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Theorem List for Metamath Proof Explorer - 10801-10900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzind2 10801* 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 9988 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 10802 Extend class notation with floor (greatest integer) function.
 class  |_
 
Definitiondf-fl 10803* Define the floor (greatest integer) function. See flval 10804 for its value, fllelt 10807 for its basic property, and flcl 10805 for its closure. For example,  ( |_ `  ( 3  /  2
) )  =  1 while  ( |_ `  -u ( 3  /  2
) )  =  -u
2 (ex-fl 20647).

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 10804* 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 10805 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 10806 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
 
Theoremfllelt 10807 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 10808 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 10809 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
 |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
 
Theoremflltp1 10810 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)
 |-  ( A  e.  RR  ->  A  <  ( ( |_ `  A )  +  1 ) )
 
Theoremfllep1 10811 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)
 |-  ( A  e.  RR  ->  A  <_  ( ( |_ `  A )  +  1 ) )
 
Theoremfraclt1 10812 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( A  -  ( |_ `  A ) )  <  1 )
 
Theoremfracle1 10813 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 10814 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
 |-  ( A  e.  RR  ->  0  <_  ( A  -  ( |_ `  A ) ) )
 
Theoremflge 10815 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 10816 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 10817 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
 
Theoremflidm 10818 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  ( |_ `  A ) )  =  ( |_ `  A ) )
 
Theoremflidz 10819 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 10820 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 10821 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 10822* 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 10823* 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 10824 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 10825 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 10826 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 10827 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 10828 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 10829 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 10830 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 10831 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 10194.) (Contributed by NM, 12-Mar-2005.)
 |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  ( 0  <_  A  <->  0 
 <_  N ) )
 
Theoremflhalf 10832 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 10833 The ceiling function returns an integer (closure law). (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( A  e.  RR  -> 
 -u ( |_ `  -u A )  e.  ZZ )
 
Theoremceige 10834 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 10835 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 10836 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 10837 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 10838 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 10839 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 10840 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 10841 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 10840. (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 10842 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 10843 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 10844 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 10845 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 10846 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 10847 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 10848 The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |- 
 sup ( RR+ ,  RR* ,  <  )  =  +oo
 
Theoremresup 10849 The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |- 
 sup ( RR ,  RR*
 ,  <  )  =  +oo
 
Theoremxrsup 10850 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 10851 Extend class notation with the modulo operation.
 class  mod
 
Definitiondf-mod 10852* Define the modulo (remainder) operation. See modval 10853 for its value. (Contributed by NM, 10-Nov-2008.)
 |- 
 mod  =  ( x  e.  RR ,  y  e.  RR+  |->  ( x  -  ( y  x.  ( |_ `  ( x  /  y ) ) ) ) )
 
Theoremmodval 10853 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 10854 Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  e.  RR )
 
Theoremmodcld 10855 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 10856  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 10857  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 10858 The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  0  <_  ( A  mod  B ) )
 
Theoremmodlt 10859 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 10860 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 10861 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 10862 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 10863 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 10864 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 10865 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  ZZ  ->  ( N  mod  1
 )  =  0 )
 
Theoremmodmulnn 10866 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 10867 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 10868 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 10869 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 10870 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 10871 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 10872 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 10873 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 10874 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 10875 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 10876 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 10877 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 10878 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 10879 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 10880 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 10881 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 10882 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 ) )
 
Theoremmodadd12d 10883 Additive 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  ->  D  e.  RR )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  ( A  mod  E )  =  ( B  mod  E ) )   &    |-  ( ph  ->  ( C  mod  E )  =  ( D  mod  E ) )   =>    |-  ( ph  ->  (
 ( A  +  C )  mod  E )  =  ( ( B  +  D )  mod  E ) )
 
Theoremmodsub12d 10884 Subtraction 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  ->  D  e.  RR )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  ( A  mod  E )  =  ( B  mod  E ) )   &    |-  ( ph  ->  ( C  mod  E )  =  ( D  mod  E ) )   =>    |-  ( ph  ->  (
 ( A  -  C )  mod  E )  =  ( ( B  -  D )  mod  E ) )
 
Theoremmoddi 10885 Distribute multiplication over a modulo operation. (Contributed by NM, 11-Nov-2008.)
 |-  ( ( A  e.  RR+  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( A  x.  ( B  mod  C ) )  =  ( ( A  x.  B )  mod  ( A  x.  C ) ) )
 
Theoremmodsubdir 10886 Distribute the modulo operation over a subtraction. (Contributed by NM, 30-Dec-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( ( B  mod  C )  <_  ( A  mod  C )  <->  ( ( A  -  B )  mod  C )  =  ( ( A  mod  C )  -  ( B  mod  C ) ) ) )
 
Theoremmodirr 10887 A number modulo an irrational multiple of it is nonzero. (Contributed by NM, 11-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( A  /  B )  e.  ( RR  \  QQ ) )  ->  ( A  mod  B )  =/=  0 )
 
5.6.3  The infinite sequence builder "seq"
 
Theoremom2uz0i 10888* The mapping  G is a one-to-one mapping from  om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number  C (normally 0 for the upper integers  NN0 or 1 for the upper integers  NN), 1 maps to  C + 1, etc. This theorem shows the value of  G at ordinal natural number zero. (This series of theorems generalizes an earlier series for  NN0 contributed by Raph Levien, 10-Apr-2004.) (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  ( G `  (/) )  =  C
 
Theoremom2uzsuci 10889* The value of  G (see om2uz0i 10888) at a successor. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  ( A  e.  om  ->  ( G `  suc  A )  =  ( ( G `  A )  +  1 ) )
 
Theoremom2uzuzi 10890* The value  G (see om2uz0i 10888) at an ordinal natural number is in the upper integers. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  ( A  e.  om  ->  ( G `  A )  e.  ( ZZ>= `  C ) )
 
Theoremom2uzlti 10891* Less-than relation for  G (see om2uz0i 10888). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  e.  B  ->  ( G `  A )  <  ( G `
  B ) ) )
 
Theoremom2uzlt2i 10892* The mapping  G (see om2uz0i 10888) preserves order. (Contributed by NM, 4-May-2005.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( A  e.  B 
 <->  ( G `  A )  <  ( G `  B ) ) )
 
Theoremom2uzrani 10893* Range of  G (see om2uz0i 10888). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |- 
 ran  G  =  ( ZZ>=
 `  C )
 
Theoremom2uzf1oi 10894*  G (see om2uz0i 10888) is a one-to-one onto mapping. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  G : om -1-1-onto-> ( ZZ>= `  C )
 
Theoremom2uzisoi 10895*  G (see om2uz0i 10888) is an isomorphism from natural ordinals to upper integers. (Contributed by NM, 9-Oct-2008.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  G  Isom  _E  ,  <  ( om ,  ( ZZ>= `  C ) )
 
Theoremom2uzoi 10896* An alternative definition of  G in terms of df-oi 7109. (Contributed by Mario Carneiro, 2-Jun-2015.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  G  = OrdIso (  <  ,  ( ZZ>= `  C )
 )
 
Theoremom2uzrdg 10897* A helper lemma for the value of a recursive definition generator on upper integers (typically either  NN or  NN0) with characteristic function  F ( x ,  y ) and initial value  A. Normally  F is a function on the partition, and  A is a member of the partition. See also comment in om2uz0i 10888. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   &    |-  A  e.  _V   &    |-  R  =  ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om )   =>    |-  ( B  e.  om  ->  ( R `  B )  =  <. ( G `
  B ) ,  ( 2nd `  ( R `  B ) )
 >. )
 
Theoremuzrdglem 10898* A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   &    |-  A  e.  _V   &    |-  R  =  ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om )   =>    |-  ( B  e.  ( ZZ>=
 `  C )  ->  <. B ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >.  e.  ran  R )
 
Theoremuzrdgfni 10899* The recursive definition generator on upper integers is a function. See comment in om2uzrdg 10897. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   &    |-  A  e.  _V   &    |-  R  =  ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om )   &    |-  S  =  ran  R   =>    |-  S  Fn  ( ZZ>= `  C )
 
Theoremuzrdg0i 10900* Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 10897. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   &    |-  A  e.  _V   &    |-  R  =  ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om )   &    |-  S  =  ran  R   =>    |-  ( S `  C )  =  A
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