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

Theoremelioomnf 11001 Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)

Theoremelicopnf 11002 Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremrepos 11003 Two ways of saying that a real number is positive. (Contributed by NM, 7-May-2007.)

Theoremioof 11004 The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremiccf 11005 The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremunirnioo 11006 The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)

Theoremdfioo2 11007* Alternate definition of the set of open intervals of extended reals. (Contributed by NM, 1-Mar-2007.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremioorebas 11008 Open intervals are elements of the set of all open intervals. (Contributed by Mario Carneiro, 26-Mar-2015.)

Theoremelrege0 11009 The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Theoremelxrge0 11010 Elementhood in the set of nonnegative extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)

Theoremge0addcl 11011 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremge0mulcl 11012 The nonnegative reals are closed under multiplication. (Contributed by Mario Carneiro, 19-Jun-2014.)

Theoremge0xaddcl 11013 The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremge0xmulcl 11014 The nonnegative extended reals are closed under multiplication. (Contributed by Mario Carneiro, 26-Aug-2015.)

Theoremlbicc2 11015 The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)

Theoremubicc2 11016 The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.)

Theorem0elunit 11017 Zero is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)

Theorem1elunit 11018 One is an element of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremiooneg 11019 Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremiccneg 11020 Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)

Theoremicoshft 11021 A shifted real is a member of a shifted, closed-below, open-above real interval. (Contributed by Paul Chapman, 25-Mar-2008.)

Theoremicoshftf1o 11022* Shifting a closed-below, open-above interval is one-to-one onto. (Contributed by Paul Chapman, 25-Mar-2008.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Theoremicoun 11023 The union of end-to-end closed-below, open-above real intervals. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Theoremicodisj 11024 End-to-end closed-below, open-above real intervals are disjoint. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremsnunioo 11025 The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Theoremsnunico 11026 The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014.)

Theoremprunioo 11027 The closure of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Theoremioodisj 11028 If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.)

Theoremioojoin 11029 Join two open intervals to create a third. (Contributed by NM, 11-Aug-2008.) (Proof shortened by Mario Carneiro, 16-Jun-2014.)

Theoremdifreicc 11030 The class difference of and a closed interval. (Contributed by FL, 18-Jun-2007.)

Theoremiccsplit 11031 Split a closed interval into the union of two closed intervals. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftr 11032 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftri 11033 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftl 11034 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccshftli 11035 Membership in a shifted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccdil 11036 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiccdili 11037 Membership in a dilated interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremicccntr 11038 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremicccntri 11039 Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremlincmb01cmp 11040 A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.)

Theoremiccf1o 11041* Describe a bijection from to an arbitrary nontrivial closed interval . (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremiccen 11042 Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremxov1plusxeqvd 11043 A complex number is positive real iff is in . Deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theoremunitssre 11044 is a subset of the reals. (Contributed by David Moews, 28-Feb-2017.)

5.5.5  Finite intervals of integers

Syntaxcfz 11045 Extend class notation to include the notation for a contiguous finite set of integers. Read " " as "the set of integers from to inclusive."

Definitiondf-fz 11046* Define an operation that produces a finite set of sequential integers. Read " " as "the set of integers from to inclusive." See fzval 11047 for its value and additional comments. (Contributed by NM, 6-Sep-2005.)

Theoremfzval 11047* 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 11048 An alternative way of expressing a finite set of sequential integers. (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremfzf 11049 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 11050 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.)

Theoremelfz 11051 Membership in a finite set of sequential integers. (Contributed by NM, 29-Sep-2005.)

Theoremelfz2 11052 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 11053 Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005.)

Theoremelfz4 11054 Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzuzb 11055 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 11056 Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzuz 11057 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 11058 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 11059 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 11060 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 11061 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 11062 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 11063 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 11064 Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremelfzle3 11065 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 11066 Membership in a finite set of sequential integers - special case. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremeluzfz2 11067 Membership in a finite set of sequential integers - special case. (Contributed by NM, 13-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremeluzfz2b 11068 Membership in a finite set of sequential integers - special case. (Contributed by NM, 14-Sep-2005.)

Theoremelfz3 11069 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 21-Jul-2005.)

Theoremelfz1eq 11070 Membership in a finite set of sequential integers containing one integer. (Contributed by NM, 19-Sep-2005.)

Theorempeano2fzr 11071 A Peano-postulate-like theorem for downward closure of a finite set of sequential integers. (Contributed by Mario Carneiro, 27-May-2014.)

Theoremfzn0 11072 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 11073 A finite set of sequential integers is empty if the bounds are reversed. (Contributed by NM, 22-Aug-2005.)

Theoremfzen 11074 A shifted finite set of sequential integers is equinumerous to the original set. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremfz1n 11075 A 1-based finite set of sequential integers is empty iff it ends at index . (Contributed by Paul Chapman, 22-Jun-2011.)

Theorem0fz1 11076 Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremfz10 11077 There are no integers between 1 and 0. (Contributed by Jeff Madsen, 16-Jun-2010.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfzsplit2 11078 Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremfzsplit 11079 Split a finite interval of integers into two parts. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 13-Apr-2016.)

Theoremfzdisj 11080 Condition for two finite intervals of integers to be disjoint. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremfz01en 11081 0-based and 1-based finite sets of sequential integers are equinumerous. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremelfznn 11082 A member of a finite set of sequential integers starting at 1 is a natural number. (Contributed by NM, 24-Aug-2005.)

Theoremelfz1end 11083 A nonempty finite range of integers contains its end point. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Theoremelfz2nn0 11084 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 11085 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 11086 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 11087 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 11088 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 11089 Membership of a sum in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)

Theoremfzsubel 11090 Membership of a difference in a finite set of sequential integers. (Contributed by NM, 30-Jul-2005.)

Theoremfzopth 11091 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 11092 Two ways to express a nondecreasing sequence of four integers. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Theoremfzss1 11093 Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfzss2 11094 Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfzssuz 11095 A finite set of sequential integers is a subset of a set of upper integers. (Contributed by NM, 28-Oct-2005.)

Theoremfzsn 11096 A finite interval of integers with one element. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfzssp1 11097 Subset relationship for finite sets of sequential integers. (Contributed by NM, 21-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfzsuc 11098 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 11099 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 11100 Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

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