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Theorem List for Metamath Proof Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem9lt10 9801 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theorem8lt10 9802 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theorem7lt10 9803 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem6lt10 9804 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem5lt10 9805 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem4lt10 9806 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem3lt10 9807 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem2lt10 9808 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)

Theorem1lt10 9809 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)

Theorem1ne2 9810 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)

Theoremhalfgt0 9811 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)

Theoremhalflt1 9812 One-half is less than one. (Contributed by NM, 24-Feb-2005.)

Theorem1mhlfehlf 9813 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)

Theorem8th4div3 9814 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)

Theoremhalfpm6th 9815 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)

Theoremhalfcl 9816 Closure of half of a number (frequently used special case). (Contributed by NM, 1-Jan-2006.)

Theoremrehalfcl 9817 Real closure of half. (Contributed by NM, 1-Jan-2006.)

Theoremhalf0 9818 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)

Theorem2halves 9819 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)

Theoremhalfpos2 9820 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)

Theoremhalfpos 9821 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremhalfnneg2 9822 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)

Theoremhalfaddsubcl 9823 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremhalfaddsub 9824 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)

Theoremlt2halves 9825 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)

Theoremaddltmul 9826 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)

Theoremnominpos 9827* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)

Theoremavglt1 9828 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremavglt2 9829 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremavgle1 9830 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)

Theoremavgle2 9831 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)

Theoremavgle 9832 The average of two numbers is less than or equal to at least one of them. (Contributed by NM, 9-Dec-2005.) (Revised by Mario Carneiro, 28-May-2014.)

Theorem2timesd 9833 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremtimes2d 9834 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremhalfcld 9835 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)

Theorem2halvesd 9836 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrehalfcld 9837 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2halvesd 9838 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrehalfcli 9839 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)

5.4.5  The Archimedean property

Theoremnnunb 9840* The set of natural numbers is unbounded above. Theorem I.28 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)

Theoremarch 9841* Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)

Theoremnnrecl 9842* There exists a natural number whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)

Theorembndndx 9843* A bounded real sequence is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)

5.4.6  Nonnegative integers (as a subset of complex numbers)

Syntaxcn0 9844 Extend class notation to include the class of nonnegative integers.

Definitiondf-n0 9845 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremelnn0 9846 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnnssnn0 9847 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0ssre 9848 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0sscn 9849 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)

Theoremnn0ex 9850 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)

Theoremnnnn0 9851 A natural number is a nonnegative integer. (Contributed by NM, 9-May-2004.)

Theoremnnnn0i 9852 A natural number is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)

Theoremnn0re 9853 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)

Theoremnn0cn 9854 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)

Theoremnn0rei 9855 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)

Theoremnn0cni 9856 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)

Theoremdfn2 9857 The set of natural numbers (positive integers) defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)

Theoremelnnne0 9858 The natural number property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Theorem0nn0 9859 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem1nn0 9860 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem2nn0 9861 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)

Theorem3nn0 9862 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem4nn0 9863 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem5nn0 9864 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem6nn0 9865 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem7nn0 9866 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem8nn0 9867 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem9nn0 9868 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theorem10nn0 9869 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)

Theoremnn0ge0 9870 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnn0nlt0 9871 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0i 9872 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0le0eq0 9873 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)

Theoremnnnn0addcl 9874 A natural number plus a nonnegative integer is a natural number. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0nnaddcl 9875 A nonnegative integer plus a natural number is a natural number. (Contributed by NM, 22-Dec-2005.)

Theoremun0addcl 9876 If is closed under addition, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.)

Theoremun0mulcl 9877 If is closed under multiplication, then so is . (Contributed by Mario Carneiro, 17-Jul-2014.)

Theoremnn0addcl 9878 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremnn0mulcl 9879 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)

Theoremnn0addcli 9880 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0mulcli 9881 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0p1nn 9882 A nonnegative integer plus 1 is a natural number. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)

Theorempeano2nn0 9883 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)

Theoremnnm1nn0 9884 A natural number minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremelnn0nn 9885 The nonnegative integer property expressed in terms of natural numbers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremelnnnn0 9886 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)

Theoremelnnnn0b 9887 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)

Theoremelnnnn0c 9888 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)

Theoremnn0addge1 9889 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge2 9890 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge1i 9891 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0addge2i 9892 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)

Theoremnn0sub 9893 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnn0le2xi 9894 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0lele2xi 9895 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremnn0supp 9896 Two ways to write the support of a function on . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremnnnn0d 9897 A natural number is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0red 9898 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0cnd 9899 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnn0ge0d 9900 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)

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