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Type | Label | Description |
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Statement | ||
Theorem | 1nn 8001 | Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) |
Theorem | peano2nn 8002 | Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | nnred 8003 | A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nncnd 8004 | A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | peano2nnd 8005 | Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnind 8006* | 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 step. See nnaddcl 8010 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) |
Theorem | nnindALT 8007* |
Principle of Mathematical Induction (inference schema). The last four
hypotheses give us the substitution instances we need; the first two are
the induction step and the basis.
This ALT version of nnind 8006 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.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | nn1m1nn 8008 | Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.) |
Theorem | nn1suc 8009* | If a statement holds for 1 and also holds for a successor, it holds for all positive integers. 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.) |
Theorem | nnaddcl 8010 | Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.) |
Theorem | nnmulcl 8011 | Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) |
Theorem | nnmulcli 8012 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | nnge1 8013 | A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnle1eq1 8014 | A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.) |
Theorem | nngt0 8015 | A positive integer is positive. (Contributed by NM, 26-Sep-1999.) |
Theorem | nnnlt1 8016 | A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | 0nnn 8017 | Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnne0 8018 | A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) |
Theorem | nnap0 8019 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.) |
# | ||
Theorem | nngt0i 8020 | A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.) |
Theorem | nnne0i 8021 | A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.) |
Theorem | nn2ge 8022* | There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.) |
Theorem | nn1gt1 8023 | A positive integer is either one or greater than one. This is for ; 0elnn 4368 is a similar theorem for (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.) |
Theorem | nngt1ne1 8024 | A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.) |
Theorem | nndivre 8025 | The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.) |
Theorem | nnrecre 8026 | The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.) |
Theorem | nnrecgt0 8027 | The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnsub 8028 | Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | nnsubi 8029 | Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.) |
Theorem | nndiv 8030* | Two ways to express " divides " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | nndivtr 8031 | 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.) |
Theorem | nnge1d 8032 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nngt0d 8033 | A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnne0d 8034 | A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnap0d 8035 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.) |
# | ||
Theorem | nnrecred 8036 | The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnaddcld 8037 | Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnmulcld 8038 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nndivred 8039 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 6954 through df-9 8056), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 6954 and df-1 6955). Integers can also be exhibited as sums of powers of 10 (e.g. the number 103 can be expressed as ; ) or as some other expression built from operations on the numbers 0 through 9. For example, the prime number 823541 can be expressed as . 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. | ||
Syntax | c2 8040 | Extend class notation to include the number 2. |
Syntax | c3 8041 | Extend class notation to include the number 3. |
Syntax | c4 8042 | Extend class notation to include the number 4. |
Syntax | c5 8043 | Extend class notation to include the number 5. |
Syntax | c6 8044 | Extend class notation to include the number 6. |
Syntax | c7 8045 | Extend class notation to include the number 7. |
Syntax | c8 8046 | Extend class notation to include the number 8. |
Syntax | c9 8047 | Extend class notation to include the number 9. |
Syntax | c10 8048 | Extend class notation to include the number 10. |
Definition | df-2 8049 | Define the number 2. (Contributed by NM, 27-May-1999.) |
Definition | df-3 8050 | Define the number 3. (Contributed by NM, 27-May-1999.) |
Definition | df-4 8051 | Define the number 4. (Contributed by NM, 27-May-1999.) |
Definition | df-5 8052 | Define the number 5. (Contributed by NM, 27-May-1999.) |
Definition | df-6 8053 | Define the number 6. (Contributed by NM, 27-May-1999.) |
Definition | df-7 8054 | Define the number 7. (Contributed by NM, 27-May-1999.) |
Definition | df-8 8055 | Define the number 8. (Contributed by NM, 27-May-1999.) |
Definition | df-9 8056 | Define the number 9. (Contributed by NM, 27-May-1999.) |
Theorem | 0ne1 8057 | (common case). See aso 1ap0 7655. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 1ne0 8058 | . See aso 1ap0 7655. (Contributed by Jim Kingdon, 9-Mar-2020.) |
Theorem | 1m1e0 8059 | (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | 2re 8060 | The number 2 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 2cn 8061 | The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) |
Theorem | 2ex 8062 | 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 2cnd 8063 | 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 3re 8064 | The number 3 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 3cn 8065 | The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) |
Theorem | 3ex 8066 | 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 4re 8067 | The number 4 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 4cn 8068 | The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | 5re 8069 | The number 5 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 5cn 8070 | The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 6re 8071 | The number 6 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 6cn 8072 | The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 7re 8073 | The number 7 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 7cn 8074 | The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 8re 8075 | The number 8 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 8cn 8076 | The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 9re 8077 | The number 9 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 9cn 8078 | The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 0le0 8079 | Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | 0le2 8080 | 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.) |
Theorem | 2pos 8081 | The number 2 is positive. (Contributed by NM, 27-May-1999.) |
Theorem | 2ne0 8082 | The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.) |
Theorem | 2ap0 8083 | The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
# | ||
Theorem | 3pos 8084 | The number 3 is positive. (Contributed by NM, 27-May-1999.) |
Theorem | 3ne0 8085 | The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
Theorem | 3ap0 8086 | The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.) |
# | ||
Theorem | 4pos 8087 | The number 4 is positive. (Contributed by NM, 27-May-1999.) |
Theorem | 4ne0 8088 | The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.) |
Theorem | 4ap0 8089 | The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.) |
# | ||
Theorem | 5pos 8090 | The number 5 is positive. (Contributed by NM, 27-May-1999.) |
Theorem | 6pos 8091 | The number 6 is positive. (Contributed by NM, 27-May-1999.) |
Theorem | 7pos 8092 | The number 7 is positive. (Contributed by NM, 27-May-1999.) |
Theorem | 8pos 8093 | The number 8 is positive. (Contributed by NM, 27-May-1999.) |
Theorem | 9pos 8094 | The number 9 is positive. (Contributed by NM, 27-May-1999.) |
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. | ||
Theorem | neg1cn 8095 | -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | neg1rr 8096 | -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.) |
Theorem | neg1ne0 8097 | -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | neg1lt0 8098 | -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | neg1ap0 8099 | -1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.) |
# | ||
Theorem | negneg1e1 8100 | is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
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