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Theorem List for Intuitionistic Logic Explorer - 10201-10300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelfzodifsumelfzo 10201 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.)
((๐‘€ โˆˆ (0...๐‘) โˆง ๐‘ โˆˆ (0...๐‘ƒ)) โ†’ (๐ผ โˆˆ (0..^(๐‘ โˆ’ ๐‘€)) โ†’ (๐ผ + ๐‘€) โˆˆ (0..^๐‘ƒ)))
 
Theoremelfzom1elp1fzo 10202 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.)
((๐‘ โˆˆ โ„ค โˆง ๐ผ โˆˆ (0..^(๐‘ โˆ’ 1))) โ†’ (๐ผ + 1) โˆˆ (0..^๐‘))
 
Theoremelfzom1elfzo 10203 Membership in a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.)
((๐‘ โˆˆ โ„ค โˆง ๐ผ โˆˆ (0..^(๐‘ โˆ’ 1))) โ†’ ๐ผ โˆˆ (0..^๐‘))
 
Theoremfzval3 10204 Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(๐‘ โˆˆ โ„ค โ†’ (๐‘€...๐‘) = (๐‘€..^(๐‘ + 1)))
 
Theoremfzosn 10205 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(๐ด โˆˆ โ„ค โ†’ (๐ด..^(๐ด + 1)) = {๐ด})
 
Theoremelfzomin 10206 Membership of an integer in the smallest open range of integers. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
(๐‘ โˆˆ โ„ค โ†’ ๐‘ โˆˆ (๐‘..^(๐‘ + 1)))
 
Theoremzpnn0elfzo 10207 Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((๐‘ โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„•0) โ†’ (๐‘ + ๐‘) โˆˆ (๐‘..^((๐‘ + ๐‘) + 1)))
 
Theoremzpnn0elfzo1 10208 Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((๐‘ โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„•0) โ†’ (๐‘ + ๐‘) โˆˆ (๐‘..^(๐‘ + (๐‘ + 1))))
 
Theoremfzosplitsnm1 10209 Removing a singleton from a half-open integer range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ (โ„คโ‰ฅโ€˜(๐ด + 1))) โ†’ (๐ด..^๐ต) = ((๐ด..^(๐ต โˆ’ 1)) โˆช {(๐ต โˆ’ 1)}))
 
Theoremelfzonlteqm1 10210 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.)
((๐ด โˆˆ (0..^๐ต) โˆง ยฌ ๐ด < (๐ต โˆ’ 1)) โ†’ ๐ด = (๐ต โˆ’ 1))
 
Theoremfzonn0p1 10211 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.)
(๐‘ โˆˆ โ„•0 โ†’ ๐‘ โˆˆ (0..^(๐‘ + 1)))
 
Theoremfzossfzop1 10212 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.)
(๐‘ โˆˆ โ„•0 โ†’ (0..^๐‘) โŠ† (0..^(๐‘ + 1)))
 
Theoremfzonn0p1p1 10213 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.)
(๐ผ โˆˆ (0..^๐‘) โ†’ (๐ผ + 1) โˆˆ (0..^(๐‘ + 1)))
 
Theoremelfzom1p1elfzo 10214 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.)
((๐‘ โˆˆ โ„• โˆง ๐‘‹ โˆˆ (0..^(๐‘ โˆ’ 1))) โ†’ (๐‘‹ + 1) โˆˆ (0..^๐‘))
 
Theoremfzo0ssnn0 10215 Half-open integer ranges starting with 0 are subsets of NN0. (Contributed by Thierry Arnoux, 8-Oct-2018.)
(0..^๐‘) โŠ† โ„•0
 
Theoremfzo01 10216 Expressing the singleton of 0 as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(0..^1) = {0}
 
Theoremfzo12sn 10217 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.)
(1..^2) = {1}
 
Theoremfzo0to2pr 10218 A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(0..^2) = {0, 1}
 
Theoremfzo0to3tp 10219 A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
(0..^3) = {0, 1, 2}
 
Theoremfzo0to42pr 10220 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.)
(0..^4) = ({0, 1} โˆช {2, 3})
 
Theoremfzo0sn0fzo1 10221 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.)
(๐‘ โˆˆ โ„• โ†’ (0..^๐‘) = ({0} โˆช (1..^๐‘)))
 
Theoremfzoend 10222 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
(๐ด โˆˆ (๐ด..^๐ต) โ†’ (๐ต โˆ’ 1) โˆˆ (๐ด..^๐ต))
 
Theoremfzo0end 10223 The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
(๐ต โˆˆ โ„• โ†’ (๐ต โˆ’ 1) โˆˆ (0..^๐ต))
 
Theoremssfzo12 10224 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
((๐พ โˆˆ โ„ค โˆง ๐ฟ โˆˆ โ„ค โˆง ๐พ < ๐ฟ) โ†’ ((๐พ..^๐ฟ) โŠ† (๐‘€..^๐‘) โ†’ (๐‘€ โ‰ค ๐พ โˆง ๐ฟ โ‰ค ๐‘)))
 
Theoremssfzo12bi 10225 Subset relationship for half-open integer ranges. (Contributed by Alexander van der Vekens, 5-Nov-2018.)
(((๐พ โˆˆ โ„ค โˆง ๐ฟ โˆˆ โ„ค) โˆง (๐‘€ โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„ค) โˆง ๐พ < ๐ฟ) โ†’ ((๐พ..^๐ฟ) โŠ† (๐‘€..^๐‘) โ†” (๐‘€ โ‰ค ๐พ โˆง ๐ฟ โ‰ค ๐‘)))
 
Theoremubmelm1fzo 10226 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.)
(๐พ โˆˆ (0..^๐‘) โ†’ ((๐‘ โˆ’ ๐พ) โˆ’ 1) โˆˆ (0..^๐‘))
 
Theoremfzofzp1 10227 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.)
(๐ถ โˆˆ (๐ด..^๐ต) โ†’ (๐ถ + 1) โˆˆ (๐ด...๐ต))
 
Theoremfzofzp1b 10228 If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.)
(๐ถ โˆˆ (โ„คโ‰ฅโ€˜๐ด) โ†’ (๐ถ โˆˆ (๐ด..^๐ต) โ†” (๐ถ + 1) โˆˆ (๐ด...๐ต)))
 
Theoremelfzom1b 10229 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.)
((๐พ โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„ค) โ†’ (๐พ โˆˆ (1..^๐‘) โ†” (๐พ โˆ’ 1) โˆˆ (0..^(๐‘ โˆ’ 1))))
 
Theoremelfzonelfzo 10230 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.)
(๐‘ โˆˆ โ„ค โ†’ ((๐พ โˆˆ (๐‘€..^๐‘…) โˆง ยฌ ๐พ โˆˆ (๐‘€..^๐‘)) โ†’ ๐พ โˆˆ (๐‘..^๐‘…)))
 
Theoremelfzomelpfzo 10231 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.)
(((๐‘€ โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„ค) โˆง (๐พ โˆˆ โ„ค โˆง ๐ฟ โˆˆ โ„ค)) โ†’ (๐พ โˆˆ ((๐‘€ โˆ’ ๐ฟ)..^(๐‘ โˆ’ ๐ฟ)) โ†” (๐พ + ๐ฟ) โˆˆ (๐‘€..^๐‘)))
 
Theorempeano2fzor 10232 A Peano-postulate-like theorem for downward closure of a half-open integer range. (Contributed by Mario Carneiro, 1-Oct-2015.)
((๐พ โˆˆ (โ„คโ‰ฅโ€˜๐‘€) โˆง (๐พ + 1) โˆˆ (๐‘€..^๐‘)) โ†’ ๐พ โˆˆ (๐‘€..^๐‘))
 
Theoremfzosplitsn 10233 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(๐ต โˆˆ (โ„คโ‰ฅโ€˜๐ด) โ†’ (๐ด..^(๐ต + 1)) = ((๐ด..^๐ต) โˆช {๐ต}))
 
Theoremfzosplitprm1 10234 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„ค โˆง ๐ด < ๐ต) โ†’ (๐ด..^(๐ต + 1)) = ((๐ด..^(๐ต โˆ’ 1)) โˆช {(๐ต โˆ’ 1), ๐ต}))
 
Theoremfzosplitsni 10235 Membership in a half-open range extended by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(๐ต โˆˆ (โ„คโ‰ฅโ€˜๐ด) โ†’ (๐ถ โˆˆ (๐ด..^(๐ต + 1)) โ†” (๐ถ โˆˆ (๐ด..^๐ต) โˆจ ๐ถ = ๐ต)))
 
Theoremfzisfzounsn 10236 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.)
(๐ต โˆˆ (โ„คโ‰ฅโ€˜๐ด) โ†’ (๐ด...๐ต) = ((๐ด..^๐ต) โˆช {๐ต}))
 
Theoremfzostep1 10237 Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
(๐ด โˆˆ (๐ต..^๐ถ) โ†’ ((๐ด + 1) โˆˆ (๐ต..^๐ถ) โˆจ (๐ด + 1) = ๐ถ))
 
Theoremfzoshftral 10238* Shift the scanning order inside of a quantification over a half-open integer range, analogous to fzshftral 10108. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((๐‘€ โˆˆ โ„ค โˆง ๐‘ โˆˆ โ„ค โˆง ๐พ โˆˆ โ„ค) โ†’ (โˆ€๐‘— โˆˆ (๐‘€..^๐‘)๐œ‘ โ†” โˆ€๐‘˜ โˆˆ ((๐‘€ + ๐พ)..^(๐‘ + ๐พ))[(๐‘˜ โˆ’ ๐พ) / ๐‘—]๐œ‘))
 
Theoremfzind2 10239* Induction on the integers from ๐‘€ to ๐‘ 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 9368 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
(๐‘ฅ = ๐‘€ โ†’ (๐œ‘ โ†” ๐œ“))    &   (๐‘ฅ = ๐‘ฆ โ†’ (๐œ‘ โ†” ๐œ’))    &   (๐‘ฅ = (๐‘ฆ + 1) โ†’ (๐œ‘ โ†” ๐œƒ))    &   (๐‘ฅ = ๐พ โ†’ (๐œ‘ โ†” ๐œ))    &   (๐‘ โˆˆ (โ„คโ‰ฅโ€˜๐‘€) โ†’ ๐œ“)    &   (๐‘ฆ โˆˆ (๐‘€..^๐‘) โ†’ (๐œ’ โ†’ ๐œƒ))    โ‡’   (๐พ โˆˆ (๐‘€...๐‘) โ†’ ๐œ)
 
Theoremexfzdc 10240* Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.)
(๐œ‘ โ†’ ๐‘€ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„ค)    &   ((๐œ‘ โˆง ๐‘› โˆˆ (๐‘€...๐‘)) โ†’ DECID ๐œ“)    โ‡’   (๐œ‘ โ†’ DECID โˆƒ๐‘› โˆˆ (๐‘€...๐‘)๐œ“)
 
Theoremfvinim0ffz 10241 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.)
((๐น:(0...๐พ)โŸถ๐‘‰ โˆง ๐พ โˆˆ โ„•0) โ†’ (((๐น โ€œ {0, ๐พ}) โˆฉ (๐น โ€œ (1..^๐พ))) = โˆ… โ†” ((๐นโ€˜0) โˆ‰ (๐น โ€œ (1..^๐พ)) โˆง (๐นโ€˜๐พ) โˆ‰ (๐น โ€œ (1..^๐พ)))))
 
Theoremsubfzo0 10242 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.)
((๐ผ โˆˆ (0..^๐‘) โˆง ๐ฝ โˆˆ (0..^๐‘)) โ†’ (-๐‘ < (๐ผ โˆ’ ๐ฝ) โˆง (๐ผ โˆ’ ๐ฝ) < ๐‘))
 
4.5.7  Rational numbers (cont.)
 
Theoremqtri3or 10243 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
((๐‘€ โˆˆ โ„š โˆง ๐‘ โˆˆ โ„š) โ†’ (๐‘€ < ๐‘ โˆจ ๐‘€ = ๐‘ โˆจ ๐‘ < ๐‘€))
 
Theoremqletric 10244 Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ๐ต โˆˆ โ„š) โ†’ (๐ด โ‰ค ๐ต โˆจ ๐ต โ‰ค ๐ด))
 
Theoremqlelttric 10245 Rational trichotomy. (Contributed by Jim Kingdon, 7-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ๐ต โˆˆ โ„š) โ†’ (๐ด โ‰ค ๐ต โˆจ ๐ต < ๐ด))
 
Theoremqltnle 10246 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 8-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ๐ต โˆˆ โ„š) โ†’ (๐ด < ๐ต โ†” ยฌ ๐ต โ‰ค ๐ด))
 
Theoremqdceq 10247 Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ๐ต โˆˆ โ„š) โ†’ DECID ๐ด = ๐ต)
 
Theoremexbtwnzlemstep 10248* Lemma for exbtwnzlemex 10250. Induction step. (Contributed by Jim Kingdon, 10-May-2022.)
(๐œ‘ โ†’ ๐พ โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   ((๐œ‘ โˆง ๐‘› โˆˆ โ„ค) โ†’ (๐‘› โ‰ค ๐ด โˆจ ๐ด < ๐‘›))    โ‡’   ((๐œ‘ โˆง โˆƒ๐‘š โˆˆ โ„ค (๐‘š โ‰ค ๐ด โˆง ๐ด < (๐‘š + (๐พ + 1)))) โ†’ โˆƒ๐‘š โˆˆ โ„ค (๐‘š โ‰ค ๐ด โˆง ๐ด < (๐‘š + ๐พ)))
 
Theoremexbtwnzlemshrink 10249* Lemma for exbtwnzlemex 10250. Shrinking the range around ๐ด. (Contributed by Jim Kingdon, 10-May-2022.)
(๐œ‘ โ†’ ๐ฝ โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   ((๐œ‘ โˆง ๐‘› โˆˆ โ„ค) โ†’ (๐‘› โ‰ค ๐ด โˆจ ๐ด < ๐‘›))    โ‡’   ((๐œ‘ โˆง โˆƒ๐‘š โˆˆ โ„ค (๐‘š โ‰ค ๐ด โˆง ๐ด < (๐‘š + ๐ฝ))) โ†’ โˆƒ๐‘ฅ โˆˆ โ„ค (๐‘ฅ โ‰ค ๐ด โˆง ๐ด < (๐‘ฅ + 1)))
 
Theoremexbtwnzlemex 10250* Existence of an integer so that a given real number is between the integer and its successor. The real number must satisfy the ๐‘› โ‰ค ๐ด โˆจ ๐ด < ๐‘› 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 ๐ด. 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 ๐‘› โ‰ค ๐ด โˆจ ๐ด < ๐‘› hypothesis, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)

(๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   ((๐œ‘ โˆง ๐‘› โˆˆ โ„ค) โ†’ (๐‘› โ‰ค ๐ด โˆจ ๐ด < ๐‘›))    โ‡’   (๐œ‘ โ†’ โˆƒ๐‘ฅ โˆˆ โ„ค (๐‘ฅ โ‰ค ๐ด โˆง ๐ด < (๐‘ฅ + 1)))
 
Theoremexbtwnz 10251* 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.)
(๐œ‘ โ†’ โˆƒ๐‘ฅ โˆˆ โ„ค (๐‘ฅ โ‰ค ๐ด โˆง ๐ด < (๐‘ฅ + 1)))    &   (๐œ‘ โ†’ ๐ด โˆˆ โ„)    โ‡’   (๐œ‘ โ†’ โˆƒ!๐‘ฅ โˆˆ โ„ค (๐‘ฅ โ‰ค ๐ด โˆง ๐ด < (๐‘ฅ + 1)))
 
Theoremqbtwnz 10252* There is a unique greatest integer less than or equal to a rational number. (Contributed by Jim Kingdon, 8-Oct-2021.)
(๐ด โˆˆ โ„š โ†’ โˆƒ!๐‘ฅ โˆˆ โ„ค (๐‘ฅ โ‰ค ๐ด โˆง ๐ด < (๐‘ฅ + 1)))
 
Theoremrebtwn2zlemstep 10253* Lemma for rebtwn2z 10255. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)
((๐พ โˆˆ (โ„คโ‰ฅโ€˜2) โˆง ๐ด โˆˆ โ„ โˆง โˆƒ๐‘š โˆˆ โ„ค (๐‘š < ๐ด โˆง ๐ด < (๐‘š + (๐พ + 1)))) โ†’ โˆƒ๐‘š โˆˆ โ„ค (๐‘š < ๐ด โˆง ๐ด < (๐‘š + ๐พ)))
 
Theoremrebtwn2zlemshrink 10254* Lemma for rebtwn2z 10255. Shrinking the range around the given real number. (Contributed by Jim Kingdon, 13-Oct-2021.)
((๐ด โˆˆ โ„ โˆง ๐ฝ โˆˆ (โ„คโ‰ฅโ€˜2) โˆง โˆƒ๐‘š โˆˆ โ„ค (๐‘š < ๐ด โˆง ๐ด < (๐‘š + ๐ฝ))) โ†’ โˆƒ๐‘ฅ โˆˆ โ„ค (๐‘ฅ < ๐ด โˆง ๐ด < (๐‘ฅ + 2)))
 
Theoremrebtwn2z 10255* 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.)

(๐ด โˆˆ โ„ โ†’ โˆƒ๐‘ฅ โˆˆ โ„ค (๐‘ฅ < ๐ด โˆง ๐ด < (๐‘ฅ + 2)))
 
Theoremqbtwnrelemcalc 10256 Lemma for qbtwnre 10257. Calculations involved in showing the constructed rational number is less than ๐ต. (Contributed by Jim Kingdon, 14-Oct-2021.)
(๐œ‘ โ†’ ๐‘€ โˆˆ โ„ค)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•)    &   (๐œ‘ โ†’ ๐ด โˆˆ โ„)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„)    &   (๐œ‘ โ†’ ๐‘€ < (๐ด ยท (2 ยท ๐‘)))    &   (๐œ‘ โ†’ (1 / ๐‘) < (๐ต โˆ’ ๐ด))    โ‡’   (๐œ‘ โ†’ ((๐‘€ + 2) / (2 ยท ๐‘)) < ๐ต)
 
Theoremqbtwnre 10257* The rational numbers are dense in โ„: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.)
((๐ด โˆˆ โ„ โˆง ๐ต โˆˆ โ„ โˆง ๐ด < ๐ต) โ†’ โˆƒ๐‘ฅ โˆˆ โ„š (๐ด < ๐‘ฅ โˆง ๐‘ฅ < ๐ต))
 
Theoremqbtwnxr 10258* The rational numbers are dense in โ„*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
((๐ด โˆˆ โ„* โˆง ๐ต โˆˆ โ„* โˆง ๐ด < ๐ต) โ†’ โˆƒ๐‘ฅ โˆˆ โ„š (๐ด < ๐‘ฅ โˆง ๐‘ฅ < ๐ต))
 
Theoremqavgle 10259 The average of two rational numbers is less than or equal to at least one of them. (Contributed by Jim Kingdon, 3-Nov-2021.)
((๐ด โˆˆ โ„š โˆง ๐ต โˆˆ โ„š) โ†’ (((๐ด + ๐ต) / 2) โ‰ค ๐ด โˆจ ((๐ด + ๐ต) / 2) โ‰ค ๐ต))
 
Theoremioo0 10260 An empty open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
((๐ด โˆˆ โ„* โˆง ๐ต โˆˆ โ„*) โ†’ ((๐ด(,)๐ต) = โˆ… โ†” ๐ต โ‰ค ๐ด))
 
Theoremioom 10261* 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.)
((๐ด โˆˆ โ„* โˆง ๐ต โˆˆ โ„*) โ†’ (โˆƒ๐‘ฅ ๐‘ฅ โˆˆ (๐ด(,)๐ต) โ†” ๐ด < ๐ต))
 
Theoremico0 10262 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
((๐ด โˆˆ โ„* โˆง ๐ต โˆˆ โ„*) โ†’ ((๐ด[,)๐ต) = โˆ… โ†” ๐ต โ‰ค ๐ด))
 
Theoremioc0 10263 An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)
((๐ด โˆˆ โ„* โˆง ๐ต โˆˆ โ„*) โ†’ ((๐ด(,]๐ต) = โˆ… โ†” ๐ต โ‰ค ๐ด))
 
Theoremdfrp2 10264 Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
โ„+ = (0(,)+โˆž)
 
Theoremelicod 10265 Membership in a left-closed right-open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„*)    &   (๐œ‘ โ†’ ๐ต โˆˆ โ„*)    &   (๐œ‘ โ†’ ๐ถ โˆˆ โ„*)    &   (๐œ‘ โ†’ ๐ด โ‰ค ๐ถ)    &   (๐œ‘ โ†’ ๐ถ < ๐ต)    โ‡’   (๐œ‘ โ†’ ๐ถ โˆˆ (๐ด[,)๐ต))
 
Theoremicogelb 10266 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.)
((๐ด โˆˆ โ„* โˆง ๐ต โˆˆ โ„* โˆง ๐ถ โˆˆ (๐ด[,)๐ต)) โ†’ ๐ด โ‰ค ๐ถ)
 
Theoremelicore 10267 A member of a left-closed right-open interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((๐ด โˆˆ โ„ โˆง ๐ถ โˆˆ (๐ด[,)๐ต)) โ†’ ๐ถ โˆˆ โ„)
 
4.6  Elementary integer functions
 
4.6.1  The floor and ceiling functions
 
Syntaxcfl 10268 Extend class notation with floor (greatest integer) function.
class โŒŠ
 
Syntaxcceil 10269 Extend class notation to include the ceiling function.
class โŒˆ
 
Definitiondf-fl 10270* Define the floor (greatest integer less than or equal to) function. See flval 10272 for its value, flqlelt 10276 for its basic property, and flqcl 10273 for its closure. For example, (โŒŠโ€˜(3 / 2)) = 1 while (โŒŠโ€˜-(3 / 2)) = -2 (ex-fl 14480).

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.)

โŒŠ = (๐‘ฅ โˆˆ โ„ โ†ฆ (โ„ฉ๐‘ฆ โˆˆ โ„ค (๐‘ฆ โ‰ค ๐‘ฅ โˆง ๐‘ฅ < (๐‘ฆ + 1))))
 
Definitiondf-ceil 10271 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 10306 for its value, ceilqge 10310 and ceilqm1lt 10312 for its basic properties, and ceilqcl 10308 for its closure. For example, (โŒˆโ€˜(3 / 2)) = 2 while (โŒˆโ€˜-(3 / 2)) = -1 (ex-ceil 14481).

As described in df-fl 10270 most theorems are only for rationals, not reals.

The symbol โŒˆ is inspired by the gamma shaped left bracket of the usual notation. (Contributed by David A. Wheeler, 19-May-2015.)

โŒˆ = (๐‘ฅ โˆˆ โ„ โ†ฆ -(โŒŠโ€˜-๐‘ฅ))
 
Theoremflval 10272* Value of the floor (greatest integer) function. The floor of ๐ด is the (unique) integer less than or equal to ๐ด whose successor is strictly greater than ๐ด. (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)
(๐ด โˆˆ โ„ โ†’ (โŒŠโ€˜๐ด) = (โ„ฉ๐‘ฅ โˆˆ โ„ค (๐‘ฅ โ‰ค ๐ด โˆง ๐ด < (๐‘ฅ + 1))))
 
Theoremflqcl 10273 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 10275. (Contributed by Jim Kingdon, 8-Oct-2021.)
(๐ด โˆˆ โ„š โ†’ (โŒŠโ€˜๐ด) โˆˆ โ„ค)
 
Theoremapbtwnz 10274* 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.)
((๐ด โˆˆ โ„ โˆง โˆ€๐‘› โˆˆ โ„ค ๐ด # ๐‘›) โ†’ โˆƒ!๐‘ฅ โˆˆ โ„ค (๐‘ฅ โ‰ค ๐ด โˆง ๐ด < (๐‘ฅ + 1)))
 
Theoremflapcl 10275* 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 12180) would satisfy this condition. (Contributed by Jim Kingdon, 11-May-2022.)
((๐ด โˆˆ โ„ โˆง โˆ€๐‘› โˆˆ โ„ค ๐ด # ๐‘›) โ†’ (โŒŠโ€˜๐ด) โˆˆ โ„ค)
 
Theoremflqlelt 10276 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(๐ด โˆˆ โ„š โ†’ ((โŒŠโ€˜๐ด) โ‰ค ๐ด โˆง ๐ด < ((โŒŠโ€˜๐ด) + 1)))
 
Theoremflqcld 10277 The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.)
(๐œ‘ โ†’ ๐ด โˆˆ โ„š)    โ‡’   (๐œ‘ โ†’ (โŒŠโ€˜๐ด) โˆˆ โ„ค)
 
Theoremflqle 10278 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(๐ด โˆˆ โ„š โ†’ (โŒŠโ€˜๐ด) โ‰ค ๐ด)
 
Theoremflqltp1 10279 A basic property of the floor (greatest integer) function. (Contributed by Jim Kingdon, 8-Oct-2021.)
(๐ด โˆˆ โ„š โ†’ ๐ด < ((โŒŠโ€˜๐ด) + 1))
 
Theoremqfraclt1 10280 The fractional part of a rational number is less than one. (Contributed by Jim Kingdon, 8-Oct-2021.)
(๐ด โˆˆ โ„š โ†’ (๐ด โˆ’ (โŒŠโ€˜๐ด)) < 1)
 
Theoremqfracge0 10281 The fractional part of a rational number is nonnegative. (Contributed by Jim Kingdon, 8-Oct-2021.)
(๐ด โˆˆ โ„š โ†’ 0 โ‰ค (๐ด โˆ’ (โŒŠโ€˜๐ด)))
 
Theoremflqge 10282 The floor function value is the greatest integer less than or equal to its argument. (Contributed by Jim Kingdon, 8-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ๐ต โˆˆ โ„ค) โ†’ (๐ต โ‰ค ๐ด โ†” ๐ต โ‰ค (โŒŠโ€˜๐ด)))
 
Theoremflqlt 10283 The floor function value is less than the next integer. (Contributed by Jim Kingdon, 8-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ๐ต โˆˆ โ„ค) โ†’ (๐ด < ๐ต โ†” (โŒŠโ€˜๐ด) < ๐ต))
 
Theoremflid 10284 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)
(๐ด โˆˆ โ„ค โ†’ (โŒŠโ€˜๐ด) = ๐ด)
 
Theoremflqidm 10285 The floor function is idempotent. (Contributed by Jim Kingdon, 8-Oct-2021.)
(๐ด โˆˆ โ„š โ†’ (โŒŠโ€˜(โŒŠโ€˜๐ด)) = (โŒŠโ€˜๐ด))
 
Theoremflqidz 10286 A rational number equals its floor iff it is an integer. (Contributed by Jim Kingdon, 9-Oct-2021.)
(๐ด โˆˆ โ„š โ†’ ((โŒŠโ€˜๐ด) = ๐ด โ†” ๐ด โˆˆ โ„ค))
 
Theoremflqltnz 10287 If A is not an integer, then the floor of A is less than A. (Contributed by Jim Kingdon, 9-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ยฌ ๐ด โˆˆ โ„ค) โ†’ (โŒŠโ€˜๐ด) < ๐ด)
 
Theoremflqwordi 10288 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ๐ต โˆˆ โ„š โˆง ๐ด โ‰ค ๐ต) โ†’ (โŒŠโ€˜๐ด) โ‰ค (โŒŠโ€˜๐ต))
 
Theoremflqword2 10289 Ordering relationship for the greatest integer function. (Contributed by Jim Kingdon, 9-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ๐ต โˆˆ โ„š โˆง ๐ด โ‰ค ๐ต) โ†’ (โŒŠโ€˜๐ต) โˆˆ (โ„คโ‰ฅโ€˜(โŒŠโ€˜๐ด)))
 
Theoremflqbi 10290 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ๐ต โˆˆ โ„ค) โ†’ ((โŒŠโ€˜๐ด) = ๐ต โ†” (๐ต โ‰ค ๐ด โˆง ๐ด < (๐ต + 1))))
 
Theoremflqbi2 10291 A condition equivalent to floor. (Contributed by Jim Kingdon, 9-Oct-2021.)
((๐‘ โˆˆ โ„ค โˆง ๐น โˆˆ โ„š) โ†’ ((โŒŠโ€˜(๐‘ + ๐น)) = ๐‘ โ†” (0 โ‰ค ๐น โˆง ๐น < 1)))
 
Theoremadddivflid 10292 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.)
((๐ด โˆˆ โ„ค โˆง ๐ต โˆˆ โ„•0 โˆง ๐ถ โˆˆ โ„•) โ†’ (๐ต < ๐ถ โ†” (โŒŠโ€˜(๐ด + (๐ต / ๐ถ))) = ๐ด))
 
Theoremflqge0nn0 10293 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
((๐ด โˆˆ โ„š โˆง 0 โ‰ค ๐ด) โ†’ (โŒŠโ€˜๐ด) โˆˆ โ„•0)
 
Theoremflqge1nn 10294 The floor of a number greater than or equal to 1 is a positive integer. (Contributed by Jim Kingdon, 10-Oct-2021.)
((๐ด โˆˆ โ„š โˆง 1 โ‰ค ๐ด) โ†’ (โŒŠโ€˜๐ด) โˆˆ โ„•)
 
Theoremfldivnn0 10295 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.)
((๐พ โˆˆ โ„•0 โˆง ๐ฟ โˆˆ โ„•) โ†’ (โŒŠโ€˜(๐พ / ๐ฟ)) โˆˆ โ„•0)
 
Theoremdivfl0 10296 The floor of a fraction is 0 iff the denominator is less than the numerator. (Contributed by AV, 8-Jul-2021.)
((๐ด โˆˆ โ„•0 โˆง ๐ต โˆˆ โ„•) โ†’ (๐ด < ๐ต โ†” (โŒŠโ€˜(๐ด / ๐ต)) = 0))
 
Theoremflqaddz 10297 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
((๐ด โˆˆ โ„š โˆง ๐‘ โˆˆ โ„ค) โ†’ (โŒŠโ€˜(๐ด + ๐‘)) = ((โŒŠโ€˜๐ด) + ๐‘))
 
Theoremflqzadd 10298 An integer can be moved in and out of the floor of a sum. (Contributed by Jim Kingdon, 10-Oct-2021.)
((๐‘ โˆˆ โ„ค โˆง ๐ด โˆˆ โ„š) โ†’ (โŒŠโ€˜(๐‘ + ๐ด)) = (๐‘ + (โŒŠโ€˜๐ด)))
 
Theoremflqmulnn0 10299 Move a nonnegative integer in and out of a floor. (Contributed by Jim Kingdon, 10-Oct-2021.)
((๐‘ โˆˆ โ„•0 โˆง ๐ด โˆˆ โ„š) โ†’ (๐‘ ยท (โŒŠโ€˜๐ด)) โ‰ค (โŒŠโ€˜(๐‘ ยท ๐ด)))
 
Theorembtwnzge0 10300 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.)
(((๐ด โˆˆ โ„ โˆง ๐‘ โˆˆ โ„ค) โˆง (๐‘ โ‰ค ๐ด โˆง ๐ด < (๐‘ + 1))) โ†’ (0 โ‰ค ๐ด โ†” 0 โ‰ค ๐‘))
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