contributed by Jim Kingdon, 8-Sep-2022.)
 |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  M  e.  X )   =>    |-  ( ph  ->  ( 1 ... ( `  X ) )  =  (
 ( 1 ... ( `  ( X  \  { M } ) ) )  u.  { ( `  X ) } ) )
 
Theoremzfz1isolemiso 10550* Lemma for zfz1iso 10552. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)
 |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  X 
 C_  ZZ )   &    |-  ( ph  ->  M  e.  X )   &    |-  ( ph  ->  A. z  e.  X  z  <_  M )   &    |-  ( ph  ->  G  Isom  <  ,  <  ( ( 1
 ... ( `  ( X  \  { M } )
 ) ) ,  ( X  \  { M }
 ) ) )   &    |-  ( ph  ->  A  e.  (
 1 ... ( `  X ) ) )   &    |-  ( ph  ->  B  e.  (
 1 ... ( `  X ) ) )   =>    |-  ( ph  ->  ( A  <  B  <->  ( ( G  u.  { <. ( `  X ) ,  M >. } ) `  A )  <  ( ( G  u.  { <. ( `  X ) ,  M >. } ) `  B ) ) )
 
Theoremzfz1isolem1 10551* Lemma for zfz1iso 10552. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)
 |-  ( ph  ->  K  e.  om )   &    |-  ( ph  ->  A. y ( ( ( y  C_  ZZ  /\  y  e.  Fin )  /\  y  ~~  K )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  y ) ) ,  y ) ) )   &    |-  ( ph  ->  X  C_  ZZ )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  X 
 ~~  suc  K )   &    |-  ( ph  ->  M  e.  X )   &    |-  ( ph  ->  A. z  e.  X  z  <_  M )   =>    |-  ( ph  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( `  X ) ) ,  X ) )
 
Theoremzfz1iso 10552* A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)
 |-  ( ( A  C_  ZZ  /\  A  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1
 ... ( `  A )
 ) ,  A ) )
 
Theoremseq3coll 10553* The function  F contains a sparse set of nonzero values to be summed. The function  G is an order isomorphism from the set of nonzero values of  F to a 1-based finite sequence, and  H collects these nonzero values together. Under these conditions, the sum over the values in  H yields the same result as the sum over the original set  F. (Contributed by Mario Carneiro, 2-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
 |-  ( ( ph  /\  k  e.  S )  ->  ( Z  .+  k )  =  k )   &    |-  ( ( ph  /\  k  e.  S ) 
 ->  ( k  .+  Z )  =  k )   &    |-  (
 ( ph  /\  ( k  e.  S  /\  n  e.  S ) )  ->  ( k  .+  n )  e.  S )   &    |-  ( ph  ->  Z  e.  S )   &    |-  ( ph  ->  G  Isom  <  ,  <  (
 ( 1 ... ( `  A ) ) ,  A ) )   &    |-  ( ph  ->  N  e.  (
 1 ... ( `  A ) ) )   &    |-  ( ph  ->  A  C_  ( ZZ>=
 `  M ) )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( F `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ZZ>= `  1 )
 )  ->  ( H `  k )  e.  S )   &    |-  ( ( ph  /\  k  e.  ( ( M ... ( G `  ( `  A ) ) )  \  A ) )  ->  ( F `  k )  =  Z )   &    |-  (
 ( ph  /\  n  e.  ( 1 ... ( `  A ) ) ) 
 ->  ( H `  n )  =  ( F `  ( G `  n ) ) )   =>    |-  ( ph  ->  ( 
 seq M (  .+  ,  F ) `  ( G `  N ) )  =  (  seq 1
 (  .+  ,  H ) `  N ) )
 
4.7  Elementary real and complex functions
 
4.7.1  The "shift" operation
 
Syntaxcshi 10554 Extend class notation with function shifter.
 class  shift
 
Definitiondf-shft 10555* 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 10565 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 10556* 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 10557* 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 ) ) )
 
Theoremshftfvalg 10558* 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.)
 |-  ( ( A  e.  CC  /\  F  e.  V )  ->  ( F  shift  A )  =  { <. x ,  y >.  |  ( x  e.  CC  /\  ( x  -  A ) F y ) }
 )
 
Theoremovshftex 10559 Existence of the result of applying shift. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  ( ( F  e.  V  /\  A  e.  CC )  ->  ( F  shift  A )  e.  _V )
 
Theoremshftfibg 10560 Value of a fiber of the relation  F. (Contributed by Jim Kingdon, 15-Aug-2021.)
 |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) " { B } )  =  ( F " { ( B  -  A ) }
 ) )
 
Theoremshftfval 10561* 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 10562* 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 10563 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 10564* 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 10565 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 10566 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 10567 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 10568 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 10569 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 10570* 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 10571 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 10572 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  F  e.  _V   =>    |-  ( F  shift  0 )  =  ( F  |`  CC )
 
Theoremshftidt 10573 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 10574 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 10575 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 ) )
 
Theoremshftvalg 10576 Value of a sequence shifted by  A. (Contributed by Scott Fenton, 16-Dec-2017.)
 |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  A ) `  B )  =  ( F `  ( B  -  A ) ) )
 
Theoremshftval4g 10577 Value of a sequence shifted by  -u A. (Contributed by Jim Kingdon, 19-Aug-2021.)
 |-  ( ( F  e.  V  /\  A  e.  CC  /\  B  e.  CC )  ->  ( ( F  shift  -u A ) `  B )  =  ( F `  ( A  +  B ) ) )
 
Theoremseq3shft 10578* Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 17-Oct-2022.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ( ph  /\  x  e.  ( ZZ>= `  ( M  -  N ) ) ) 
 ->  ( F `  x )  e.  S )   &    |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  ( x  .+  y )  e.  S )   =>    |-  ( ph  ->  seq
 M (  .+  ,  ( F  shift  N ) )  =  (  seq ( M  -  N ) (  .+  ,  F )  shift  N ) )
 
4.7.2  Real and imaginary parts; conjugate
 
Syntaxccj 10579 Extend class notation to include complex conjugate function.
 class  *
 
Syntaxcre 10580 Extend class notation to include real part of a complex number.
 class  Re
 
Syntaxcim 10581 Extend class notation to include imaginary part of a complex number.
 class  Im
 
Definitiondf-cj 10582* Define the complex conjugate function. See cjcli 10653 for its closure and cjval 10585 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 10583 Define a function whose value is the real part of a complex number. See reval 10589 for its value, recli 10651 for its closure, and replim 10599 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Re  =  ( x  e.  CC  |->  ( ( x  +  ( * `
  x ) ) 
 /  2 ) )
 
Definitiondf-im 10584 Define a function whose value is the imaginary part of a complex number. See imval 10590 for its value, imcli 10652 for its closure, and replim 10599 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)
 |-  Im  =  ( x  e.  CC  |->  ( Re
 `  ( x  /  _i ) ) )
 
Theoremcjval 10585* 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 10586 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 10587 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)
 |-  * : CC --> CC
 
Theoremcjcl 10588 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 10589 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 10590 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 10591 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 10592 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 10593 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 10594 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 10595 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 10596 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 10597 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 10598 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 )
 
Theoremreplim 10599 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( A  e.  CC  ->  A  =  ( ( Re `  A )  +  ( _i  x.  ( Im `  A ) ) ) )
 
Theoremremim 10600 Value of the conjugate of a complex number. The value is the real part minus  _i times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
 |-  ( A  e.  CC  ->  ( * `  A )  =  ( ( Re `  A )  -  ( _i  x.  ( Im `  A ) ) ) )
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