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Theorem List for Metamath Proof Explorer - 11201-11300   *Has distinct variable group(s)
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5.6  Elementary integer functions

5.6.1  The floor (greatest integer) function

Syntaxcfl 11201 Extend class notation with floor (greatest integer) function.

Definitiondf-fl 11202* Define the floor (greatest integer) function. See flval 11203 for its value, fllelt 11206 for its basic property, and flcl 11204 for its closure. For example, while (ex-fl 21755).

The term "floor" was coined by Ken Iverson. He also invented a mathematical notation for floor, consisting of an L-shaped left bracket and its reflection as a right bracket. In APL, the left-bracket alone is used, and we borrow this idea. (Thanks to Paul Chapman for this information.) (Contributed by NM, 14-Nov-2004.)

Theoremflval 11203* Value of the floor (greatest integer) function. The floor of is the (unique) integer less than or equal to whose successor is strictly greater than . (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremflcl 11204 The floor (greatest integer) function is an integer (closure law). (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremreflcl 11205 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)

Theoremfllelt 11206 A basic property of the floor (greatest integer) function. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremflcld 11207 The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.)

Theoremflle 11208 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)

Theoremflltp1 11209 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)

Theoremfllep1 11210 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremfraclt1 11211 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)

Theoremfracle1 11212 The fractional part of a real number is less than or equal to one. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremfracge0 11213 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)

Theoremflge 11214 The floor function value is the greatest integer less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Fan Zheng, 14-Jul-2016.)

Theoremfllt 11215 The floor function value is less than the next integer. (Contributed by NM, 24-Feb-2005.)

Theoremflid 11216 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)

Theoremflidm 11217 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)

Theoremflidz 11218 A real number equals its floor iff it is an integer. (Contributed by NM, 11-Nov-2008.)

Theoremflwordi 11219 Ordering relationship for the greatest integer function. (Contributed by NM, 31-Dec-2005.) (Proof shortened by Fan Zheng, 14-Jul-2016.)

Theoremflword2 11220 Ordering relationship for the greatest integer function. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theoremflval2 11221* An alternate way to define the floor (greatest integer) function. (Contributed by NM, 16-Nov-2004.)

Theoremflval3 11222* An alternate way to define the floor (greatest integer) function, as the supremum of all integers less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 6-Sep-2014.)

Theoremflbi 11223 A condition equivalent to floor. (Contributed by NM, 11-Mar-2005.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremflbi2 11224 A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.)

Theoremflge0nn0 11225 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by NM, 26-Apr-2005.)

Theoremflge1nn 11226 The floor of a number greater than or equal to 1 is a natural number. (Contributed by NM, 26-Apr-2005.)

Theoremfladdz 11227 An integer can be moved in and out of the floor of a sum. (Contributed by NM, 27-Apr-2005.) (Proof shortened by Fan Zheng, 16-Jun-2016.)

Theoremflzadd 11228 An integer can be moved in and out of the floor of a sum. (Contributed by NM, 2-Jan-2009.)

Theoremflmulnn0 11229 Move a nonnegative integer in and out of a floor. (Contributed by NM, 2-Jan-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)

Theorembtwnzge0 11230 A real bounded between an integer and its successor is nonnegative iff the integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217. (For the first half see rebtwnz 10573.) (Contributed by NM, 12-Mar-2005.)

Theoremflhalf 11231 Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)

Theoremceicl 11232 The ceiling function returns an integer (closure law). (Contributed by Jeffrey Hankins, 10-Jun-2007.)

Theoremceige 11233 The ceiling of a real number is greater than or equal to that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)

Theoremceim1l 11234 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)

Theoremceile 11235 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jeffrey Hankins, 10-Jun-2007.)

Theoremquoremz 11236 Quotient and remainder of an integer divided by a natural number. TO DO - is this really needed for anything? Should we use to simplify it? (Contributed by NM, 14-Aug-2008.)

Theoremquoremnn0 11237 Quotient and remainder of a nonnegative integer divided by a natural number. (Contributed by NM, 14-Aug-2008.)

Theoremquoremnn0ALT 11238 Quotient and remainder of a nonnegative integer divided by a natural number. TO DO - Keep either quoremnn0ALT 11238 ((if we don't keep quoremz 11236) or quoremnn0 11237 (Contributed by NM, 14-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremintfrac2 11239 Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 11263? (Contributed by NM, 16-Aug-2008.)

Theoremintfracq 11240 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 11239. (Contributed by NM, 16-Aug-2008.)

Theoremfldiv 11241 Cancellation of the embedded floor of a real divided by an integer. (Contributed by NM, 16-Aug-2008.)

Theoremfldiv2 11242 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 11243 Finite set of sequential integers starting at 1 and ending at a real number. (Contributed by Mario Carneiro, 3-May-2016.)

Theoremuzsup 11244 A set of upper integers is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremioopnfsup 11245 A set of upper reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremicopnfsup 11246 A set of upper reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremrpsup 11247 The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremresup 11248 The real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremxrsup 11249 The extended real numbers are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)

5.6.2  The modulo (remainder) operation

Syntaxcmo 11250 Extend class notation with the modulo operation.

Definitiondf-mod 11251* Define the modulo (remainder) operation. See modval 11252 for its value. (Contributed by NM, 10-Nov-2008.)

Theoremmodval 11252 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.)

Theoremmodcl 11253 Closure law for the modulo operation. (Contributed by NM, 10-Nov-2008.)

Theoremmodcld 11254 Closure law for the modulo operation. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmod0 11255 is zero iff is evenly divisible by . (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)

Theoremnegmod0 11256 is divisible by iff its negative is. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)

Theoremmodge0 11257 The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008.)

Theoremmodlt 11258 The modulo operation is less than its second argument. (Contributed by NM, 10-Nov-2008.)

Theoremmoddiffl 11259 The modulo operation differs from by an integer multiple of . (Contributed by Mario Carneiro, 6-Sep-2016.)

Theoremmoddifz 11260 The modulo operation differs from by an integer multiple of . (Contributed by Mario Carneiro, 15-Jul-2014.)

Theoremmodfrac 11261 The fractional part of a number is the number modulo 1. (Contributed by NM, 11-Nov-2008.)

Theoremflmod 11262 The floor function expressed in terms of the modulo operation. (Contributed by NM, 11-Nov-2008.)

Theoremintfrac 11263 Break a number into its integer part and its fractional part. (Contributed by NM, 31-Dec-2008.)

Theoremzmod10 11264 An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmodmulnn 11265 Move a natural number in and out of a floor in the first argument of a modulo operation. (Contributed by NM, 2-Jan-2009.)

Theoremzmodcl 11266 Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.)

Theoremzmodcld 11267 Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzmodfz 11268 An integer mod lies in the first nonnegative integers. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremzmodfzo 11269 An integer mod lies in the first nonnegative integers. (Contributed by Stefan O'Rear, 6-Sep-2015.)
..^

Theoremmodid 11270 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)

Theoremmodid2 11271 Identity law for modulo. (Contributed by NM, 29-Dec-2008.)

Theorem0mod 11272 Special case: 0 modulo a positive real number is 0. (Contributed by Mario Carneiro, 22-Feb-2014.)

Theorem1mod 11273 Special case: 1 modulo a real number greater than 1 is 1. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremmodabs 11274 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)

Theoremmodabs2 11275 Absorption law for modulo. (Contributed by NM, 29-Dec-2008.)

Theoremmodcyc 11276 The modulo operation is periodic. (Contributed by NM, 10-Nov-2008.)

Theoremmodcyc2 11277 The modulo operation is periodic. (Contributed by NM, 12-Nov-2008.)

Theoremmodadd1 11278 Addition property of the modulo operation. (Contributed by NM, 12-Nov-2008.)

Theoremmodmul1 11279 Multiplication property of the modulo operation. Note that the multiplier must be an integer. (Contributed by NM, 12-Nov-2008.)

Theoremmodmul12d 11280 Multiplication property of the modulo operation. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremmodnegd 11281 Negation property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremmodadd12d 11282 Additive property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremmodsub12d 11283 Subtraction property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)

Theoremmoddi 11284 Distribute multiplication over a modulo operation. (Contributed by NM, 11-Nov-2008.)

Theoremmodsubdir 11285 Distribute the modulo operation over a subtraction. (Contributed by NM, 30-Dec-2008.)

Theoremmodirr 11286 A number modulo an irrational multiple of it is nonzero. (Contributed by NM, 11-Nov-2008.)

5.6.3  The infinite sequence builder "seq"

Theoremom2uz0i 11287* 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 11288* The value of (see om2uz0i 11287) at a successor. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzuzi 11289* The value (see om2uz0i 11287) at an ordinal natural number is in the upper integers. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzlti 11290* Less-than relation for (see om2uz0i 11287). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzlt2i 11291* The mapping (see om2uz0i 11287) preserves order. (Contributed by NM, 4-May-2005.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzrani 11292* Range of (see om2uz0i 11287). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzf1oi 11293* (see om2uz0i 11287) is a one-to-one onto mapping. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzisoi 11294* (see om2uz0i 11287) is an isomorphism from natural ordinals to upper integers. (Contributed by NM, 9-Oct-2008.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremom2uzoi 11295* An alternative definition of in terms of df-oi 7479. (Contributed by Mario Carneiro, 2-Jun-2015.)
OrdIso

Theoremom2uzrdg 11296* A helper lemma for the value of a recursive definition generator on upper integers (typically either or ) with characteristic function and initial value . Normally is a function on the partition, and is a member of the partition. See also comment in om2uz0i 11287. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremuzrdglem 11297* 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.)

Theoremuzrdgfni 11298* The recursive definition generator on upper integers is a function. See comment in om2uzrdg 11296. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.)

Theoremuzrdg0i 11299* Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 11296. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Theoremuzrdgsuci 11300* Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 11296. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)

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