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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fzosplitsn 10201 | Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzosplitprm1 10202 | Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
..^ ..^ | ||
Theorem | fzosplitsni 10203 | Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzisfzounsn 10204 | A finite interval of integers as union of a half-open integer range and a singleton. (Contributed by Alexander van der Vekens, 15-Jun-2018.) |
..^ | ||
Theorem | fzostep1 10205 | Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzoshftral 10206* | Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 10076. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
..^ ..^ | ||
Theorem | fzind2 10207* | Induction on the integers from to inclusive. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Version of fzind 9339 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.) |
..^ | ||
Theorem | exfzdc 10208* | Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.) |
DECID DECID | ||
Theorem | fvinim0ffz 10209 | The function values for the borders of a finite interval of integers, which is the domain of the function, are not in the image of the interior of the interval iff the intersection of the images of the interior and the borders is empty. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 5-Feb-2021.) |
..^ ..^ ..^ | ||
Theorem | subfzo0 10210 | The difference between two elements in a half-open range of nonnegative integers is greater than the negation of the upper bound and less than the upper bound of the range. (Contributed by AV, 20-Mar-2021.) |
..^ ..^ | ||
Theorem | qtri3or 10211 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
Theorem | qletric 10212 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
Theorem | qlelttric 10213 | Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.) |
Theorem | qltnle 10214 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qdceq 10215 | Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.) |
DECID | ||
Theorem | exbtwnzlemstep 10216* | Lemma for exbtwnzlemex 10218. Induction step. (Contributed by Jim Kingdon, 10-May-2022.) |
Theorem | exbtwnzlemshrink 10217* | Lemma for exbtwnzlemex 10218. Shrinking the range around . (Contributed by Jim Kingdon, 10-May-2022.) |
Theorem | exbtwnzlemex 10218* |
Existence of an integer so that a given real number is between the
integer and its successor. The real number must satisfy the
hypothesis. For example
either a rational number or
a number which is irrational (in the sense of being apart from any
rational number) will meet this condition.
The proof starts by finding two integers which are less than and greater than . Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on the hypothesis, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | exbtwnz 10219* | If a real number is between an integer and its successor, there is a unique greatest integer less than or equal to the real number. (Contributed by Jim Kingdon, 10-May-2022.) |
Theorem | qbtwnz 10220* | There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | rebtwn2zlemstep 10221* | Lemma for rebtwn2z 10223. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | rebtwn2zlemshrink 10222* | Lemma for rebtwn2z 10223. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | rebtwn2z 10223* |
A real number can be bounded by integers above and below which are two
apart.
The proof starts by finding two integers which are less than and greater than the given real number. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | qbtwnrelemcalc 10224 | Lemma for qbtwnre 10225. Calculations involved in showing the constructed rational number is less than . (Contributed by Jim Kingdon, 14-Oct-2021.) |
Theorem | qbtwnre 10225* | The rational numbers are dense in : any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.) |
Theorem | qbtwnxr 10226* | The rational numbers are dense in : any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
Theorem | qavgle 10227 | The average of two rational numbers is less than or equal to at least one of them. (Contributed by Jim Kingdon, 3-Nov-2021.) |
Theorem | ioo0 10228 | An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.) |
Theorem | ioom 10229* | An open interval of extended reals is inhabited iff the lower argument is less than the upper argument. (Contributed by Jim Kingdon, 27-Nov-2021.) |
Theorem | ico0 10230 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
Theorem | ioc0 10231 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
Theorem | dfrp2 10232 | Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.) |
Theorem | elicod 10233 | Membership in a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | icogelb 10234 | An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | elicore 10235 | A member of a left-closed right-open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Syntax | cfl 10236 | Extend class notation with floor (greatest integer) function. |
Syntax | cceil 10237 | Extend class notation to include the ceiling function. |
⌈ | ||
Definition | df-fl 10238* |
Define the floor (greatest integer less than or equal to) function. See
flval 10240 for its value, flqlelt 10244 for its basic property, and flqcl 10241 for
its closure. For example, while
(ex-fl 14017).
Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible for all real numbers. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision. 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.) |
Definition | df-ceil 10239 |
The ceiling (least integer greater than or equal to) function. Defined in
ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of
Mathematical Functions" , front introduction, "Common Notations
and
Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4.
See ceilqval 10274 for its value, ceilqge 10278 and ceilqm1lt 10280 for its basic
properties, and ceilqcl 10276 for its closure. For example,
⌈ while ⌈
(ex-ceil 14018).
As described in df-fl 10238 most theorems are only for rationals, not reals. The symbol ⌈ is inspired by the gamma shaped left bracket of the usual notation. (Contributed by David A. Wheeler, 19-May-2015.) |
⌈ | ||
Theorem | flval 10240* | 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.) |
Theorem | flqcl 10241 | The floor (greatest integer) function yields an integer when applied to a rational (closure law). For a similar closure law for real numbers apart from any integer, see flapcl 10243. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | apbtwnz 10242* | There is a unique greatest integer less than or equal to a real number which is apart from all integers. (Contributed by Jim Kingdon, 11-May-2022.) |
# | ||
Theorem | flapcl 10243* | The floor (greatest integer) function yields an integer when applied to a real number apart from any integer. For example, an irrational number (see for example sqrt2irrap 12145) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.) |
# | ||
Theorem | flqlelt 10244 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqcld 10245 | The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqle 10246 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqltp1 10247 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qfraclt1 10248 | The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qfracge0 10249 | The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqge 10250 | The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqlt 10251 | The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flid 10252 | An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
Theorem | flqidm 10253 | The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqidz 10254 | A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqltnz 10255 | If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqwordi 10256 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqword2 10257 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqbi 10258 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqbi2 10259 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | adddivflid 10260 | The floor of a sum of an integer and a fraction is equal to the integer iff the denominator of the fraction is less than the numerator. (Contributed by AV, 14-Jul-2021.) |
Theorem | flqge0nn0 10261 | The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqge1nn 10262 | The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | fldivnn0 10263 | The floor function of a division of a nonnegative integer by a positive integer is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
Theorem | divfl0 10264 | The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.) |
Theorem | flqaddz 10265 | An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqzadd 10266 | An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqmulnn0 10267 | Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | btwnzge0 10268 | A real bounded between an integer and its successor is nonnegative iff the integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217. (Contributed by NM, 12-Mar-2005.) |
Theorem | 2tnp1ge0ge0 10269 | Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.) |
Theorem | flhalf 10270 | Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.) |
Theorem | fldivnn0le 10271 | The floor function of a division of a nonnegative integer by a positive integer is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
Theorem | flltdivnn0lt 10272 | The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
Theorem | fldiv4p1lem1div2 10273 | The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.) |
Theorem | ceilqval 10274 | The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⌈ | ||
Theorem | ceiqcl 10275 | The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqcl 10276 | Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqge 10277 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqge 10278 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqm1l 10279 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqm1lt 10280 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqle 10281 | The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqle 10282 | The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceilid 10283 | An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.) |
⌈ | ||
Theorem | ceilqidz 10284 | A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | flqleceil 10285 | The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | flqeqceilz 10286 | A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | intqfrac2 10287 | Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | intfracq 10288 | Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10287. (Contributed by NM, 16-Aug-2008.) |
Theorem | flqdiv 10289 | Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Syntax | cmo 10290 | Extend class notation with the modulo operation. |
Definition | df-mod 10291* | Define the modulo (remainder) operation. See modqval 10292 for its value. For example, and . As with df-fl 10238 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.) |
Theorem | modqval 10292 | 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 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 10241 we only prove this for rationals although other particular kinds of real numbers may be possible. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modqvalr 10293 | The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modqcl 10294 | Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | flqpmodeq 10295 | Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modqcld 10296 | Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modq0 10297 | is zero iff is evenly divisible by . (Contributed by Jim Kingdon, 17-Oct-2021.) |
Theorem | mulqmod0 10298 | The product of an integer and a positive rational number is 0 modulo the positive real number. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | negqmod0 10299 | is divisible by iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | modqge0 10300 | The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.) |
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