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

Theoremfzpr 11101 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfztp 11102 A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)

Theoremfz0tp 11103 An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremfzsuc2 11104 Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)

Theoremfzp1disj 11105 is the disjoint union of with . (Contributed by Mario Carneiro, 7-Mar-2014.)

Theoremfzprval 11106* Two ways of defining the first two values of a sequence on . (Contributed by NM, 5-Sep-2011.)

Theoremfztpval 11107* Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.)

Theoremfzrev 11108 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev2 11109 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev2i 11110 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev3 11111 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfzrev3i 11112 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfznn0 11113 Finite set of sequential integers starting at 0. (Contributed by NM, 1-Aug-2005.)

Theoremnn0fz0 11114 A non-negative integer is always part of its zero-based finite sequence. (Contributed by Scott Fenton, 21-Mar-2018.)

Theoremfznn 11115 Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)

Theoremelfzm11 11116 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremfzctr 11117 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)

Theoremuzsplit 11118 Express an upper integer set as the disjoint (see uzdisj 11119) union of the first values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theoremuzdisj 11119 The first elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theorem1fv 11120 A one value function. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Theorem4fvwrd4 11121* The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)

Theoremfseq1p1m1 11122 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 11123 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremfz1sbc 11124* Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)

Theoremelfzm1b 11125 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 11126 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)

Theoremfzm1 11127 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfzneuz 11128 No finite set of sequential integers equals a set of upper integers. (Contributed by NM, 11-Dec-2005.)

Theoremfznuz 11129 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)

Theoremuznfz 11130 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)

Theoremfzrevral 11131* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrevral2 11132* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrevral3 11133* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfzshftral 11134* 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 11135 Syntax for half-open integer ranges.
..^

Definitiondf-fzo 11136* 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 11044, which includes . Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 11166 with fzsplit 11077, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremfzof 11137 Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoel1 11138 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoel2 11139 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzoelz 11140 Reverse closure for half-open integer sets. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremfzoval 11141 Value of the half-open integer set in terms of the closed integer set. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Theoremelfzo 11142 Membership in a half-open finite set of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzo2 11143 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzouz 11144 Membership in a half-open integer interval. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzolb 11145 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 11146 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 11147 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 11148 A member in a half-open integer interval is less than the upper bound. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzolt3 11149 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremelfzolt2b 11150 A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremelfzolt3b 11151 Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzonel 11152 A half-open range does not contain its right endpoint. (Contributed by Stefan O'Rear, 25-Aug-2015.)
..^

Theoremelfzouz2 11153 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 11154 A half-open range is contained in the corresponding closed range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^

Theoremelfzo3 11155 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 11156 A half-open integer interval is nonempty iff it contains its left endpoint. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzossfz 11157 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 11158 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 11159 Half-open sets with equal endpoints are empty. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^

Theoremfzonnsub 11160 If then is a positive integer. (Contributed by Mario Carneiro, 29-Sep-2015.) (Revised by Mario Carneiro, 1-Jan-2017.)
..^

Theoremfzonnsub2 11161 If then is a positive integer. (Contributed by Mario Carneiro, 1-Jan-2017.)
..^

Theoremfzoss1 11162 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzoss2 11163 Subset relationship for half-open sequences of integers. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzossrbm1 11164 Subset of a half open range. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
..^ ..^

Theoremfzospliti 11165 One direction of splitting a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ ..^ ..^

Theoremfzosplit 11166 Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ ..^ ..^

Theoremfzodisj 11167 Abutting half-open integer ranges are disjoint. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^ ..^

Theoremfzouzsplit 11168 Split an upper integer set into a half-open integer range and another upper integer set. (Contributed by Mario Carneiro, 21-Sep-2016.)
..^

Theoremfzouzdisj 11169 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 11170 An integer is strictly greater than zero iff it is a member of . (Contributed by Mario Carneiro, 29-Sep-2015.)
..^

Theoremelfzo0 11171 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 11172 Half-opened integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
..^

Theoremelfzo1 11173 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
..^

Theoremfzo0n0 11174 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 11175 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzoaddel2 11176 Translate membership in a shifted-down half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzosubel 11177 Translate membership in a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzosubel2 11178 Membership in a translated half-open integer range implies translated membership in the original range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzosubel3 11179 Membership in a translated half-open integer range when the original range is zero-based. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^ ..^

Theoremfzval3 11180 Expressing a closed integer range as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremfzosn 11181 Expressing a singleton as a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^

Theoremfzo01 11182 Expressing the singleton of as a half-open integer range. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremfzo12sn 11183 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 11184 A half-open integer range from 0 to 2 is an unordered pair. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
..^

Theoremfzo0to3tp 11185 A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
..^

Theoremfzo0to42pr 11186 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 11187 The endpoint of a half-open integer range. (Contributed by Mario Carneiro, 29-Sep-2015.)
..^ ..^

Theoremfzo0end 11188 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 11189 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 11190 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 11191 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 11192 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 11193 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 11194 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 1-Oct-2015.)
..^ ..^

Theoremfzosplitsn 11195 Extending a half-open range by a singleton on the end. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzosplitsni 11196 Membership in a half-open range extende by a singleton. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzostep1 11197 Two possibilities for a number one greater than a number in a half-open range. (Contributed by Stefan O'Rear, 23-Aug-2015.)
..^ ..^

Theoremfzind2 11198* 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 10368 using integer range definitions. (Contributed by Mario Carneiro, 6-Feb-2016.)
..^

Theoreminjresinjlem 11199 Lemma for injresinj 11200. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
..^ ..^

Theoreminjresinj 11200 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.)
..^ ..^

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