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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lbinfle 8701* | 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 8702* | 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 8703* | The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.) |
Theorem | suprnubex 8704* | 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 8705* | 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 8706 | Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | dfinfre 8707* | The infimum of a set of reals . (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
inf | ||
Theorem | sup3exmid 8708* | 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 8709 | 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 8710* | 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 8711* | 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 8712* | The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.) |
Syntax | cn 8713 | Extend class notation to include the class of positive integers. |
Definition | df-inn 8714* | Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8715 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.) |
Theorem | dfnn2 8715* | Definition of the set of positive integers. Another name for df-inn 8714. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) |
Theorem | peano5nni 8716* | 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 8717 | The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) |
Theorem | nnsscn 8718 | The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
Theorem | nnex 8719 | The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | nnre 8720 | A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nncn 8721 | A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nnrei 8722 | A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nncni 8723 | A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) |
Theorem | 1nn 8724 | Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) |
Theorem | peano2nn 8725 | 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 8726 | A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nncnd 8727 | A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | peano2nnd 8728 | Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnind 8729* | 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 8733 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 8730* |
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 8729 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 8731 | Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.) |
Theorem | nn1suc 8732* | 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 8733 | Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.) |
Theorem | nnmulcl 8734 | Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) |
Theorem | nnmulcli 8735 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | nnge1 8736 | A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnle1eq1 8737 | A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.) |
Theorem | nngt0 8738 | A positive integer is positive. (Contributed by NM, 26-Sep-1999.) |
Theorem | nnnlt1 8739 | A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | 0nnn 8740 | Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnne0 8741 | A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) |
Theorem | nnap0 8742 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.) |
# | ||
Theorem | nngt0i 8743 | A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.) |
Theorem | nnap0i 8744 | A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.) |
# | ||
Theorem | nnne0i 8745 | A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.) |
Theorem | nn2ge 8746* | There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.) |
Theorem | nn1gt1 8747 | A positive integer is either one or greater than one. This is for ; 0elnn 4527 is a similar theorem for (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.) |
Theorem | nngt1ne1 8748 | A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.) |
Theorem | nndivre 8749 | The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.) |
Theorem | nnrecre 8750 | The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.) |
Theorem | nnrecgt0 8751 | The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnsub 8752 | Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | nnsubi 8753 | Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.) |
Theorem | nndiv 8754* | Two ways to express " divides " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | nndivtr 8755 | 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 8756 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nngt0d 8757 | A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnne0d 8758 | A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnap0d 8759 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.) |
# | ||
Theorem | nnrecred 8760 | The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnaddcld 8761 | Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnmulcld 8762 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nndivred 8763 | 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 7620 through df-9 8779), 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 7620 and df-1 7621). 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 8764 | Extend class notation to include the number 2. |
Syntax | c3 8765 | Extend class notation to include the number 3. |
Syntax | c4 8766 | Extend class notation to include the number 4. |
Syntax | c5 8767 | Extend class notation to include the number 5. |
Syntax | c6 8768 | Extend class notation to include the number 6. |
Syntax | c7 8769 | Extend class notation to include the number 7. |
Syntax | c8 8770 | Extend class notation to include the number 8. |
Syntax | c9 8771 | Extend class notation to include the number 9. |
Definition | df-2 8772 | Define the number 2. (Contributed by NM, 27-May-1999.) |
Definition | df-3 8773 | Define the number 3. (Contributed by NM, 27-May-1999.) |
Definition | df-4 8774 | Define the number 4. (Contributed by NM, 27-May-1999.) |
Definition | df-5 8775 | Define the number 5. (Contributed by NM, 27-May-1999.) |
Definition | df-6 8776 | Define the number 6. (Contributed by NM, 27-May-1999.) |
Definition | df-7 8777 | Define the number 7. (Contributed by NM, 27-May-1999.) |
Definition | df-8 8778 | Define the number 8. (Contributed by NM, 27-May-1999.) |
Definition | df-9 8779 | Define the number 9. (Contributed by NM, 27-May-1999.) |
Theorem | 0ne1 8780 | (common case). See aso 1ap0 8345. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 1ne0 8781 | . See aso 1ap0 8345. (Contributed by Jim Kingdon, 9-Mar-2020.) |
Theorem | 1m1e0 8782 | (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | 2re 8783 | The number 2 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 2cn 8784 | The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) |
Theorem | 2ex 8785 | 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 2cnd 8786 | 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 3re 8787 | The number 3 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 3cn 8788 | The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) |
Theorem | 3ex 8789 | 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 4re 8790 | The number 4 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 4cn 8791 | The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | 5re 8792 | The number 5 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 5cn 8793 | The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 6re 8794 | The number 6 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 6cn 8795 | The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 7re 8796 | The number 7 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 7cn 8797 | The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 8re 8798 | The number 8 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 8cn 8799 | The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 9re 8800 | The number 9 is real. (Contributed by NM, 27-May-1999.) |
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