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Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlbinfle 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

Theoremsuprubex 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.)

Theoremsuprlubex 8703* The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)

Theoremsuprnubex 8704* An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.)

Theoremsuprleubex 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.)

Theoremnegiso 8706 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremdfinfre 8707* The infimum of a set of reals . (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
inf

Theoremsup3exmid 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

4.3.11  Imaginary and complex number properties

Theoremcrap0 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.)
# # #

Theoremcreur 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.)

Theoremcreui 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.)

Theoremcju 8712* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)

4.4  Integer sets

4.4.1  Positive integers (as a subset of complex numbers)

Syntaxcn 8713 Extend class notation to include the class of positive integers.

Definitiondf-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.)

Theoremdfnn2 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.)

Theorempeano5nni 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.)

Theoremnnssre 8717 The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremnnsscn 8718 The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremnnex 8719 The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnre 8720 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncn 8721 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)

Theoremnnrei 8722 A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncni 8723 A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)

Theorem1nn 8724 Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)

Theorempeano2nn 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.)

Theoremnnred 8726 A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnncnd 8727 A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempeano2nnd 8728 Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)

4.4.2  Principle of mathematical induction

Theoremnnind 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.)

TheoremnnindALT 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.)

Theoremnn1m1nn 8731 Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremnn1suc 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.)

Theoremnnaddcl 8733 Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)

Theoremnnmulcl 8734 Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)

Theoremnnmulcli 8735 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnnge1 8736 A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)

Theoremnnle1eq1 8737 A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)

Theoremnngt0 8738 A positive integer is positive. (Contributed by NM, 26-Sep-1999.)

Theoremnnnlt1 8739 A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theorem0nnn 8740 Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)

Theoremnnne0 8741 A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)

Theoremnnap0 8742 A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
#

Theoremnngt0i 8743 A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)

Theoremnnap0i 8744 A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
#

Theoremnnne0i 8745 A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)

Theoremnn2ge 8746* There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)

Theoremnn1gt1 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.)

Theoremnngt1ne1 8748 A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)

Theoremnndivre 8749 The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)

Theoremnnrecre 8750 The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)

Theoremnnrecgt0 8751 The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)

Theoremnnsub 8752 Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnnsubi 8753 Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)

Theoremnndiv 8754* Two ways to express " divides " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnndivtr 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.)

Theoremnnge1d 8756 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnngt0d 8757 A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnne0d 8758 A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnap0d 8759 A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
#

Theoremnnrecred 8760 The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnaddcld 8761 Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnmulcld 8762 Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnndivred 8763 A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)

4.4.3  Decimal representation of numbers

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.

Syntaxc2 8764 Extend class notation to include the number 2.

Syntaxc3 8765 Extend class notation to include the number 3.

Syntaxc4 8766 Extend class notation to include the number 4.

Syntaxc5 8767 Extend class notation to include the number 5.

Syntaxc6 8768 Extend class notation to include the number 6.

Syntaxc7 8769 Extend class notation to include the number 7.

Syntaxc8 8770 Extend class notation to include the number 8.

Syntaxc9 8771 Extend class notation to include the number 9.

Definitiondf-2 8772 Define the number 2. (Contributed by NM, 27-May-1999.)

Definitiondf-3 8773 Define the number 3. (Contributed by NM, 27-May-1999.)

Definitiondf-4 8774 Define the number 4. (Contributed by NM, 27-May-1999.)

Definitiondf-5 8775 Define the number 5. (Contributed by NM, 27-May-1999.)

Definitiondf-6 8776 Define the number 6. (Contributed by NM, 27-May-1999.)

Definitiondf-7 8777 Define the number 7. (Contributed by NM, 27-May-1999.)

Definitiondf-8 8778 Define the number 8. (Contributed by NM, 27-May-1999.)

Definitiondf-9 8779 Define the number 9. (Contributed by NM, 27-May-1999.)

Theorem0ne1 8780 (common case). See aso 1ap0 8345. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem1ne0 8781 . See aso 1ap0 8345. (Contributed by Jim Kingdon, 9-Mar-2020.)

Theorem1m1e0 8782 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2re 8783 The number 2 is real. (Contributed by NM, 27-May-1999.)

Theorem2cn 8784 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)

Theorem2ex 8785 2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem2cnd 8786 2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem3re 8787 The number 3 is real. (Contributed by NM, 27-May-1999.)

Theorem3cn 8788 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)

Theorem3ex 8789 3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem4re 8790 The number 4 is real. (Contributed by NM, 27-May-1999.)

Theorem4cn 8791 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem5re 8792 The number 5 is real. (Contributed by NM, 27-May-1999.)

Theorem5cn 8793 The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem6re 8794 The number 6 is real. (Contributed by NM, 27-May-1999.)

Theorem6cn 8795 The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem7re 8796 The number 7 is real. (Contributed by NM, 27-May-1999.)

Theorem7cn 8797 The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem8re 8798 The number 8 is real. (Contributed by NM, 27-May-1999.)

Theorem8cn 8799 The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)

Theorem9re 8800 The number 9 is real. (Contributed by NM, 27-May-1999.)

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