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Theorem List for Intuitionistic Logic Explorer - 9701-9800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelfz1eq 9701 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
 |-  ( K  e.  ( N ... N )  ->  K  =  N )
 
Theoremelfzubelfz 9702 If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ( M ... N ) )
 
Theorempeano2fzr 9703 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  ( K  +  1 )  e.  ( M
 ... N ) ) 
 ->  K  e.  ( M
 ... N ) )
 
Theoremfzm 9704* Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)
 |-  ( E. x  x  e.  ( M ... N )  <->  N  e.  ( ZZ>=
 `  M ) )
 
Theoremfztri3or 9705 Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  <  M  \/  K  e.  ( M
 ... N )  \/  N  <  K ) )
 
Theoremfzdcel 9706 Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  K  e.  ( M ... N ) )
 
Theoremfznlem 9707 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M 
 ->  ( M ... N )  =  (/) ) )
 
Theoremfzn 9708 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  <-> 
 ( M ... N )  =  (/) ) )
 
Theoremfzen 9709 A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( M ... N )  ~~  ( ( M  +  K ) ... ( N  +  K ) ) )
 
Theoremfz1n 9710 A 1-based finite set of sequential integers is empty iff it ends at index  0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  NN0  ->  ( ( 1 ...
 N )  =  (/)  <->  N  =  0 ) )
 
Theorem0fz1 9711 Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
 |-  ( ( N  e.  NN0  /\  F  Fn  ( 1
 ... N ) ) 
 ->  ( F  =  (/)  <->  N  =  0 ) )
 
Theoremfz10 9712 There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( 1 ... 0
 )  =  (/)
 
Theoremuzsubsubfz 9713 Membership of an integer greater than L decreased by ( L - M ) in an M based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 |-  ( ( L  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  L ) )  ->  ( N  -  ( L  -  M ) )  e.  ( M ... N ) )
 
Theoremuzsubsubfz1 9714 Membership of an integer greater than L decreased by ( L - 1 ) in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 |-  ( ( L  e.  NN  /\  N  e.  ( ZZ>=
 `  L ) ) 
 ->  ( N  -  ( L  -  1 ) )  e.  ( 1 ...
 N ) )
 
Theoremige3m2fz 9715 Membership of an integer greater than 2 decreased by 2 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 |-  ( N  e.  ( ZZ>=
 `  3 )  ->  ( N  -  2
 )  e.  ( 1
 ... N ) )
 
Theoremfzsplit2 9716 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)
 |-  ( ( ( K  +  1 )  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K ) )  ->  ( M ... N )  =  ( ( M
 ... K )  u.  ( ( K  +  1 ) ... N ) ) )
 
Theoremfzsplit 9717 Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)
 |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =  ( ( M
 ... K )  u.  ( ( K  +  1 ) ... N ) ) )
 
Theoremfzdisj 9718 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( K  <  M  ->  ( ( J ... K )  i^i  ( M
 ... N ) )  =  (/) )
 
Theoremfz01en 9719 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)
 |-  ( N  e.  ZZ  ->  ( 0 ... ( N  -  1 ) ) 
 ~~  ( 1 ...
 N ) )
 
Theoremelfznn 9720 A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.)
 |-  ( K  e.  (
 1 ... N )  ->  K  e.  NN )
 
Theoremelfz1end 9721 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  NN  <->  A  e.  ( 1 ... A ) )
 
Theoremfz1ssnn 9722 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  ( 1 ... A )  C_  NN
 
Theoremfznn0sub 9723 Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  ( N  -  K )  e.  NN0 )
 
Theoremfzmmmeqm 9724 Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)
 |-  ( M  e.  ( L ... N )  ->  ( ( N  -  L )  -  ( M  -  L ) )  =  ( N  -  M ) )
 
Theoremfzaddel 9725 Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( J  e.  ( M ... N )  <->  ( J  +  K )  e.  (
 ( M  +  K ) ... ( N  +  K ) ) ) )
 
Theoremfzsubel 9726 Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( J  e.  ( M ... N )  <->  ( J  -  K )  e.  (
 ( M  -  K ) ... ( N  -  K ) ) ) )
 
Theoremfzopth 9727 A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( ( M ... N )  =  ( J
 ... K )  <->  ( M  =  J  /\  N  =  K ) ) )
 
Theoremfzass4 9728 Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( B  e.  ( A ... D ) 
 /\  C  e.  ( B ... D ) )  <-> 
 ( B  e.  ( A ... C )  /\  C  e.  ( A ... D ) ) )
 
Theoremfzss1 9729 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( ZZ>=
 `  M )  ->  ( K ... N ) 
 C_  ( M ... N ) )
 
Theoremfzss2 9730 Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( N  e.  ( ZZ>=
 `  K )  ->  ( M ... K ) 
 C_  ( M ... N ) )
 
Theoremfzssuz 9731 A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)
 |-  ( M ... N )  C_  ( ZZ>= `  M )
 
Theoremfzsn 9732 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
 
Theoremfzssp1 9733 Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( M ... N )  C_  ( M ... ( N  +  1
 ) )
 
Theoremfzssnn 9734 Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
 |-  ( M  e.  NN  ->  ( M ... N )  C_  NN )
 
Theoremfzsuc 9735 Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( M ... ( N  +  1 ) )  =  ( ( M
 ... N )  u. 
 { ( N  +  1 ) } )
 )
 
Theoremfzpred 9736 Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( M ... N )  =  ( { M }  u.  ( ( M  +  1 ) ... N ) ) )
 
Theoremfzpreddisj 9737 A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( { M }  i^i  ( ( M  +  1 ) ... N ) )  =  (/) )
 
Theoremelfzp1 9738 Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( K  e.  ( M ... ( N  +  1 ) )  <->  ( K  e.  ( M ... N )  \/  K  =  ( N  +  1 ) ) ) )
 
Theoremfzp1ss 9739 Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( M  e.  ZZ  ->  ( ( M  +  1 ) ... N )  C_  ( M ... N ) )
 
Theoremfzelp1 9740 Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  e.  ( M ... ( N  +  1 ) ) )
 
Theoremfzp1elp1 9741 Add one to an element of a finite set of integers. (Contributed by Jeff Madsen, 6-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  ( K  +  1
 )  e.  ( M
 ... ( N  +  1 ) ) )
 
Theoremfznatpl1 9742 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)
 |-  ( ( N  e.  NN  /\  I  e.  (
 1 ... ( N  -  1 ) ) ) 
 ->  ( I  +  1 )  e.  ( 1
 ... N ) )
 
Theoremfzpr 9743 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ZZ  ->  ( M ... ( M  +  1 )
 )  =  { M ,  ( M  +  1 ) } )
 
Theoremfztp 9744 A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)
 |-  ( M  e.  ZZ  ->  ( M ... ( M  +  2 )
 )  =  { M ,  ( M  +  1 ) ,  ( M  +  2 ) }
 )
 
Theoremfzsuc2 9745 Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ( ZZ>=
 `  ( M  -  1 ) ) ) 
 ->  ( M ... ( N  +  1 )
 )  =  ( ( M ... N )  u.  { ( N  +  1 ) }
 ) )
 
Theoremfzp1disj 9746  ( M ... ( N  +  1 ) ) is the disjoint union of  ( M ... N ) with  { ( N  +  1 ) }. (Contributed by Mario Carneiro, 7-Mar-2014.)
 |-  ( ( M ... N )  i^i  { ( N  +  1 ) } )  =  (/)
 
Theoremfzdifsuc 9747 Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( M ... N )  =  ( ( M
 ... ( N  +  1 ) )  \  { ( N  +  1 ) } )
 )
 
Theoremfzprval 9748* Two ways of defining the first two values of a sequence on  NN. (Contributed by NM, 5-Sep-2011.)
 |-  ( A. x  e.  ( 1 ... 2
 ) ( F `  x )  =  if ( x  =  1 ,  A ,  B )  <-> 
 ( ( F `  1 )  =  A  /\  ( F `  2
 )  =  B ) )
 
Theoremfztpval 9749* Two ways of defining the first three values of a sequence on  NN. (Contributed by NM, 13-Sep-2011.)
 |-  ( A. x  e.  ( 1 ... 3
 ) ( F `  x )  =  if ( x  =  1 ,  A ,  if ( x  =  2 ,  B ,  C )
 ) 
 <->  ( ( F `  1 )  =  A  /\  ( F `  2
 )  =  B  /\  ( F `  3 )  =  C ) )
 
Theoremfzrev 9750 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( K  e.  (
 ( J  -  N ) ... ( J  -  M ) )  <->  ( J  -  K )  e.  ( M ... N ) ) )
 
Theoremfzrev2 9751 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( J  e.  ZZ  /\  K  e.  ZZ ) )  ->  ( K  e.  ( M ... N )  <->  ( J  -  K )  e.  (
 ( J  -  N ) ... ( J  -  M ) ) ) )
 
Theoremfzrev2i 9752 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 |-  ( ( J  e.  ZZ  /\  K  e.  ( M ... N ) ) 
 ->  ( J  -  K )  e.  ( ( J  -  N ) ... ( J  -  M ) ) )
 
Theoremfzrev3 9753 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
 |-  ( K  e.  ZZ  ->  ( K  e.  ( M ... N )  <->  ( ( M  +  N )  -  K )  e.  ( M ... N ) ) )
 
Theoremfzrev3i 9754 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
 |-  ( K  e.  ( M ... N )  ->  ( ( M  +  N )  -  K )  e.  ( M ... N ) )
 
Theoremfznn 9755 Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)
 |-  ( N  e.  ZZ  ->  ( K  e.  (
 1 ... N )  <->  ( K  e.  NN  /\  K  <_  N ) ) )
 
Theoremelfz1b 9756 Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)
 |-  ( N  e.  (
 1 ... M )  <->  ( N  e.  NN  /\  M  e.  NN  /\  N  <_  M )
 )
 
Theoremelfzm11 9757 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... ( N  -  1 ) )  <-> 
 ( K  e.  ZZ  /\  M  <_  K  /\  K  <  N ) ) )
 
Theoremuzsplit 9758 Express an upper integer set as the disjoint (see uzdisj 9759) union of the first  N values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( ZZ>= `  M )  =  ( ( M ... ( N  -  1
 ) )  u.  ( ZZ>=
 `  N ) ) )
 
Theoremuzdisj 9759 The first  N elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)
 |-  ( ( M ... ( N  -  1
 ) )  i^i  ( ZZ>=
 `  N ) )  =  (/)
 
Theoremfseq1p1m1 9760 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.)
 |-  H  =  { <. ( N  +  1 ) ,  B >. }   =>    |-  ( N  e.  NN0 
 ->  ( ( F :
 ( 1 ... N )
 --> A  /\  B  e.  A  /\  G  =  ( F  u.  H ) )  <->  ( G :
 ( 1 ... ( N  +  1 )
 ) --> A  /\  ( G `  ( N  +  1 ) )  =  B  /\  F  =  ( G  |`  ( 1
 ... N ) ) ) ) )
 
Theoremfseq1m1p1 9761 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  H  =  { <. N ,  B >. }   =>    |-  ( N  e.  NN  ->  ( ( F : ( 1 ... ( N  -  1
 ) ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H ) )  <->  ( G :
 ( 1 ... N )
 --> A  /\  ( G `
  N )  =  B  /\  F  =  ( G  |`  ( 1
 ... ( N  -  1 ) ) ) ) ) )
 
Theoremfz1sbc 9762* Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)
 |-  ( N  e.  ZZ  ->  ( A. k  e.  ( N ... N ) ph  <->  [. N  /  k ]. ph ) )
 
Theoremelfzp1b 9763 An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 0 ... ( N  -  1 ) )  <-> 
 ( K  +  1 )  e.  ( 1
 ... N ) ) )
 
Theoremelfzm1b 9764 An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 1 ... N ) 
 <->  ( K  -  1
 )  e.  ( 0
 ... ( N  -  1 ) ) ) )
 
Theoremelfzp12 9765 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( K  e.  ( M ... N )  <->  ( K  =  M  \/  K  e.  (
 ( M  +  1 ) ... N ) ) ) )
 
Theoremfzm1 9766 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  ( K  e.  ( M ... N )  <->  ( K  e.  ( M ... ( N  -  1 ) )  \/  K  =  N ) ) )
 
Theoremfzneuz 9767 No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.)
 |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  ZZ )  ->  -.  ( M ... N )  =  ( ZZ>= `  K ) )
 
Theoremfznuz 9768 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)
 |-  ( K  e.  ( M ... N )  ->  -.  K  e.  ( ZZ>= `  ( N  +  1
 ) ) )
 
Theoremuznfz 9769 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)
 |-  ( K  e.  ( ZZ>=
 `  N )  ->  -.  K  e.  ( M
 ... ( N  -  1 ) ) )
 
Theoremfzp1nel 9770 One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.)
 |- 
 -.  ( N  +  1 )  e.  ( M ... N )
 
Theoremfzrevral 9771* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( A. j  e.  ( M ... N ) ph  <->  A. k  e.  (
 ( K  -  N ) ... ( K  -  M ) ) [. ( K  -  k
 )  /  j ]. ph ) )
 
Theoremfzrevral2 9772* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( A. j  e.  ( ( K  -  N ) ... ( K  -  M ) )
 ph 
 <-> 
 A. k  e.  ( M ... N ) [. ( K  -  k
 )  /  j ]. ph ) )
 
Theoremfzrevral3 9773* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A. j  e.  ( M ... N ) ph  <->  A. k  e.  ( M ... N ) [. ( ( M  +  N )  -  k
 )  /  j ]. ph ) )
 
Theoremfzshftral 9774* Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( A. j  e.  ( M ... N ) ph  <->  A. k  e.  (
 ( M  +  K ) ... ( N  +  K ) ) [. ( k  -  K )  /  j ]. ph )
 )
 
Theoremige2m1fz1 9775 Membership of an integer greater than 1 decreased by 1 in a 1 based finite set of sequential integers (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 |-  ( N  e.  ( ZZ>=
 `  2 )  ->  ( N  -  1
 )  e.  ( 1
 ... N ) )
 
Theoremige2m1fz 9776 Membership in a 0 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander van der Vekens, 15-Sep-2018.)
 |-  ( ( N  e.  NN0  /\  2  <_  N ) 
 ->  ( N  -  1
 )  e.  ( 0
 ... N ) )
 
Theoremfz01or 9777 An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.)
 |-  ( A  e.  (
 0 ... 1 )  <->  ( A  =  0  \/  A  =  1 ) )
 
3.5.5  Finite intervals of nonnegative integers

Finite intervals of nonnegative integers (or "finite sets of sequential nonnegative integers") are finite intervals of integers with 0 as lower bound:  ( 0 ... N ), usually abbreviated by "fz0".

 
Theoremelfz2nn0 9778 Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  (
 0 ... N )  <->  ( K  e.  NN0  /\  N  e.  NN0  /\  K  <_  N ) )
 
Theoremfznn0 9779 Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)
 |-  ( N  e.  NN0  ->  ( K  e.  (
 0 ... N )  <->  ( K  e.  NN0  /\  K  <_  N )
 ) )
 
Theoremelfznn0 9780 A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  (
 0 ... N )  ->  K  e.  NN0 )
 
Theoremelfz3nn0 9781 The upper bound of a nonempty finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  (
 0 ... N )  ->  N  e.  NN0 )
 
Theoremfz0ssnn0 9782 Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)
 |-  ( 0 ... N )  C_  NN0
 
Theoremfz1ssfz0 9783 Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( 1 ... N )  C_  ( 0 ...
 N )
 
Theorem0elfz 9784 0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.)
 |-  ( N  e.  NN0  -> 
 0  e.  ( 0
 ... N ) )
 
Theoremnn0fz0 9785 A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.)
 |-  ( N  e.  NN0  <->  N  e.  ( 0 ... N ) )
 
Theoremelfz0add 9786 An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( N  e.  (
 0 ... A )  ->  N  e.  ( 0 ... ( A  +  B ) ) ) )
 
Theoremfz0tp 9787 An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  ( 0 ... 2
 )  =  { 0 ,  1 ,  2 }
 
Theoremelfz0ubfz0 9788 An element of a finite set of sequential nonnegative integers is an element of a finite set of sequential nonnegative integers with the upper bound being an element of the finite set of sequential nonnegative integers with the same lower bound as for the first interval and the element under consideration as upper bound. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
 |-  ( ( K  e.  ( 0 ... N )  /\  L  e.  ( K ... N ) ) 
 ->  K  e.  ( 0
 ... L ) )
 
Theoremelfz0fzfz0 9789 A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.)
 |-  ( ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... X ) ) 
 ->  M  e.  ( 0
 ... N ) )
 
Theoremfz0fzelfz0 9790 If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.)
 |-  ( ( N  e.  ( 0 ... R )  /\  M  e.  ( N ... R ) ) 
 ->  M  e.  ( 0
 ... R ) )
 
Theoremfznn0sub2 9791 Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  -  K )  e.  ( 0 ... N ) )
 
Theoremuzsubfz0 9792 Membership of an integer greater than L decreased by L in a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
 |-  ( ( L  e.  NN0  /\  N  e.  ( ZZ>= `  L ) )  ->  ( N  -  L )  e.  ( 0 ... N ) )
 
Theoremfz0fzdiffz0 9793 The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.)
 |-  ( ( M  e.  ( 0 ... N )  /\  K  e.  ( M ... N ) ) 
 ->  ( K  -  M )  e.  ( 0 ... N ) )
 
Theoremelfzmlbm 9794 Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)
 |-  ( K  e.  ( M ... N )  ->  ( K  -  M )  e.  ( 0 ... ( N  -  M ) ) )
 
Theoremelfzmlbp 9795 Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
 |-  ( ( N  e.  ZZ  /\  K  e.  ( M ... ( M  +  N ) ) ) 
 ->  ( K  -  M )  e.  ( 0 ... N ) )
 
Theoremfzctr 9796 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)
 |-  ( N  e.  NN0  ->  N  e.  ( 0 ... ( 2  x.  N ) ) )
 
Theoremdifelfzle 9797 The difference of two integers from a finite set of sequential nonnegative integers is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
 |-  ( ( K  e.  ( 0 ... N )  /\  M  e.  (
 0 ... N )  /\  K  <_  M )  ->  ( M  -  K )  e.  ( 0 ... N ) )
 
Theoremdifelfznle 9798 The difference of two integers from a finite set of sequential nonnegative integers increased by the upper bound is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.)
 |-  ( ( K  e.  ( 0 ... N )  /\  M  e.  (
 0 ... N )  /\  -.  K  <_  M )  ->  ( ( M  +  N )  -  K )  e.  ( 0 ... N ) )
 
Theoremnn0split 9799 Express the set of nonnegative integers as the disjoint (see nn0disj 9801) union of the first  N  +  1 values and the rest. (Contributed by AV, 8-Nov-2019.)
 |-  ( N  e.  NN0  ->  NN0  =  ( ( 0
 ... N )  u.  ( ZZ>= `  ( N  +  1 ) ) ) )
 
Theoremnnsplit 9800 Express the set of positive integers as the disjoint union of the first  N values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  ( N  e.  NN  ->  NN  =  ( ( 1 ... N )  u.  ( ZZ>= `  ( N  +  1 )
 ) ) )
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