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Theorem List for Metamath Proof Explorer - 10701-10800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelfz3nn0 10701 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 10702 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 10703 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 10704 Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)

Theoremfzsubel 10705 Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)

Theoremfzopth 10706 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 10707 Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Theoremfzss1 10708 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfzss2 10709 Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfzssuz 10710 A finite set of sequential integers is a subset of a set of upper integers. (Contributed by NM, 28-Oct-2005.)

Theoremfzsn 10711 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfzssp1 10712 Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzsuc 10713 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 10714 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 10715 Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzelp1 10716 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 10717 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 10718 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfztp 10719 A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)

Theoremfzsuc2 10720 Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)

Theoremfzp1disj 10721 is the disjoint union of with . (Contributed by Mario Carneiro, 7-Mar-2014.)

Theoremfzprval 10722* Two ways of defining the first two values of a sequence on . (Contributed by NM, 5-Sep-2011.)

Theoremfztpval 10723* Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.)

Theoremfzrev 10724 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev2 10725 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev2i 10726 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev3 10727 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfzrev3i 10728 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfznn0 10729 Finite set of sequential integers starting at 0. (Contributed by NM, 1-Aug-2005.)

Theoremfznn 10730 Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)

Theoremelfzm11 10731 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremfzctr 10732 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)

Theoremuzsplit 10733 Express an upper integer set as the disjoint (see uzdisj 10734) union of the first values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theoremuzdisj 10734 The first elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theoremfseq1p1m1 10735 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 10736 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremfz1sbc 10737* Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)

Theoremelfzm1b 10738 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 10739 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)

Theoremfzm1 10740 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfzneuz 10741 No finite set of sequential integers equals a set of upper integers. (Contributed by NM, 11-Dec-2005.)

Theoremfznuz 10742 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)

Theoremuznfz 10743 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)

Theoremfzrevral 10744* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrevral2 10745* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrevral3 10746* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfzshftral 10747* 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 10748 Syntax for half-open integer ranges.
..^

Definitiondf-fzo 10749* 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 10661, which includes . Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 10777 with fzsplit 10694, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremfzof 10750 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoel1 10751 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoel2 10752 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoelz 10753 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremfzoval 10754 Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzo 10755 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzo2 10756 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzouz 10757 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzolb 10758 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 10759 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 10760 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.)
..^

Theoremelfzolt2 10761 A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzolt3 10762 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzolt2b 10763 A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremelfzolt3b 10764 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzonel 10765 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
..^

Theoremelfzouz2 10766 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 10767 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^

Theoremelfzo3 10768 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 10769 A half-open integer interval is nonempty iff it contains its left endpoint. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzossfz 10770 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.)
..^

Theoremfzo0 10771 Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzonnsub 10772 If then is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
..^

Theoremfzonnsub2 10773 If then is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
..^

Theoremfzoss1 10774 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzoss2 10775 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzospliti 10776 One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ ..^ ..^

Theoremfzosplit 10777 Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ ..^ ..^

Theoremfzodisj 10778 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ ..^

Theoremfzouzsplit 10779 Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
..^

Theoremfzouzdisj 10780 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 10781 An integer is strictly greater than zero iff it is a member of . (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzo0 10782 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.)
..^

Theoremfzo0n0 10783 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 10784 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzoaddel2 10785 Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzosubel 10786 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzosubel2 10787 Membership in a translated half-open integer range implies translated membership in the original range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzosubel3 10788 Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzval3 10789 Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremfzosn 10790 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^

Theoremfzo01 10791 Expressing the singleton of as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremfzoend 10792 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzo0end 10793 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 10794 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 10795 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 10796 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.)
..^ ..^

Theorempeano2fzor 10797 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 1-Oct-2015.)
..^ ..^

Theoremfzosplitsn 10798 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzosplitsni 10799 Membership in a half-open range extende by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzostep1 10800 Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

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