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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ssfzo12 10001 | Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.) |
..^ ..^ | ||
Theorem | ssfzo12bi 10002 | Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.) |
..^ ..^ | ||
Theorem | ubmelm1fzo 10003 | The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.) |
..^ ..^ | ||
Theorem | fzofzp1 10004 | If a point is in a half-open range, the next point is in the closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ | ||
Theorem | fzofzp1b 10005 | If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.) |
..^ | ||
Theorem | elfzom1b 10006 | An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Mario Carneiro, 27-Sep-2015.) |
..^ ..^ | ||
Theorem | elfzonelfzo 10007 | If an element of a half-open integer range is not contained in the lower subrange, it must be in the upper subrange. (Contributed by Alexander van der Vekens, 30-Mar-2018.) |
..^ ..^ ..^ | ||
Theorem | elfzomelpfzo 10008 | An integer increased by another integer is an element of a half-open integer range if and only if the integer is contained in the half-open integer range with bounds decreased by the other integer. (Contributed by Alexander van der Vekens, 30-Mar-2018.) |
..^ ..^ | ||
Theorem | peano2fzor 10009 | A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.) |
..^ ..^ | ||
Theorem | fzosplitsn 10010 | Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzosplitprm1 10011 | Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
..^ ..^ | ||
Theorem | fzosplitsni 10012 | Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzisfzounsn 10013 | 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 10014 | 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 10015* | Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 9888. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
..^ ..^ | ||
Theorem | fzind2 10016* | 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 9166 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.) |
..^ | ||
Theorem | exfzdc 10017* | Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.) |
DECID DECID | ||
Theorem | fvinim0ffz 10018 | 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 10019 | 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 10020 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
Theorem | qletric 10021 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
Theorem | qlelttric 10022 | Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.) |
Theorem | qltnle 10023 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qdceq 10024 | Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.) |
DECID | ||
Theorem | exbtwnzlemstep 10025* | Lemma for exbtwnzlemex 10027. Induction step. (Contributed by Jim Kingdon, 10-May-2022.) |
Theorem | exbtwnzlemshrink 10026* | Lemma for exbtwnzlemex 10027. Shrinking the range around . (Contributed by Jim Kingdon, 10-May-2022.) |
Theorem | exbtwnzlemex 10027* |
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 10028* | 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 10029* | There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | rebtwn2zlemstep 10030* | Lemma for rebtwn2z 10032. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | rebtwn2zlemshrink 10031* | Lemma for rebtwn2z 10032. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | rebtwn2z 10032* |
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 10033 | Lemma for qbtwnre 10034. Calculations involved in showing the constructed rational number is less than . (Contributed by Jim Kingdon, 14-Oct-2021.) |
Theorem | qbtwnre 10034* | 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 10035* | 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 10036 | 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 10037 | An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.) |
Theorem | ioom 10038* | 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 10039 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
Theorem | ioc0 10040 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
Syntax | cfl 10041 | Extend class notation with floor (greatest integer) function. |
Syntax | cceil 10042 | Extend class notation to include the ceiling function. |
⌈ | ||
Definition | df-fl 10043* |
Define the floor (greatest integer less than or equal to) function. See
flval 10045 for its value, flqlelt 10049 for its basic property, and flqcl 10046 for
its closure. For example, while
(ex-fl 12937).
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 10044 |
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 10079 for its value, ceilqge 10083 and ceilqm1lt 10085 for its basic
properties, and ceilqcl 10081 for its closure. For example,
⌈ while ⌈
(ex-ceil 12938).
As described in df-fl 10043 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 10045* | 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 10046 | 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 10048. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | apbtwnz 10047* | 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 10048* | 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 11858) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.) |
# | ||
Theorem | flqlelt 10049 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqcld 10050 | The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqle 10051 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqltp1 10052 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qfraclt1 10053 | The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qfracge0 10054 | The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqge 10055 | The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqlt 10056 | The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flid 10057 | An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
Theorem | flqidm 10058 | The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqidz 10059 | A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqltnz 10060 | If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqwordi 10061 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqword2 10062 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqbi 10063 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqbi2 10064 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | adddivflid 10065 | 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 10066 | The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqge1nn 10067 | The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | fldivnn0 10068 | 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 10069 | The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.) |
Theorem | flqaddz 10070 | An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqzadd 10071 | An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqmulnn0 10072 | Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | btwnzge0 10073 | 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 10074 | Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.) |
Theorem | flhalf 10075 | 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 10076 | 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 10077 | 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 10078 | 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 10079 | The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⌈ | ||
Theorem | ceiqcl 10080 | The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqcl 10081 | Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqge 10082 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqge 10083 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqm1l 10084 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqm1lt 10085 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqle 10086 | 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 10087 | 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 10088 | An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.) |
⌈ | ||
Theorem | ceilqidz 10089 | A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | flqleceil 10090 | The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | flqeqceilz 10091 | A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | intqfrac2 10092 | Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | intfracq 10093 | Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 10092. (Contributed by NM, 16-Aug-2008.) |
Theorem | flqdiv 10094 | Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Syntax | cmo 10095 | Extend class notation with the modulo operation. |
Definition | df-mod 10096* | Define the modulo (remainder) operation. See modqval 10097 for its value. For example, and . As with df-fl 10043 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 10097 | 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 10046 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 10098 | The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modqcl 10099 | Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | flqpmodeq 10100 | Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.) |
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