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Theorem List for Metamath Proof Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltned 9201 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremne0gt0d 9202 A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  0  <  A )
 
Theoremlttrid 9203 Ordering on reals satisfies strict trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
 
Theoremlttri2d 9204 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  =/=  B  <->  ( A  <  B  \/  B  <  A ) ) )
 
Theoremlttri3d 9205 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
 
Theoremlttri4d 9206 Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  \/  A  =  B  \/  B  <  A ) )
 
Theoremletri3d 9207 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  =  B  <->  ( A  <_  B 
 /\  B  <_  A ) ) )
 
Theoremleloed 9208 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  <  B  \/  A  =  B ) ) )
 
Theoremeqleltd 9209 Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  =  B  <->  ( A  <_  B 
 /\  -.  A  <  B ) ) )
 
Theoremltlend 9210 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  <_  B 
 /\  B  =/=  A ) ) )
 
Theoremlenltd 9211 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  -.  B  <  A ) )
 
Theoremltnled 9212 'Less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  -.  B  <_  A ) )
 
Theoremltled 9213 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremltnsymd 9214 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  -.  B  <  A )
 
Theoremletrid 9215 Trichotomy law for 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  \/  B  <_  A ) )
 
Theoremleltned 9216 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( A  <  B  <->  B  =/=  A ) )
 
Theoremmulgt0d 9217 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <  A )   &    |-  ( ph  ->  0  <  B )   =>    |-  ( ph  ->  0  <  ( A  x.  B ) )
 
Theoremltadd2d 9218 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) ) )
 
Theoremletrd 9219 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  A  <_  C )
 
Theoremlelttrd 9220 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremltadd2dd 9221 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( C  +  A )  <  ( C  +  B ) )
 
Theoremltletrd 9222 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremlttrd 9223 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  A  <  C )
 
5.2.5  Initial properties of the complex numbers
 
Theoremmul12 9224 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C ) ) )
 
Theoremmul32 9225 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B ) )
 
Theoremmul31 9226 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( ( C  x.  B )  x.  A ) )
 
Theoremmul4 9227 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )
 
Theoremmuladd11 9228 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  ( 1  +  B ) )  =  (
 ( 1  +  A )  +  ( B  +  ( A  x.  B ) ) ) )
 
Theorem1p1times 9229 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( ( 1  +  1 )  x.  A )  =  ( A  +  A ) )
 
Theorempeano2cn 9230 A theorem for complex numbers analogous the second Peano postulate peano2nn 10004. (Contributed by NM, 17-Aug-2005.)
 |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
 
Theorempeano2re 9231 A theorem for reals analogous the second Peano postulate peano2nn 10004. (Contributed by NM, 5-Jul-2005.)
 |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
 
Theoremreaddcan 9232 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( C  +  A )  =  ( C  +  B )  <->  A  =  B ) )
 
Theorem00id 9233  0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( 0  +  0 )  =  0
 
Theoremmul02lem1 9234 Lemma for mul02 9236. If any real does not produce  0 when multiplied by  0, then any complex is equal to double itself. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( ( ( A  e.  RR  /\  (
 0  x.  A )  =/=  0 )  /\  B  e.  CC )  ->  B  =  ( B  +  B ) )
 
Theoremmul02lem2 9235 Lemma for mul02 9236. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  RR  ->  ( 0  x.  A )  =  0 )
 
Theoremmul02 9236 Multiplication by  0. 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.)
 |-  ( A  e.  CC  ->  ( 0  x.  A )  =  0 )
 
Theoremmul01 9237 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( A  x.  0
 )  =  0 )
 
Theoremaddid1 9238  0 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.)
 |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
 
Theoremcnegex 9239* 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.)
 |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
 
Theoremcnegex2 9240* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
 
Theoremaddid2 9241  0 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.)
 |-  ( A  e.  CC  ->  ( 0  +  A )  =  A )
 
Theoremaddcan 9242 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.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
 
Theoremaddcan2 9243 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B ) )
 
Theoremaddcom 9244 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.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  =  ( B  +  A )
 )
 
Theoremaddid1i 9245  0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( A  +  0 )  =  A
 
Theoremaddid2i 9246  0 is a left identity for addition. (Contributed by Mario Carneiro, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( 0  +  A )  =  A
 
Theoremmul02i 9247 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( 0  x.  A )  =  0
 
Theoremmul01i 9248 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( A  x.  0 )  =  0
 
Theoremaddcomi 9249 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  B )  =  ( B  +  A )
 
Theoremaddcomli 9250 Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  +  B )  =  C   =>    |-  ( B  +  A )  =  C
 
Theoremaddcani 9251 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  =  ( A  +  C )  <->  B  =  C )
 
Theoremaddcan2i 9252 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  C )  =  ( B  +  C )  <->  A  =  B )
 
Theoremmul12i 9253 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.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C ) )
 
Theoremmul32i 9254 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B )
 
Theoremmul4i 9255 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D ) )
 
Theoremmul02d 9256 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 0  x.  A )  =  0 )
 
Theoremmul01d 9257 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  0 )  =  0 )
 
Theoremaddid1d 9258  0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  0 )  =  A )
 
Theoremaddid2d 9259  0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 0  +  A )  =  A )
 
Theoremaddcomd 9260 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  B )  =  ( B  +  A ) )
 
Theoremaddcand 9261 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
 
Theoremaddcan2d 9262 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  C )  =  ( B  +  C )  <->  A  =  B ) )
 
Theoremaddcanad 9263 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 9261. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  ( A  +  C ) )   =>    |-  ( ph  ->  B  =  C )
 
Theoremaddcan2ad 9264 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 9262. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  +  C )  =  ( B  +  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremaddneintrd 9265 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 9263. Consequence of addcand 9261. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  ( A  +  B )  =/=  ( A  +  C ) )
 
Theoremaddneintr2d 9266 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 9264. Consequence of addcan2d 9262. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  +  C )  =/=  ( B  +  C ) )
 
Theoremmul12d 9267 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C ) ) )
 
Theoremmul32d 9268 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B ) )
 
Theoremmul31d 9269 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  x.  C )  =  ( ( C  x.  B )  x.  A ) )
 
Theoremmul4d 9270 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )
 
5.3  Real and complex numbers - basic operations
 
5.3.1  Addition
 
Theoremadd12 9271 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C )
 )  =  ( B  +  ( A  +  C ) ) )
 
Theoremadd32 9272 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( ( A  +  C )  +  B ) )
 
Theoremadd4 9273 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( B  +  D ) ) )
 
Theoremadd42 9274 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( D  +  B ) ) )
 
Theoremadd12i 9275 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( A  +  ( B  +  C ) )  =  ( B  +  ( A  +  C )
 )
 
Theoremadd32i 9276 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  +  C )  =  ( ( A  +  C )  +  B )
 
Theoremadd4i 9277 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( B  +  D ) )
 
Theoremadd42i 9278 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( D  +  B ) )
 
Theoremadd12d 9279 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  +  ( B  +  C ) )  =  ( B  +  ( A  +  C )
 ) )
 
Theoremadd32d 9280 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  +  C )  =  ( ( A  +  C )  +  B ) )
 
Theoremadd4d 9281 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( B  +  D )
 ) )
 
Theoremadd42d 9282 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  +  ( C  +  D ) )  =  ( ( A  +  C )  +  ( D  +  B )
 ) )
 
5.3.2  Subtraction
 
Syntaxcmin 9283 Extend class notation to include subtraction.
 class  -
 
Syntaxcneg 9284 Extend class notation to include unary minus. The symbol  -u 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 ( -u) and subtraction cmin 9283 ( -) to prevent syntax ambiguity. For example, looking at the syntax definition co 6073, if we used the same symbol then " (  -  A  -  B ) " could mean either " -  A " minus " B", or it could represent the (meaningless) operation of classes " - " and " -  B " connected with "operation" " A". On the other hand, " ( -u A  -  B ) " is unambiguous.
 class  -u A
 
Definitiondf-sub 9285* Define subtraction. Theorem subval 9289 shows its value (and describes how this definition works), theorem subaddi 9379 relates it to addition, and theorems subcli 9368 and resubcli 9355 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
 |- 
 -  =  ( x  e.  CC ,  y  e.  CC  |->  ( iota_ z  e. 
 CC ( y  +  z )  =  x ) )
 
Definitiondf-neg 9286 Define the negative of a number (unary minus). We use different symbols for unary minus ( -u) and subtraction ( -) to prevent syntax ambiguity. See cneg 9284 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
 |-  -u A  =  (
 0  -  A )
 
Theorem0cnALT 9287 0 is a complex number. (Proved without referencing ax-1cn 9040. Compare 0cn 9076.) (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  0  e.  CC
 
Theoremnegeu 9288* Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E! x  e. 
 CC  ( A  +  x )  =  B )
 
Theoremsubval 9289* Value of subtraction, which is the (unique) element  x such that  B  +  x  =  A. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B )  =  ( iota_ x  e.  CC ( B  +  x )  =  A ) )
 
Theoremnegeq 9290 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
 |-  ( A  =  B  -> 
 -u A  =  -u B )
 
Theoremnegeqi 9291 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
 |-  A  =  B   =>    |-  -u A  =  -u B
 
Theoremnegeqd 9292 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  -u A  =  -u B )
 
Theoremnfnegd 9293 Deduction version of nfneg 9294. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x -u A )
 
Theoremnfneg 9294 Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x -u A
 
Theoremcsbnegg 9295 Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ -u B  =  -u [_ A  /  x ]_ B )
 
Theoremnegex 9296 A negative is a set. (Contributed by NM, 4-Apr-2005.)
 |-  -u A  e.  _V
 
Theoremsubcl 9297 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B )  e.  CC )
 
Theoremnegcl 9298 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
 |-  ( A  e.  CC  -> 
 -u A  e.  CC )
 
Theoremsubf 9299 Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |- 
 -  : ( CC 
 X.  CC ) --> CC
 
Theoremsubadd 9300 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  =  C  <->  ( B  +  C )  =  A ) )
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