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Theorem List for Metamath Proof Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfraclt1 10901 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)
 |-  ( A  e.  RR  ->  ( A  -  ( |_ `  A ) )  <  1 )
 
Theoremfracle1 10902 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 10903 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)
 |-  ( A  e.  RR  ->  0  <_  ( A  -  ( |_ `  A ) ) )
 
Theoremflge 10904 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 10905 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 10906 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
 |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
 
Theoremflidm 10907 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)
 |-  ( A  e.  RR  ->  ( |_ `  ( |_ `  A ) )  =  ( |_ `  A ) )
 
Theoremflidz 10908 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 10909 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 10910 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 10911* 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 10912* 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 10913 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 10914 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 10915 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 10916 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 10917 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 10918 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 10919 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 10920 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 10283.) (Contributed by NM, 12-Mar-2005.)
 |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  ( 0  <_  A  <->  0 
 <_  N ) )
 
Theoremflhalf 10921 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 10922 The ceiling function returns an integer (closure law). (Contributed by Jeffrey Hankins, 10-Jun-2007.)
 |-  ( A  e.  RR  -> 
 -u ( |_ `  -u A )  e.  ZZ )
 
Theoremceige 10923 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 10924 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 10925 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 10926 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 10927 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 10928 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 10929 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 10930 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 10929. (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 10931 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 10932 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 10933 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 10934 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 10935 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 10936 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 10937 The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |- 
 sup ( RR+ ,  RR* ,  <  )  =  +oo
 
Theoremresup 10938 The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
 |- 
 sup ( RR ,  RR*
 ,  <  )  =  +oo
 
Theoremxrsup 10939 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 10940 Extend class notation with the modulo operation.
 class  mod
 
Definitiondf-mod 10941* Define the modulo (remainder) operation. See modval 10942 for its value. (Contributed by NM, 10-Nov-2008.)
 |- 
 mod  =  ( x  e.  RR ,  y  e.  RR+  |->  ( x  -  ( y  x.  ( |_ `  ( x  /  y ) ) ) ) )
 
Theoremmodval 10942 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 10943 Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  e.  RR )
 
Theoremmodcld 10944 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 10945  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 10946  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 10947 The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  0  <_  ( A  mod  B ) )
 
Theoremmodlt 10948 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 10949 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 10950 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 10951 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 10952 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 10953 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 10954 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  ZZ  ->  ( N  mod  1
 )  =  0 )
 
Theoremmodmulnn 10955 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 10956 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 10957 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 10958 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 10959 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 10960 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 10961 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 10962 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 10963 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 10964 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 10965 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 10966 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 10967 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 10968 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 10969 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 10970 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 10971 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 10972 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 10973 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 10974 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 10975 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 10976 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 10977* 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 10978* The value of  G (see om2uz0i 10977) 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 10979* The value  G (see om2uz0i 10977) 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 10980* Less-than relation for  G (see om2uz0i 10977). (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 10981* The mapping  G (see om2uz0i 10977) 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 10982* Range of  G (see om2uz0i 10977). (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 10983*  G (see om2uz0i 10977) 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 10984*  G (see om2uz0i 10977) 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 10985* An alternative definition of  G in terms of df-oi 7193. (Contributed by Mario Carneiro, 2-Jun-2015.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  G  = OrdIso (  <  ,  ( ZZ>= `  C )
 )
 
Theoremom2uzrdg 10986* 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 10977. (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 10987* 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 10988* The recursive definition generator on upper integers is a function. See comment in om2uzrdg 10986. (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 10989* Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 10986. (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
 
Theoremuzrdgsuci 10990* Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 10986. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  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   =>    |-  ( B  e.  ( ZZ>=
 `  C )  ->  ( S `  ( B  +  1 ) )  =  ( B F ( S `  B ) ) )
 
Theoremltweuz 10991  < is a well-founded relation on any sequence of upper integers. (Contributed by Andrew Salmon, 13-Nov-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |- 
 <  We  ( ZZ>= `  A )
 
Theoremltwenn 10992 Less than well orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
 |- 
 <  We  NN
 
Theoremltwefz 10993 Less than well orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |- 
 <  We  ( M ... N )
 
Theoremuzenom 10994 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  Z  ~~  om )
 
Theoremuzinf 10995 An upper integer set is infinite. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  -.  Z  e.  Fin )
 
Theoremuzrdgxfr 10996* Transfer the value of the recursive sequence builder from one base to another. (Contributed by Mario Carneiro, 1-Apr-2014.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  A )  |`  om )   &    |-  H  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  B )  |`  om )   &    |-  A  e.  ZZ   &    |-  B  e.  ZZ   =>    |-  ( N  e.  om  ->  ( G `  N )  =  ( ( H `  N )  +  ( A  -  B ) ) )
 
Theoremfzennn 10997 The cardinality of a finite set of sequential integers. (See om2uz0i 10977 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  ( N  e.  NN0  ->  (
 1 ... N )  ~~  ( `' G `  N ) )
 
Theoremfzen2 10998 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Mario Carneiro, 13-Feb-2014.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  ( N  e.  ( ZZ>= `  M )  ->  ( M
 ... N )  ~~  ( `' G `  ( ( N  +  1 )  -  M ) ) )
 
Theoremcardfz 10999 The cardinality of a finite set of sequential integers. (See om2uz0i 10977 for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  ( N  e.  NN0  ->  ( card `  ( 1 ...
 N ) )  =  ( `' G `  N ) )
 
Theoremhashgf1o 11000  G maps  om one-to-one onto  NN0. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 13-Sep-2013.)
 |-  G  =  ( rec ( ( x  e. 
 _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )   =>    |-  G : om
 -1-1-onto-> NN0
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