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Theorem List for Intuitionistic Logic Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremflqdiv 10101 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 10102 Extend class notation with the modulo operation.
 class  mod
 
Definitiondf-mod 10103* Define the modulo (remainder) operation. See modqval 10104 for its value. For example,  ( 5  mod  3 )  =  2 and  ( -u 7  mod  2 )  =  1. As with df-fl 10050 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 10104 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 10053 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 10105 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 10106 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 10107 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 10108 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 10109  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 10110 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 10111  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 10112 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 10113 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 10114 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 10115 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 10116 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 10117 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 10118 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 10119 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 10120 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  ZZ  ->  ( N  mod  1
 )  =  0 )
 
Theoremzmod1congr 10121 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 10122 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 10123 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 10124 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 10125 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 10126 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 10127 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 10128 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 10129 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 10130 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 10131 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 10132 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 10133 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 10134 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
 |-  ( M  e.  (
 0..^ N )  ->  ( M  mod  N )  =  M )
 
Theoremq0mod 10135 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 10136 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 10137 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 10138 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 10139 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 10140 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 10141 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 10142 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 10143 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 10144 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 10145 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 10146* 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 10147* 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 10148* 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 10149 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 10150 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 10151 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 10152 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 10153 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 10154 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 10155 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 10156 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 10157 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 10158 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 10159 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 10160 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 10161 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 10162 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 10163 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 10164 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 10165 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 10166 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 ) ) )
 
Theoremmodaddmodup 10167 The sum of an integer modulo a positive integer and another integer minus the positive integer equals the sum of the two integers modulo the positive integer if the other integer is in the upper part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
 |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( ( M  -  ( A  mod  M ) )..^ M )  ->  ( ( B  +  ( A  mod  M ) )  -  M )  =  ( ( B  +  A )  mod  M ) ) )
 
Theoremmodaddmodlo 10168 The sum of an integer modulo a positive integer and another integer equals the sum of the two integers modulo the positive integer if the other integer is in the lower part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.)
 |-  ( ( A  e.  ZZ  /\  M  e.  NN )  ->  ( B  e.  ( 0..^ ( M  -  ( A  mod  M ) ) )  ->  ( B  +  ( A  mod  M ) )  =  ( ( B  +  A )  mod  M ) ) )
 
Theoremmodqmulmod 10169 The product of a rational number modulo a modulus and an integer equals the product of the rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 25-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( ( A 
 mod  M )  x.  B )  mod  M )  =  ( ( A  x.  B )  mod  M ) )
 
Theoremmodqmulmodr 10170 The product of an integer and a rational number modulo a modulus equals the product of the integer and the rational number modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( ( A  e.  ZZ  /\  B  e.  QQ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  x.  ( B  mod  M ) )  mod  M )  =  ( ( A  x.  B )  mod  M ) )
 
Theoremmodqaddmulmod 10171 The sum of a rational number and the product of a second rational number modulo a modulus and an integer equals the sum of the rational number and the product of the other rational number and the integer modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  C  e.  ZZ )  /\  ( M  e.  QQ  /\  0  <  M ) )  ->  ( ( A  +  ( ( B  mod  M )  x.  C ) )  mod  M )  =  ( ( A  +  ( B  x.  C ) )  mod  M ) )
 
Theoremmodqdi 10172 Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  0  <  A )  /\  B  e.  QQ  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( A  x.  ( B  mod  C ) )  =  ( ( A  x.  B )  mod  ( A  x.  C ) ) )
 
Theoremmodqsubdir 10173 Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( C  e.  QQ  /\  0  <  C ) )  ->  ( ( B  mod  C )  <_  ( A  mod  C )  <->  ( ( A  -  B )  mod  C )  =  ( ( A  mod  C )  -  ( B  mod  C ) ) ) )
 
Theoremmodqeqmodmin 10174 A rational number equals the difference of the rational number and a modulus modulo the modulus. (Contributed by Jim Kingdon, 26-Oct-2021.)
 |-  ( ( A  e.  QQ  /\  M  e.  QQ  /\  0  <  M ) 
 ->  ( A  mod  M )  =  ( ( A  -  M )  mod  M ) )
 
Theoremmodfzo0difsn 10175* For a number within a half-open range of nonnegative integers with one excluded integer there is a positive integer so that the number is equal to the sum of the positive integer and the excluded integer modulo the upper bound of the range. (Contributed by AV, 19-Mar-2021.)
 |-  ( ( J  e.  ( 0..^ N )  /\  K  e.  ( (
 0..^ N )  \  { J } ) ) 
 ->  E. i  e.  (
 1..^ N ) K  =  ( ( i  +  J )  mod  N ) )
 
Theoremmodsumfzodifsn 10176 The sum of a number within a half-open range of positive integers is an element of the corresponding open range of nonnegative integers with one excluded integer modulo the excluded integer. (Contributed by AV, 19-Mar-2021.)
 |-  ( ( J  e.  ( 0..^ N )  /\  K  e.  ( 1..^ N ) )  ->  ( ( K  +  J )  mod  N )  e.  ( ( 0..^ N )  \  { J } ) )
 
Theoremmodlteq 10177 Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
 |-  ( ( I  e.  ( 0..^ N ) 
 /\  J  e.  (
 0..^ N ) ) 
 ->  ( ( I  mod  N )  =  ( J 
 mod  N )  <->  I  =  J ) )
 
Theoremaddmodlteq 10178 Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. (Contributed by AV, 20-Mar-2021.)
 |-  ( ( I  e.  ( 0..^ N ) 
 /\  J  e.  (
 0..^ N )  /\  S  e.  ZZ )  ->  ( ( ( I  +  S )  mod  N )  =  ( ( J  +  S ) 
 mod  N )  <->  I  =  J ) )
 
4.6.3  Miscellaneous theorems about integers
 
Theoremfrec2uz0d 10179* 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. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  ( G `  (/) )  =  C )
 
Theoremfrec2uzzd 10180* The value of  G (see frec2uz0d 10179) is an integer. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   =>    |-  ( ph  ->  ( G `  A )  e. 
 ZZ )
 
Theoremfrec2uzsucd 10181* The value of  G (see frec2uz0d 10179) at a successor. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   =>    |-  ( ph  ->  ( G `  suc  A )  =  ( ( G `
  A )  +  1 ) )
 
Theoremfrec2uzuzd 10182* The value  G (see frec2uz0d 10179) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   =>    |-  ( ph  ->  ( G `  A )  e.  ( ZZ>= `  C )
 )
 
Theoremfrec2uzltd 10183* Less-than relation for  G (see frec2uz0d 10179). (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( A  e.  B  ->  ( G `  A )  <  ( G `  B ) ) )
 
Theoremfrec2uzlt2d 10184* The mapping  G (see frec2uz0d 10179) preserves order. (Contributed by Jim Kingdon, 16-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  om )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( A  e.  B  <->  ( G `  A )  <  ( G `
  B ) ) )
 
Theoremfrec2uzrand 10185* Range of  G (see frec2uz0d 10179). (Contributed by Jim Kingdon, 17-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  ran  G  =  ( ZZ>= `  C )
 )
 
Theoremfrec2uzf1od 10186*  G (see frec2uz0d 10179) is a one-to-one onto mapping. (Contributed by Jim Kingdon, 17-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  G : om
 -1-1-onto-> ( ZZ>= `  C )
 )
 
Theoremfrec2uzisod 10187*  G (see frec2uz0d 10179) is an isomorphism from natural ordinals to upper integers. (Contributed by Jim Kingdon, 17-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  G  Isom  _E  ,  <  ( om ,  ( ZZ>= `  C ) ) )
 
Theoremfrecuzrdgrrn 10188* The function  R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of 
S. (Contributed by Jim Kingdon, 28-Mar-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ( ph  /\  D  e.  om )  ->  ( R `  D )  e.  ( ( ZZ>= `  C )  X.  S ) )
 
Theoremfrec2uzrdg 10189* 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. This lemma shows that evaluating  R at an element of  om gives an ordered pair whose first element is the index (translated from  om to  ( ZZ>= `  C )). See comment in frec2uz0d 10179 which describes  G and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  B  e.  om )   =>    |-  ( ph  ->  ( R `  B )  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B ) ) >. )
 
Theoremfrecuzrdgrcl 10190* The function  R (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 1-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  R : om --> ( ( ZZ>= `  C )  X.  S ) )
 
Theoremfrecuzrdglem 10191* A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  B  e.  ( ZZ>= `  C ) )   =>    |-  ( ph  ->  <. B ,  ( 2nd `  ( R `  ( `' G `  B ) ) )
 >.  e.  ran  R )
 
Theoremfrecuzrdgtcl 10192* The recursive definition generator on upper integers is a function. See comment in frec2uz0d 10179 for the description of  G as the mapping from  om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 26-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  T  =  ran  R )   =>    |-  ( ph  ->  T :
 ( ZZ>= `  C ) --> S )
 
Theoremfrecuzrdg0 10193* Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10179 for the description of  G as the mapping from  om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 27-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  T  =  ran  R )   =>    |-  ( ph  ->  ( T `  C )  =  A )
 
Theoremfrecuzrdgsuc 10194* Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10179 for the description of  G as the mapping from 
om to  ( ZZ>= `  C
). (Contributed by Jim Kingdon, 28-May-2020.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>=
 `  C )  /\  y  e.  S )
 )  ->  ( x F y )  e.  S )   &    |-  R  = frec (
 ( x  e.  ( ZZ>=
 `  C ) ,  y  e.  S  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  T  =  ran  R )   =>    |-  ( ( ph  /\  B  e.  ( ZZ>= `  C )
 )  ->  ( T `  ( B  +  1 ) )  =  ( B F ( T `
  B ) ) )
 
Theoremfrecuzrdgrclt 10195* The function  R (used in the definition of the recursive definition generator on upper integers) yields ordered pairs of integers and elements of  S. Similar to frecuzrdgrcl 10190 except that  S and  T need not be the same. (Contributed by Jim Kingdon, 22-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  R : om --> ( ( ZZ>= `  C )  X.  S ) )
 
Theoremfrecuzrdgg 10196* Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating  R at a natural number gives an ordered pair whose first element is the mapping of that natural number via  G. (Contributed by Jim Kingdon, 23-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  ( ph  ->  N  e.  om )   &    |-  G  = frec ( ( x  e. 
 ZZ  |->  ( x  +  1 ) ) ,  C )   =>    |-  ( ph  ->  ( 1st `  ( R `  N ) )  =  ( G `  N ) )
 
Theoremfrecuzrdgdomlem 10197* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C ) )
 
Theoremfrecuzrdgdom 10198* The domain of the result of the recursive definition generator on upper integers. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  dom  ran  R  =  ( ZZ>= `  C ) )
 
Theoremfrecuzrdgfunlem 10199* The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   &    |-  G  = frec (
 ( x  e.  ZZ  |->  ( x  +  1
 ) ) ,  C )   =>    |-  ( ph  ->  Fun  ran  R )
 
Theoremfrecuzrdgfun 10200* The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
 |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  A  e.  S )   &    |-  ( ph  ->  S  C_  T )   &    |-  ( ( ph  /\  ( x  e.  ( ZZ>= `  C )  /\  y  e.  S ) )  ->  ( x F y )  e.  S )   &    |-  R  = frec ( ( x  e.  ( ZZ>= `  C ) ,  y  e.  T  |->  <.
 ( x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )   =>    |-  ( ph  ->  Fun  ran  R )
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