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Theorem List for Intuitionistic Logic Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelfzomelpfzo 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.)
..^ ..^

Theorempeano2fzor 9902 A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
..^ ..^

Theoremfzosplitsn 9903 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzosplitprm1 9904 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
..^ ..^

Theoremfzosplitsni 9905 Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzisfzounsn 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.)
..^

Theoremfzostep1 9907 Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzoshftral 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.)
..^ ..^

Theoremfzind2 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.)
..^

Theoremexfzdc 9910* Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.)
DECID        DECID

Theoremfvinim0ffz 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.)
..^ ..^ ..^

Theoremsubfzo0 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.)
..^ ..^

4.5.7  Rational numbers (cont.)

Theoremqtri3or 9913 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)

Theoremqletric 9914 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)

Theoremqlelttric 9915 Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)

Theoremqltnle 9916 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremqdceq 9917 Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
DECID

Theoremexbtwnzlemstep 9918* Lemma for exbtwnzlemex 9920. Induction step. (Contributed by Jim Kingdon, 10-May-2022.)

Theoremexbtwnzlemshrink 9919* Lemma for exbtwnzlemex 9920. Shrinking the range around . (Contributed by Jim Kingdon, 10-May-2022.)

Theoremexbtwnzlemex 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.)

Theoremexbtwnz 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.)

Theoremqbtwnz 9922* There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremrebtwn2zlemstep 9923* Lemma for rebtwn2z 9925. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)

Theoremrebtwn2zlemshrink 9924* Lemma for rebtwn2z 9925. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)

Theoremrebtwn2z 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.)

Theoremqbtwnrelemcalc 9926 Lemma for qbtwnre 9927. Calculations involved in showing the constructed rational number is less than . (Contributed by Jim Kingdon, 14-Oct-2021.)

Theoremqbtwnre 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.)

Theoremqbtwnxr 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.)

Theoremqavgle 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.)

Theoremioo0 9930 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremioom 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.)

Theoremico0 9932 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

Theoremioc0 9933 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

4.6  Elementary integer functions

4.6.1  The floor and ceiling functions

Syntaxcfl 9934 Extend class notation with floor (greatest integer) function.

Syntaxcceil 9935 Extend class notation to include the ceiling function.

Definitiondf-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.)

Definitiondf-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.)

Theoremflval 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.)

Theoremflqcl 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.)

Theoremapbtwnz 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.)
#

Theoremflapcl 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.)
#

Theoremflqlelt 9942 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremflqcld 9943 The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremflqle 9944 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremflqltp1 9945 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremqfraclt1 9946 The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremqfracge0 9947 The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremflqge 9948 The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremflqlt 9949 The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremflid 9950 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)

Theoremflqidm 9951 The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)

Theoremflqidz 9952 A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)

Theoremflqltnz 9953 If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)

Theoremflqwordi 9954 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)

Theoremflqword2 9955 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)

Theoremflqbi 9956 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)

Theoremflqbi2 9957 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)

Theoremadddivflid 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.)

Theoremflqge0nn0 9959 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)

Theoremflqge1nn 9960 The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)

Theoremfldivnn0 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.)

Theoremdivfl0 9962 The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)

Theoremflqaddz 9963 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)

Theoremflqzadd 9964 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)

Theoremflqmulnn0 9965 Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)

Theorembtwnzge0 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.)

Theorem2tnp1ge0ge0 9967 Two times an integer plus one is not negative iff the integer is not negative. (Contributed by AV, 19-Jun-2021.)

Theoremflhalf 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.)

Theoremfldivnn0le 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.)

Theoremflltdivnn0lt 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.)

Theoremfldiv4p1lem1div2 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.)

Theoremceilqval 9972 The value of the ceiling function. (Contributed by Jim Kingdon, 10-Oct-2021.)

Theoremceiqcl 9973 The ceiling function returns an integer (closure law). (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremceilqcl 9974 Closure of the ceiling function. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremceiqge 9975 The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremceilqge 9976 The ceiling of a real number is greater than or equal to that number. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremceiqm1l 9977 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremceilqm1lt 9978 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremceiqle 9979 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremceilqle 9980 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremceilid 9981 An integer is its own ceiling. (Contributed by AV, 30-Nov-2018.)

Theoremceilqidz 9982 A rational number equals its ceiling iff it is an integer. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremflqleceil 9983 The floor of a rational number is less than or equal to its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremflqeqceilz 9984 A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.)

Theoremintqfrac2 9985 Decompose a real into integer and fractional parts. (Contributed by Jim Kingdon, 18-Oct-2021.)

Theoremintfracq 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.)

Theoremflqdiv 9987 Cancellation of the embedded floor of a real divided by an integer. (Contributed by Jim Kingdon, 18-Oct-2021.)

4.6.2  The modulo (remainder) operation

Syntaxcmo 9988 Extend class notation with the modulo operation.

Definitiondf-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.)

Theoremmodqval 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.)

Theoremmodqvalr 9991 The value of the modulo operation (multiplication in reversed order). (Contributed by Jim Kingdon, 16-Oct-2021.)

Theoremmodqcl 9992 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)

Theoremflqpmodeq 9993 Partition of a division into its integer part and the remainder. (Contributed by Jim Kingdon, 16-Oct-2021.)

Theoremmodqcld 9994 Closure law for the modulo operation. (Contributed by Jim Kingdon, 16-Oct-2021.)

Theoremmodq0 9995 is zero iff is evenly divisible by . (Contributed by Jim Kingdon, 17-Oct-2021.)

Theoremmulqmod0 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.)

Theoremnegqmod0 9997 is divisible by iff its negative is. (Contributed by Jim Kingdon, 18-Oct-2021.)

Theoremmodqge0 9998 The modulo operation is nonnegative. (Contributed by Jim Kingdon, 18-Oct-2021.)

Theoremmodqlt 9999 The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.)

Theoremmodqelico 10000 Modular reduction produces a half-open interval. (Contributed by Jim Kingdon, 18-Oct-2021.)

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