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Theorem List for Metamath Proof Explorer - 11001-11100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfzval 11001* The value of a finite set of sequential integers. E.g., means the set . A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where _k means our ; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremfzval2 11002 An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremfzf 11003 Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremelfz1 11004 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)

Theoremelfz 11005 Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)

Theoremelfz2 11006 Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show and . (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfz5 11007 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)

Theoremelfz4 11008 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzuzb 11009 Membership in a finite set of sequential integers in terms of sets of upper integers. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremeluzfz 11010 Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzuz 11011 A member of a finite set of sequential integers belongs to a set of upper integers. (Contributed by NM, 17-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzuz3 11012 Membership in a finite set of sequential integers implies membership in a set of upper integers. (Contributed by NM, 28-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzel2 11013 Membership in a finite set of sequential integer implies the upper bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzel1 11014 Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzelz 11015 A member of a finite set of sequential integer is an integer. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzle1 11016 A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzle2 11017 A member of a finite set of sequential integer is less than or equal to the upper bound. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzuz2 11018 Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzle3 11019 Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremeluzfz1 11020 Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremeluzfz2 11021 Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremeluzfz2b 11022 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)

Theoremelfz3 11023 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)

Theoremelfz1eq 11024 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)

Theorempeano2fzr 11025 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)

Theoremfzn0 11026 Properties of a finite interval of integers which is non-empty. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzn 11027 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)

Theoremfzen 11028 A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremfz1n 11029 A 1-based finite set of sequential integers is empty iff it ends at index . (Contributed by Paul Chapman, 22-Jun-2011.)

Theorem0fz1 11030 Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremfz10 11031 There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfzsplit2 11032 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremfzsplit 11033 Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)

Theoremfzdisj 11034 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremfz01en 11035 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremelfznn 11036 A member of a finite set of sequential integers starting at 1 is a natural number. (Contributed by NM, 24-Aug-2005.)

Theoremelfz1end 11037 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Theoremelfz2nn0 11038 Membership in a finite set of sequential integers starting at 0. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfznn0 11039 A member of a finite set of sequential integers starting at 0 is a nonnegative integer. (Contributed by NM, 5-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfz3nn0 11040 The upper bound of a nonempty finite set of sequential integers starting at 0 is a nonnegative integer. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfznn0sub 11041 Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfznn0sub2 11042 Subtraction closure for a member of a finite set of sequential integers. (Contributed by NM, 26-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzaddel 11043 Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)

Theoremfzsubel 11044 Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)

Theoremfzopth 11045 A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzass4 11046 Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Theoremfzss1 11047 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfzss2 11048 Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfzssuz 11049 A finite set of sequential integers is a subset of a set of upper integers. (Contributed by NM, 28-Oct-2005.)

Theoremfzsn 11050 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfzssp1 11051 Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzsuc 11052 Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzp1 11053 Append an element to a finite set of sequential integers. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfzp1ss 11054 Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzelp1 11055 Membership in a set of sequential integers with an appended element. (Contributed by NM, 7-Dec-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzp1elp1 11056 Add one to an element of a finite set of integers. (Contributed by Jeff Madsen, 6-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzpr 11057 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfztp 11058 A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)

Theoremfz0tp 11059 An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremfzsuc2 11060 Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)

Theoremfzp1disj 11061 is the disjoint union of with . (Contributed by Mario Carneiro, 7-Mar-2014.)

Theoremfzprval 11062* Two ways of defining the first two values of a sequence on . (Contributed by NM, 5-Sep-2011.)

Theoremfztpval 11063* Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.)

Theoremfzrev 11064 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev2 11065 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev2i 11066 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev3 11067 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfzrev3i 11068 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfznn0 11069 Finite set of sequential integers starting at 0. (Contributed by NM, 1-Aug-2005.)

Theoremfznn 11070 Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)

Theoremelfzm11 11071 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremfzctr 11072 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)

Theoremuzsplit 11073 Express an upper integer set as the disjoint (see uzdisj 11074) union of the first values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theoremuzdisj 11074 The first elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theorem1fv 11075 A one value function. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Theorem4fvwrd4 11076* The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)

Theoremfseq1p1m1 11077 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.)

Theoremfseq1m1p1 11078 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremfz1sbc 11079* Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)

Theoremelfzm1b 11080 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.)

Theoremelfzp12 11081 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)

Theoremfzm1 11082 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfzneuz 11083 No finite set of sequential integers equals a set of upper integers. (Contributed by NM, 11-Dec-2005.)

Theoremfznuz 11084 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)

Theoremuznfz 11085 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)

Theoremfzrevral 11086* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrevral2 11087* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrevral3 11088* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfzshftral 11089* Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)

5.5.6  Half-open integer ranges

Syntaxcfzo 11090 Syntax for half-open integer ranges.
..^

Definitiondf-fzo 11091* 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 11000, which includes . Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 11121 with fzsplit 11033, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremfzof 11092 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoel1 11093 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoel2 11094 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoelz 11095 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremfzoval 11096 Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzo 11097 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzo2 11098 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzouz 11099 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzolb 11100 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.)
..^

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