Home | Intuitionistic Logic Explorer Theorem List (p. 102 of 139) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fzossnn0 10101 | A half-open integer range starting at a nonnegative integer is a subset of the nonnegative integers. (Contributed by Alexander van der Vekens, 13-May-2018.) |
..^ | ||
Theorem | fzospliti 10102 | One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ ..^ ..^ | ||
Theorem | fzosplit 10103 | Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ ..^ ..^ | ||
Theorem | fzodisj 10104 | Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ ..^ | ||
Theorem | fzouzsplit 10105 | Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.) |
..^ | ||
Theorem | fzouzdisj 10106 | A half-open integer range does not overlap the upper integer range starting at the endpoint of the first range. (Contributed by Mario Carneiro, 21-Sep-2016.) |
..^ | ||
Theorem | lbfzo0 10107 | An integer is strictly greater than zero iff it is a member of . (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | elfzo0 10108 | Membership in a half-open integer range based at 0. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | fzo1fzo0n0 10109 | An integer between 1 and an upper bound of a half-open integer range is not 0 and between 0 and the upper bound of the half-open integer range. (Contributed by Alexander van der Vekens, 21-Mar-2018.) |
..^ ..^ | ||
Theorem | elfzo0z 10110 | Membership in a half-open range of nonnegative integers, generalization of elfzo0 10108 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
..^ | ||
Theorem | elfzo0le 10111 | A member in a half-open range of nonnegative integers is less than or equal to the upper bound of the range. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
..^ | ||
Theorem | elfzonn0 10112 | A member of a half-open range of nonnegative integers is a nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.) |
..^ | ||
Theorem | fzonmapblen 10113 | The result of subtracting a nonnegative integer from a positive integer and adding another nonnegative integer which is less than the first one is less then the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.) |
..^ ..^ | ||
Theorem | fzofzim 10114 | If a nonnegative integer in a finite interval of integers is not the upper bound of the interval, it is contained in the corresponding half-open integer range. (Contributed by Alexander van der Vekens, 15-Jun-2018.) |
..^ | ||
Theorem | fzossnn 10115 | Half-open integer ranges starting with 1 are subsets of . (Contributed by Thierry Arnoux, 28-Dec-2016.) |
..^ | ||
Theorem | elfzo1 10116 | Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
..^ | ||
Theorem | fzo0m 10117* | A half-open integer range based at 0 is inhabited precisely if the upper bound is a positive integer. (Contributed by Jim Kingdon, 20-Apr-2020.) |
..^ | ||
Theorem | fzoaddel 10118 | Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ ..^ | ||
Theorem | fzoaddel2 10119 | Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ ..^ | ||
Theorem | fzosubel 10120 | Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ ..^ | ||
Theorem | fzosubel2 10121 | Membership in a translated half-open integer range implies translated membership in the original range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ ..^ | ||
Theorem | fzosubel3 10122 | Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ ..^ | ||
Theorem | eluzgtdifelfzo 10123 | Membership of the difference of integers in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 17-Sep-2018.) |
..^ | ||
Theorem | ige2m2fzo 10124 | Membership of an integer greater than 1 decreased by 2 in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 3-Oct-2018.) |
..^ | ||
Theorem | fzocatel 10125 | Translate membership in a half-open integer range. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
..^ ..^ ..^ | ||
Theorem | ubmelfzo 10126 | If an integer in a 1 based finite set of sequential integers is subtracted from the upper bound of this finite set of sequential integers, the result is contained in a half-open range of nonnegative integers with the same upper bound. (Contributed by AV, 18-Mar-2018.) (Revised by AV, 30-Oct-2018.) |
..^ | ||
Theorem | elfzodifsumelfzo 10127 | If an integer is in a half-open range of nonnegative integers with a difference as upper bound, the sum of the integer with the subtrahend of the difference is in the a half-open range of nonnegative integers containing the minuend of the difference. (Contributed by AV, 13-Nov-2018.) |
..^ ..^ | ||
Theorem | elfzom1elp1fzo 10128 | Membership of an integer incremented by one in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Proof shortened by AV, 5-Jan-2020.) |
..^ ..^ | ||
Theorem | elfzom1elfzo 10129 | Membership in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) |
..^ ..^ | ||
Theorem | fzval3 10130 | Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | fzosn 10131 | Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ | ||
Theorem | elfzomin 10132 | Membership of an integer in the smallest open range of integers. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
..^ | ||
Theorem | zpnn0elfzo 10133 | Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
..^ | ||
Theorem | zpnn0elfzo1 10134 | Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
..^ | ||
Theorem | fzosplitsnm1 10135 | Removing a singleton from a half-open integer range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
..^ ..^ | ||
Theorem | elfzonlteqm1 10136 | If an element of a half-open integer range is not less than the upper bound of the range decreased by 1, it must be equal to the upper bound of the range decreased by 1. (Contributed by AV, 3-Nov-2018.) |
..^ | ||
Theorem | fzonn0p1 10137 | A nonnegative integer is element of the half-open range of nonnegative integers with the element increased by one as an upper bound. (Contributed by Alexander van der Vekens, 5-Aug-2018.) |
..^ | ||
Theorem | fzossfzop1 10138 | A half-open range of nonnegative integers is a subset of a half-open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.) |
..^ ..^ | ||
Theorem | fzonn0p1p1 10139 | If a nonnegative integer is element of a half-open range of nonnegative integers, increasing this integer by one results in an element of a half- open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018.) |
..^ ..^ | ||
Theorem | elfzom1p1elfzo 10140 | Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, 1-Aug-2018.) |
..^ ..^ | ||
Theorem | fzo0ssnn0 10141 | Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.) |
..^ | ||
Theorem | fzo01 10142 | Expressing the singleton of as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | fzo12sn 10143 | A 1-based half-open integer interval up to, but not including, 2 is a singleton. (Contributed by Alexander van der Vekens, 31-Jan-2018.) |
..^ | ||
Theorem | fzo0to2pr 10144 | A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
..^ | ||
Theorem | fzo0to3tp 10145 | A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.) |
..^ | ||
Theorem | fzo0to42pr 10146 | A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.) |
..^ | ||
Theorem | fzo0sn0fzo1 10147 | A half-open range of nonnegative integers is the union of the singleton set containing 0 and a half-open range of positive integers. (Contributed by Alexander van der Vekens, 18-May-2018.) |
..^ ..^ | ||
Theorem | fzoend 10148 | The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ ..^ | ||
Theorem | fzo0end 10149 | The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | ssfzo12 10150 | Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.) |
..^ ..^ | ||
Theorem | ssfzo12bi 10151 | Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.) |
..^ ..^ | ||
Theorem | ubmelm1fzo 10152 | 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 10153 | 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 10154 | 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 10155 | 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 10156 | 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 10157 | 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 10158 | A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.) |
..^ ..^ | ||
Theorem | fzosplitsn 10159 | Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzosplitprm1 10160 | Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
..^ ..^ | ||
Theorem | fzosplitsni 10161 | Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzisfzounsn 10162 | 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 10163 | 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 10164* | Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 10034. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
..^ ..^ | ||
Theorem | fzind2 10165* | 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 9298 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.) |
..^ | ||
Theorem | exfzdc 10166* | Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.) |
DECID DECID | ||
Theorem | fvinim0ffz 10167 | 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 10168 | 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 10169 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
Theorem | qletric 10170 | Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
Theorem | qlelttric 10171 | Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.) |
Theorem | qltnle 10172 | 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | qdceq 10173 | Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.) |
DECID | ||
Theorem | exbtwnzlemstep 10174* | Lemma for exbtwnzlemex 10176. Induction step. (Contributed by Jim Kingdon, 10-May-2022.) |
Theorem | exbtwnzlemshrink 10175* | Lemma for exbtwnzlemex 10176. Shrinking the range around . (Contributed by Jim Kingdon, 10-May-2022.) |
Theorem | exbtwnzlemex 10176* |
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 10177* | 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 10178* | There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | rebtwn2zlemstep 10179* | Lemma for rebtwn2z 10181. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | rebtwn2zlemshrink 10180* | Lemma for rebtwn2z 10181. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.) |
Theorem | rebtwn2z 10181* |
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 10182 | Lemma for qbtwnre 10183. Calculations involved in showing the constructed rational number is less than . (Contributed by Jim Kingdon, 14-Oct-2021.) |
Theorem | qbtwnre 10183* | 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 10184* | 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 10185 | 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 10186 | An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.) |
Theorem | ioom 10187* | 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 10188 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
Theorem | ioc0 10189 | An empty open interval of extended reals. (Contributed by FL, 30-May-2014.) |
Theorem | dfrp2 10190 | Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.) |
Theorem | elicod 10191 | Membership in a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | icogelb 10192 | 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 10193 | A member of a left-closed right-open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Syntax | cfl 10194 | Extend class notation with floor (greatest integer) function. |
Syntax | cceil 10195 | Extend class notation to include the ceiling function. |
⌈ | ||
Definition | df-fl 10196* |
Define the floor (greatest integer less than or equal to) function. See
flval 10198 for its value, flqlelt 10202 for its basic property, and flqcl 10199 for
its closure. For example, while
(ex-fl 13469).
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 10197 |
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 10232 for its value, ceilqge 10236 and ceilqm1lt 10238 for its basic
properties, and ceilqcl 10234 for its closure. For example,
⌈ while ⌈
(ex-ceil 13470).
As described in df-fl 10196 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 10198* | 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 10199 | 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 10201. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Theorem | apbtwnz 10200* | 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.) |
# |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |