Home | Intuitionistic Logic Explorer Theorem List (p. 88 of 133) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lemul12bd 8701 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
Theorem | mulle0r 8702 | Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.) |
Theorem | lbreu 8703* | If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.) |
Theorem | lbcl 8704* | 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 8705* | 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 8706* | 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 8707* | 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 8708* | 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 8709* | 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 8710* | The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.) |
Theorem | suprnubex 8711* | 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 8712* | 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 8713 | Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | dfinfre 8714* | The infimum of a set of reals . (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
inf | ||
Theorem | sup3exmid 8715* | 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 8716 | 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 8717* | 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 8718* | 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 8719* | The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.) |
Syntax | cn 8720 | Extend class notation to include the class of positive integers. |
Definition | df-inn 8721* | Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8722 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.) |
Theorem | dfnn2 8722* | Definition of the set of positive integers. Another name for df-inn 8721. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) |
Theorem | peano5nni 8723* | 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 8724 | The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) |
Theorem | nnsscn 8725 | The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
Theorem | nnex 8726 | The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Theorem | nnre 8727 | A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nncn 8728 | A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nnrei 8729 | A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Theorem | nncni 8730 | A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) |
Theorem | 1nn 8731 | Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) |
Theorem | peano2nn 8732 | 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 8733 | A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nncnd 8734 | A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | peano2nnd 8735 | Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnind 8736* | 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 8740 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 8737* |
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 8736 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 8738 | Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.) |
Theorem | nn1suc 8739* | 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 8740 | Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.) |
Theorem | nnmulcl 8741 | Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) |
Theorem | nnmulcli 8742 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Theorem | nnge1 8743 | A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnle1eq1 8744 | A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.) |
Theorem | nngt0 8745 | A positive integer is positive. (Contributed by NM, 26-Sep-1999.) |
Theorem | nnnlt1 8746 | A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | 0nnn 8747 | Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnne0 8748 | A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) |
Theorem | nnap0 8749 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.) |
# | ||
Theorem | nngt0i 8750 | A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.) |
Theorem | nnap0i 8751 | A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.) |
# | ||
Theorem | nnne0i 8752 | A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.) |
Theorem | nn2ge 8753* | There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.) |
Theorem | nn1gt1 8754 | A positive integer is either one or greater than one. This is for ; 0elnn 4532 is a similar theorem for (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.) |
Theorem | nngt1ne1 8755 | A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.) |
Theorem | nndivre 8756 | The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.) |
Theorem | nnrecre 8757 | The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.) |
Theorem | nnrecgt0 8758 | The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.) |
Theorem | nnsub 8759 | Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.) |
Theorem | nnsubi 8760 | Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.) |
Theorem | nndiv 8761* | Two ways to express " divides " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Theorem | nndivtr 8762 | 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 8763 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nngt0d 8764 | A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnne0d 8765 | A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnap0d 8766 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.) |
# | ||
Theorem | nnrecred 8767 | The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnaddcld 8768 | Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nnmulcld 8769 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | nndivred 8770 | 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 7627 through df-9 8786), 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 7627 and df-1 7628). 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 8771 | Extend class notation to include the number 2. |
Syntax | c3 8772 | Extend class notation to include the number 3. |
Syntax | c4 8773 | Extend class notation to include the number 4. |
Syntax | c5 8774 | Extend class notation to include the number 5. |
Syntax | c6 8775 | Extend class notation to include the number 6. |
Syntax | c7 8776 | Extend class notation to include the number 7. |
Syntax | c8 8777 | Extend class notation to include the number 8. |
Syntax | c9 8778 | Extend class notation to include the number 9. |
Definition | df-2 8779 | Define the number 2. (Contributed by NM, 27-May-1999.) |
Definition | df-3 8780 | Define the number 3. (Contributed by NM, 27-May-1999.) |
Definition | df-4 8781 | Define the number 4. (Contributed by NM, 27-May-1999.) |
Definition | df-5 8782 | Define the number 5. (Contributed by NM, 27-May-1999.) |
Definition | df-6 8783 | Define the number 6. (Contributed by NM, 27-May-1999.) |
Definition | df-7 8784 | Define the number 7. (Contributed by NM, 27-May-1999.) |
Definition | df-8 8785 | Define the number 8. (Contributed by NM, 27-May-1999.) |
Definition | df-9 8786 | Define the number 9. (Contributed by NM, 27-May-1999.) |
Theorem | 0ne1 8787 | (common case). See aso 1ap0 8352. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 1ne0 8788 | . See aso 1ap0 8352. (Contributed by Jim Kingdon, 9-Mar-2020.) |
Theorem | 1m1e0 8789 | (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | 2re 8790 | The number 2 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 2cn 8791 | The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) |
Theorem | 2ex 8792 | 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 2cnd 8793 | 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 3re 8794 | The number 3 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 3cn 8795 | The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) |
Theorem | 3ex 8796 | 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
Theorem | 4re 8797 | The number 4 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 4cn 8798 | The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) |
Theorem | 5re 8799 | The number 5 is real. (Contributed by NM, 27-May-1999.) |
Theorem | 5cn 8800 | The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |