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Theorem List for Metamath Proof Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltleii 9201 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)

Theoremltnei 9202 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)

Theoremletrii 9203 Trichotomy law for 'less than or equal to'. (Contributed by NM, 2-Aug-1999.)

Theoremlttri 9204 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremlelttri 9205 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)

Theoremltletri 9206 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)

Theoremletri 9207 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)

Theoremle2tri3i 9208 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)

Theoremltadd2i 9209 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Paul Chapman, 27-Jan-2008.)

Theoremmulgt0i 9210 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremmulgt0ii 9211 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltnrd 9212 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremgtned 9213 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltned 9214 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremne0gt0d 9215 A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttrid 9216 Ordering on reals satisfies strict trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri2d 9217 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri3d 9218 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri4d 9219 Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremletri3d 9220 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleloed 9221 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremeqleltd 9222 Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)

Theoremltlend 9223 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenltd 9224 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnled 9225 'Less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltled 9226 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnsymd 9227 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrid 9228 Trichotomy law for 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleltned 9229 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulgt0d 9230 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd2d 9231 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrd 9232 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)

Theoremlelttrd 9233 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)

Theoremltadd2dd 9234 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltletrd 9235 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)

Theoremlttrd 9236 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)

5.2.5  Initial properties of the complex numbers

Theoremmul12 9237 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)

Theoremmul32 9238 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)

Theoremmul31 9239 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul4 9240 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)

Theoremmuladd11 9241 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)

Theorem1p1times 9242 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempeano2cn 9243 A theorem for complex numbers analogous the second Peano postulate peano2nn 10017. (Contributed by NM, 17-Aug-2005.)

Theorempeano2re 9244 A theorem for reals analogous the second Peano postulate peano2nn 10017. (Contributed by NM, 5-Jul-2005.)

Theoremreaddcan 9245 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theorem00id 9246 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul02lem1 9247 Lemma for mul02 9249. 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 9248 Lemma for mul02 9249. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul02 9249 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 9250 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddid1 9251 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 9252* 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 9253* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddid2 9254 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 9255 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 9256 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcom 9257 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 9258 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddid2i 9259 is a left identity for addition. (Contributed by Mario Carneiro, 3-Jan-2013.)

Theoremmul02i 9260 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)

Theoremmul01i 9261 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcomi 9262 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddcani 9264 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 9265 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 9266 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 9267 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)

Theoremmul4i 9268 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)

Theoremmul02d 9269 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul01d 9270 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddid2d 9272 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcomd 9273 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddcand 9274 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcan2d 9275 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcanad 9276 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 9274. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddcan2ad 9277 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 9275. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintrd 9278 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 9276. Consequence of addcand 9274. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintr2d 9279 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 9277. Consequence of addcan2d 9275. (Contributed by David Moews, 28-Feb-2017.)

Theoremmul12d 9280 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul32d 9281 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul31d 9282 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul4d 9283 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)

5.3  Real and complex numbers - basic operations

Theoremadd12 9284 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)

Theoremadd32 9285 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)

Theoremadd4 9286 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremadd42 9287 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)

Theoremadd12i 9288 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd32i 9289 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd4i 9290 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)

Theoremadd42i 9291 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)

Theoremadd12d 9292 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd32d 9293 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd4d 9294 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd42d 9295 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

5.3.2  Subtraction

Syntaxcmin 9296 Extend class notation to include subtraction.

Syntaxcneg 9297 Extend class notation to include unary minus. The symbol is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus () and subtraction cmin 9296 () to prevent syntax ambiguity. For example, looking at the syntax definition co 6084, if we used the same symbol then " " could mean either " " minus "", or it could represent the (meaningless) operation of classes " " and " " connected with "operation" "". On the other hand, " " is unambiguous.

Definitiondf-sub 9298* Define subtraction. Theorem subval 9302 shows its value (and describes how this definition works), theorem subaddi 9392 relates it to addition, and theorems subcli 9381 and resubcli 9368 prove its closure laws. (Contributed by NM, 26-Nov-1994.)

Definitiondf-neg 9299 Define the negative of a number (unary minus). We use different symbols for unary minus () and subtraction () to prevent syntax ambiguity. See cneg 9297 for a discussion of this. (Contributed by NM, 10-Feb-1995.)

Theorem0cnALT 9300 0 is a complex number. (Proved without referencing ax-1cn 9053. Compare 0cn 9089.) (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

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