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Theorem List for Metamath Proof Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzo0n0 10901 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 10902 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 10903 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 10904 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 10905 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 10906 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 10907 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 10908 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  ( A  e.  ZZ  ->  ( A..^ ( A  +  1 ) )  =  { A }
 )
 
Theoremfzo01 10909 Expressing the singleton of  0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( 0..^ 1 )  =  { 0 }
 
Theoremfzoend 10910 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 10911 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 10912 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 10913 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 10914 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 10915 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 10916 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 10917 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 10918 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 10919* 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 10105 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 10920 Extend class notation with floor (greatest integer) function.
 class  |_
 
Definitiondf-fl 10921* Define the floor (greatest integer) function. See flval 10922 for its value, fllelt 10925 for its basic property, and flcl 10923 for its closure. For example,  ( |_ `  ( 3  /  2
) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 20811).

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 10922* 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 10923 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 10924 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
 
Theoremfllelt 10925 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 10926 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 10927 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
 
Theoremflltp1 10928 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
 |-  ( A  e.  RR  ->  A  <  ( ( |_ `  A )  +  1 ) )
 
Theoremfllep1 10929 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( A  e.  RR  ->  A  <_  ( ( |_ `  A )  +  1 ) )
 
Theoremfraclt1 10930 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( A  -  ( |_ `  A ) )  <  1 )
 
Theoremfracle1 10931 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 10932 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
 |-  ( A  e.  RR  ->  0  <_  ( A  -  ( |_ `  A ) ) )
 
Theoremflge 10933 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 10934 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 10935 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
 
Theoremflidm 10936 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  ( |_ `  A ) )  =  ( |_ `  A ) )
 
Theoremflidz 10937 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 10938 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 10939 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 10940* 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 10941* 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 10942 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 10943 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 10944 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 10945 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 10946 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 10947 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 10948 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 10949 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 10311.) (Contributed by NM, 12-Mar-2005.)
 |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  ( 0  <_  A  <->  0 
 <_  N ) )
 
Theoremflhalf 10950 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 10951 The ceiling function returns an integer (closure law). (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( A  e.  RR  -> 
 -u ( |_ `  -u A )  e.  ZZ )
 
Theoremceige 10952 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 10953 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 10954 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 10955 Quotient and remainder of an integer divided by a natural number. (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 ) ) ) )
 
Theoremquoremnn0ALT 10956 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 ) ) ) )
 
Theoremquoremnn0 10957 Quotient and remainder of a nonnegative integer divided by a natural number. (Contributed by NM, 14-Aug-2008.) (Proof modification 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 10958 Decompose a real into integer and fractional parts. (Contributed by NM, 16-Aug-2008.)
 |-  Z  =  ( |_ `  A )   &    |-  F  =  ( A  -  Z )   =>    |-  ( A  e.  RR  ->  ( 0  <_  F  /\  F  <  1  /\  A  =  ( Z  +  F ) ) )
 
Theoremintfracq 10959 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 10958. (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 10960 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 10961 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 10962 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 10963 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 10964 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 10965 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 10966 The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |- 
 sup ( RR+ ,  RR* ,  <  )  =  +oo
 
Theoremresup 10967 The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |- 
 sup ( RR ,  RR*
 ,  <  )  =  +oo
 
Theoremxrsup 10968 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 10969 Extend class notation with the modulo operation.
 class  mod
 
Definitiondf-mod 10970* Define the modulo (remainder) operation. See modval 10971 for its value. (Contributed by NM, 10-Nov-2008.)
 |- 
 mod  =  ( x  e.  RR ,  y  e.  RR+  |->  ( x  -  ( y  x.  ( |_ `  ( x  /  y ) ) ) ) )
 
Theoremmodval 10971 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 10972 Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  e.  RR )
 
Theoremmodcld 10973 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 10974  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 10975  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 10976 The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  0  <_  ( A  mod  B ) )
 
Theoremmodlt 10977 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 10978 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 10979 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 10980 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 ) ) )
 
Theoremflmod 10981 The floor function expressed in terms of the modulo operation. (Contributed by NM, 11-Nov-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  =  ( A  -  ( A  mod  1
 ) ) )
 
Theoremintfrac 10982 Break a number into its integer part and its fractional part. (Contributed by NM, 31-Dec-2008.)
 |-  ( A  e.  RR  ->  A  =  ( ( |_ `  A )  +  ( A  mod  1 ) ) )
 
Theoremzmod10 10983 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  ZZ  ->  ( N  mod  1
 )  =  0 )
 
Theoremmodmulnn 10984 Move a natural number in and out of a floor in the first argument of a modulo operation. (Contributed by NM, 2-Jan-2009.)
 |-  ( ( N  e.  NN  /\  A  e.  RR  /\  M  e.  NN )  ->  ( ( N  x.  ( |_ `  A ) )  mod  ( N  x.  M ) ) 
 <_  ( ( |_ `  ( N  x.  A ) ) 
 mod  ( N  x.  M ) ) )
 
Theoremzmodcl 10985 Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  mod  B )  e.  NN0 )
 
Theoremzmodcld 10986 Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  NN )   =>    |-  ( ph  ->  ( A  mod  B )  e.  NN0 )
 
Theoremzmodfz 10987 An integer mod  B lies in the first  B nonnegative integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  mod  B )  e.  ( 0
 ... ( B  -  1 ) ) )
 
Theoremzmodfzo 10988 An integer mod  B lies in the first  B nonnegative integers. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  mod  B )  e.  ( 0..^ B ) )
 
Theoremmodid 10989 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  (
 0  <_  A  /\  A  <  B ) ) 
 ->  ( A  mod  B )  =  A )
 
Theoremmodid2 10990 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A 
 mod  B )  =  A  <->  ( 0  <_  A  /\  A  <  B ) ) )
 
Theorem0mod 10991 Special case: 0 modulo a positive real number is 0. (Contributed by Mario Carneiro, 22-Feb-2014.)
 |-  ( N  e.  RR+  ->  ( 0  mod  N )  =  0 )
 
Theorem1mod 10992 Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  ( ( N  e.  RR  /\  1  <  N )  ->  ( 1  mod 
 N )  =  1 )
 
Theoremmodabs 10993 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  C  e.  RR+ )  /\  B  <_  C )  ->  ( ( A  mod  B )  mod  C )  =  ( A 
 mod  B ) )
 
Theoremmodabs2 10994 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A 
 mod  B )  mod  B )  =  ( A  mod  B ) )
 
Theoremmodcyc 10995 The modulo operation is periodic. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  N  e.  ZZ )  ->  ( ( A  +  ( N  x.  B ) )  mod  B )  =  ( A  mod  B ) )
 
Theoremmodcyc2 10996 The modulo operation is periodic. (Contributed by NM, 12-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  N  e.  ZZ )  ->  ( ( A  -  ( B  x.  N ) )  mod  B )  =  ( A  mod  B ) )
 
Theoremmodadd1 10997 Addition property of the modulo operation. (Contributed by NM, 12-Nov-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D ) )  ->  ( ( A  +  C ) 
 mod  D )  =  ( ( B  +  C )  mod  D ) )
 
Theoremmodmul1 10998 Multiplication property of the modulo operation. Note that the multiplier  C must be an integer. (Contributed by NM, 12-Nov-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ZZ  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D ) )  ->  ( ( A  x.  C ) 
 mod  D )  =  ( ( B  x.  C )  mod  D ) )
 
Theoremmodmul12d 10999 Multiplication property of the modulo operation. (Contributed by Mario Carneiro, 5-Feb-2015.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  ZZ )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  ( A  mod  E )  =  ( B  mod  E ) )   &    |-  ( ph  ->  ( C  mod  E )  =  ( D  mod  E ) )   =>    |-  ( ph  ->  (
 ( A  x.  C )  mod  E )  =  ( ( B  x.  D )  mod  E ) )
 
Theoremmodnegd 11000 Negation property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  ( A  mod  C )  =  ( B  mod  C ) )   =>    |-  ( ph  ->  ( -u A  mod  C )  =  ( -u B  mod  C ) )
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