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Theorem List for Intuitionistic Logic Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfz01en 9801 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 9802 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 9803 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 9804 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 9805 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 9806 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 9807 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 9808 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 9809 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 9810 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 9811 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 9812 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 9813 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 9814 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( M  e.  ZZ  ->  ( M ... M )  =  { M } )
 
Theoremfzssp1 9815 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 9816 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 9817 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 9818 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 9819 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 9820 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 9821 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 9822 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 9823 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 9824 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 9825 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 9826 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 9827 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 9828  ( 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 9829 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 9830* 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 9831* 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 9832 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 9833 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 9834 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 9835 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 9836 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 9837 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 9838 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 9839 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 9840 Express an upper integer set as the disjoint (see uzdisj 9841) 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 9841 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 9842 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 9843 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 9844* 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 9845 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 9846 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 9847 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 9848 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 9849 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 9850 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 9851 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 9852 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 9853* 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 9854* 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 9855* 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 9856* 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 9857 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 9858 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 9859 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 ) )
 
4.5.5  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 9860 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 9861 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 9862 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 9863 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 9864 Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.)
 |-  ( 0 ... N )  C_  NN0
 
Theoremfz1ssfz0 9865 Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  ( 1 ... N )  C_  ( 0 ...
 N )
 
Theorem0elfz 9866 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 9867 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 9868 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 9869 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 9870 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 9871 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 9872 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 9873 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 9874 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 9875 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 9876 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 9877 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 9878 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 9879 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 9880 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 9881 Express the set of nonnegative integers as the disjoint (see nn0disj 9883) 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 9882 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 )
 ) ) )
 
Theoremnn0disj 9883 The first  N  +  1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)
 |-  ( ( 0 ...
 N )  i^i  ( ZZ>=
 `  ( N  +  1 ) ) )  =  (/)
 
Theorem1fv 9884 A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |-  ( ( N  e.  V  /\  P  =  { <. 0 ,  N >. } )  ->  ( P : ( 0 ... 0 ) --> V  /\  ( P `  0 )  =  N ) )
 
Theorem4fvwrd4 9885* The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
 |-  ( ( L  e.  ( ZZ>= `  3 )  /\  P : ( 0
 ... L ) --> V ) 
 ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( P `
  0 )  =  a  /\  ( P `
  1 )  =  b )  /\  (
 ( P `  2
 )  =  c  /\  ( P `  3 )  =  d ) ) )
 
Theorem2ffzeq 9886* Two functions over 0 based finite set of sequential integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
 |-  ( ( M  e.  NN0  /\  F : ( 0
 ... M ) --> X  /\  P : ( 0 ...
 N ) --> Y ) 
 ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0
 ... M ) ( F `  i )  =  ( P `  i ) ) ) )
 
4.5.6  Half-open integer ranges
 
Syntaxcfzo 9887 Syntax for half-open integer ranges.
 class ..^
 
Definitiondf-fzo 9888* 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 9759, which includes  N. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 9922 with fzsplit 9799, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ 
 =  ( m  e. 
 ZZ ,  n  e. 
 ZZ  |->  ( m ... ( n  -  1
 ) ) )
 
Theoremfzof 9889 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ : ( ZZ  X.  ZZ ) --> ~P ZZ
 
Theoremelfzoel1 9890 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  B  e.  ZZ )
 
Theoremelfzoel2 9891 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  C  e.  ZZ )
 
Theoremelfzoelz 9892 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  A  e.  ZZ )
 
Theoremfzoval 9893 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 9894 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 9895 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 9896 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( ZZ>= `  M )
 )
 
Theoremfzodcel 9897 Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.)
 |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  K  e.  ( M..^ N ) )
 
Theoremfzolb 9898 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 alternate 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 9899 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 alternate 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 9900 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 )
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