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Type | Label | Description |
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Statement | ||
Theorem | fznatpl1 10001 | Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.) |
Theorem | fzpr 10002 | A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | fztp 10003 | A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.) |
Theorem | fzsuc2 10004 | Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.) |
Theorem | fzp1disj 10005 | is the disjoint union of with . (Contributed by Mario Carneiro, 7-Mar-2014.) |
Theorem | fzdifsuc 10006 | Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.) |
Theorem | fzprval 10007* | Two ways of defining the first two values of a sequence on . (Contributed by NM, 5-Sep-2011.) |
Theorem | fztpval 10008* | Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.) |
Theorem | fzrev 10009 | Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrev2 10010 | Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrev2i 10011 | Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrev3 10012 | The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
Theorem | fzrev3i 10013 | The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
Theorem | fznn 10014 | Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.) |
Theorem | elfz1b 10015 | Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.) |
Theorem | elfzm11 10016 | Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
Theorem | uzsplit 10017 | Express an upper integer set as the disjoint (see uzdisj 10018) union of the first values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.) |
Theorem | uzdisj 10018 | The first elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.) |
Theorem | fseq1p1m1 10019 | Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.) |
Theorem | fseq1m1p1 10020 | Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) |
Theorem | fz1sbc 10021* | Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.) |
Theorem | elfzp1b 10022 | An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | elfzm1b 10023 | 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 Paul Chapman, 22-Jun-2011.) |
Theorem | elfzp12 10024 | Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.) |
Theorem | fzm1 10025 | Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | fzneuz 10026 | No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.) |
Theorem | fznuz 10027 | Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.) |
Theorem | uznfz 10028 | Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.) |
Theorem | fzp1nel 10029 | One plus the upper bound of a finite set of integers is not a member of that set. (Contributed by Scott Fenton, 16-Dec-2017.) |
Theorem | fzrevral 10030* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrevral2 10031* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.) |
Theorem | fzrevral3 10032* | Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.) |
Theorem | fzshftral 10033* | Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.) |
Theorem | ige2m1fz1 10034 | Membership of an integer greater than 1 decreased by 1 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.) |
Theorem | ige2m1fz 10035 | Membership in a 0 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 18-Jun-2018.) (Proof shortened by Alexander van der Vekens, 15-Sep-2018.) |
Theorem | fz01or 10036 | An integer is in the integer range from zero to one iff it is either zero or one. (Contributed by Jim Kingdon, 11-Nov-2021.) |
Finite intervals of nonnegative integers (or "finite sets of sequential nonnegative integers") are finite intervals of integers with 0 as lower bound: , usually abbreviated by "fz0". | ||
Theorem | elfz2nn0 10037 | Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fznn0 10038 | Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.) |
Theorem | elfznn0 10039 | A member of a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | elfz3nn0 10040 | The upper bound of a nonempty finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | fz0ssnn0 10041 | Finite sets of sequential nonnegative integers starting with 0 are subsets of NN0. (Contributed by JJ, 1-Jun-2021.) |
Theorem | fz1ssfz0 10042 | Subset relationship for finite sets of sequential integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Theorem | 0elfz 10043 | 0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018.) |
Theorem | nn0fz0 10044 | A nonnegative integer is always part of the finite set of sequential nonnegative integers with this integer as upper bound. (Contributed by Scott Fenton, 21-Mar-2018.) |
Theorem | elfz0add 10045 | An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.) |
Theorem | fz0sn 10046 | An integer range from 0 to 0 is a singleton. (Contributed by AV, 18-Apr-2021.) |
Theorem | fz0tp 10047 | An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
Theorem | fz0to3un2pr 10048 | An integer range from 0 to 3 is the union of two unordered pairs. (Contributed by AV, 7-Feb-2021.) |
Theorem | fz0to4untppr 10049 | An integer range from 0 to 4 is the union of a triple and a pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.) |
Theorem | elfz0ubfz0 10050 | An element of a finite set of sequential nonnegative integers is an element of a finite set of sequential nonnegative integers with the upper bound being an element of the finite set of sequential nonnegative integers with the same lower bound as for the first interval and the element under consideration as upper bound. (Contributed by Alexander van der Vekens, 3-Apr-2018.) |
Theorem | elfz0fzfz0 10051 | A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.) |
Theorem | fz0fzelfz0 10052 | If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.) |
Theorem | fznn0sub2 10053 | Subtraction closure for a member of a finite set of sequential nonnegative integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Theorem | uzsubfz0 10054 | Membership of an integer greater than L decreased by L in a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
Theorem | fz0fzdiffz0 10055 | The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.) |
Theorem | elfzmlbm 10056 | Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.) |
Theorem | elfzmlbp 10057 | Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) |
Theorem | fzctr 10058 | Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.) |
Theorem | difelfzle 10059 | The difference of two integers from a finite set of sequential nonnegative integers is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.) |
Theorem | difelfznle 10060 | The difference of two integers from a finite set of sequential nonnegative integers increased by the upper bound is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.) |
Theorem | nn0split 10061 | Express the set of nonnegative integers as the disjoint (see nn0disj 10063) union of the first values and the rest. (Contributed by AV, 8-Nov-2019.) |
Theorem | nnsplit 10062 | Express the set of positive integers as the disjoint union of the first values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Theorem | nn0disj 10063 | The first elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.) |
Theorem | 1fv 10064 | A function on a singleton. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
Theorem | 4fvwrd4 10065* | The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.) |
Theorem | 2ffzeq 10066* | Two functions over 0 based finite set of sequential integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 30-Jun-2018.) |
Syntax | cfzo 10067 | Syntax for half-open integer ranges. |
..^ | ||
Definition | df-fzo 10068* | Define a function generating sets of integers using a half-open range. Read ..^ as the integers from up to, but not including, ; contrast with df-fz 9936, which includes . Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 10102 with fzsplit 9976, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | fzof 10069 | Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | elfzoel1 10070 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | elfzoel2 10071 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | elfzoelz 10072 | Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | fzoval 10073 | Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
..^ | ||
Theorem | elfzo 10074 | Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | elfzo2 10075 | Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | elfzouz 10076 | Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | fzodcel 10077 | Decidability of membership in a half-open integer interval. (Contributed by Jim Kingdon, 25-Aug-2022.) |
DECID ..^ | ||
Theorem | fzolb 10078 | The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with . This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate . (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | fzolb2 10079 | The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with . This provides an alternate notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate . (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | elfzole1 10080 | 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.) |
..^ | ||
Theorem | elfzolt2 10081 | A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | elfzolt3 10082 | Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
..^ | ||
Theorem | elfzolt2b 10083 | A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ ..^ | ||
Theorem | elfzolt3b 10084 | Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.) |
..^ ..^ | ||
Theorem | fzonel 10085 | A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
..^ | ||
Theorem | elfzouz2 10086 | 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.) |
..^ | ||
Theorem | elfzofz 10087 | A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
..^ | ||
Theorem | elfzo3 10088 | 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.) |
..^ ..^ | ||
Theorem | fzom 10089* | A half-open integer interval is inhabited iff it contains its left endpoint. (Contributed by Jim Kingdon, 20-Apr-2020.) |
..^ ..^ | ||
Theorem | fzossfz 10090 | 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.) |
..^ | ||
Theorem | fzon 10091 | 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.) |
..^ | ||
Theorem | fzonlt0 10092 | A half-open integer range is empty if the bounds are equal or reversed. (Contributed by AV, 20-Oct-2018.) |
..^ | ||
Theorem | fzo0 10093 | Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
..^ | ||
Theorem | fzonnsub 10094 | If then is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.) |
..^ | ||
Theorem | fzonnsub2 10095 | If then is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.) |
..^ | ||
Theorem | fzoss1 10096 | Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
..^ ..^ | ||
Theorem | fzoss2 10097 | Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
..^ ..^ | ||
Theorem | fzossrbm1 10098 | Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.) |
..^ ..^ | ||
Theorem | fzo0ss1 10099 | Subset relationship for half-open integer ranges with lower bounds 0 and 1. (Contributed by Alexander van der Vekens, 18-Mar-2018.) |
..^ ..^ | ||
Theorem | fzossnn0 10100 | 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.) |
..^ |
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