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Theorem List for Metamath Proof Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnn0addcld 9901 Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0mulcld 9902 Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

5.4.7  Integers (as a subset of complex numbers)

Syntaxcz 9903 Extend class notation to include the class of integers.

Definitiondf-z 9904 Define the set of integers, which are the positive and negative natural numbers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.)

Theoremelz 9905 Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)

Theoremnnnegz 9906 The negative of a natural number is an integer. (Contributed by NM, 12-Jan-2002.)

Theoremzre 9907 An integer is a real. (Contributed by NM, 8-Jan-2002.)

Theoremzcn 9908 An integer is a complex number. (Contributed by NM, 9-May-2004.)

Theoremzrei 9909 An integer is a real number. (Contributed by NM, 14-Jul-2005.)

Theoremzssre 9910 The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)

Theoremzsscn 9911 The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremzex 9912 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremelnnz 9913 Natural number property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)

Theorem0z 9914 Zero is an integer. (Contributed by NM, 12-Jan-2002.)

Theoremelnn0z 9915 Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)

Theoremelznn0nn 9916 Integer property expressed in terms nonnegative integers and natural numbers. (Contributed by NM, 10-May-2004.)

Theoremelznn0 9917 Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremelznn 9918 Integer property expressed in terms natural numbers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)

Theoremelz2 9919* 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 9920 Alternative definition of the integers, based on elz2 9919. (Contributed by Mario Carneiro, 16-May-2014.)

TheoremzexALT 9921 The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 16-May-2014.) (Proof modification is discouraged.)

Theoremnnssz 9922 Natural numbers are a subset of integers. (Contributed by NM, 9-Jan-2002.)

Theoremnn0ssz 9923 Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)

Theoremnnz 9924 A natural number is an integer. (Contributed by NM, 9-May-2004.)

Theoremnn0z 9925 A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)

Theoremnnzi 9926 A natural number is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnn0zi 9927 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremelnnz1 9928 Natural number property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremznnnlt1 9929 An integer is not a natural number iff it is less than one. (Contributed by NM, 13-Jul-2005.)

Theoremnnzrab 9930 Natural numbers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)

Theoremnn0zrab 9931 Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)

Theorem1z 9932 One is an integer. (Contributed by NM, 10-May-2004.)

Theorem2z 9933 Two is an integer. (Contributed by NM, 10-May-2004.)

Theoremznegcl 9934 Closure law for negative integers. (Contributed by NM, 9-May-2004.)

Theoremznegclb 9935 A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremnn0negz 9936 The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)

Theoremnn0negzi 9937 The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremzaddcl 9938 Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theorempeano2z 9939 Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)

Theoremzsubcl 9940 Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)

Theorempeano2zm 9941 "Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)

Theoremzrevaddcl 9942 Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)

Theoremznnsub 9943 The positive difference of unequal integers is a natural number. (Generalization of nnsub 9664.) (Contributed by NM, 11-May-2004.)

Theoremznn0sub 9944 The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9893.) (Contributed by NM, 14-Jul-2005.)

Theoremzmulcl 9945 Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)

Theoremzltp1le 9946 Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremzleltp1 9947 Integer ordering relation. (Contributed by NM, 10-May-2004.)

Theoremzlem1lt 9948 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremzltlem1 9949 Integer ordering relation. (Contributed by NM, 13-Nov-2004.)

Theoremnnleltp1 9950 Natural number ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnnltp1le 9951 Natural number ordering relation. (Contributed by NM, 19-Aug-2001.)

Theoremnnaddm1cl 9952 Closure of addition of natural numbers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0ltp1le 9953 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0leltp1 9954 Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremnn0ltlem1 9955 Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0sub2 9956 Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)

Theoremnn0lt10b 9957 A nonnegative integer less than is . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremnn0lem1lt 9958 Nonnegative integer ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnlem1lt 9959 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremnnltlem1 9960 Natural number ordering relation. (Contributed by NM, 21-Jun-2005.)

Theoremzdiv 9961* Two ways to express " divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivadd 9962 Property of divisibility: if divides and then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzdivmul 9963 Property of divisibility: if divides then it divides . (Contributed by NM, 3-Oct-2008.)

Theoremzextle 9964* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremzextlt 9965* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)

Theoremrecnz 9966 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)

Theorembtwnnz 9967 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)

Theoremgtndiv 9968 A larger number does not divide a smaller natural number. (Contributed by NM, 3-May-2005.)

Theoremhalfnz 9969 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)

Theoremsuprzcl 9970* 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 9971* Two ways to express " is a prime number (or 1)." See also isprm 12634. (Contributed by NM, 4-May-2005.)

Theoremmsqznn 9972 The square of a nonzero integer is a natural number. (Contributed by NM, 2-Aug-2004.)

Theoremzneo 9973 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 9974 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 9975 A natural number is even or odd but not both. (Contributed by NM, 20-Aug-2001.)

Theoremzeo 9976 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)

Theoremzeo2 9977 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theorempeano2uz2 9978* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)

Theorempeano5uzi 9979* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)

Theorempeano5uzti 9980* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)

Theoremdfuzi 9981* An expression for the upper integers that start at that is analogous to df-n 9627 for natural numbers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)

Theoremuzind 9982* 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 9983* 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 9984* 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 9985* 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 9986* 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 9987* 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 9988* 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 9989* 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 9990* 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 9991* 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 9992* 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 9993* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)

Theoremnn0zd 9994 A natural number is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremnnzd 9995 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzred 9996 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzcnd 9997 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremznegcld 9998 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theorempeano2zd 9999 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremzaddcld 10000 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)

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