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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lt2msqi 8801 | The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.) |
Theorem | le2msqi 8802 | The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.) |
Theorem | msq11i 8803 | The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) |
Theorem | divgt0i2i 8804 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
Theorem | ltrecii 8805 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
Theorem | divgt0ii 8806 | The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
Theorem | ltmul1i 8807 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) |
Theorem | ltdiv1i 8808 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
Theorem | ltmuldivi 8809 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) |
Theorem | ltmul2i 8810 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) |
Theorem | lemul1i 8811 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.) |
Theorem | lemul2i 8812 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.) |
Theorem | ltdiv23i 8813 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
Theorem | ltdiv23ii 8814 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
Theorem | ltmul1ii 8815 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.) |
Theorem | ltdiv1ii 8816 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
Theorem | ltp1d 8817 | A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lep1d 8818 | A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | ltm1d 8819 | A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lem1d 8820 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | recgt0d 8821 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | divgt0d 8822 | The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | mulgt1d 8823 | The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemulge11d 8824 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemulge12d 8825 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemul1ad 8826 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemul2ad 8827 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | ltmul12ad 8828 | Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemul12ad 8829 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | lemul12bd 8830 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | mulle0r 8831 | Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.) |
Theorem | lbreu 8832* | If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.) |
Theorem | lbcl 8833* | If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Theorem | lble 8834* | If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Theorem | lbinf 8835* | If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
inf | ||
Theorem | lbinfcl 8836* | If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
inf | ||
Theorem | lbinfle 8837* | If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
inf | ||
Theorem | suprubex 8838* | A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.) |
Theorem | suprlubex 8839* | The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.) |
Theorem | suprnubex 8840* | An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.) |
Theorem | suprleubex 8841* | The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.) |
Theorem | negiso 8842 | Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | dfinfre 8843* | The infimum of a set of reals . (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
inf | ||
Theorem | sup3exmid 8844* | If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.) |
DECID | ||
Theorem | crap0 8845 | The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.) |
# # # | ||
Theorem | creur 8846* | The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Theorem | creui 8847* | The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Theorem | cju 8848* | The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.) |
Syntax | cn 8849 | Extend class notation to include the class of positive integers. |
Definition | df-inn 8850* | Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8851 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.) |
Theorem | dfnn2 8851* | Definition of the set of positive integers. Another name for df-inn 8850. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) |
Theorem | peano5nni 8852* | 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.) |
Theorem | nnssre 8853 | The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) |
Theorem | nnsscn 8854 | The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
Theorem | nnex 8855 | The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | nnre 8856 | A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nncn 8857 | A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nnrei 8858 | A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nncni 8859 | A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) |
Theorem | 1nn 8860 | Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) |
Theorem | peano2nn 8861 | 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 8862 | A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nncnd 8863 | A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | peano2nnd 8864 | Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnind 8865* | 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 8869 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 8866* |
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 8865 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 8867 | Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.) |
Theorem | nn1suc 8868* | 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 8869 | Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.) |
Theorem | nnmulcl 8870 | Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) |
Theorem | nnmulcli 8871 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | nnge1 8872 | A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnle1eq1 8873 | A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.) |
Theorem | nngt0 8874 | A positive integer is positive. (Contributed by NM, 26-Sep-1999.) |
Theorem | nnnlt1 8875 | A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | 0nnn 8876 | Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnne0 8877 | A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) |
Theorem | nnap0 8878 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.) |
# | ||
Theorem | nngt0i 8879 | A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.) |
Theorem | nnap0i 8880 | A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.) |
# | ||
Theorem | nnne0i 8881 | A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.) |
Theorem | nn2ge 8882* | There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.) |
Theorem | nn1gt1 8883 | A positive integer is either one or greater than one. This is for ; 0elnn 4591 is a similar theorem for (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.) |
Theorem | nngt1ne1 8884 | A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.) |
Theorem | nndivre 8885 | The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.) |
Theorem | nnrecre 8886 | The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.) |
Theorem | nnrecgt0 8887 | The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnsub 8888 | Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | nnsubi 8889 | Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.) |
Theorem | nndiv 8890* | Two ways to express " divides " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | nndivtr 8891 | 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 8892 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nngt0d 8893 | A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnne0d 8894 | A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnap0d 8895 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.) |
# | ||
Theorem | nnrecred 8896 | The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnaddcld 8897 | Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnmulcld 8898 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nndivred 8899 | 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 7752 through df-9 8915), 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 7752 and df-1 7753). 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 8900 | Extend class notation to include the number 2. |
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