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Theorem List for Metamath Proof Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzval2 10801 An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  =  ( ( M [,] N )  i^i  ZZ ) )
 
Theoremfzf 10802 Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |- 
 ... : ( ZZ  X.  ZZ ) --> ~P ZZ
 
Theoremelfz1 10803 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <-> 
 ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N ) ) )
 
Theoremelfz 10804 Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <->  ( M  <_  K 
 /\  K  <_  N ) ) )
 
Theoremelfz2 10805 Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show  M  e.  ZZ and  N  e.  ZZ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M 
 <_  K  /\  K  <_  N ) ) )
 
Theoremelfz5 10806 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <->  K  <_  N ) )
 
Theoremelfz4 10807 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M 
 <_  K  /\  K  <_  N ) )  ->  K  e.  ( M ... N ) )
 
Theoremelfzuzb 10808 Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  <->  ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K ) ) )
 
Theoremeluzfz 10809 Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K ) )  ->  K  e.  ( M ... N ) )
 
Theoremelfzuz 10810 A member of a finite set of sequential integers belongs to a set of upper integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  e.  ( ZZ>= `  M ) )
 
Theoremelfzuz3 10811 Membership in a finite set of sequential integers implies membership in a set of upper integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  K ) )
 
Theoremelfzel2 10812 Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ZZ )
 
Theoremelfzel1 10813 Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  M  e.  ZZ )
 
Theoremelfzelz 10814 A member of a finite set of sequential integer is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  e.  ZZ )
 
Theoremelfzle1 10815 A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  M  <_  K )
 
Theoremelfzle2 10816 A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  K  <_  N )
 
Theoremelfzuz2 10817 Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  M ) )
 
Theoremelfzle3 10818 Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  ( M ... N )  ->  M  <_  N )
 
Theoremeluzfz1 10819 Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  M  e.  ( M ... N ) )
 
Theoremeluzfz2 10820 Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( N  e.  ( ZZ>=
 `  M )  ->  N  e.  ( M ... N ) )
 
Theoremeluzfz2b 10821 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)
 |-  ( N  e.  ( ZZ>=
 `  M )  <->  N  e.  ( M ... N ) )
 
Theoremelfz3 10822 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)
 |-  ( N  e.  ZZ  ->  N  e.  ( N
 ... N ) )
 
Theoremelfz1eq 10823 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)
 |-  ( K  e.  ( N ... N )  ->  K  =  N )
 
Theorempeano2fzr 10824 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 ) )
 
Theoremfzn0 10825 Properties of a finite interval of integers which is non-empty. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
 `  M ) )
 
Theoremfzn 10826 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 10827 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 10828 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 10829 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 10830 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
 )  =  (/)
 
Theoremfzsplit2 10831 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 10832 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 10833 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 10834 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 10835 A member of a finite set of sequential integers starting at 1 is a natural number. (Contributed by NM, 24-Aug-2005.)
 |-  ( K  e.  (
 1 ... N )  ->  K  e.  NN )
 
Theoremelfz1end 10836 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 ) )
 
Theoremelfz2nn0 10837 Membership in a finite set of sequential integers starting at 0. (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 ) )
 
Theoremelfznn0 10838 A member of a finite set of sequential integers starting at 0 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 10839 The upper bound of a nonempty finite set of sequential integers starting at 0 is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  (
 0 ... N )  ->  N  e.  NN0 )
 
Theoremfznn0sub 10840 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 )
 
Theoremfznn0sub2 10841 Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( K  e.  (
 0 ... N )  ->  ( N  -  K )  e.  ( 0 ... N ) )
 
Theoremfzaddel 10842 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 10843 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 10844 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 10845 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 10846 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 10847 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 10848 A finite set of sequential integers is a subset of a set of upper integers. (Contributed by NM, 28-Oct-2005.)
 |-  ( M ... N )  C_  ( ZZ>= `  M )
 
Theoremfzsn 10849 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
 
Theoremfzssp1 10850 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
 ) )
 
Theoremfzsuc 10851 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 ) } )
 )
 
Theoremelfzp1 10852 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 10853 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 10854 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 10855 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 ) ) )
 
Theoremfzpr 10856 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 10857 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 10858 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 10859  ( 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 ) } )  =  (/)
 
Theoremfzprval 10860* 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 10861* 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 10862 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 10863 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 10864 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 10865 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 10866 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 ) )
 
Theoremfznn0 10867 Finite set of sequential integers starting at 0. (Contributed by NM, 1-Aug-2005.)
 |-  ( N  e.  NN0  ->  ( K  e.  (
 0 ... N )  <->  ( K  e.  NN0  /\  K  <_  N )
 ) )
 
Theoremfznn 10868 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 ) ) )
 
Theoremelfzm11 10869 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 ) ) )
 
Theoremfzctr 10870 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 ) ) )
 
Theoremuzsplit 10871 Express an upper integer set as the disjoint (see uzdisj 10872) 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 10872 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 10873 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 10874 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 10875* 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 ) )
 
Theoremelfzm1b 10876 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 10877 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 10878 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 10879 No finite set of sequential integers equals a set of upper integers. (Contributed by NM, 11-Dec-2005.)
 |-  ( ( N  e.  ( ZZ>= `  M )  /\  K  e.  ZZ )  ->  -.  ( M ... N )  =  ( ZZ>= `  K ) )
 
Theoremfznuz 10880 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 10881 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 ) ) )
 
Theoremfzrevral 10882* 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 10883* 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 10884* 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 10885* 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 )
 )
 
5.5.6  Half-open integer ranges
 
Syntaxcfzo 10886 Syntax for half-open integer ranges.
 class ..^
 
Definitiondf-fzo 10887* Define a function generating sets of integers using a half-open range. Read  ( M..^ N
) as the integers from 
M up to, but not including,  N; contrast with  ( M ... N ) df-fz 10799, which includes  N. Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 10915 with fzsplit 10832, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ 
 =  ( m  e. 
 ZZ ,  n  e. 
 ZZ  |->  ( m ... ( n  -  1
 ) ) )
 
Theoremfzof 10888 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ : ( ZZ  X.  ZZ ) --> ~P ZZ
 
Theoremelfzoel1 10889 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  B  e.  ZZ )
 
Theoremelfzoel2 10890 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  C  e.  ZZ )
 
Theoremelfzoelz 10891 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  A  e.  ZZ )
 
Theoremfzoval 10892 Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( N  e.  ZZ  ->  ( M..^ N )  =  ( M ... ( N  -  1
 ) ) )
 
Theoremelfzo 10893 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M..^ N )  <->  ( M  <_  K 
 /\  K  <  N ) ) )
 
Theoremelfzo2 10894 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  <->  ( K  e.  ( ZZ>= `  M )  /\  N  e.  ZZ  /\  K  <  N ) )
 
Theoremelfzouz 10895 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( ZZ>= `  M )
 )
 
Theoremfzolb 10896 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with  M  <  N. This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 
M  e.  ( ZZ>= `  N ). (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( M  e.  ( M..^ N )  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N ) )
 
Theoremfzolb2 10897 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with  M  <  N. This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate 
M  e.  ( ZZ>= `  N ). (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  e.  ( M..^ N )  <->  M  <  N ) )
 
Theoremelfzole1 10898 A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  M  <_  K )
 
Theoremelfzolt2 10899 A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  <  N )
 
Theoremelfzolt3 10900 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  M  <  N )
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