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Theorem List for Metamath Proof Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelfzolt2b 10901 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 10902 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 10903 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |- 
 -.  B  e.  ( A..^ B )
 
Theoremelfzouz2 10904 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 10905 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 10906 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 10907 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 10908 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 10909 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 10910 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 10911 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 10912 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 10913 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 10914 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 10915 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 10916 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
 |-  ( ( A..^ B )  i^i  ( B..^ C ) )  =  (/)
 
Theoremfzouzsplit 10917 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 10918 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 10919 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 10920 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 10921 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 10922 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 10923 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 10924 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 10925 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 10926 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 10927 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 10928 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( A  e.  ZZ  ->  ( A..^ ( A  +  1 ) )  =  { A }
 )
 
Theoremfzo01 10929 Expressing the singleton of  0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( 0..^ 1 )  =  { 0 }
 
Theoremfzoend 10930 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 10931 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 10932 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 10933 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 10934 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 10935 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 10936 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 10937 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 10938 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 10939* 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 10126 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 10940 Extend class notation with floor (greatest integer) function.
 class  |_
 
Definitiondf-fl 10941* Define the floor (greatest integer) function. See flval 10942 for its value, fllelt 10945 for its basic property, and flcl 10943 for its closure. For example,  ( |_ `  ( 3  /  2
) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 20850).

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 10942* 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 10943 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 10944 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
 
Theoremfllelt 10945 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 10946 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 10947 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
 
Theoremflltp1 10948 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
 |-  ( A  e.  RR  ->  A  <  ( ( |_ `  A )  +  1 ) )
 
Theoremfllep1 10949 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( A  e.  RR  ->  A  <_  ( ( |_ `  A )  +  1 ) )
 
Theoremfraclt1 10950 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( A  -  ( |_ `  A ) )  <  1 )
 
Theoremfracle1 10951 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 10952 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
 |-  ( A  e.  RR  ->  0  <_  ( A  -  ( |_ `  A ) ) )
 
Theoremflge 10953 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 10954 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 10955 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
 
Theoremflidm 10956 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  ( |_ `  A ) )  =  ( |_ `  A ) )
 
Theoremflidz 10957 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 10958 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 10959 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 10960* 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 10961* 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 10962 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 10963 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 10964 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 )
 
Theoremflge1nn 10965 The floor of a number greater than or equal to 1 is a natural number. (Contributed by NM, 26-Apr-2005.)
 |-  ( ( A  e.  RR  /\  1  <_  A )  ->  ( |_ `  A )  e.  NN )
 
Theoremfladdz 10966 An integer can be moved in and out of the floor of a sum. (Contributed by NM, 27-Apr-2005.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
 |-  ( ( A  e.  RR  /\  N  e.  ZZ )  ->  ( |_ `  ( A  +  N )
 )  =  ( ( |_ `  A )  +  N ) )
 
Theoremflzadd 10967 An integer can be moved in and out of the floor of a sum. (Contributed by NM, 2-Jan-2009.)
 |-  ( ( N  e.  ZZ  /\  A  e.  RR )  ->  ( |_ `  ( N  +  A )
 )  =  ( N  +  ( |_ `  A ) ) )
 
Theoremflmulnn0 10968 Move a nonnegative integer in and out of a floor. (Contributed by NM, 2-Jan-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)
 |-  ( ( N  e.  NN0  /\  A  e.  RR )  ->  ( N  x.  ( |_ `  A ) ) 
 <_  ( |_ `  ( N  x.  A ) ) )
 
Theorembtwnzge0 10969 A real bounded between an integer and its successor is nonnegative iff the integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217. (For the first half see rebtwnz 10331.) (Contributed by NM, 12-Mar-2005.)
 |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  ( 0  <_  A  <->  0 
 <_  N ) )
 
Theoremflhalf 10970 Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
 |-  ( N  e.  ZZ  ->  N  <_  ( 2  x.  ( |_ `  (
 ( N  +  1 )  /  2 ) ) ) )
 
Theoremceicl 10971 The ceiling function returns an integer (closure law). (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( A  e.  RR  -> 
 -u ( |_ `  -u A )  e.  ZZ )
 
Theoremceige 10972 The ceiling of a real number is greater than or equal to that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( A  e.  RR  ->  A  <_  -u ( |_ `  -u A ) )
 
Theoremceim1l 10973 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( A  e.  RR  ->  ( -u ( |_ `  -u A )  -  1 )  <  A )
 
Theoremceile 10974 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  ZZ  /\  A  <_  B )  -> 
 -u ( |_ `  -u A )  <_  B )
 
Theoremquoremz 10975 Quotient and remainder of an integer divided by a natural number. TO DO - is this really needed for anything? Should we use  mod to simplify it? (Contributed by NM, 14-Aug-2008.)
 |-  Q  =  ( |_ `  ( A  /  B ) )   &    |-  R  =  ( A  -  ( B  x.  Q ) )   =>    |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( ( Q  e.  ZZ  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
 
Theoremquoremnn0 10976 Quotient and remainder of a nonnegative integer divided by a natural number. (Contributed by NM, 14-Aug-2008.)
 |-  Q  =  ( |_ `  ( A  /  B ) )   &    |-  R  =  ( A  -  ( B  x.  Q ) )   =>    |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( Q  e.  NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
 
Theoremquoremnn0ALT 10977 Quotient and remainder of a nonnegative integer divided by a natural number. TO DO - Keep either quoremnn0ALT 10977 ((if we don't keep quoremz 10975) or quoremnn0 10976 (Contributed by NM, 14-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  Q  =  ( |_ `  ( A  /  B ) )   &    |-  R  =  ( A  -  ( B  x.  Q ) )   =>    |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( ( Q  e.  NN0  /\  R  e.  NN0 )  /\  ( R  <  B  /\  A  =  ( ( B  x.  Q )  +  R ) ) ) )
 
Theoremintfrac2 10978 Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 11002? (Contributed by NM, 16-Aug-2008.)
 |-  Z  =  ( |_ `  A )   &    |-  F  =  ( A  -  Z )   =>    |-  ( A  e.  RR  ->  ( 0  <_  F  /\  F  <  1  /\  A  =  ( Z  +  F ) ) )
 
Theoremintfracq 10979 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 10978. (Contributed by NM, 16-Aug-2008.)
 |-  Z  =  ( |_ `  ( M  /  N ) )   &    |-  F  =  ( ( M  /  N )  -  Z )   =>    |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  (
 0  <_  F  /\  F  <_  ( ( N  -  1 )  /  N )  /\  ( M 
 /  N )  =  ( Z  +  F ) ) )
 
Theoremfldiv 10980 Cancellation of the embedded floor of a real divided by an integer. (Contributed by NM, 16-Aug-2008.)
 |-  ( ( A  e.  RR  /\  N  e.  NN )  ->  ( |_ `  (
 ( |_ `  A )  /  N ) )  =  ( |_ `  ( A  /  N ) ) )
 
Theoremfldiv2 10981 Cancellation of an embedded floor of a ratio. Generalization of Equation 2.4 in [CormenLeisersonRivest] p. 33 (where  A must be an integer). (Contributed by NM, 9-Nov-2008.)
 |-  ( ( A  e.  RR  /\  M  e.  NN  /\  N  e.  NN )  ->  ( |_ `  (
 ( |_ `  ( A  /  M ) ) 
 /  N ) )  =  ( |_ `  ( A  /  ( M  x.  N ) ) ) )
 
Theoremfznnfl 10982 Finite set of sequential integers starting at 1 and ending at a real number. (Contributed by Mario Carneiro, 3-May-2016.)
 |-  ( N  e.  RR  ->  ( K  e.  (
 1 ... ( |_ `  N ) )  <->  ( K  e.  NN  /\  K  <_  N ) ) )
 
Theoremuzsup 10983 A set of upper integers is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  sup ( Z ,  RR*
 ,  <  )  =  +oo )
 
Theoremioopnfsup 10984 A set of upper reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  sup ( ( A (,)  +oo ) ,  RR* ,  <  )  =  +oo )
 
Theoremicopnfsup 10985 A set of upper reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  sup ( ( A [,)  +oo ) ,  RR* ,  <  )  =  +oo )
 
Theoremrpsup 10986 The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |- 
 sup ( RR+ ,  RR* ,  <  )  =  +oo
 
Theoremresup 10987 The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |- 
 sup ( RR ,  RR*
 ,  <  )  =  +oo
 
Theoremxrsup 10988 The extended real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |- 
 sup ( RR* ,  RR* ,  <  )  =  +oo
 
5.6.2  The modulo (remainder) operation
 
Syntaxcmo 10989 Extend class notation with the modulo operation.
 class  mod
 
Definitiondf-mod 10990* Define the modulo (remainder) operation. See modval 10991 for its value. (Contributed by NM, 10-Nov-2008.)
 |- 
 mod  =  ( x  e.  RR ,  y  e.  RR+  |->  ( x  -  ( y  x.  ( |_ `  ( x  /  y ) ) ) ) )
 
Theoremmodval 10991 The value of the modulo operation. The modulo congruence notation of number theory,  J  ==  K ( modulo  N ), can be expressed in our notation as  ( J  mod  N )  =  ( K  mod  N ). Definition 1 in Knuth, The Art of Computer Programming, Vol. I (1972), p. 38. Knuth uses "mod" for the operation and "modulo" for the congruence. Unlike Knuth, we restrict the second argument to positive reals to simplify certain theorems. (This also gives us future flexibility to extend it to any one of several different conventions for a zero or negative second argument, should there be an advantage in doing so.) (Contributed by NM, 10-Nov-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A 
 /  B ) ) ) ) )
 
Theoremmodcl 10992 Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  e.  RR )
 
Theoremmodcld 10993 Closure law for the modulo operation. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   =>    |-  ( ph  ->  ( A  mod  B )  e. 
 RR )
 
Theoremmod0 10994  A  mod  B is zero iff  A is evenly divisible by  B. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A 
 mod  B )  =  0  <-> 
 ( A  /  B )  e.  ZZ )
 )
 
Theoremnegmod0 10995  A is divisible by  B iff its negative is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A 
 mod  B )  =  0  <-> 
 ( -u A  mod  B )  =  0 )
 )
 
Theoremmodge0 10996 The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  0  <_  ( A  mod  B ) )
 
Theoremmodlt 10997 The modulo operation is less than its second argument. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  <  B )
 
Theoremmoddiffl 10998 The modulo operation differs from 
A by an integer multiple of  B. (Contributed by Mario Carneiro, 6-Sep-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  -  ( A  mod  B ) )  /  B )  =  ( |_ `  ( A  /  B ) ) )
 
Theoremmoddifz 10999 The modulo operation differs from 
A by an integer multiple of  B. (Contributed by Mario Carneiro, 15-Jul-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  -  ( A  mod  B ) )  /  B )  e.  ZZ )
 
Theoremmodfrac 11000 The fractional part of a number is the number modulo 1. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR  ->  ( A  mod  1
 )  =  ( A  -  ( |_ `  A ) ) )
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