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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | qletric 9201 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
Theorem | qlelttric 9202 | Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.) |
Theorem | qltnle 9203 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qdceq 9204 | Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.) |
DECID | ||
Theorem | qbtwnzlemstep 9205* | Lemma for qbtwnz 9208. Induction step. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qbtwnzlemshrink 9206* | Lemma for qbtwnz 9208. Shrinking the range around the given rational number. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qbtwnzlemex 9207* |
Lemma for qbtwnz 9208. Existence of the integer.
The proof starts by finding two integers which are less than and greater than the given rational 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 rational number trichotomy, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qbtwnz 9208* | There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | rebtwn2zlemstep 9209* | Lemma for rebtwn2z 9211. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | rebtwn2zlemshrink 9210* | Lemma for rebtwn2z 9211. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | rebtwn2z 9211* |
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 9212 | Lemma for qbtwnre 9213. Calculations involved in showing the constructed rational number is less than . (Contributed by Jim Kingdon, 14-Oct-2021.) |
Theorem | qbtwnre 9213* | 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 9214* | 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 9215 | 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 9216 | An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.) |
Theorem | ioom 9217* | 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 9218 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
Theorem | ioc0 9219 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
Syntax | cfl 9220 | Extend class notation with floor (greatest integer) function. |
Syntax | cceil 9221 | Extend class notation to include the ceiling function. |
⌈ | ||
Definition | df-fl 9222* |
Define the floor (greatest integer less than or equal to) function. See
flval 9224 for its value, flqlelt 9226 for its basic property, and flqcl 9225 for
its closure. For example, while
(ex-fl 10279).
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 beyond the rationals. 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 9223 |
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 9256 for its value, ceilqge 9260 and ceilqm1lt 9262 for its basic
properties, and ceilqcl 9258 for its closure. For example,
⌈ while ⌈
(ex-ceil 10280).
As described in df-fl 9222 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 9224* | 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 9225 | The floor (greatest integer) function yields an integer when applied to a rational (closure law). It would presumably be possible to prove a similar result for some real numbers (for example, those apart from any integer), but not real numbers in general. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqlelt 9226 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqcld 9227 | The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqle 9228 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqltp1 9229 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qfraclt1 9230 | The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qfracge0 9231 | The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqge 9232 | The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqlt 9233 | The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flid 9234 | An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
Theorem | flqidm 9235 | The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqidz 9236 | A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqltnz 9237 | If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqwordi 9238 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqword2 9239 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqbi 9240 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqbi2 9241 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | adddivflid 9242 | 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 9243 | The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqge1nn 9244 | The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | fldivnn0 9245 | 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 9246 | The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.) |
Theorem | flqaddz 9247 | An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqzadd 9248 | An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqmulnn0 9249 | Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | btwnzge0 9250 | 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 9251 | Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.) |
Theorem | flhalf 9252 | 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 9253 | 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 9254 | 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 9255 | 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 9256 | The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⌈ | ||
Theorem | ceiqcl 9257 | The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqcl 9258 | Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqge 9259 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqge 9260 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqm1l 9261 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqm1lt 9262 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqle 9263 | 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 9264 | 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 9265 | An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.) |
⌈ | ||
Theorem | ceilqidz 9266 | A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | flqleceil 9267 | The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | flqeqceilz 9268 | A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | intqfrac2 9269 | Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | intfracq 9270 | Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 9269. (Contributed by NM, 16-Aug-2008.) |
Theorem | flqdiv 9271 | Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Syntax | cmo 9272 | Extend class notation with the modulo operation. |
Definition | df-mod 9273* | Define the modulo (remainder) operation. See modqval 9274 for its value. For example, and . As with df-fl 9222 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 9274 | 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 9225 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 9275 | The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modqcl 9276 | Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | flqpmodeq 9277 | Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modqcld 9278 | Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modq0 9279 | is zero iff is evenly divisible by . (Contributed by Jim Kingdon, 17-Oct-2021.) |
Theorem | mulqmod0 9280 | 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 9281 | is divisible by iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | modqge0 9282 | The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | modqlt 9283 | The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | modqelico 9284 | Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | modqdiffl 9285 | The modulo operation differs from by an integer multiple of . (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | modqdifz 9286 | The modulo operation differs from by an integer multiple of . (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | modqfrac 9287 | The fractional part of a number is the number modulo 1. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | flqmod 9288 | The floor function expressed in terms of the modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | intqfrac 9289 | Break a number into its integer part and its fractional part. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | zmod10 9290 | An integer modulo 1 is 0. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | zmod1congr 9291 | Two arbitrary integers are congruent modulo 1, see example 4 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.) |
Theorem | modqmulnn 9292 | 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.) |
Theorem | modqvalp1 9293 | The value of the modulo operation (expressed with sum of denominator and nominator). (Contributed by Jim Kingdon, 20-Oct-2021.) |
Theorem | zmodcl 9294 | Closure law for the modulo operation restricted to integers. (Contributed by NM, 27-Nov-2008.) |
Theorem | zmodcld 9295 | Closure law for the modulo operation restricted to integers. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | zmodfz 9296 | An integer mod lies in the first nonnegative integers. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Theorem | zmodfzo 9297 | An integer mod lies in the first nonnegative integers. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
..^ | ||
Theorem | zmodfzp1 9298 | An integer mod lies in the first nonnegative integers. (Contributed by AV, 27-Oct-2018.) |
Theorem | modqid 9299 | Identity law for modulo. (Contributed by Jim Kingdon, 21-Oct-2021.) |
Theorem | modqid0 9300 | A positive real number modulo itself is 0. (Contributed by Jim Kingdon, 21-Oct-2021.) |
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