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Theorem List for Metamath Proof Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelnnz 10001 Natural number property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)

Theorem0z 10002 Zero is an integer. (Contributed by NM, 12-Jan-2002.)

Theoremelnn0z 10003 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)

Theoremelznn0nn 10004 Integer property expressed in terms nonnegative integers and natural numbers. (Contributed by NM, 10-May-2004.)

Theoremelznn0 10005 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremelznn 10006 Integer property expressed in terms natural numbers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)

Theoremelz2 10007* Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the natural numbers. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremdfz2 10008 Alternative definition of the integers, based on elz2 10007. (Contributed by Mario Carneiro, 16-May-2014.)

TheoremzexALT 10009 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-May-2014.) (Proof modification is discouraged.)

Theoremnnssz 10010 Natural numbers are a subset of integers. (Contributed by NM, 9-Jan-2002.)

Theoremnn0ssz 10011 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)

Theoremnnz 10012 A natural number is an integer. (Contributed by NM, 9-May-2004.)

Theoremnn0z 10013 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)

Theoremnnzi 10014 A natural number is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnn0zi 10015 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremelnnz1 10016 Natural number property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremznnnlt1 10017 An integer is not a natural number iff it is less than one. (Contributed by NM, 13-Jul-2005.)

Theoremnnzrab 10018 Natural numbers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)

Theoremnn0zrab 10019 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)

Theorem1z 10020 One is an integer. (Contributed by NM, 10-May-2004.)

Theorem2z 10021 Two is an integer. (Contributed by NM, 10-May-2004.)

Theoremznegcl 10022 Closure law for negative integers. (Contributed by NM, 9-May-2004.)

Theoremznegclb 10023 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremnn0negz 10024 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)

Theoremnn0negzi 10025 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremzaddcl 10026 Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theorempeano2z 10027 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)

Theoremzsubcl 10028 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)

Theorempeano2zm 10029 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)

Theoremzrevaddcl 10030 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)

Theoremznnsub 10031 The positive difference of unequal integers is a natural number. (Generalization of nnsub 9752.) (Contributed by NM, 11-May-2004.)

Theoremznn0sub 10032 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9981.) (Contributed by NM, 14-Jul-2005.)

Theoremzmulcl 10033 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)

Theoremzltp1le 10034 Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremzleltp1 10035 Integer ordering relation. (Contributed by NM, 10-May-2004.)

Theoremzlem1lt 10036 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremzltlem1 10037 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremnnleltp1 10038 Natural number ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnnltp1le 10039 Natural number ordering relation. (Contributed by NM, 19-Aug-2001.)

Theoremnnaddm1cl 10040 Closure of addition of natural numbers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0ltp1le 10041 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0leltp1 10042 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremnn0ltlem1 10043 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0sub2 10044 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)

Theoremnn0lt10b 10045 A nonnegative integer less than is . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremnn0lem1lt 10046 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnlem1lt 10047 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnltlem1 10048 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremzdiv 10049* Two ways to express " divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivadd 10050 Property of divisibility: if divides and then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivmul 10051 Property of divisibility: if divides then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzextle 10052* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremzextlt 10053* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremrecnz 10054 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)

Theorembtwnnz 10055 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)

Theoremgtndiv 10056 A larger number does not divide a smaller natural number. (Contributed by NM, 3-May-2005.)

Theoremhalfnz 10057 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)

Theoremsuprzcl 10058* The supremum of a bounded-above set of integers is a member of the set. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremprime 10059* Two ways to express " is a prime number (or 1)." See also isprm 12722. (Contributed by NM, 4-May-2005.)

Theoremmsqznn 10060 The square of a nonzero integer is a natural number. (Contributed by NM, 2-Aug-2004.)

Theoremzneo 10061 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.)

Theoremnneo 10062 A natural number is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)

Theoremnneoi 10063 A natural number is even or odd but not both. (Contributed by NM, 20-Aug-2001.)

Theoremzeo 10064 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)

Theoremzeo2 10065 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theorempeano2uz2 10066* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)

Theorempeano5uzi 10067* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)

Theorempeano5uzti 10068* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)

Theoremdfuzi 10069* An expression for the upper integers that start at that is analogous to df-n 9715 for natural numbers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)

Theoremuzind 10070* 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 hypothesis. (Contributed by NM, 5-Jul-2005.)

Theoremuzind2 10071* 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 hypothesis. (Contributed by NM, 25-Jul-2005.)

Theoremuzind3 10072* 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 hypothesis. (Contributed by NM, 26-Jul-2005.)

TheoremuzindOLD 10073* Induction on the upper integers that start at an integer . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis.

Warning: The HTML proof page is 3/4 megabyte in size. An attempt to shorten it is on my to-do list. Anyone is welcome to try. (Contributed by NM, 11-May-2004.) (New usage is discouraged.)

Theoremuzind3OLD 10074* Induction on the set of upper integers that starts at . The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 9-Nov-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnn0ind 10075* 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 hypothesis. (Contributed by NM, 13-May-2004.)

Theoremfzind 10076* 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. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremfnn0ind 10077* 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. (Contributed by Paul Chapman, 31-Mar-2011.)

Theoremnn0indALT 10078* 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 hypothesis. (Contributed by NM, 28-Nov-2005.)

Theoremnn0ind-raph 10079* 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 hypothesis. 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 10080* 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 10081* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)

Theoremnn0zd 10082 A natural number is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnzd 10083 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzred 10084 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzcnd 10085 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremznegcld 10086 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theorempeano2zd 10087 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzaddcld 10088 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzsubcld 10089 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzmulcld 10090 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)

5.4.8  Decimal arithmetic

Syntaxcdc 10091 Constant used for decimal constructor.
;

Definitiondf-dec 10092 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, ;;; ;;; ;;; 1kp2ke3k 20776. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdecex 10093 A decimal number is a set. (Contributed by Mario Carneiro, 17-Apr-2015.)
;

Theoremdeceq1 10094 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2 10095 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq1i 10096 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq2i 10097 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremdeceq12i 10098 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; ;

Theoremnumnncl 10099 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnum0u 10100 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)

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