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Theorem List for Metamath Proof Explorer - 11101-11200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelfzo 11101 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzo2 11102 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzouz 11103 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzolb 11104 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with . This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate . (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzolb2 11105 The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with . This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate . (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzole1 11106 A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
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Theoremelfzolt2 11107 A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzolt3 11108 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzolt2b 11109 A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremelfzolt3b 11110 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzonel 11111 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
..^

Theoremelfzouz2 11112 The upper bound of a half-open range is greater or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzofz 11113 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^

Theoremelfzo3 11114 Express membership in a half-open integer interval in terms of the "less than or equal" and "less than" predicates on integers, resp. , ..^ . (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzon0 11115 A half-open integer interval is nonempty iff it contains its left endpoint. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzossfz 11116 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzon 11117 A half-open set of sequential integers is empty if the bounds are equal or reversed. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
..^

Theoremfzo0 11118 Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzonnsub 11119 If then is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
..^

Theoremfzonnsub2 11120 If then is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
..^

Theoremfzoss1 11121 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzoss2 11122 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzossrbm1 11123 Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
..^ ..^

Theoremfzospliti 11124 One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ ..^ ..^

Theoremfzosplit 11125 Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ ..^ ..^

Theoremfzodisj 11126 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ ..^

Theoremfzouzsplit 11127 Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
..^

Theoremfzouzdisj 11128 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.)
..^

Theoremlbfzo0 11129 An integer is strictly greater than zero iff it is a member of . (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzo0 11130 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.)
..^

Theoremfzossnn 11131 Half-opened integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
..^

Theoremelfzo1 11132 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
..^

Theoremfzo0n0 11133 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.)
..^

Theoremfzoaddel 11134 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzoaddel2 11135 Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzosubel 11136 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzosubel2 11137 Membership in a translated half-open integer range implies translated membership in the original range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzosubel3 11138 Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzval3 11139 Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremfzosn 11140 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^

Theoremfzo01 11141 Expressing the singleton of as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremfzo12sn 11142 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.)
..^

Theoremfzo0to2pr 11143 A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
..^

Theoremfzo0to3tp 11144 A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
..^

Theoremfzo0to42pr 11145 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.)
..^

Theoremfzoend 11146 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzo0end 11147 The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzofzp1 11148 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.)
..^

Theoremfzofzp1b 11149 If a point is in a half-open range, the next point is in the closed range. (Contributed by Mario Carneiro, 27-Sep-2015.)
..^

Theoremelfzom1b 11150 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.)
..^ ..^

Theoremelfznelfzo 11151 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.)
..^

Theoremelfznelfzob 11152 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 integerss. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
..^

Theorempeano2fzor 11153 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 1-Oct-2015.)
..^ ..^

Theoremfzosplitsn 11154 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzosplitsni 11155 Membership in a half-open range extende by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzostep1 11156 Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzind2 11157* 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 hypothesis. Version of fzind 10328 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
..^

Theoreminjresinjlem 11158 Lemma for injresinj 11159. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
..^ ..^

Theoreminjresinj 11159 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.)
..^ ..^

5.6  Elementary integer functions

5.6.1  The floor (greatest integer) function

Syntaxcfl 11160 Extend class notation with floor (greatest integer) function.

Definitiondf-fl 11161* Define the floor (greatest integer) function. See flval 11162 for its value, fllelt 11165 for its basic property, and flcl 11163 for its closure. For example, while (ex-fl 21712).

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.)

Theoremflval 11162* 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.)

Theoremflcl 11163 The floor (greatest integer) function is an integer (closure law). (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremreflcl 11164 The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.)

Theoremfllelt 11165 A basic property of the floor (greatest integer) function. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremflcld 11166 The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016.)

Theoremflle 11167 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)

Theoremflltp1 11168 A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.)

Theoremfllep1 11169 A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremfraclt1 11170 The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008.)

Theoremfracle1 11171 The fractional part of a real number is less than or equal to one. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremfracge0 11172 The fractional part of a real number is nonnegative. (Contributed by NM, 17-Jul-2008.)

Theoremflge 11173 The floor function value is the greatest integer less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Fan Zheng, 14-Jul-2016.)

Theoremfllt 11174 The floor function value is less than the next integer. (Contributed by NM, 24-Feb-2005.)

Theoremflid 11175 An integer is its own floor. (Contributed by NM, 15-Nov-2004.)

Theoremflidm 11176 The floor function is idempotent. (Contributed by NM, 17-Aug-2008.)

Theoremflidz 11177 A real number equals its floor iff it is an integer. (Contributed by NM, 11-Nov-2008.)

Theoremflwordi 11178 Ordering relationship for the greatest integer function. (Contributed by NM, 31-Dec-2005.) (Proof shortened by Fan Zheng, 14-Jul-2016.)

Theoremflword2 11179 Ordering relationship for the greatest integer function. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theoremflval2 11180* An alternate way to define the floor (greatest integer) function. (Contributed by NM, 16-Nov-2004.)

Theoremflval3 11181* An alternate way to define the floor (greatest integer) function, as the supremum of all integers less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 6-Sep-2014.)

Theoremflbi 11182 A condition equivalent to floor. (Contributed by NM, 11-Mar-2005.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremflbi2 11183 A condition equivalent to floor. (Contributed by NM, 15-Aug-2008.)

Theoremflge0nn0 11184 The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by NM, 26-Apr-2005.)

Theoremflge1nn 11185 The floor of a number greater than or equal to 1 is a natural number. (Contributed by NM, 26-Apr-2005.)

Theoremfladdz 11186 An integer can be moved in and out of the floor of a sum. (Contributed by NM, 27-Apr-2005.) (Proof shortened by Fan Zheng, 16-Jun-2016.)

Theoremflzadd 11187 An integer can be moved in and out of the floor of a sum. (Contributed by NM, 2-Jan-2009.)

Theoremflmulnn0 11188 Move a nonnegative integer in and out of a floor. (Contributed by NM, 2-Jan-2009.) (Proof shortened by Fan Zheng, 7-Jun-2016.)

Theorembtwnzge0 11189 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. (For the first half see rebtwnz 10533.) (Contributed by NM, 12-Mar-2005.)

Theoremflhalf 11190 Ordering relation for the floor of half of an integer. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)

Theoremceicl 11191 The ceiling function returns an integer (closure law). (Contributed by Jeffrey Hankins, 10-Jun-2007.)

Theoremceige 11192 The ceiling of a real number is greater than or equal to that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)

Theoremceim1l 11193 One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)

Theoremceile 11194 The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jeffrey Hankins, 10-Jun-2007.)

Theoremquoremz 11195 Quotient and remainder of an integer divided by a natural number. TO DO - is this really needed for anything? Should we use to simplify it? (Contributed by NM, 14-Aug-2008.)

Theoremquoremnn0 11196 Quotient and remainder of a nonnegative integer divided by a natural number. (Contributed by NM, 14-Aug-2008.)

Theoremquoremnn0ALT 11197 Quotient and remainder of a nonnegative integer divided by a natural number. TO DO - Keep either quoremnn0ALT 11197 ((if we don't keep quoremz 11195) or quoremnn0 11196 (Contributed by NM, 14-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremintfrac2 11198 Decompose a real into integer and fractional parts. TODO - should we replace this with intfrac 11222? (Contributed by NM, 16-Aug-2008.)

Theoremintfracq 11199 Decompose a rational number, expressed as a ratio, into integer and fractional parts. The fractional part has a tighter bound than that of intfrac2 11198. (Contributed by NM, 16-Aug-2008.)

Theoremfldiv 11200 Cancellation of the embedded floor of a real divided by an integer. (Contributed by NM, 16-Aug-2008.)

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