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Type | Label | Description |
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Statement | ||
Theorem | divgt0ii 8701 | The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
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Theorem | ltmul1i 8702 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) |
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Theorem | ltdiv1i 8703 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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Theorem | ltmuldivi 8704 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) |
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Theorem | ltmul2i 8705 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) |
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Theorem | lemul1i 8706 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.) |
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Theorem | lemul2i 8707 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.) |
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Theorem | ltdiv23i 8708 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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Theorem | ltdiv23ii 8709 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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Theorem | ltmul1ii 8710 | 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.) |
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Theorem | ltdiv1ii 8711 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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Theorem | ltp1d 8712 | A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | lep1d 8713 | A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | ltm1d 8714 | A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | lem1d 8715 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | recgt0d 8716 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | divgt0d 8717 | The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | mulgt1d 8718 | The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | lemulge11d 8719 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | lemulge12d 8720 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | lemul1ad 8721 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | lemul2ad 8722 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | ltmul12ad 8723 | Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | lemul12ad 8724 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | lemul12bd 8725 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
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Theorem | mulle0r 8726 | Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.) |
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Theorem | lbreu 8727* | If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.) |
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Theorem | lbcl 8728* | 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.) |
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Theorem | lble 8729* | 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.) |
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Theorem | lbinf 8730* | 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.) |
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Theorem | lbinfcl 8731* | 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.) |
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Theorem | lbinfle 8732* | 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.) |
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Theorem | suprubex 8733* | 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.) |
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Theorem | suprlubex 8734* | The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.) |
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Theorem | suprnubex 8735* | An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.) |
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Theorem | suprleubex 8736* | 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.) |
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Theorem | negiso 8737 | Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.) |
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Theorem | dfinfre 8738* |
The infimum of a set of reals ![]() |
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Theorem | sup3exmid 8739* | If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.) |
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Theorem | crap0 8740 | 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.) |
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Theorem | creur 8741* | 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.) |
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Theorem | creui 8742* | 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.) |
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Theorem | cju 8743* | The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.) |
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Syntax | cn 8744 | Extend class notation to include the class of positive integers. |
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Definition | df-inn 8745* | Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8746 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.) |
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Theorem | dfnn2 8746* | Definition of the set of positive integers. Another name for df-inn 8745. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) |
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Theorem | peano5nni 8747* | 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.) |
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Theorem | nnssre 8748 | The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.) |
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Theorem | nnsscn 8749 | The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.) |
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Theorem | nnex 8750 | The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.) |
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Theorem | nnre 8751 | A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
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Theorem | nncn 8752 | A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) |
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Theorem | nnrei 8753 | A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
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Theorem | nncni 8754 | A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.) |
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Theorem | 1nn 8755 | Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.) |
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Theorem | peano2nn 8756 | 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.) |
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Theorem | nnred 8757 | A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nncnd 8758 | A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | peano2nnd 8759 | Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nnind 8760* | 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 8764 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.) |
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Theorem | nnindALT 8761* |
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 8760 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.) |
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Theorem | nn1m1nn 8762 | Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.) |
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Theorem | nn1suc 8763* | 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.) |
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Theorem | nnaddcl 8764 | Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.) |
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Theorem | nnmulcl 8765 | Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.) |
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Theorem | nnmulcli 8766 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.) |
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Theorem | nnge1 8767 | A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
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Theorem | nnle1eq1 8768 | A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.) |
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Theorem | nngt0 8769 | A positive integer is positive. (Contributed by NM, 26-Sep-1999.) |
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Theorem | nnnlt1 8770 | A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | 0nnn 8771 | Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) |
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Theorem | nnne0 8772 | A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) |
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Theorem | nnap0 8773 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.) |
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Theorem | nngt0i 8774 | A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.) |
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Theorem | nnap0i 8775 | A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.) |
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Theorem | nnne0i 8776 | A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.) |
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Theorem | nn2ge 8777* | There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.) |
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Theorem | nn1gt1 8778 |
A positive integer is either one or greater than one. This is for
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Theorem | nngt1ne1 8779 | A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.) |
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Theorem | nndivre 8780 | The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.) |
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Theorem | nnrecre 8781 | The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.) |
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Theorem | nnrecgt0 8782 | The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.) |
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Theorem | nnsub 8783 | Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.) |
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Theorem | nnsubi 8784 | Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.) |
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Theorem | nndiv 8785* |
Two ways to express "![]() ![]() |
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Theorem | nndivtr 8786 |
Transitive property of divisibility: if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nnge1d 8787 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nngt0d 8788 | A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nnne0d 8789 | A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nnap0d 8790 | A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.) |
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Theorem | nnrecred 8791 | The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nnaddcld 8792 | Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nnmulcld 8793 | Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | nndivred 8794 | A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
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The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7651 through df-9 8810), 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 7651 and df-1 7652).
Integers can also be exhibited as sums of powers of 10 (e.g. the number 103
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 8795 | Extend class notation to include the number 2. |
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Syntax | c3 8796 | Extend class notation to include the number 3. |
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Syntax | c4 8797 | Extend class notation to include the number 4. |
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Syntax | c5 8798 | Extend class notation to include the number 5. |
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Syntax | c6 8799 | Extend class notation to include the number 6. |
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Syntax | c7 8800 | Extend class notation to include the number 7. |
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