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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elfzomelpfzo 9901 | 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 9902 | A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.) |
..^ ..^ | ||
Theorem | fzosplitsn 9903 | Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzosplitprm1 9904 | Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
..^ ..^ | ||
Theorem | fzosplitsni 9905 | Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzisfzounsn 9906 | 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 9907 | 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 9908* | Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 9781. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
..^ ..^ | ||
Theorem | fzind2 9909* | 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 9070 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.) |
..^ | ||
Theorem | exfzdc 9910* | Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.) |
DECID DECID | ||
Theorem | fvinim0ffz 9911 | 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 9912 | 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 9913 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
Theorem | qletric 9914 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
Theorem | qlelttric 9915 | Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.) |
Theorem | qltnle 9916 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qdceq 9917 | Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.) |
DECID | ||
Theorem | exbtwnzlemstep 9918* | Lemma for exbtwnzlemex 9920. Induction step. (Contributed by Jim Kingdon, 10-May-2022.) |
Theorem | exbtwnzlemshrink 9919* | Lemma for exbtwnzlemex 9920. Shrinking the range around . (Contributed by Jim Kingdon, 10-May-2022.) |
Theorem | exbtwnzlemex 9920* |
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 9921* | 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 9922* | There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | rebtwn2zlemstep 9923* | Lemma for rebtwn2z 9925. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | rebtwn2zlemshrink 9924* | Lemma for rebtwn2z 9925. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | rebtwn2z 9925* |
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 9926 | Lemma for qbtwnre 9927. Calculations involved in showing the constructed rational number is less than . (Contributed by Jim Kingdon, 14-Oct-2021.) |
Theorem | qbtwnre 9927* | 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 9928* | 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 9929 | 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 9930 | An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.) |
Theorem | ioom 9931* | 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 9932 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
Theorem | ioc0 9933 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
Syntax | cfl 9934 | Extend class notation with floor (greatest integer) function. |
Syntax | cceil 9935 | Extend class notation to include the ceiling function. |
⌈ | ||
Definition | df-fl 9936* |
Define the floor (greatest integer less than or equal to) function. See
flval 9938 for its value, flqlelt 9942 for its basic property, and flqcl 9939 for
its closure. For example, while
(ex-fl 12630).
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 9937 |
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 9972 for its value, ceilqge 9976 and ceilqm1lt 9978 for its basic
properties, and ceilqcl 9974 for its closure. For example,
⌈ while ⌈
(ex-ceil 12631).
As described in df-fl 9936 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 9938* | 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 9939 | 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 9941. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | apbtwnz 9940* | 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 9941* | 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 11703) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.) |
# | ||
Theorem | flqlelt 9942 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqcld 9943 | The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqle 9944 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqltp1 9945 | A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qfraclt1 9946 | The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qfracge0 9947 | The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqge 9948 | The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqlt 9949 | The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flid 9950 | An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
Theorem | flqidm 9951 | The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | flqidz 9952 | A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqltnz 9953 | If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqwordi 9954 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqword2 9955 | Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqbi 9956 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | flqbi2 9957 | A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.) |
Theorem | adddivflid 9958 | 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 9959 | The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqge1nn 9960 | The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | fldivnn0 9961 | 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 9962 | The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.) |
Theorem | flqaddz 9963 | An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqzadd 9964 | An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | flqmulnn0 9965 | Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.) |
Theorem | btwnzge0 9966 | 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 9967 | Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.) |
Theorem | flhalf 9968 | 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 9969 | 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 9970 | 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 9971 | 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 9972 | The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.) |
⌈ | ||
Theorem | ceiqcl 9973 | The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqcl 9974 | Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqge 9975 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqge 9976 | The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqm1l 9977 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Theorem | ceilqm1lt 9978 | One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | ceiqle 9979 | 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 9980 | 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 9981 | An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.) |
⌈ | ||
Theorem | ceilqidz 9982 | A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | flqleceil 9983 | The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | flqeqceilz 9984 | A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.) |
⌈ | ||
Theorem | intqfrac2 9985 | Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | intfracq 9986 | Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intqfrac2 9985. (Contributed by NM, 16-Aug-2008.) |
Theorem | flqdiv 9987 | Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Syntax | cmo 9988 | Extend class notation with the modulo operation. |
Definition | df-mod 9989* | Define the modulo (remainder) operation. See modqval 9990 for its value. For example, and . As with df-fl 9936 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 9990 | 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 9939 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 9991 | The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modqcl 9992 | Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | flqpmodeq 9993 | Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modqcld 9994 | Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.) |
Theorem | modq0 9995 | is zero iff is evenly divisible by . (Contributed by Jim Kingdon, 17-Oct-2021.) |
Theorem | mulqmod0 9996 | 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 9997 | is divisible by iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | modqge0 9998 | The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | modqlt 9999 | The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Theorem | modqelico 10000 | Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.) |
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