Home Intuitionistic Logic ExplorerTheorem List (p. 90 of 102) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempeano2fzr 8901 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)

Theoremfzm 8902* Properties of a finite interval of integers which is inhabited. (Contributed by Jim Kingdon, 15-Apr-2020.)

Theoremfztri3or 8903 Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)

Theoremfzdcel 8904 Decidability of membership in a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
DECID

Theoremfznlem 8905 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)

Theoremfzn 8906 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)

Theoremfzen 8907 A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremfz1n 8908 A 1-based finite set of sequential integers is empty iff it ends at index . (Contributed by Paul Chapman, 22-Jun-2011.)

Theorem0fz1 8909 Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremfz10 8910 There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremuzsubsubfz 8911 Membership of an integer greater than L decreased by ( L - M ) in an M based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)

Theoremuzsubsubfz1 8912 Membership of an integer greater than L decreased by ( L - 1 ) in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)

Theoremige3m2fz 8913 Membership of an integer greater than 2 decreased by 2 in a 1 based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018.)

Theoremfzsplit2 8914 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremfzsplit 8915 Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)

Theoremfzdisj 8916 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremfz01en 8917 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremelfznn 8918 A member of a finite set of sequential integers starting at 1 is a positive integer. (Contributed by NM, 24-Aug-2005.)

Theoremelfz1end 8919 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Theoremfznn0sub 8920 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.)

Theoremfzmmmeqm 8921 Subtracting the difference of a member of a finite range of integers and the lower bound of the range from the difference of the upper bound and the lower bound of the range results in the difference of the upper bound of the range and the member. (Contributed by Alexander van der Vekens, 27-May-2018.)

Theoremfzaddel 8922 Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)

Theoremfzsubel 8923 Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)

Theoremfzopth 8924 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 8925 Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Theoremfzss1 8926 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfzss2 8927 Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfzssuz 8928 A finite set of sequential integers is a subset of an upper set of integers. (Contributed by NM, 28-Oct-2005.)

Theoremfzsn 8929 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfzssp1 8930 Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzsuc 8931 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.)

Theoremfzpred 8932 Join a predecessor to the beginning of a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)

Theoremfzpreddisj 8933 A finite set of sequential integers is disjoint with its predecessor. (Contributed by AV, 24-Aug-2019.)

Theoremelfzp1 8934 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 8935 Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzelp1 8936 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 8937 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.)

Theoremfznatpl1 8938 Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.)

Theoremfzpr 8939 A finite interval of integers with two elements. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfztp 8940 A finite interval of integers with three elements. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 7-Mar-2014.)

Theoremfzsuc2 8941 Join a successor to the end of a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Mar-2014.)

Theoremfzp1disj 8942 is the disjoint union of with . (Contributed by Mario Carneiro, 7-Mar-2014.)

Theoremfzdifsuc 8943 Remove a successor from the end of a finite set of sequential integers. (Contributed by AV, 4-Sep-2019.)

Theoremfzprval 8944* Two ways of defining the first two values of a sequence on . (Contributed by NM, 5-Sep-2011.)

Theoremfztpval 8945* Two ways of defining the first three values of a sequence on . (Contributed by NM, 13-Sep-2011.)

Theoremfzrev 8946 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev2 8947 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev2i 8948 Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrev3 8949 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfzrev3i 8950 The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfznn 8951 Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011.) (Revised by Mario Carneiro, 18-Jun-2015.)

Theoremelfz1b 8952 Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.)

Theoremelfzm11 8953 Membership in a finite set of sequential integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremuzsplit 8954 Express an upper integer set as the disjoint (see uzdisj 8955) union of the first values and the rest. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theoremuzdisj 8955 The first elements of an upper integer set are distinct from any later members. (Contributed by Mario Carneiro, 24-Apr-2014.)

Theoremfseq1p1m1 8956 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 8957 Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremfz1sbc 8958* Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.)

Theoremelfzp1b 8959 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.)

Theoremelfzm1b 8960 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 8961 Options for membership in a finite interval of integers. (Contributed by Jeff Madsen, 18-Jun-2010.)

Theoremfzm1 8962 Choices for an element of a finite interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfzneuz 8963 No finite set of sequential integers equals an upper set of integers. (Contributed by NM, 11-Dec-2005.)

Theoremfznuz 8964 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 30-Jun-2013.) (Revised by Mario Carneiro, 24-Aug-2013.)

Theoremuznfz 8965 Disjointness of the upper integers and a finite sequence. (Contributed by Mario Carneiro, 24-Aug-2013.)

Theoremfzp1nel 8966 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.)

Theoremfzrevral 8967* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrevral2 8968* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 25-Nov-2005.)

Theoremfzrevral3 8969* Reversal of scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 20-Nov-2005.)

Theoremfzshftral 8970* Shift the scanning order inside of a quantification over a finite set of sequential integers. (Contributed by NM, 27-Nov-2005.)

Theoremige2m1fz1 8971 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.)

Theoremige2m1fz 8972 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.)

3.5.5  Finite intervals of nonnegative integers

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".

Theoremelfz2nn0 8973 Membership in a finite set of sequential nonnegative integers. (Contributed by NM, 16-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfznn0 8974 Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.)

Theoremelfznn0 8975 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.)

Theoremelfz3nn0 8976 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.)

Theorem0elfz 8977 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.)

Theoremnn0fz0 8978 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.)

Theoremelfz0add 8979 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.)

Theoremelfz0addOLD 8980 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.) Obsolete version of elfz0add 8979 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremfz0tp 8981 An integer range from 0 to 2 is an unordered triple. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremelfz0ubfz0 8982 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.)

Theoremelfz0fzfz0 8983 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.)

Theoremfz0fzelfz0 8984 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.)

Theoremfznn0sub2 8985 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.)

Theoremuzsubfz0 8986 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.)

Theoremfz0fzdiffz0 8987 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.)

Theoremelfzmlbm 8988 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.)

TheoremelfzmlbmOLD 8989 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.) Obsolete version of elfzmlbm 8988 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremelfzmlbp 8990 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.)

Theoremfzctr 8991 Lemma for theorems about the central binomial coefficient. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 2-Aug-2014.)

Theoremdifelfzle 8992 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.)

Theoremdifelfznle 8993 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.)

Theoremnn0split 8994 Express the set of nonnegative integers as the disjoint (see nn0disj 8995) union of the first values and the rest. (Contributed by AV, 8-Nov-2019.)

Theoremnn0disj 8995 The first elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)

Theorem1fv 8996 A one value function. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Theorem4fvwrd4 8997* The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)

Theorem2ffzeq 8998* 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.)

3.5.6  Half-open integer ranges

Syntaxcfzo 8999 Syntax for half-open integer ranges.
..^

Definitiondf-fzo 9000* 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 8875, which includes . Not including the endpoint simplifies a number of formulae related to cardinality and splitting; contrast fzosplit 9033 with fzsplit 8915, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.)
..^

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10124
 Copyright terms: Public domain < Previous  Next >