Home | Metamath
Proof Explorer Theorem List (p. 126 of 426) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-27775) |
Hilbert Space Explorer
(27776-29300) |
Users' Mathboxes
(29301-42551) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fzosplit 12501 | Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ ..^ ..^ | ||
Theorem | fzodisj 12502 | Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ ..^ | ||
Theorem | fzouzsplit 12503 | 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 12504 | 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 | fzodisjsn 12505 | A half-open integer range and the singleton of its upper bound are disjoint. (Contributed by AV, 7-Mar-2021.) |
..^ | ||
Theorem | prinfzo0 12506 | The intersection of a half-open integer range and the pair of its outer left borders is empty. (Contributed by AV, 9-Jan-2021.) |
..^ | ||
Theorem | lbfzo0 12507 | An integer is strictly greater than zero iff it is a member of . (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | elfzo0 12508 | 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 | elfzo0z 12509 | Membership in a half-open range of nonnegative integers, generalization of elfzo0 12508 requiring the upper bound to be an integer only. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
..^ | ||
Theorem | nn0p1elfzo 12510 | A nonnegative integer increased by 1 which is less than or equal to another integer is an element of a half-open range of integers. (Contributed by AV, 27-Feb-2021.) |
..^ | ||
Theorem | elfzo0le 12511 | 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 12512 | 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 12513 | 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 than the positive integer. (Contributed by Alexander van der Vekens, 19-May-2018.) |
..^ ..^ | ||
Theorem | fzofzim 12514 | 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 | fz1fzo0m1 12515 | Translation of one between closed and open integer ranges. (Contributed by Thierry Arnoux, 28-Jul-2020.) |
..^ | ||
Theorem | fzossnn 12516 | Half-open integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.) |
..^ | ||
Theorem | elfzo1 12517 | Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
..^ | ||
Theorem | fzo1fzo0n0 12518 | 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 | fzo0n0 12519 | A half-open integer range based at 0 is nonempty precisely if the upper bound is a positive integer. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | fzoaddel 12520 | Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ ..^ | ||
Theorem | fzo0addel 12521 | Translate membership in a 0 based half-open integer range. (Contributed by AV, 30-Apr-2020.) |
..^ ..^ | ||
Theorem | fzo0addelr 12522 | Translate membership in a 0 based half-open integer range. (Contributed by AV, 30-Apr-2020.) |
..^ ..^ | ||
Theorem | fzoaddel2 12523 | Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ ..^ | ||
Theorem | elfzoext 12524 | Membership of an integer in an extended open range of integers. (Contributed by AV, 30-Apr-2020.) |
..^ ..^ | ||
Theorem | elincfzoext 12525 | Membership of an increased integer in a correspondingly extended half-open range of integers. (Contributed by AV, 30-Apr-2020.) |
..^ ..^ | ||
Theorem | fzosubel 12526 | Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ ..^ | ||
Theorem | fzosubel2 12527 | 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 12528 | 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 12529 | 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 12530 | 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 12531 | Translate membership in a half-open integer range. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
..^ ..^ ..^ | ||
Theorem | ubmelfzo 12532 | 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 12533 | 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 12534 | 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 12535 | Membership in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) |
..^ ..^ | ||
Theorem | fzval3 12536 | Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | fz0add1fz1 12537 | Translate membership in a 0 based half-open integer range into membership in a 1 based finite sequence of integers. (Contributed by Alexander van der Vekens, 23-Nov-2017.) |
..^ | ||
Theorem | fzosn 12538 | Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ | ||
Theorem | elfzomin 12539 | Membership of an integer in the smallest open range of integers. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
..^ | ||
Theorem | zpnn0elfzo 12540 | 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 12541 | 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 12542 | Removing a singleton from a half-open integer range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
..^ ..^ | ||
Theorem | elfzonlteqm1 12543 | 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 12544 | 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 12545 | 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 12546 | 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 12547 | 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 12548 | Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.) (Proof shortened by JJ, 1-Jun-2021.) |
..^ | ||
Theorem | fzo0ssnn0OLD 12549 | Obsolete proof of fzo0ssnn0 12548 as of 1-Jun-2021. Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
..^ | ||
Theorem | fzo01 12550 | Expressing the singleton of as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | fzo12sn 12551 | 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 | fzo13pr 12552 | A 1-based half-open integer interval up to, but not including, 3 is a pair. (Contributed by Thierry Arnoux, 11-Jul-2020.) |
..^ | ||
Theorem | fzo0to2pr 12553 | A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
..^ | ||
Theorem | fzo0to3tp 12554 | A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.) |
..^ | ||
Theorem | fzo0to42pr 12555 | 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 | fzo1to4tp 12556 | A half-open integer range from 1 to 4 is an unordered triple. (Contributed by AV, 28-Jul-2021.) |
..^ | ||
Theorem | fzo0sn0fzo1 12557 | 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 | elfzo0l 12558 | A member of a half-open range of nonnegative integers is either 0 or a member of the corresponding half-open range of positive integers. (Contributed by AV, 5-Feb-2021.) |
..^ ..^ | ||
Theorem | fzoend 12559 | The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ ..^ | ||
Theorem | fzo0end 12560 | 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 12561 | Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.) |
..^ ..^ | ||
Theorem | ssfzoulel 12562 | If a half-open integer range is a subset of a half-open range of nonnegative integers, but its lower bound is greater than or equal to the upper bound of the containing range, or its upper bound is less than or equal to 0, then its upper bound is less than or equal to its lower bound (and therefore it is actually empty). (Contributed by Alexander van der Vekens, 24-May-2018.) |
..^ ..^ | ||
Theorem | ssfzo12bi 12563 | Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.) |
..^ ..^ | ||
Theorem | ubmelm1fzo 12564 | 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 12565 | 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 12566 | 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 12567 | 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 | elfzom1elp1fzo1 12568 | Membership of a nonnegative integer incremented by one in a half-open range of positive integers. (Contributed by AV, 20-Mar-2021.) |
..^ ..^ | ||
Theorem | elfzo1elm1fzo0 12569 | Membership of a positive integer decremented by one in a half-open range of nonnegative integers. (Contributed by AV, 20-Mar-2021.) |
..^ ..^ | ||
Theorem | elfzonelfzo 12570 | 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 | fzonfzoufzol 12571 | If an element of a half-open integer range is not in the upper part of the range, it is in the lower part of the range. (Contributed by Alexander van der Vekens, 29-Oct-2018.) |
..^ ..^ ..^ | ||
Theorem | elfzomelpfzo 12572 | 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 | elfznelfzo 12573 | A value in a finite set of sequential integers is a border value if it is not contained in the half-open integer range contained in the finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by Thierry Arnoux, 22-Dec-2021.) |
..^ | ||
Theorem | elfznelfzob 12574 | A value in a finite set of sequential integers is a border value if and only if it is not contained in the half-open integer range contained in the finite set of sequential integers. (Contributed by Alexander van der Vekens, 17-Jan-2018.) (Revised by Thierry Arnoux, 22-Dec-2021.) |
..^ | ||
Theorem | peano2fzor 12575 | A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.) |
..^ ..^ | ||
Theorem | fzosplitsn 12576 | Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzosplitpr 12577 | Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
..^ ..^ | ||
Theorem | fzosplitprm1 12578 | Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 25-Jun-2022.) |
..^ ..^ | ||
Theorem | fzosplitsni 12579 | Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ ..^ | ||
Theorem | fzisfzounsn 12580 | 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 | elfzr 12581 | A member of a finite interval of integers is either a member of the corresponding half-open integer range or the upper bound of the interval. (Contributed by AV, 5-Feb-2021.) |
..^ | ||
Theorem | elfzlmr 12582 | A member of a finite interval of integers is either its lower bound or its upper bound or an element of its interior. (Contributed by AV, 5-Feb-2021.) |
..^ | ||
Theorem | elfz0lmr 12583 | A member of a finite interval of nonnegative integers is either 0 or its upper bound or an element of its interior. (Contributed by AV, 5-Feb-2021.) |
..^ | ||
Theorem | fzostep1 12584 | 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 12585* | Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 12428. (Contributed by Alexander van der Vekens, 23-Sep-2018.) |
..^ ..^ | ||
Theorem | fzind2 12586* | 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 11475 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.) |
..^ | ||
Theorem | fvinim0ffz 12587 | 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 | injresinjlem 12588 | Lemma for injresinj 12589. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Proof shortened by AV, 14-Feb-2021.) (Revised by Thierry Arnoux, 23-Dec-2021.) |
..^ ..^ | ||
Theorem | injresinj 12589 | A function whose restriction is injective and the values of the remaining arguments are different from all other values is injective itself. (Contributed by Alexander van der Vekens, 31-Oct-2017.) |
..^ ..^ | ||
Theorem | subfzo0 12590 | 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.) |
..^ ..^ | ||
Syntax | cfl 12591 | Extend class notation with floor (greatest integer) function. |
Syntax | cceil 12592 | Extend class notation to include the ceiling function. |
⌈ | ||
Definition | df-fl 12593* |
Define the floor (greatest integer less than or equal to) function. See
flval 12595 for its value, fllelt 12598 for its basic property, and flcl 12596
for
its closure. For example, while
(ex-fl 27304).
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 12594 |
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 ceilval 12639 for its value, ceilge 12645 and ceilm1lt 12647 for its basic
properties, and ceilcl 12643 for its closure. For example,
⌈ while ⌈
(ex-ceil 27305).
The symbol ⌈ is inspired by the gamma shaped left bracket of the usual notation. (Contributed by David A. Wheeler, 19-May-2015.) |
⌈ | ||
Theorem | flval 12595* | 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 | flcl 12596 | The floor (greatest integer) function is an integer (closure law). (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.) |
Theorem | reflcl 12597 | The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.) |
Theorem | fllelt 12598 | A basic property of the floor (greatest integer) function. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.) |
Theorem | flcld 12599 | The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | flle 12600 | A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |