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Theorem List for Intuitionistic Logic Explorer - 10301-10400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremflqltnz 10301 If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  -.  A  e.  ZZ )  ->  ( |_ `  A )  <  A )
 
Theoremflqwordi 10302 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  A  <_  B )  ->  ( |_ `  A )  <_  ( |_ `  B ) )
 
Theoremflqword2 10303 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  A  <_  B )  ->  ( |_ `  B )  e.  ( ZZ>= `  ( |_ `  A ) ) )
 
Theoremflqbi 10304 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  ZZ )  ->  ( ( |_ `  A )  =  B  <->  ( B  <_  A  /\  A  <  ( B  +  1 ) ) ) )
 
Theoremflqbi2 10305 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
 |-  ( ( N  e.  ZZ  /\  F  e.  QQ )  ->  ( ( |_ `  ( N  +  F ) )  =  N  <->  ( 0  <_  F  /\  F  <  1 ) ) )
 
Theoremadddivflid 10306 The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN0  /\  C  e.  NN )  ->  ( B  <  C  <->  ( |_ `  ( A  +  ( B  /  C ) ) )  =  A ) )
 
Theoremflqge0nn0 10307 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  0  <_  A )  ->  ( |_ `  A )  e.  NN0 )
 
Theoremflqge1nn 10308 The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  1  <_  A )  ->  ( |_ `  A )  e.  NN )
 
Theoremfldivnn0 10309 The floor function of a division of a nonnegative integer by a positive integer is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) )  e.  NN0 )
 
Theoremdivfl0 10310 The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN )  ->  ( A  <  B  <->  ( |_ `  ( A 
 /  B ) )  =  0 ) )
 
Theoremflqaddz 10311 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  N  e.  ZZ )  ->  ( |_ `  ( A  +  N )
 )  =  ( ( |_ `  A )  +  N ) )
 
Theoremflqzadd 10312 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( ( N  e.  ZZ  /\  A  e.  QQ )  ->  ( |_ `  ( N  +  A )
 )  =  ( N  +  ( |_ `  A ) ) )
 
Theoremflqmulnn0 10313 Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( ( N  e.  NN0  /\  A  e.  QQ )  ->  ( N  x.  ( |_ `  A ) ) 
 <_  ( |_ `  ( N  x.  A ) ) )
 
Theorembtwnzge0 10314 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. (Contributed by NM, 12-Mar-2005.)
 |-  ( ( ( A  e.  RR  /\  N  e.  ZZ )  /\  ( N  <_  A  /\  A  <  ( N  +  1 ) ) )  ->  ( 0  <_  A  <->  0 
 <_  N ) )
 
Theorem2tnp1ge0ge0 10315 Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.)
 |-  ( N  e.  ZZ  ->  ( 0  <_  (
 ( 2  x.  N )  +  1 )  <->  0 
 <_  N ) )
 
Theoremflhalf 10316 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 ) ) ) )
 
Theoremfldivnn0le 10317 The floor function of a division of a nonnegative integer by a positive integer is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  L  e.  NN )  ->  ( |_ `  ( K  /  L ) ) 
 <_  ( K  /  L ) )
 
Theoremflltdivnn0lt 10318 The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( K  e.  NN0  /\  N  e.  NN0  /\  L  e.  NN )  ->  ( K  <  N  ->  ( |_ `  ( K  /  L ) )  < 
 ( N  /  L ) ) )
 
Theoremfldiv4p1lem1div2 10319 The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
 |-  ( ( N  =  3  \/  N  e.  ( ZZ>=
 `  5 ) ) 
 ->  ( ( |_ `  ( N  /  4 ) )  +  1 )  <_  ( ( N  -  1 )  /  2
 ) )
 
Theoremceilqval 10320 The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( `  A )  =  -u ( |_ `  -u A ) )
 
Theoremceiqcl 10321 The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  -> 
 -u ( |_ `  -u A )  e.  ZZ )
 
Theoremceilqcl 10322 Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( `  A )  e.  ZZ )
 
Theoremceiqge 10323 The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  A  <_  -u ( |_ `  -u A ) )
 
Theoremceilqge 10324 The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  A  <_  ( `  A ) )
 
Theoremceiqm1l 10325 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( -u ( |_ `  -u A )  -  1 )  <  A )
 
Theoremceilqm1lt 10326 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( ( `  A )  -  1 )  <  A )
 
Theoremceiqle 10327 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  A  <_  B )  -> 
 -u ( |_ `  -u A )  <_  B )
 
Theoremceilqle 10328 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  ZZ  /\  A  <_  B )  ->  ( `  A )  <_  B )
 
Theoremceilid 10329 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)
 |-  ( A  e.  ZZ  ->  ( `  A )  =  A )
 
Theoremceilqidz 10330 A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  ( `  A )  =  A ) )
 
Theoremflqleceil 10331 The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  A )  <_  ( `  A )
 )
 
Theoremflqeqceilz 10332 A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( A  e.  ZZ  <->  ( |_ `  A )  =  ( `  A )
 ) )
 
Theoremintqfrac2 10333 Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  Z  =  ( |_ `  A )   &    |-  F  =  ( A  -  Z )   =>    |-  ( A  e.  QQ  ->  ( 0  <_  F  /\  F  <  1  /\  A  =  ( Z  +  F ) ) )
 
Theoremintfracq 10334 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10333. (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 ) ) )
 
Theoremflqdiv 10335 Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  N  e.  NN )  ->  ( |_ `  (
 ( |_ `  A )  /  N ) )  =  ( |_ `  ( A  /  N ) ) )
 
4.6.2  The modulo (remainder) operation
 
Syntaxcmo 10336 Extend class notation with the modulo operation.
 class  mod
 
Definitiondf-mod 10337* Define the modulo (remainder) operation. See modqval 10338 for its value. For example,  ( 5  mod  3 )  =  2 and  ( -u 7  mod  2 )  =  1. As with df-fl 10284 we define this for first and second arguments which are real and positive real, respectively, even though many theorems will need to be more restricted (for example, specify rational arguments). (Contributed by NM, 10-Nov-2008.)
 |- 
 mod  =  ( x  e.  RR ,  y  e.  RR+  |->  ( x  -  ( y  x.  ( |_ `  ( x  /  y ) ) ) ) )
 
Theoremmodqval 10338 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 numbers 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.) As with flqcl 10287 we only prove this for rationals although other particular kinds of real numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( A  mod  B )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
 
Theoremmodqvalr 10339 The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( A  mod  B )  =  ( A  -  ( ( |_ `  ( A  /  B ) )  x.  B ) ) )
 
Theoremmodqcl 10340 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( A  mod  B )  e.  QQ )
 
Theoremflqpmodeq 10341 Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( ( |_ `  ( A  /  B ) )  x.  B )  +  ( A  mod  B ) )  =  A )
 
Theoremmodqcld 10342 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  ( A  mod  B )  e. 
 QQ )
 
Theoremmodq0 10343  A  mod  B is zero iff  A is evenly divisible by  B. (Contributed by Jim Kingdon, 17-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  mod  B )  =  0  <->  ( A  /  B )  e.  ZZ ) )
 
Theoremmulqmod0 10344 The product of an integer and a positive rational number is 0 modulo the positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M ) 
 ->  ( ( A  x.  M )  mod  M )  =  0 )
 
Theoremnegqmod0 10345  A is divisible by  B iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  mod  B )  =  0  <->  ( -u A  mod  B )  =  0 ) )
 
Theoremmodqge0 10346 The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  0  <_  ( A 
 mod  B ) )
 
Theoremmodqlt 10347 The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( A  mod  B )  <  B )
 
Theoremmodqelico 10348 Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( A  mod  B )  e.  ( 0 [,) B ) )
 
Theoremmodqdiffl 10349 The modulo operation differs from 
A by an integer multiple of  B. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  -  ( A  mod  B ) )  /  B )  =  ( |_ `  ( A  /  B ) ) )
 
Theoremmodqdifz 10350 The modulo operation differs from 
A by an integer multiple of  B. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  -  ( A  mod  B ) )  /  B )  e.  ZZ )
 
Theoremmodqfrac 10351 The fractional part of a number is the number modulo 1. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( A  mod  1
 )  =  ( A  -  ( |_ `  A ) ) )
 
Theoremflqmod 10352 The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( A  e.  QQ  ->  ( |_ `  A )  =  ( A  -  ( A  mod  1
 ) ) )
 
Theoremintqfrac 10353 Break a number into its integer part and its fractional part. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( A  e.  QQ  ->  A  =  ( ( |_ `  A )  +  ( A  mod  1 ) ) )
 
Theoremzmod10 10354 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  ZZ  ->  ( N  mod  1
 )  =  0 )
 
Theoremzmod1congr 10355 Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  mod  1 )  =  ( B  mod  1 ) )
 
Theoremmodqmulnn 10356 Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)
 |-  ( ( N  e.  NN  /\  A  e.  QQ  /\  M  e.  NN )  ->  ( ( N  x.  ( |_ `  A ) )  mod  ( N  x.  M ) ) 
 <_  ( ( |_ `  ( N  x.  A ) ) 
 mod  ( N  x.  M ) ) )
 
Theoremmodqvalp1 10357 The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Jim Kingdon, 20-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  +  B )  -  (
 ( ( |_ `  ( A  /  B ) )  +  1 )  x.  B ) )  =  ( A  mod  B ) )
 
Theoremzmodcl 10358 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 10359 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 10360 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 10361 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 ) )
 
Theoremzmodfzp1 10362 An integer mod  B lies in the first  B  +  1 nonnegative integers. (Contributed by AV, 27-Oct-2018.)
 |-  ( ( A  e.  ZZ  /\  B  e.  NN )  ->  ( A  mod  B )  e.  ( 0
 ... B ) )
 
Theoremmodqid 10363 Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  (
 0  <_  A  /\  A  <  B ) ) 
 ->  ( A  mod  B )  =  A )
 
Theoremmodqid0 10364 A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( N  e.  QQ  /\  0  <  N )  ->  ( N  mod  N )  =  0 )
 
Theoremmodqid2 10365 Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  mod  B )  =  A  <->  ( 0  <_  A  /\  A  <  B ) ) )
 
Theoremzmodid2 10366 Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M 
 mod  N )  =  M  <->  M  e.  ( 0 ... ( N  -  1
 ) ) ) )
 
Theoremzmodidfzo 10367 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
 |-  ( ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M 
 mod  N )  =  M  <->  M  e.  ( 0..^ N ) ) )
 
Theoremzmodidfzoimp 10368 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
 |-  ( M  e.  (
 0..^ N )  ->  ( M  mod  N )  =  M )
 
Theoremq0mod 10369 Special case: 0 modulo a positive real number is 0. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( N  e.  QQ  /\  0  <  N )  ->  ( 0  mod 
 N )  =  0 )
 
Theoremq1mod 10370 Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( N  e.  QQ  /\  1  <  N )  ->  ( 1  mod 
 N )  =  1 )
 
Theoremmodqabs 10371 Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  0  <  B )   &    |-  ( ph  ->  C  e.  QQ )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  (
 ( A  mod  B )  mod  C )  =  ( A  mod  B ) )
 
Theoremmodqabs2 10372 Absorption law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B ) 
 ->  ( ( A  mod  B )  mod  B )  =  ( A  mod  B ) )
 
Theoremmodqcyc 10373 The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  <  B ) )  ->  ( ( A  +  ( N  x.  B ) )  mod  B )  =  ( A  mod  B ) )
 
Theoremmodqcyc2 10374 The modulo operation is periodic. (Contributed by Jim Kingdon, 21-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  N  e.  ZZ )  /\  ( B  e.  QQ  /\  0  <  B ) )  ->  ( ( A  -  ( B  x.  N ) )  mod  B )  =  ( A  mod  B ) )
 
Theoremmodqadd1 10375 Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  C  e.  QQ )   &    |-  ( ph  ->  D  e.  QQ )   &    |-  ( ph  ->  0  <  D )   &    |-  ( ph  ->  ( A  mod  D )  =  ( B 
 mod  D ) )   =>    |-  ( ph  ->  ( ( A  +  C )  mod  D )  =  ( ( B  +  C )  mod  D ) )
 
Theoremmodqaddabs 10376 Absorption law for modulo. (Contributed by Jim Kingdon, 22-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( ( A 
 mod  C )  +  ( B  mod  C ) ) 
 mod  C )  =  ( ( A  +  B )  mod  C ) )
 
Theoremmodqaddmod 10377 The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the same modulus. (Contributed by Jim Kingdon, 23-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( ( A 
 mod  M )  +  B )  mod  M )  =  ( ( A  +  B )  mod  M ) )
 
Theoremmulqaddmodid 10378 The sum of a positive rational number less than an upper bound and the product of an integer and the upper bound is the positive rational number modulo the upper bound. (Contributed by Jim Kingdon, 23-Oct-2021.)
 |-  ( ( ( N  e.  ZZ  /\  M  e.  QQ )  /\  ( A  e.  QQ  /\  A  e.  ( 0 [,) M ) ) )  ->  ( ( ( N  x.  M )  +  A )  mod  M )  =  A )
 
Theoremmulp1mod1 10379 The product of an integer and an integer greater than 1 increased by 1 is 1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.)
 |-  ( ( A  e.  ZZ  /\  N  e.  ( ZZ>=
 `  2 ) ) 
 ->  ( ( ( N  x.  A )  +  1 )  mod  N )  =  1 )
 
Theoremmodqmuladd 10380* Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  B  e.  (
 0 [,) M ) )   &    |-  ( ph  ->  M  e.  QQ )   &    |-  ( ph  ->  0  <  M )   =>    |-  ( ph  ->  ( ( A  mod  M )  =  B  <->  E. k  e.  ZZ  A  =  ( (
 k  x.  M )  +  B ) ) )
 
Theoremmodqmuladdim 10381* Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
 |-  ( ( A  e.  ZZ  /\  M  e.  QQ  /\  0  <  M ) 
 ->  ( ( A  mod  M )  =  B  ->  E. k  e.  ZZ  A  =  ( ( k  x.  M )  +  B ) ) )
 
Theoremmodqmuladdnn0 10382* Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)
 |-  ( ( A  e.  NN0  /\  M  e.  QQ  /\  0  <  M )  ->  ( ( A  mod  M )  =  B  ->  E. k  e.  NN0  A  =  ( ( k  x.  M )  +  B ) ) )
 
Theoremqnegmod 10383 The negation of a number modulo a positive number is equal to the difference of the modulus and the number modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  N  e.  QQ  /\  0  <  N ) 
 ->  ( -u A  mod  N )  =  ( ( N  -  A )  mod  N ) )
 
Theoremm1modnnsub1 10384 Minus one modulo a positive integer is equal to the integer minus one. (Contributed by AV, 14-Jul-2021.)
 |-  ( M  e.  NN  ->  ( -u 1  mod  M )  =  ( M  -  1 ) )
 
Theoremm1modge3gt1 10385 Minus one modulo an integer greater than two is greater than one. (Contributed by AV, 14-Jul-2021.)
 |-  ( M  e.  ( ZZ>=
 `  3 )  -> 
 1  <  ( -u 1  mod  M ) )
 
Theoremaddmodid 10386 The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof shortened by AV, 5-Jul-2020.)
 |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  ( ( M  +  A )  mod  M )  =  A )
 
Theoremaddmodidr 10387 The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by AV, 19-Mar-2021.)
 |-  ( ( A  e.  NN0  /\  M  e.  NN  /\  A  <  M )  ->  ( ( A  +  M )  mod  M )  =  A )
 
Theoremmodqadd2mod 10388 The sum of a number modulo a modulus and another number equals the sum of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  A )  mod  M ) )
 
Theoremmodqm1p1mod0 10389 If a number modulo a modulus equals the modulus decreased by 1, the first number increased by 1 modulo the modulus equals 0. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M ) 
 ->  ( ( A  mod  M )  =  ( M  -  1 )  ->  ( ( A  +  1 )  mod  M )  =  0 ) )
 
Theoremmodqltm1p1mod 10390 If a number modulo a modulus is less than the modulus decreased by 1, the first number increased by 1 modulo the modulus equals the first number modulo the modulus, increased by 1. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  ( A  mod  M )  < 
 ( M  -  1
 ) )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  +  1 )  mod  M )  =  ( ( A 
 mod  M )  +  1 ) )
 
Theoremmodqmul1 10391 Multiplication property of the modulo operation. Note that the multiplier  C must be an integer. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  QQ )   &    |-  ( ph  ->  0  <  D )   &    |-  ( ph  ->  ( A  mod  D )  =  ( B 
 mod  D ) )   =>    |-  ( ph  ->  ( ( A  x.  C )  mod  D )  =  ( ( B  x.  C )  mod  D ) )
 
Theoremmodqmul12d 10392 Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ph  ->  A  e.  ZZ )   &    |-  ( ph  ->  B  e.  ZZ )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  ZZ )   &    |-  ( ph  ->  E  e.  QQ )   &    |-  ( ph  ->  0  <  E )   &    |-  ( 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 ) )
 
Theoremmodqnegd 10393 Negation property of the modulo operation. (Contributed by Jim Kingdon, 24-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  C  e.  QQ )   &    |-  ( ph  ->  0  <  C )   &    |-  ( ph  ->  ( A  mod  C )  =  ( B  mod  C ) )   =>    |-  ( ph  ->  ( -u A  mod  C )  =  ( -u B  mod  C ) )
 
Theoremmodqadd12d 10394 Additive property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  C  e.  QQ )   &    |-  ( ph  ->  D  e.  QQ )   &    |-  ( ph  ->  E  e.  QQ )   &    |-  ( ph  ->  0  <  E )   &    |-  ( ph  ->  ( A  mod  E )  =  ( B  mod  E ) )   &    |-  ( ph  ->  ( C  mod  E )  =  ( D  mod  E ) )   =>    |-  ( ph  ->  (
 ( A  +  C )  mod  E )  =  ( ( B  +  D )  mod  E ) )
 
Theoremmodqsub12d 10395 Subtraction property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ph  ->  A  e.  QQ )   &    |-  ( ph  ->  B  e.  QQ )   &    |-  ( ph  ->  C  e.  QQ )   &    |-  ( ph  ->  D  e.  QQ )   &    |-  ( ph  ->  E  e.  QQ )   &    |-  ( ph  ->  0  <  E )   &    |-  ( ph  ->  ( A  mod  E )  =  ( B  mod  E ) )   &    |-  ( ph  ->  ( C  mod  E )  =  ( D  mod  E ) )   =>    |-  ( ph  ->  (
 ( A  -  C )  mod  E )  =  ( ( B  -  D )  mod  E ) )
 
Theoremmodqsubmod 10396 The difference of a number modulo a modulus and another number equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( ( A 
 mod  M )  -  B )  mod  M )  =  ( ( A  -  B )  mod  M ) )
 
Theoremmodqsubmodmod 10397 The difference of a number modulo a modulus and another number modulo the same modulus equals the difference of the two numbers modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( ( A 
 mod  M )  -  ( B  mod  M ) ) 
 mod  M )  =  ( ( A  -  B )  mod  M ) )
 
Theoremq2txmodxeq0 10398 Two times a positive number modulo the number is zero. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( X  e.  QQ  /\  0  <  X )  ->  ( ( 2  x.  X )  mod  X )  =  0 )
 
Theoremq2submod 10399 If a number is between a modulus and twice the modulus, the first number modulo the modulus equals the first number minus the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B 
 <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )
 
Theoremmodifeq2int 10400 If a nonnegative integer is less than twice a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  ->  ( A  mod  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
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