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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxrsup 12101 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 12102 Extend class notation with the modulo operation.
 class  mod
 
Definitiondf-mod 12103* Define the modulo (remainder) operation. See modval 12104 for its value. (Contributed by NM, 10-Nov-2008.)
 |- 
 mod  =  ( x  e.  RR ,  y  e.  RR+  |->  ( x  -  ( y  x.  ( |_ `  ( x  /  y ) ) ) ) )
 
Theoremmodval 12104 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 ) ) ) ) )
 
Theoremmodvalr 12105 The value of the modulo operation (multiplication in reversed order). (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  =  ( A  -  ( ( |_ `  ( A  /  B ) )  x.  B ) ) )
 
Theoremmodcl 12106 Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  e.  RR )
 
Theoremflpmodeq 12107 Partition of a division into its integer part and the remainder. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( ( |_ `  ( A 
 /  B ) )  x.  B )  +  ( A  mod  B ) )  =  A )
 
Theoremmodcld 12108 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 12109  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 )
 )
 
Theoremmulmod0 12110 The product of an integer and a positive real number is 0 modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.) (Revised by AV, 5-Jul-2020.)
 |-  ( ( A  e.  ZZ  /\  M  e.  RR+ )  ->  ( ( A  x.  M )  mod  M )  =  0 )
 
Theoremnegmod0 12111  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 12112 The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  0  <_  ( A  mod  B ) )
 
Theoremmodlt 12113 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 12114 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 12115 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 12116 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 12117 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 12118 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 12119 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( N  e.  ZZ  ->  ( N  mod  1
 )  =  0 )
 
Theoremmodmulnn 12120 Move a positive integer 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 ) ) )
 
Theoremmodvalp1 12121 The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  +  B )  -  ( ( ( |_ `  ( A  /  B ) )  +  1
 )  x.  B ) )  =  ( A 
 mod  B ) )
 
Theoremzmodcl 12122 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 12123 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 12124 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 12125 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 12126 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 ) )
 
Theoremmodid 12127 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  (
 0  <_  A  /\  A  <  B ) ) 
 ->  ( A  mod  B )  =  A )
 
Theoremmodid0 12128 A positive real number modulo itself is 0. (Contributed by Alexander van der Vekens, 15-May-2018.)
 |-  ( N  e.  RR+  ->  ( N  mod  N )  =  0 )
 
Theoremmodidmul0OLD 12129 The product of an integer and a positive integer is 0 modulo the positive integer. (Contributed by Alexander van der Vekens, 17-May-2018.) Obsolete version of mulmod0 12110 as of 5-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( A  e.  ZZ  /\  N  e.  NN )  ->  ( ( A  x.  N )  mod  N )  =  0 )
 
Theoremmodid2 12130 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A 
 mod  B )  =  A  <->  ( 0  <_  A  /\  A  <  B ) ) )
 
Theoremzmodid2 12131 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 12132 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 12133 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
 |-  ( M  e.  (
 0..^ N )  ->  ( M  mod  N )  =  M )
 
Theorem0mod 12134 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 12135 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 12136 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 12137 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 12138 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 12139 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 12140 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 ) )
 
Theoremmodaddabs 12141 Absorption law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( ( ( A 
 mod  C )  +  ( B  mod  C ) ) 
 mod  C )  =  ( ( A  +  B )  mod  C ) )
 
Theoremmodaddmod 12142 The sum of a real number modulo a positive real number and another real number equals the sum of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  M  e.  RR+ )  ->  ( ( ( A 
 mod  M )  +  B )  mod  M )  =  ( ( A  +  B )  mod  M ) )
 
Theoremmuladdmodid 12143 The sum of a positive real number less than an upper bound and the product of an integer and the upper bound is the positive real number modulo the upper bound. (Contributed by AV, 5-Jul-2020.)
 |-  ( ( N  e.  ZZ  /\  M  e.  RR+  /\  A  e.  ( 0 [,) M ) ) 
 ->  ( ( ( N  x.  M )  +  A )  mod  M )  =  A )
 
Theoremnegmod 12144 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 AV, 5-Jul-2020.)
 |-  ( ( A  e.  RR  /\  N  e.  RR+ )  ->  ( -u A  mod  N )  =  ( ( N  -  A )  mod  N ) )
 
Theoremaddmodid 12145 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 )
 
Theoremmodadd2mod 12146 The sum of a real number modulo a positive real number and another real number equals the sum of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  M  e.  RR+ )  ->  ( ( B  +  ( A  mod  M ) )  mod  M )  =  ( ( B  +  A )  mod  M ) )
 
Theoremmodm1p1mod0 12147 If an real number modulo a positive real number equals the positive real number decreased by 1, the real number increased by 1 modulo the positive real number equals 0. (Contributed by AV, 2-Nov-2018.)
 |-  ( ( A  e.  RR  /\  M  e.  RR+ )  ->  ( ( A 
 mod  M )  =  ( M  -  1 ) 
 ->  ( ( A  +  1 )  mod  M )  =  0 ) )
 
Theoremmodltm1p1mod 12148 If a real number modulo a positive real number is less than the positive real number decreased by 1, the real number increased by 1 modulo the positive real number equals the real number modulo the positive real number increased by 1. (Contributed by AV, 2-Nov-2018.)
 |-  ( ( A  e.  RR  /\  M  e.  RR+  /\  ( A  mod  M )  <  ( M  -  1 ) )  ->  ( ( A  +  1 )  mod  M )  =  ( ( A 
 mod  M )  +  1 ) )
 
Theoremmodmul1 12149 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 12150 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 12151 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 12152 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 12153 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 ) )
 
Theoremmodsubmod 12154 The difference of a real number modulo a positive real number and another real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  M  e.  RR+ )  ->  ( ( ( A 
 mod  M )  -  B )  mod  M )  =  ( ( A  -  B )  mod  M ) )
 
Theoremmodsubmodmod 12155 The difference of a real number modulo a positive real number and another real number modulo this positive real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  M  e.  RR+ )  ->  ( ( ( A 
 mod  M )  -  ( B  mod  M ) ) 
 mod  M )  =  ( ( A  -  B )  mod  M ) )
 
Theorem2txmodxeq0 12156 Two times a positive real number modulo the real number is zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( X  e.  RR+  ->  ( ( 2  x.  X )  mod  X )  =  0 )
 
Theorem2submod 12157 If a real number is between a positive real number and the double of the positive real number, the real number modulo the positive real number equals the real number minus the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )
 
Theoremmodifeq2int 12158 If a nonnegative integer is less than the double of 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 12159 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 12160 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 ) ) )
 
Theoremmodmulmod 12161 The product of a real number modulo a positive real number and an integer equals the product of the real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  ZZ  /\  M  e.  RR+ )  ->  ( ( ( A 
 mod  M )  x.  B )  mod  M )  =  ( ( A  x.  B )  mod  M ) )
 
Theoremmodaddmulmod 12162 The sum of a real number and the product of a second real number modulo a positive real number and an integer equals the sum of the real number and the product of the other real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  ZZ )  /\  M  e.  RR+ )  ->  ( ( A  +  ( ( B  mod  M )  x.  C ) )  mod  M )  =  ( ( A  +  ( B  x.  C ) ) 
 mod  M ) )
 
Theoremmoddi 12163 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 12164 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 ) ) ) )
 
Theoremmodeqmodmin 12165 A real number equals the difference of the real number and a positive real number modulo the positive real number. (Contributed by AV, 3-Nov-2018.)
 |-  ( ( A  e.  RR  /\  M  e.  RR+ )  ->  ( A  mod  M )  =  ( ( A  -  M ) 
 mod  M ) )
 
Theoremmodirr 12166 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  Miscellaneous theorems about integers
 
Theoremom2uz0i 12167* 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 12168* The value of  G (see om2uz0i 12167) 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 12169* The value  G (see om2uz0i 12167) 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 12170* Less-than relation for  G (see om2uz0i 12167). (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 12171* The mapping  G (see om2uz0i 12167) 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 12172* Range of  G (see om2uz0i 12167). (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 12173*  G (see om2uz0i 12167) 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 12174*  G (see om2uz0i 12167) 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 12175* An alternative definition of  G in terms of df-oi 8034. (Contributed by Mario Carneiro, 2-Jun-2015.)
 |-  C  e.  ZZ   &    |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )   =>    |-  G  = OrdIso (  <  ,  ( ZZ>= `  C )
 )
 
Theoremom2uzrdg 12176* 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 12167. (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 12177* 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 12178* The recursive definition generator on upper integers is a function. See comment in om2uzrdg 12176. (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 12179* Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 12176. (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 12180* Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 12176. (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 12181  < 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 12182 Less than well-orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
 |- 
 <  We  NN
 
Theoremltwefz 12183 Less than well-orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)
 |- 
 <  We  ( M ... N )
 
Theoremuzenom 12184 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  Z  ~~  om )
 
Theoremuzinf 12185 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 12186* 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 12187 The cardinality of a finite set of sequential integers. (See om2uz0i 12167 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 12188 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 12189 The cardinality of a finite set of sequential integers. (See om2uz0i 12167 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 12190  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
 
Theoremfzfi 12191 A finite interval of integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
 |-  ( M ... N )  e.  Fin
 
Theoremfzfid 12192 Commonly used special case of fzfi 12191. (Contributed by Mario Carneiro, 25-May-2014.)
 |-  ( ph  ->  ( M ... N )  e. 
 Fin )
 
Theoremfzofi 12193 Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  ( M..^ N )  e.  Fin
 
Theoremfsequb 12194* The values of a finite real sequence have an upper bound. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( A. k  e.  ( M ... N ) ( F `  k )  e.  RR  ->  E. x  e.  RR  A. k  e.  ( M
 ... N ) ( F `  k )  <  x )
 
Theoremfsequb2 12195* The values of a finite real sequence have an upper bound. (Contributed by NM, 20-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
 |-  ( F : ( M ... N ) --> RR  ->  E. x  e.  RR  A. y  e. 
 ran  F  y  <_  x )
 
Theoremfseqsupcl 12196 The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( N  e.  ( ZZ>= `  M )  /\  F : ( M
 ... N ) --> RR )  ->  sup ( ran  F ,  RR ,  <  )  e.  RR )
 
Theoremfseqsupubi 12197 The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005.)
 |-  ( ( K  e.  ( M ... N ) 
 /\  F : ( M ... N ) --> RR )  ->  ( F `  K )  <_  sup ( ran  F ,  RR ,  <  ) )
 
Theoremnn0ennn 12198 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
 |- 
 NN0  ~~  NN
 
Theoremnnenom 12199 The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
 |- 
 NN  ~~  om
 
Theoremuzindi 12200* Indirect strong induction on the upper integers. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  T  e.  ( ZZ>= `  L ) )   &    |-  ( ( ph  /\  R  e.  ( L
 ... T )  /\  A. y ( S  e.  ( L..^ R )  ->  ch ) )  ->  ps )   &    |-  ( x  =  y  ->  ( ps  <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( x  =  y  ->  R  =  S )   &    |-  ( x  =  A  ->  R  =  T )   =>    |-  ( ph  ->  th )
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