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

Theoremnnrecred 10001 The reciprocal of a natural number is real. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnaddcld 10002 Closure of addition of natural numbers. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnmulcld 10003 Closure of multiplication of natural numbers. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnndivred 10004 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 8953 and df-1 8954).

Only the digits 0 through 9 (df-0 8953 through df-9 10021) and the number 10 (df-10 10022) are explicitly defined.

We will later define the decimal constructor df-dec 10339, which will allow us to easily express larger integers in base 10. See deccl 10352 and the theorems that follow it. See also 4001prm 13419 (4001 is prime) and the proof of bpos 21030. 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 12744.

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 10005 Extend class notation to include the number 2.

Syntaxc3 10006 Extend class notation to include the number 3.

Syntaxc4 10007 Extend class notation to include the number 4.

Syntaxc5 10008 Extend class notation to include the number 5.

Syntaxc6 10009 Extend class notation to include the number 6.

Syntaxc7 10010 Extend class notation to include the number 7.

Syntaxc8 10011 Extend class notation to include the number 8.

Syntaxc9 10012 Extend class notation to include the number 9.

Syntaxc10 10013 Extend class notation to include the number 10.

Definitiondf-2 10014 Define the number 2. (Contributed by NM, 27-May-1999.)

Definitiondf-3 10015 Define the number 3. (Contributed by NM, 27-May-1999.)

Definitiondf-4 10016 Define the number 4. (Contributed by NM, 27-May-1999.)

Definitiondf-5 10017 Define the number 5. (Contributed by NM, 27-May-1999.)

Definitiondf-6 10018 Define the number 6. (Contributed by NM, 27-May-1999.)

Definitiondf-7 10019 Define the number 7. (Contributed by NM, 27-May-1999.)

Definitiondf-8 10020 Define the number 8. (Contributed by NM, 27-May-1999.)

Definitiondf-9 10021 Define the number 9. (Contributed by NM, 27-May-1999.)

Definitiondf-10 10022 Define the number 10. See remarks under df-2 10014. (Contributed by NM, 5-Feb-2007.)

Theoremneg1cn 10023 -1 is a complex number. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1m1e0 10024 . Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2re 10025 The number 2 is real. (Contributed by NM, 27-May-1999.)

Theorem2cn 10026 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)

Theorem3re 10027 The number 3 is real. (Contributed by NM, 27-May-1999.)

Theorem3cn 10028 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)

Theorem4re 10029 The number 4 is real. (Contributed by NM, 27-May-1999.)

Theorem4cn 10030 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem5re 10031 The number 5 is real. (Contributed by NM, 27-May-1999.)

Theorem6re 10032 The number 6 is real. (Contributed by NM, 27-May-1999.)

Theorem7re 10033 The number 7 is real. (Contributed by NM, 27-May-1999.)

Theorem8re 10034 The number 8 is real. (Contributed by NM, 27-May-1999.)

Theorem9re 10035 The number 9 is real. (Contributed by NM, 27-May-1999.)

Theorem10re 10036 The number 10 is real. (Contributed by NM, 5-Feb-2007.)

Theorem0le0 10037 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2pos 10038 The number 2 is positive. (Contributed by NM, 27-May-1999.)

Theorem2ne0 10039 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)

Theorem3pos 10040 The number 3 is positive. (Contributed by NM, 27-May-1999.)

Theorem3ne0 10041 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Theorem4pos 10042 The number 4 is positive. (Contributed by NM, 27-May-1999.)

Theorem5pos 10043 The number 5 is positive. (Contributed by NM, 27-May-1999.)

Theorem6pos 10044 The number 6 is positive. (Contributed by NM, 27-May-1999.)

Theorem7pos 10045 The number 7 is positive. (Contributed by NM, 27-May-1999.)

Theorem8pos 10046 The number 8 is positive. (Contributed by NM, 27-May-1999.)

Theorem9pos 10047 The number 9 is positive. (Contributed by NM, 27-May-1999.)

Theorem10pos 10048 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 10049 Zero plus one equals one. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1p1e2 10050 One plus one equals two. (Contributed by NM, 1-Apr-2008.)

Theorem2m1e1 10051 Prove that 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 10071. (Contributed by David A. Wheeler, 4-Jan-2017.)

Theorem3m1e2 10052 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)

Theorem3m1e2OLD 10053 equals minus . (Contributed by FL, 17-Oct-2010.) Obsolete version of 3m1e2 10052 as of 10-Dec-2017. (New usage is discouraged.) (Proof modification is discouraged.)

Theorem2p2e4 10054 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)

Theorem2times 10055 Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremtimes2 10056 A number times 2. (Contributed by NM, 16-Oct-2007.)

Theorem2timesi 10057 Two times a number. (Contributed by NM, 1-Aug-1999.)

Theoremtimes2i 10058 A number times 2. (Contributed by NM, 11-May-2004.)

Theorem2p1e3 10059 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem3p1e4 10060 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem4p1e5 10061 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem5p1e6 10062 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem6p1e7 10063 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem7p1e8 10064 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem8p1e9 10065 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem9p1e10 10066 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.)

Theorem3p2e5 10067 3 + 2 = 5. (Contributed by NM, 11-May-2004.)

Theorem3p3e6 10068 3 + 3 = 6. (Contributed by NM, 11-May-2004.)

Theorem4p2e6 10069 4 + 2 = 6. (Contributed by NM, 11-May-2004.)

Theorem4p3e7 10070 4 + 3 = 7. (Contributed by NM, 11-May-2004.)

Theorem4p4e8 10071 4 + 4 = 8. (Contributed by NM, 11-May-2004.)

Theorem5p2e7 10072 5 + 2 = 7. (Contributed by NM, 11-May-2004.)

Theorem5p3e8 10073 5 + 3 = 8. (Contributed by NM, 11-May-2004.)

Theorem5p4e9 10074 5 + 4 = 9. (Contributed by NM, 11-May-2004.)

Theorem5p5e10 10075 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem6p2e8 10076 6 + 2 = 8. (Contributed by NM, 11-May-2004.)

Theorem6p3e9 10077 6 + 3 = 9. (Contributed by NM, 11-May-2004.)

Theorem6p4e10 10078 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem7p2e9 10079 7 + 2 = 9. (Contributed by NM, 11-May-2004.)

Theorem7p3e10 10080 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem8p2e10 10081 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.)

Theorem1t1e1 10082 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2t2e4 10083 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)

Theorem3t2e6 10084 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)

Theorem3t3e9 10085 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)

Theorem4t2e8 10086 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)

Theorem5t2e10 10087 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.)

Theorem4d2e2 10088 One half of four is two. (Contributed by NM, 3-Sep-1999.)

Theorem2nn 10089 2 is a natural number. (Contributed by NM, 20-Aug-2001.)

Theorem3nn 10090 3 is a natural number. (Contributed by NM, 8-Jan-2006.)

Theorem4nn 10091 4 is a natural number. (Contributed by NM, 8-Jan-2006.)

Theorem5nn 10092 5 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem6nn 10093 6 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem7nn 10094 7 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem8nn 10095 8 is a natural number. (Contributed by Mario Carneiro, 15-Sep-2013.)

Theorem9nn 10096 9 is a natural number. (Contributed by NM, 21-Oct-2012.)

Theorem10nn 10097 10 is a natural number. (Contributed by NM, 8-Nov-2012.)

Theorem1lt2 10098 1 is less than 2. (Contributed by NM, 24-Feb-2005.)

Theorem2lt3 10099 2 is less than 3. (Contributed by NM, 26-Sep-2010.)

Theorem1lt3 10100 1 is less than 3. (Contributed by NM, 26-Sep-2010.)

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