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

Theoremcju 10001* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)

5.3.10  Function operation analogue theorems

Theoremofsubeq0 10002 Function analog of subeq0 9332. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremofnegsub 10003 Function analog of negsub 9354. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremofsubge0 10004 Function analog of subge0 9546. (Contributed by Mario Carneiro, 24-Jul-2014.)

5.4  Integer sets

5.4.1  Natural numbers (as a subset of complex numbers)

Syntaxcn 10005 Extend class notation to include the class of positive integers.

Definitiondf-nn 10006 The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set , df-om 4849, which start at zero). This is the convention used by most analysis books, and it is often convenient in proofs because we don't have to worry about division by zero. See nnind 10023 for the principle of mathematical induction. See dfnn2 10018 for a slight variant. See df-n0 10227 for the set of nonnegative integers starting at zero. See dfn2 10239 for defined in terms of .

This is a technical definition that helps us avoid the Axiom of Infinity in certain proofs. For a more conventional and intuitive definition ("the smallest set of reals containing as well as the successor of every member") see dfnn3 10019. (Contributed by NM, 10-Jan-1997.)

TheoremnnexALT 10007 The set of natural numbers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 3-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorempeano5nni 10008* Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnssre 10009 The natural numbers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremnnsscn 10010 The natural numbers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremnnex 10011 The set of natural numbers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnre 10012 A natural number is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncn 10013 A natural number is a complex number. (Contributed by NM, 18-Aug-1999.)

Theoremnnrei 10014 A natural number is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncni 10015 A natural number is a complex number. (Contributed by NM, 18-Aug-1999.)

Theorem1nn 10016 Peano postulate: 1 is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theorempeano2nn 10017 Peano postulate: a successor of a natural number is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremdfnn2 10018* Alternate definition of the set of natural numbers. This was our original definition, before the current df-nn 10006 replaced it. This definition requires the axiom of infinity to ensure it has the properties we expect. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)

Theoremdfnn3 10019* Alternate definition of the set of natural numbers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.)

Theoremnnred 10020 A natural number is a real number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnncnd 10021 A natural number is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempeano2nnd 10022 Peano postulate: a successor of a natural number is a natural number. (Contributed by Mario Carneiro, 27-May-2016.)

5.4.2  Principle of mathematical induction

Theoremnnind 10023* Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. See nnaddcl 10027 for an example of its use. See nn0ind 10371 for induction on nonnegative integers and uzind 10366, uzind4 10539 for induction on an arbitrary set of upper integers. See indstr 10550 for strong induction. See also nnindALT 10024. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

TheoremnnindALT 10024* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction hypothesis and the basis.

This ALT version of nnind 10023 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.)

Theoremnn1m1nn 10025 Every natural number is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremnn1suc 10026* If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnnaddcl 10027 Closure of addition of natural numbers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)

Theoremnnmulcl 10028 Closure of multiplication of natural numbers. (Contributed by NM, 12-Jan-1997.)

Theoremnnmulcli 10029 Closure of multiplication of natural numbers. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnn2ge 10030* There exists a natural number greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)

Theoremnnge1 10031 A natural number is one or greater. (Contributed by NM, 25-Aug-1999.)

Theoremnngt1ne1 10032 A natural number is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)

Theoremnnle1eq1 10033 A natural number is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)

Theoremnngt0 10034 A natural number is positive. (Contributed by NM, 26-Sep-1999.)

Theoremnnnlt1 10035 A natural number is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theorem0nnn 10036 Zero is not a natural number. (Contributed by NM, 25-Aug-1999.)

Theoremnnne0 10037 A natural number is nonzero. (Contributed by NM, 27-Sep-1999.)

Theoremnngt0i 10038 A natural number is positive (inference version). (Contributed by NM, 17-Sep-1999.)

Theoremnnne0i 10039 A natural number is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)

Theoremnndivre 10040 The quotient of a real and a natural number is real. (Contributed by NM, 28-Nov-2008.)

Theoremnnrecre 10041 The reciprocal of a natural number is real. (Contributed by NM, 8-Feb-2008.)

Theoremnnrecgt0 10042 The reciprocal of a natural number is positive. (Contributed by NM, 25-Aug-1999.)

Theoremnnsub 10043 Subtraction of natural numbers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnnsubi 10044 Subtraction of natural numbers. (Contributed by NM, 19-Aug-2001.)

Theoremnndiv 10045* Two ways to express " divides " for natural numbers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnndivtr 10046 Transitive property of divisibility: if divides and divides , then divides . Typically, would be an integer, although the theorem holds for complex . (Contributed by NM, 3-May-2005.)

Theoremnnge1d 10047 A natural number is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnngt0d 10048 A natural number is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnne0d 10049 A natural number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnrecred 10050 The reciprocal of a natural number is real. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnaddcld 10051 Closure of addition of natural numbers. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnmulcld 10052 Closure of multiplication of natural numbers. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnndivred 10053 A natural number is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)

5.4.3  Decimal representation of numbers

Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 9002 and df-1 9003).

Only the digits 0 through 9 (df-0 9002 through df-9 10070) and the number 10 (df-10 10071) are explicitly defined.

We will later define the decimal constructor df-dec 10388, which will allow us to easily express larger integers in base 10. See deccl 10401 and the theorems that follow it. See also 4001prm 13469 (4001 is prime) and the proof of bpos 21082. Note that the decimal constructor builds on the definitions in this section.

Integers can also be exhibited as sums of powers of 10 or as some other expression built from operations on the numbers 0 through 10. For example, the prime number 823541 can be expressed as . Decimals can be expressed as ratios of integers, as in cos2bnd 12794.

Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12.

Syntaxc2 10054 Extend class notation to include the number 2.

Syntaxc3 10055 Extend class notation to include the number 3.

Syntaxc4 10056 Extend class notation to include the number 4.

Syntaxc5 10057 Extend class notation to include the number 5.

Syntaxc6 10058 Extend class notation to include the number 6.

Syntaxc7 10059 Extend class notation to include the number 7.

Syntaxc8 10060 Extend class notation to include the number 8.

Syntaxc9 10061 Extend class notation to include the number 9.

Syntaxc10 10062 Extend class notation to include the number 10.

Definitiondf-2 10063 Define the number 2. (Contributed by NM, 27-May-1999.)

Definitiondf-3 10064 Define the number 3. (Contributed by NM, 27-May-1999.)

Definitiondf-4 10065 Define the number 4. (Contributed by NM, 27-May-1999.)

Definitiondf-5 10066 Define the number 5. (Contributed by NM, 27-May-1999.)

Definitiondf-6 10067 Define the number 6. (Contributed by NM, 27-May-1999.)

Definitiondf-7 10068 Define the number 7. (Contributed by NM, 27-May-1999.)

Definitiondf-8 10069 Define the number 8. (Contributed by NM, 27-May-1999.)

Definitiondf-9 10070 Define the number 9. (Contributed by NM, 27-May-1999.)

Definitiondf-10 10071 Define the number 10. See remarks under df-2 10063. (Contributed by NM, 5-Feb-2007.)

Theoremneg1cn 10072 -1 is a complex number. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1m1e0 10073 . Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2re 10074 The number 2 is real. (Contributed by NM, 27-May-1999.)

Theorem2cn 10075 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)

Theorem3re 10076 The number 3 is real. (Contributed by NM, 27-May-1999.)

Theorem3cn 10077 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)

Theorem4re 10078 The number 4 is real. (Contributed by NM, 27-May-1999.)

Theorem4cn 10079 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem5re 10080 The number 5 is real. (Contributed by NM, 27-May-1999.)

Theorem6re 10081 The number 6 is real. (Contributed by NM, 27-May-1999.)

Theorem7re 10082 The number 7 is real. (Contributed by NM, 27-May-1999.)

Theorem8re 10083 The number 8 is real. (Contributed by NM, 27-May-1999.)

Theorem9re 10084 The number 9 is real. (Contributed by NM, 27-May-1999.)

Theorem10re 10085 The number 10 is real. (Contributed by NM, 5-Feb-2007.)

Theorem0le0 10086 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2pos 10087 The number 2 is positive. (Contributed by NM, 27-May-1999.)

Theorem2ne0 10088 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)

Theorem3pos 10089 The number 3 is positive. (Contributed by NM, 27-May-1999.)

Theorem3ne0 10090 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theorem4pos 10091 The number 4 is positive. (Contributed by NM, 27-May-1999.)

Theorem5pos 10092 The number 5 is positive. (Contributed by NM, 27-May-1999.)

Theorem6pos 10093 The number 6 is positive. (Contributed by NM, 27-May-1999.)

Theorem7pos 10094 The number 7 is positive. (Contributed by NM, 27-May-1999.)

Theorem8pos 10095 The number 8 is positive. (Contributed by NM, 27-May-1999.)

Theorem9pos 10096 The number 9 is positive. (Contributed by NM, 27-May-1999.)

Theorem10pos 10097 The number 10 is positive. (Contributed by NM, 5-Feb-2007.)

5.4.4  Some properties of specific numbers

This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.

Theorem0p1e1 10098 Zero plus one equals one. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1p1e2 10099 One plus one equals two. (Contributed by NM, 1-Apr-2008.)

Theorem2m1e1 10100 Prove that 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 10120. (Contributed by David A. Wheeler, 4-Jan-2017.)

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