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Theorem List for Metamath Proof Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremletric 8801 Trichotomy law. (Contributed by NM, 18-Aug-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltlen 8802 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.)

Theoremeqle 8803 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)

Theoremltadd2 8804 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremne0gt0 8805 A nonzero nonnegative number is positive. (Contributed by NM, 20-Nov-2007.)

Theoremlecasei 8806 Ordering elimination by cases. (Contributed by NM, 6-Jul-2007.)

Theoremlelttric 8807 Trichotomy law. (Contributed by NM, 4-Apr-2005.)

Theoremltlecasei 8808 Ordering elimination by cases. (Contributed by NM, 1-Jul-2007.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltnri 8809 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)

Theoremgtneii 8810 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.)

Theoremltneii 8811 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)

Theoremlttri2i 8812 Consequence of trichotomy. (Contributed by NM, 19-Jan-1997.)

Theoremlttri3i 8813 Consequence of trichotomy. (Contributed by NM, 14-May-1999.)

Theoremletri3i 8814 Consequence of trichotomy. (Contributed by NM, 14-May-1999.)

Theoremleloei 8815 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 14-May-1999.)

Theoremltleni 8816 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.)

Theoremltnsymi 8817 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)

Theoremlenlti 8818 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)

Theoremltnlei 8819 'Less than' in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.)

Theoremltlei 8820 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)

Theoremltleii 8821 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)

Theoremeqlei 8822 Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.)

Theoremltnei 8823 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)

Theoremletrii 8824 Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999.)

Theoremlttri 8825 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremlelttri 8826 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)

Theoremltletri 8827 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)

Theoremletri 8828 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)

Theoremle2tri3i 8829 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)

Theoremltadd2i 8830 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Paul Chapman, 27-Jan-2008.)

Theoremmulgt0i 8831 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremmulgt0ii 8832 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltnrd 8833 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremgtned 8834 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltned 8835 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremne0gt0d 8836 A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttrid 8837 Ordering on reals satisfies strict trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri2d 8838 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri3d 8839 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri4d 8840 Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremletri3d 8841 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleloed 8842 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremeqleltd 8843 Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)

Theoremltlend 8844 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenltd 8845 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnled 8846 'Less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltled 8847 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnsymd 8848 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrid 8849 Trichotomy law for 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleltned 8850 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulgt0d 8851 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd2d 8852 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrd 8853 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)

Theoremlelttrd 8854 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)

Theoremltadd2dd 8855 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltletrd 8856 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)

Theoremlttrd 8857 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)

5.2.5  Initial properties of the complex numbers

Theoremmul12 8858 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)

Theoremmul32 8859 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)

Theoremmul31 8860 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul4 8861 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)

Theoremmuladd11 8862 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)

Theorem1p1times 8863 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempeano2cn 8864 A theorem for complex numbers analogous the second Peano postulate peano2nn 9638. (Contributed by NM, 17-Aug-2005.)

Theorempeano2re 8865 A theorem for reals analogous the second Peano postulate peano2nn 9638. (Contributed by NM, 5-Jul-2005.)

Theoremreaddcan 8866 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theorem00id 8867 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul02lem1 8868 Lemma for mul02 8870. If any real does not produce when multiplied by , then any complex is equal to double itself. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul02lem2 8869 Lemma for mul02 8870. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul02 8870 Multiplication by . Theorem I.6 of [Apostol] p. 18. Based on ideas by Eric Schmidt. (Contributed by NM, 10-Aug-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmul01 8871 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddid1 8872 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcnegex 8873* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcnegex2 8874* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddid2 8875 is a left identity for addition. This used to be one of our complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddcan 8876 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremaddcan2 8877 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcom 8878 Addition commutes. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddid1i 8879 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddid2i 8880 is a left identity for addition. (Contributed by Mario Carneiro, 3-Jan-2013.)

Theoremmul02i 8881 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)

Theoremmul01i 8882 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcomi 8883 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddcani 8885 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcan2i 8886 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmul12i 8887 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremmul32i 8888 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)

Theoremmul4i 8889 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)

Theoremmul02d 8890 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul01d 8891 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddid2d 8893 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcomd 8894 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddcand 8895 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcan2d 8896 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcanad 8897 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 8895. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddcan2ad 8898 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 8896. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintrd 8899 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 8897. Consequence of addcand 8895. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintr2d 8900 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 8898. Consequence of addcan2d 8896. (Contributed by David Moews, 28-Feb-2017.)

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