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Theorem List for Metamath Proof Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlenlt 9101 'Less than or equal to' expressed in terms of 'less than'. (Contributed by NM, 13-May-1999.)

Theoremltnle 9102 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.)

Theoremltso 9103 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)

Theoremlttri2 9104 Consequence of trichotomy. (Contributed by NM, 9-Oct-1999.)

Theoremlttri3 9105 Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999.)

Theoremlttri4 9106 Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremletri3 9107 Trichotomy law. (Contributed by NM, 14-May-1999.)

Theoremleloe 9108 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by NM, 13-May-1999.)

Theoremeqlelt 9109 Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)

Theoremltle 9110 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)

Theoremleltne 9111 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by NM, 27-Jul-1999.)

Theoremlelttr 9112 Transitive law. (Contributed by NM, 23-May-1999.)

Theoremltletr 9113 Transitive law. (Contributed by NM, 25-Aug-1999.)

Theoremletr 9114 Transitive law. (Contributed by NM, 12-Nov-1999.)

Theoremltnr 9115 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)

Theoremleid 9116 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)

Theoremltne 9117 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)

TheoremltneOLD 9118 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremltnsym 9119 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)

Theoremltnsym2 9120 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremletric 9121 Trichotomy law. (Contributed by NM, 18-Aug-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltlen 9122 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.)

Theoremeqle 9123 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)

Theoremltadd2 9124 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremne0gt0 9125 A nonzero nonnegative number is positive. (Contributed by NM, 20-Nov-2007.)

Theoremlecasei 9126 Ordering elimination by cases. (Contributed by NM, 6-Jul-2007.)

Theoremlelttric 9127 Trichotomy law. (Contributed by NM, 4-Apr-2005.)

Theoremltlecasei 9128 Ordering elimination by cases. (Contributed by NM, 1-Jul-2007.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltnri 9129 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)

Theoremgtneii 9130 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.)

Theoremltneii 9131 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)

Theoremlttri2i 9132 Consequence of trichotomy. (Contributed by NM, 19-Jan-1997.)

Theoremlttri3i 9133 Consequence of trichotomy. (Contributed by NM, 14-May-1999.)

Theoremletri3i 9134 Consequence of trichotomy. (Contributed by NM, 14-May-1999.)

Theoremleloei 9135 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 14-May-1999.)

Theoremltleni 9136 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 27-Oct-1999.)

Theoremltnsymi 9137 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)

Theoremlenlti 9138 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)

Theoremltnlei 9139 'Less than' in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.)

Theoremltlei 9140 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)

Theoremltleii 9141 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)

Theoremeqlei 9142 Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.)

Theoremltnei 9143 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)

Theoremletrii 9144 Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999.)

Theoremlttri 9145 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremlelttri 9146 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)

Theoremltletri 9147 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)

Theoremletri 9148 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)

Theoremle2tri3i 9149 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)

Theoremltadd2i 9150 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Paul Chapman, 27-Jan-2008.)

Theoremmulgt0i 9151 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremmulgt0ii 9152 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltnrd 9153 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremgtned 9154 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltned 9155 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremne0gt0d 9156 A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttrid 9157 Ordering on reals satisfies strict trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri2d 9158 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri3d 9159 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri4d 9160 Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremletri3d 9161 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleloed 9162 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremeqleltd 9163 Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)

Theoremltlend 9164 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenltd 9165 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnled 9166 'Less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltled 9167 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnsymd 9168 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrid 9169 Trichotomy law for 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleltned 9170 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulgt0d 9171 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd2d 9172 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrd 9173 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)

Theoremlelttrd 9174 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)

Theoremltadd2dd 9175 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltletrd 9176 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)

Theoremlttrd 9177 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)

5.2.5  Initial properties of the complex numbers

Theoremmul12 9178 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)

Theoremmul32 9179 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)

Theoremmul31 9180 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul4 9181 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)

Theoremmuladd11 9182 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)

Theorem1p1times 9183 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempeano2cn 9184 A theorem for complex numbers analogous the second Peano postulate peano2nn 9958. (Contributed by NM, 17-Aug-2005.)

Theorempeano2re 9185 A theorem for reals analogous the second Peano postulate peano2nn 9958. (Contributed by NM, 5-Jul-2005.)

Theoremreaddcan 9186 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theorem00id 9187 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul02lem1 9188 Lemma for mul02 9190. 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 9189 Lemma for mul02 9190. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul02 9190 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 9191 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddid1 9192 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 9193* 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 9194* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddid2 9195 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 9196 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 9197 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcom 9198 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 9199 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddid2i 9200 is a left identity for addition. (Contributed by Mario Carneiro, 3-Jan-2013.)

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