Home Intuitionistic Logic ExplorerTheorem List (p. 91 of 130) < 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 - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremztri3or 9001 Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)

Theoremzletric 9002 Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)

Theoremzlelttric 9003 Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)

Theoremzltnle 9004 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)

Theoremzleloe 9005 Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)

Theoremznnnlt1 9006 An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.)

Theoremzletr 9007 Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)

Theoremzrevaddcl 9008 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)

Theoremznnsub 9009 The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8669.) (Contributed by NM, 11-May-2004.)

Theoremnzadd 9010 The sum of a real number not being an integer and an integer is not an integer. Note that "not being an integer" in this case means "the negation of is an integer" rather than "is apart from any integer" (given excluded middle, those two would be equivalent). (Contributed by AV, 19-Jul-2021.)

Theoremzmulcl 9011 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)

Theoremzltp1le 9012 Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremzleltp1 9013 Integer ordering relation. (Contributed by NM, 10-May-2004.)

Theoremzlem1lt 9014 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremzltlem1 9015 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremzgt0ge1 9016 An integer greater than is greater than or equal to . (Contributed by AV, 14-Oct-2018.)

Theoremnnleltp1 9017 Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnnltp1le 9018 Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)

Theoremnnaddm1cl 9019 Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0ltp1le 9020 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0leltp1 9021 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremnn0ltlem1 9022 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremznn0sub 9023 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9024.) (Contributed by NM, 14-Jul-2005.)

Theoremnn0sub 9024 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremnn0n0n1ge2 9025 A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)

Theoremelz2 9026* Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the positive integers. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremdfz2 9027 Alternate definition of the integers, based on elz2 9026. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremnn0sub2 9028 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)

Theoremzapne 9029 Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
#

Theoremzdceq 9030 Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)
DECID

Theoremzdcle 9031 Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
DECID

Theoremzdclt 9032 Integer is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
DECID

Theoremzltlen 9033 Integer 'Less than' expressed in terms of 'less than or equal to'. Also see ltleap 8311 which is a similar result for real numbers. (Contributed by Jim Kingdon, 14-Mar-2020.)

Theoremnn0n0n1ge2b 9034 A nonnegative integer is neither 0 nor 1 if and only if it is greater than or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)

Theoremnn0lt10b 9035 A nonnegative integer less than is . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremnn0lt2 9036 A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.)

Theoremnn0le2is012 9037 A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)

Theoremnn0lem1lt 9038 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnlem1lt 9039 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnltlem1 9040 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnm1ge0 9041 A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)

Theoremnn0ge0div 9042 Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)

Theoremzdiv 9043* Two ways to express " divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivadd 9044 Property of divisibility: if divides and then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivmul 9045 Property of divisibility: if divides then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzextle 9046* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremzextlt 9047* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremrecnz 9048 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)

Theorembtwnnz 9049 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)

Theoremgtndiv 9050 A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)

Theoremhalfnz 9051 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)

Theorem3halfnz 9052 Three halves is not an integer. (Contributed by AV, 2-Jun-2020.)

Theoremsuprzclex 9053* The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)

Theoremprime 9054* Two ways to express " is a prime number (or 1)." (Contributed by NM, 4-May-2005.)

Theoremmsqznn 9055 The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)

Theoremzneo 9056 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)

Theoremnneoor 9057 A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)

Theoremnneo 9058 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)

Theoremnneoi 9059 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)

Theoremzeo 9060 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)

Theoremzeo2 9061 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theorempeano2uz2 9062* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)

Theorempeano5uzti 9063* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)

Theorempeano5uzi 9064* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)

Theoremdfuzi 9065* An expression for the upper integers that start at that is analogous to dfnn2 8632 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)

Theoremuzind 9066* Induction on the upper integers that start at . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 5-Jul-2005.)

Theoremuzind2 9067* Induction on the upper integers that start after an integer . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.)

Theoremuzind3 9068* Induction on the upper integers that start at an integer . The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 26-Jul-2005.)

Theoremnn0ind 9069* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 13-May-2004.)

Theoremfzind 9070* 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 step. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremfnn0ind 9071* 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 step. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremnn0ind-raph 9072* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)

Theoremzindd 9073* Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)

Theorembtwnz 9074* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)

Theoremnn0zd 9075 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnzd 9076 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzred 9077 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzcnd 9078 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremznegcld 9079 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theorempeano2zd 9080 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzaddcld 9081 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzsubcld 9082 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzmulcld 9083 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzadd2cl 9084 Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)

Theorembtwnapz 9085 A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
#

4.4.10  Decimal arithmetic

Syntaxcdc 9086 Constant used for decimal constructor.
;

Definitiondf-dec 9087 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base . For example, ;;; ;;; ;;; 1kp2ke3k 12629. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 1-Aug-2021.)
;

Theorem9p1e10 9088 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
;

Theoremdfdec10 9089 Version of the definition of the "decimal constructor" using ; instead of the symbol 10. Of course, this statement cannot be used as definition, because it uses the "decimal constructor". (Contributed by AV, 1-Aug-2021.)
; ;

Theoremdeceq1 9090 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdeceq2 9091 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
; ;

Theoremdeceq1i 9092 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2i 9093 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq12i 9094 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremnumnncl 9095 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0u 9096 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0h 9097 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumcl 9098 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnumsuc 9099 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdeccl 9100 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
;

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 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-12939
 Copyright terms: Public domain < Previous  Next >