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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfceil2 12101* Alternative definition of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)

Theoremceilval2 12102* The value of the ceiling function using restricted iota. (Contributed by AV, 1-Dec-2018.)

Theoremceicl 12103 The ceiling function returns an integer (closure law). (Contributed by Jeff Hankins, 10-Jun-2007.)

Theoremceilcl 12104 Closure of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)

Theoremceige 12105 The ceiling of a real number is greater than or equal to that number. (Contributed by Jeff Hankins, 10-Jun-2007.)

Theoremceilge 12106 The ceiling of a real number is greater than or equal to that number. (Contributed by AV, 30-Nov-2018.)

Theoremceim1l 12107 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeff Hankins, 10-Jun-2007.)

Theoremceilm1lt 12108 One less than the ceiling of a real number is strictly less than that number. (Contributed by AV, 30-Nov-2018.)

Theoremceile 12109 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jeff Hankins, 10-Jun-2007.)

Theoremceille 12110 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by AV, 30-Nov-2018.)

Theoremceilid 12111 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)

Theoremceilidz 12112 A real number equals its ceiling iff it is an integer. (Contributed by AV, 30-Nov-2018.)

Theoremflleceil 12113 The floor of a real number is less than or equal to its ceiling. (Contributed by AV, 30-Nov-2018.)

Theoremfleqceilz 12114 A real number is an integer iff its floor equals its ceiling. (Contributed by AV, 30-Nov-2018.)

Theoremquoremz 12115 Quotient and remainder of an integer divided by a positive integer. TO DO - is this really needed for anything? Should we use to simplify it? (Contributed by NM, 14-Aug-2008.)

Theoremquoremnn0 12116 Quotient and remainder of a nonnegative integer divided by a positive integer. (Contributed by NM, 14-Aug-2008.)

Theoremquoremnn0ALT 12117 Alternate proof of quoremnn0 12116 not using quoremz 12115. TODO - Keep either quoremnn0ALT 12117 (if we don't keep quoremz 12115) or quoremnn0 12116. (Contributed by NM, 14-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremintfrac2 12118 Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 12145? (Contributed by NM, 16-Aug-2008.)

Theoremintfracq 12119 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 12118. (Contributed by NM, 16-Aug-2008.)

Theoremfldiv 12120 Cancellation of the embedded floor of a real divided by an integer. (Contributed by NM, 16-Aug-2008.)

Theoremfldiv2 12121 Cancellation of an embedded floor of a ratio. Generalization of Equation 2.4 in [CormenLeisersonRivest] p. 33 (where must be an integer). (Contributed by NM, 9-Nov-2008.)

Theoremfznnfl 12122 Finite set of sequential integers starting at 1 and ending at a real number. (Contributed by Mario Carneiro, 3-May-2016.)

Theoremuzsup 12123 An upper set of integers is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremioopnfsup 12124 An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremicopnfsup 12125 An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremrpsup 12126 The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremresup 12127 The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremxrsup 12128 The extended real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

5.6.2  The modulo (remainder) operation

Syntaxcmo 12129 Extend class notation with the modulo operation.

Definitiondf-mod 12130* Define the modulo (remainder) operation. See modval 12131 for its value. (Contributed by NM, 10-Nov-2008.)

Theoremmodval 12131 The value of the modulo operation. The modulo congruence notation of number theory, modulo , can be expressed in our notation as . 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.)

Theoremmodvalr 12132 The value of the modulo operation (multiplication in reversed order). (Contributed by Alexander van der Vekens, 14-Apr-2018.)

Theoremmodcl 12133 Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)

Theoremflpmodeq 12134 Partition of a division into its integer part and the remainder. (Contributed by Alexander van der Vekens, 14-Apr-2018.)

Theoremmodcld 12135 Closure law for the modulo operation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmod0 12136 is zero iff is evenly divisible by . (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)

Theoremmulmod0 12137 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.)

Theoremnegmod0 12138 is divisible by iff its negative is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)

Theoremmodge0 12139 The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)

Theoremmodlt 12140 The modulo operation is less than its second argument. (Contributed by NM, 10-Nov-2008.)

Theoremmoddiffl 12141 The modulo operation differs from by an integer multiple of . (Contributed by Mario Carneiro, 6-Sep-2016.)

Theoremmoddifz 12142 The modulo operation differs from by an integer multiple of . (Contributed by Mario Carneiro, 15-Jul-2014.)

Theoremmodfrac 12143 The fractional part of a number is the number modulo 1. (Contributed by NM, 11-Nov-2008.)

Theoremflmod 12144 The floor function expressed in terms of the modulo operation. (Contributed by NM, 11-Nov-2008.)

Theoremintfrac 12145 Break a number into its integer part and its fractional part. (Contributed by NM, 31-Dec-2008.)

Theoremzmod10 12146 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmodmulnn 12147 Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by NM, 2-Jan-2009.)

Theoremmodvalp1 12148 The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Alexander van der Vekens, 14-Apr-2018.)

Theoremzmodcl 12149 Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)

Theoremzmodcld 12150 Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzmodfz 12151 An integer mod lies in the first nonnegative integers. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremzmodfzo 12152 An integer mod lies in the first nonnegative integers. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^

Theoremzmodfzp1 12153 An integer mod lies in the first nonnegative integers. (Contributed by AV, 27-Oct-2018.)

Theoremmodid 12154 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)

Theoremmodid0 12155 A positive real number modulo itself is 0. (Contributed by Alexander van der Vekens, 15-May-2018.)

Theoremmodidmul0OLD 12156 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 12137 as of 5-Jul-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremmodid2 12157 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)

Theoremzmodid2 12158 Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremzmodidfzo 12159 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
..^

Theoremzmodidfzoimp 12160 Identity law for modulo restricted to integers. (Contributed by AV, 27-Oct-2018.)
..^

Theorem0mod 12161 Special case: 0 modulo a positive real number is 0. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorem1mod 12162 Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremmodabs 12163 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)

Theoremmodabs2 12164 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)

Theoremmodcyc 12165 The modulo operation is periodic. (Contributed by NM, 10-Nov-2008.)

Theoremmodcyc2 12166 The modulo operation is periodic. (Contributed by NM, 12-Nov-2008.)

Theoremmodadd1 12167 Addition property of the modulo operation. (Contributed by NM, 12-Nov-2008.)

Theoremmodaddabs 12168 Absorption law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmodaddmod 12169 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.)

Theoremmuladdmodid 12170 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.)

Theoremnegmod 12171 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.)

Theoremaddmodid 12172 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.)

Theoremaddmodidr 12173 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.)

Theoremmodadd2mod 12174 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.)

Theoremmodm1p1mod0 12175 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.)

Theoremmodltm1p1mod 12176 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.)

Theoremmodmul1 12177 Multiplication property of the modulo operation. Note that the multiplier must be an integer. (Contributed by NM, 12-Nov-2008.)

Theoremmodmul12d 12178 Multiplication property of the modulo operation. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremmodnegd 12179 Negation property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremmodadd12d 12180 Additive property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremmodsub12d 12181 Subtraction property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremmodsubmod 12182 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.)

Theoremmodsubmodmod 12183 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.)

Theorem2txmodxeq0 12184 Two times a positive real number modulo the real number is zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)

Theorem2submod 12185 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.)

Theoremmodifeq2int 12186 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.)

Theoremmodaddmodup 12187 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.)
..^

Theoremmodaddmodlo 12188 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.)
..^

Theoremmodmulmod 12189 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.)

Theoremmodaddmulmod 12190 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.)

Theoremmoddi 12191 Distribute multiplication over a modulo operation. (Contributed by NM, 11-Nov-2008.)

Theoremmodsubdir 12192 Distribute the modulo operation over a subtraction. (Contributed by NM, 30-Dec-2008.)

Theoremmodeqmodmin 12193 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.)

Theoremmodirr 12194 A number modulo an irrational multiple of it is nonzero. (Contributed by NM, 11-Nov-2008.)

Theoremmodfzo0difsn 12195* 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.)
..^ ..^ ..^

Theoremmodsumfzodifsn 12196 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.)
..^ ..^ ..^

Theoremmodlteq 12197 Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.)
..^ ..^

Theoremaddmodlteq 12198 Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. A much shorter proof exists if the "divides" relation can be used, see addmodlteqALT 14437. (Contributed by AV, 20-Mar-2021.)
..^ ..^

5.6.3  Miscellaneous theorems about integers

Theoremom2uz0i 12199* The mapping is a one-to-one mapping from onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number (normally 0 for the upper integers or 1 for the upper integers ), 1 maps to + 1, etc. This theorem shows the value of at ordinal natural number zero. (This series of theorems generalizes an earlier series for contributed by Raph Levien, 10-Apr-2004.) (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzsuci 12200* The value of (see om2uz0i 12199) at a successor. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

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