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Theorem List for Metamath Proof Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzrev3 10801 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 10802 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 10803 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 10804 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 10805 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 10806 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 10807 Express an upper integer set as the disjoint (see uzdisj 10808) 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 10808 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 10809 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 10810 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 10811* 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 10812 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 10813 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 10814 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 10815 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 10816 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 10817 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 10818* 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 10819* 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 10820* 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 10821* 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 10822 Syntax for half-open integer ranges.
 class ..^
 
Definitiondf-fzo 10823* 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 10735, which includes  N. Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 10851 with fzsplit 10768, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ 
 =  ( m  e. 
 ZZ ,  n  e. 
 ZZ  |->  ( m ... ( n  -  1
 ) ) )
 
Theoremfzof 10824 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |- ..^ : ( ZZ  X.  ZZ ) --> ~P ZZ
 
Theoremelfzoel1 10825 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  B  e.  ZZ )
 
Theoremelfzoel2 10826 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  C  e.  ZZ )
 
Theoremelfzoelz 10827 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  A  e.  ZZ )
 
Theoremfzoval 10828 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 10829 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 10830 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 10831 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( ZZ>= `  M )
 )
 
Theoremfzolb 10832 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 10833 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 10834 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 10835 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 10836 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 )
 
Theoremelfzolt2b 10837 A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( K..^ N ) )
 
Theoremelfzolt3b 10838 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  M  e.  ( M..^ N ) )
 
Theoremfzonel 10839 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |- 
 -.  B  e.  ( A..^ B )
 
Theoremelfzouz2 10840 The upper bound of a half-open range is greater or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  ->  N  e.  ( ZZ>= `  K )
 )
 
Theoremelfzofz 10841 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( K  e.  ( M..^ N )  ->  K  e.  ( M ... N ) )
 
Theoremelfzo3 10842 Express membership in a half-open integer interval in terms of the "less than or equal" and "less than" predicates on integers, resp.  K  e.  (
ZZ>= `  M )  <->  M  <_  K,  K  e.  ( K..^ N )  <->  K  <  N. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( M..^ N )  <->  ( K  e.  ( ZZ>= `  M )  /\  K  e.  ( K..^ N ) ) )
 
Theoremfzon0 10843 A half-open integer interval is nonempty iff it contains its left endpoint. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( ( M..^ N )  =/=  (/)  <->  M  e.  ( M..^ N ) )
 
Theoremfzossfz 10844 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( A..^ B ) 
 C_  ( A ... B )
 
Theoremfzo0 10845 Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( A..^ A )  =  (/)
 
Theoremfzonnsub 10846 If  K  <  N then 
N  -  K is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
 |-  ( K  e.  ( M..^ N )  ->  ( N  -  K )  e. 
 NN )
 
Theoremfzonnsub2 10847 If  M  <  N then 
N  -  M is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( K  e.  ( M..^ N )  ->  ( N  -  M )  e. 
 NN )
 
Theoremfzoss1 10848 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( K  e.  ( ZZ>=
 `  M )  ->  ( K..^ N )  C_  ( M..^ N ) )
 
Theoremfzoss2 10849 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( N  e.  ( ZZ>=
 `  K )  ->  ( M..^ K )  C_  ( M..^ N ) )
 
Theoremfzospliti 10850 One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( ( A  e.  ( B..^ C )  /\  D  e.  ZZ )  ->  ( A  e.  ( B..^ D )  \/  A  e.  ( D..^ C ) ) )
 
Theoremfzosplit 10851 Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( D  e.  ( B ... C )  ->  ( B..^ C )  =  ( ( B..^ D )  u.  ( D..^ C ) ) )
 
Theoremfzodisj 10852 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( ( A..^ B )  i^i  ( B..^ C ) )  =  (/)
 
Theoremfzouzsplit 10853 Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( ZZ>= `  A )  =  ( ( A..^ B )  u.  ( ZZ>= `  B ) ) )
 
Theoremfzouzdisj 10854 A half-open integer range does not overlap the upper integer range starting at the endpoint of the first range. (Contributed by Mario Carneiro, 21-Sep-2016.)
 |-  ( ( A..^ B )  i^i  ( ZZ>= `  B ) )  =  (/)
 
Theoremlbfzo0 10855 An integer is strictly greater than zero iff it is a member of  NN. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( 0  e.  (
 0..^ A )  <->  A  e.  NN )
 
Theoremelfzo0 10856 Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( A  e.  (
 0..^ B )  <->  ( A  e.  NN0  /\  B  e.  NN  /\  A  <  B ) )
 
Theoremfzo0n0 10857 A half-open integer range based at 0 is nonempty precisely if the upper bound is a positive integer. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( ( 0..^ A )  =/=  (/)  <->  A  e.  NN )
 
Theoremfzoaddel 10858 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( B..^ C )  /\  D  e.  ZZ )  ->  ( A  +  D )  e.  ( ( B  +  D )..^ ( C  +  D ) ) )
 
Theoremfzoaddel2 10859 Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( 0..^ ( B  -  C ) )  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  +  C )  e.  ( C..^ B ) )
 
Theoremfzosubel 10860 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( B..^ C )  /\  D  e.  ZZ )  ->  ( A  -  D )  e.  ( ( B  -  D )..^ ( C  -  D ) ) )
 
Theoremfzosubel2 10861 Membership in a translated half-open integer range implies translated membership in the original range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( ( B  +  C )..^ ( B  +  D ) )  /\  ( B  e.  ZZ  /\  C  e.  ZZ  /\  D  e.  ZZ )
 )  ->  ( A  -  B )  e.  ( C..^ D ) )
 
Theoremfzosubel3 10862 Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( ( A  e.  ( B..^ ( B  +  D ) )  /\  D  e.  ZZ )  ->  ( A  -  B )  e.  ( 0..^ D ) )
 
Theoremfzval3 10863 Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( N  e.  ZZ  ->  ( M ... N )  =  ( M..^ ( N  +  1
 ) ) )
 
Theoremfzosn 10864 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( A  e.  ZZ  ->  ( A..^ ( A  +  1 ) )  =  { A }
 )
 
Theoremfzo01 10865 Expressing the singleton of  0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( 0..^ 1 )  =  { 0 }
 
Theoremfzoend 10866 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
 |-  ( A  e.  ( A..^ B )  ->  ( B  -  1 )  e.  ( A..^ B ) )
 
Theoremfzo0end 10867 The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
 |-  ( B  e.  NN  ->  ( B  -  1
 )  e.  ( 0..^ B ) )
 
Theoremfzofzp1 10868 If a point is in a half-open range, the next point is in the closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( C  e.  ( A..^ B )  ->  ( C  +  1 )  e.  ( A ... B ) )
 
Theoremfzofzp1b 10869 If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( C  e.  ( ZZ>=
 `  A )  ->  ( C  e.  ( A..^ B )  <->  ( C  +  1 )  e.  ( A ... B ) ) )
 
Theoremelfzom1b 10870 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 Mario Carneiro, 27-Sep-2015.)
 |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( 1..^ N )  <->  ( K  -  1 )  e.  (
 0..^ ( N  -  1 ) ) ) )
 
Theorempeano2fzor 10871 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( K  e.  ( ZZ>= `  M )  /\  ( K  +  1 )  e.  ( M..^ N ) )  ->  K  e.  ( M..^ N ) )
 
Theoremfzosplitsn 10872 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( A..^ ( B  +  1 ) )  =  ( ( A..^ B )  u.  { B }
 ) )
 
Theoremfzosplitsni 10873 Membership in a half-open range extende by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( B  e.  ( ZZ>=
 `  A )  ->  ( C  e.  ( A..^ ( B  +  1 ) )  <->  ( C  e.  ( A..^ B )  \/  C  =  B ) ) )
 
Theoremfzostep1 10874 Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( A  e.  ( B..^ C )  ->  (
 ( A  +  1 )  e.  ( B..^ C )  \/  ( A  +  1 )  =  C ) )
 
Theoremfzind2 10875* Induction on the integers from  M to  N inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. Version of fzind 10062 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
 |-  ( x  =  M  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  K  ->  (
 ph 
 <->  ta ) )   &    |-  ( N  e.  ( ZZ>= `  M )  ->  ps )   &    |-  (
 y  e.  ( M..^ N )  ->  ( ch  ->  th ) )   =>    |-  ( K  e.  ( M ... N ) 
 ->  ta )
 
5.6  Elementary integer functions
 
5.6.1  The floor (greatest integer) function
 
Syntaxcfl 10876 Extend class notation with floor (greatest integer) function.
 class  |_
 
Definitiondf-fl 10877* Define the floor (greatest integer) function. See flval 10878 for its value, fllelt 10881 for its basic property, and flcl 10879 for its closure. For example,  ( |_ `  ( 3  /  2
) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 20763).

The term "floor" was coined by Ken Iverson. He also invented a mathematical notation for floor, consisting of an L-shaped left bracket and its reflection as a right bracket. In APL, the left-bracket alone is used, and we borrow this idea. (Thanks to Paul Chapman for this information.) (Contributed by NM, 14-Nov-2004.)

 |- 
 |_  =  ( x  e.  RR  |->  ( iota_ y  e.  ZZ ( y 
 <_  x  /\  x  < 
 ( y  +  1 ) ) ) )
 
Theoremflval 10878* Value of the floor (greatest integer) function. The floor of  A is the (unique) integer less than or equal to  A whose successor is strictly greater than  A. (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  =  ( iota_ x  e. 
 ZZ ( x  <_  A  /\  A  <  ( x  +  1 )
 ) ) )
 
Theoremflcl 10879 The floor (greatest integer) function is an integer (closure law). (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
 
Theoremreflcl 10880 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
 
Theoremfllelt 10881 A basic property of the floor (greatest integer) function. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
 |-  ( A  e.  RR  ->  ( ( |_ `  A )  <_  A  /\  A  <  ( ( |_ `  A )  +  1 )
 ) )
 
Theoremflcld 10882 The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( |_ `  A )  e. 
 ZZ )
 
Theoremflle 10883 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
 
Theoremflltp1 10884 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
 |-  ( A  e.  RR  ->  A  <  ( ( |_ `  A )  +  1 ) )
 
Theoremfllep1 10885 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( A  e.  RR  ->  A  <_  ( ( |_ `  A )  +  1 ) )
 
Theoremfraclt1 10886 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( A  -  ( |_ `  A ) )  <  1 )
 
Theoremfracle1 10887 The fractional part of a real number is less than or equal to one. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( A  e.  RR  ->  ( A  -  ( |_ `  A ) ) 
 <_  1 )
 
Theoremfracge0 10888 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
 |-  ( A  e.  RR  ->  0  <_  ( A  -  ( |_ `  A ) ) )
 
Theoremflge 10889 The floor function value is the greatest integer less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( B  <_  A  <->  B  <_  ( |_ `  A ) ) )
 
Theoremfllt 10890 The floor function value is less than the next integer. (Contributed by NM, 24-Feb-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( A  <  B  <-> 
 ( |_ `  A )  <  B ) )
 
Theoremflid 10891 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
 
Theoremflidm 10892 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  ( |_ `  A ) )  =  ( |_ `  A ) )
 
Theoremflidz 10893 A real number equals its floor iff it is an integer. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR  ->  ( ( |_ `  A )  =  A  <->  A  e.  ZZ ) )
 
Theoremflwordi 10894 Ordering relationship for the greatest integer function. (Contributed by NM, 31-Dec-2005.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( |_ `  A )  <_  ( |_ `  B ) )
 
Theoremflword2 10895 Ordering relationship for the greatest integer function. (Contributed by Mario Carneiro, 7-Jun-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) ) )
 
Theoremflval2 10896* An alternate way to define the floor (greatest integer) function. (Contributed by NM, 16-Nov-2004.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  =  ( iota_ x  e. 
 ZZ ( x  <_  A  /\  A. y  e. 
 ZZ  ( y  <_  A  ->  y  <_  x ) ) ) )
 
Theoremflval3 10897* An alternate way to define the floor (greatest integer) function, as the supremum of all integers less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 6-Sep-2014.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  =  sup ( { x  e.  ZZ  |  x  <_  A } ,  RR ,  <  ) )
 
Theoremflbi 10898 A condition equivalent to floor. (Contributed by NM, 11-Mar-2005.) (Revised by Mario Carneiro, 2-Nov-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( ( |_ `  A )  =  B  <->  ( B  <_  A  /\  A  <  ( B  +  1 ) ) ) )
 
Theoremflbi2 10899 A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.)
 |-  ( ( N  e.  ZZ  /\  F  e.  RR )  ->  ( ( |_ `  ( N  +  F ) )  =  N  <->  ( 0  <_  F  /\  F  <  1 ) ) )
 
Theoremflge0nn0 10900 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by NM, 26-Apr-2005.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( |_ `  A )  e.  NN0 )
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