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Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2addsub 8201 Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( ( A  +  B )  +  C )  -  D )  =  ( (
 ( A  +  C )  -  D )  +  B ) )
 
Theoremaddsubeq4 8202 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  =  ( C  +  D )  <->  ( C  -  A )  =  ( B  -  D ) ) )
 
Theorempncan3oi 8203 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 8264 and pncan 8193, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 8299. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  +  B )  -  B )  =  A
 
Theoremmvrraddi 8204 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  B  e.  CC   &    |-  C  e.  CC   &    |-  A  =  ( B  +  C )   =>    |-  ( A  -  C )  =  B
 
Theoremmvlladdi 8205 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  +  B )  =  C   =>    |-  B  =  ( C  -  A )
 
Theoremsubid 8206 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  -  A )  =  0 )
 
Theoremsubid1 8207 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  -  0
 )  =  A )
 
Theoremnpncan 8208 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  +  ( B  -  C ) )  =  ( A  -  C ) )
 
Theoremnppcan 8209 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  -  B )  +  C )  +  B )  =  ( A  +  C ) )
 
Theoremnnpcan 8210 Cancellation law for subtraction: ((a-b)-c)+b = a-c holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  -  B )  -  C )  +  B )  =  ( A  -  C ) )
 
Theoremnppcan3 8211 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  +  ( C  +  B )
 )  =  ( A  +  C ) )
 
Theoremsubcan2 8212 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =  ( B  -  C )  <->  A  =  B ) )
 
Theoremsubeq0 8213 If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  =  0  <->  A  =  B ) )
 
Theoremnpncan2 8214 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  ( B  -  A ) )  =  0
 )
 
Theoremsubsub2 8215 Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  -  C ) )  =  ( A  +  ( C  -  B ) ) )
 
Theoremnncan 8216 Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  ( A  -  B ) )  =  B )
 
Theoremsubsub 8217 Law for double subtraction. (Contributed by NM, 13-May-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  -  C ) )  =  ( ( A  -  B )  +  C ) )
 
Theoremnppcan2 8218 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  ( B  +  C ) )  +  C )  =  ( A  -  B ) )
 
Theoremsubsub3 8219 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  -  C ) )  =  ( ( A  +  C )  -  B ) )
 
Theoremsubsub4 8220 Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  -  C )  =  ( A  -  ( B  +  C ) ) )
 
Theoremsub32 8221 Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  -  C )  =  ( ( A  -  C )  -  B ) )
 
Theoremnnncan 8222 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  ( B  -  C ) )  -  C )  =  ( A  -  B ) )
 
Theoremnnncan1 8223 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  -  ( A  -  C ) )  =  ( C  -  B ) )
 
Theoremnnncan2 8224 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  -  ( B  -  C ) )  =  ( A  -  B ) )
 
Theoremnpncan3 8225 Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  +  ( C  -  A ) )  =  ( C  -  B ) )
 
Theorempnpcan 8226 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  -  ( A  +  C )
 )  =  ( B  -  C ) )
 
Theorempnpcan2 8227 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  C )  -  ( B  +  C )
 )  =  ( A  -  B ) )
 
Theorempnncan 8228 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  C ) )  =  ( B  +  C ) )
 
Theoremppncan 8229 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  ( C  -  B ) )  =  ( A  +  C ) )
 
Theoremaddsub4 8230 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  -  ( C  +  D )
 )  =  ( ( A  -  C )  +  ( B  -  D ) ) )
 
Theoremsubadd4 8231 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  -  ( C  -  D ) )  =  ( ( A  +  D )  -  ( B  +  C ) ) )
 
Theoremsub4 8232 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  -  ( C  -  D ) )  =  ( ( A  -  C )  -  ( B  -  D ) ) )
 
Theoremneg0 8233 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
 |-  -u 0  =  0
 
Theoremnegid 8234 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
 |-  ( A  e.  CC  ->  ( A  +  -u A )  =  0 )
 
Theoremnegsub 8235 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
 
Theoremsubneg 8236 Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  -u B )  =  ( A  +  B ) )
 
Theoremnegneg 8237 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  -> 
 -u -u A  =  A )
 
Theoremneg11 8238 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  =  -u B  <->  A  =  B ) )
 
Theoremnegcon1 8239 Negative contraposition law. (Contributed by NM, 9-May-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  =  B  <->  -u B  =  A ) )
 
Theoremnegcon2 8240 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  -u B  <->  B  =  -u A ) )
 
Theoremnegeq0 8241 A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
 
Theoremsubcan 8242 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  =  ( A  -  C )  <->  B  =  C ) )
 
Theoremnegsubdi 8243 Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  -  B )  =  ( -u A  +  B ) )
 
Theoremnegdi 8244 Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  +  B )  =  ( -u A  +  -u B ) )
 
Theoremnegdi2 8245 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  +  B )  =  ( -u A  -  B ) )
 
Theoremnegsubdi2 8246 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  -  B )  =  ( B  -  A ) )
 
Theoremneg2sub 8247 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  -  -u B )  =  ( B  -  A ) )
 
Theoremrenegcl 8248 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
 |-  ( A  e.  RR  -> 
 -u A  e.  RR )
 
Theoremrenegcli 8249 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 8248 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  RR   =>    |-  -u A  e.  RR
 
Theoremresubcli 8250 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  -  B )  e.  RR
 
Theoremresubcl 8251 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B )  e.  RR )
 
Theoremnegreb 8252 The negative of a real is real. (Contributed by NM, 11-Aug-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( -u A  e.  RR  <->  A  e.  RR ) )
 
Theorempeano2cnm 8253 "Reverse" second Peano postulate analog for complex numbers: A complex number minus 1 is a complex number. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  ( N  e.  CC  ->  ( N  -  1
 )  e.  CC )
 
Theorempeano2rem 8254 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( N  e.  RR  ->  ( N  -  1
 )  e.  RR )
 
Theoremnegcli 8255 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  -u A  e.  CC
 
Theoremnegidi 8256 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  +  -u A )  =  0
 
Theoremnegnegi 8257 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 8-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  CC   =>    |-  -u -u A  =  A
 
Theoremsubidi 8258 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  -  A )  =  0
 
Theoremsubid1i 8259 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  -  0 )  =  A
 
Theoremnegne0bi 8260 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0 
 <->  -u A  =/=  0
 )
 
Theoremnegrebi 8261 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( -u A  e.  RR  <->  A  e.  RR )
 
Theoremnegne0i 8262 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  -u A  =/=  0
 
Theoremsubcli 8263 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  -  B )  e.  CC
 
Theorempncan3i 8264 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  ( B  -  A ) )  =  B
 
Theoremnegsubi 8265 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  -u B )  =  ( A  -  B )
 
Theoremsubnegi 8266 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  -  -u B )  =  ( A  +  B )
 
Theoremsubeq0i 8267 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  -  B )  =  0  <->  A  =  B )
 
Theoremneg11i 8268 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  =  -u B 
 <->  A  =  B )
 
Theoremnegcon1i 8269 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  =  B  <->  -u B  =  A )
 
Theoremnegcon2i 8270 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  =  -u B 
 <->  B  =  -u A )
 
Theoremnegdii 8271 Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  +  B )  =  ( -u A  +  -u B )
 
Theoremnegsubdii 8272 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  -  B )  =  ( -u A  +  B )
 
Theoremnegsubdi2i 8273 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  -  B )  =  ( B  -  A )
 
Theoremsubaddi 8274 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  C  <->  ( B  +  C )  =  A )
 
Theoremsubadd2i 8275 Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  C  <->  ( C  +  B )  =  A )
 
Theoremsubaddrii 8276 Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  ( B  +  C )  =  A   =>    |-  ( A  -  B )  =  C
 
Theoremsubsub23i 8277 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  C  <->  ( A  -  C )  =  B )
 
Theoremaddsubassi 8278 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  -  C )  =  ( A  +  ( B  -  C ) )
 
Theoremaddsubi 8279 Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  -  C )  =  ( ( A  -  C )  +  B )
 
Theoremsubcani 8280 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  ( A  -  C )  <->  B  =  C )
 
Theoremsubcan2i 8281 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  C )  =  ( B  -  C )  <->  A  =  B )
 
Theorempnncani 8282 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  -  ( A  -  C ) )  =  ( B  +  C )
 
Theoremaddsub4i 8283 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  -  ( C  +  D )
 )  =  ( ( A  -  C )  +  ( B  -  D ) )
 
Theorem0reALT 8284 Alternate proof of 0re 7987. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  0  e.  RR
 
Theoremnegcld 8285 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  -u A  e.  CC )
 
Theoremsubidd 8286 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  -  A )  =  0 )
 
Theoremsubid1d 8287 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  -  0 )  =  A )
 
Theoremnegidd 8288 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  -u A )  =  0 )
 
Theoremnegnegd 8289 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  -u -u A  =  A )
 
Theoremnegeq0d 8290 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  0  <->  -u A  =  0 ) )
 
Theoremnegne0bd 8291 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =/=  0  <->  -u A  =/=  0
 ) )
 
Theoremnegcon1d 8292 Contraposition law for unary minus. Deduction form of negcon1 8239. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  =  B  <->  -u B  =  A )
 )
 
Theoremnegcon1ad 8293 Contraposition law for unary minus. One-way deduction form of negcon1 8239. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  -u A  =  B )   =>    |-  ( ph  ->  -u B  =  A )
 
Theoremneg11ad 8294 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 8238. Generalization of neg11d 8310. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  =  -u B 
 <->  A  =  B ) )
 
Theoremnegned 8295 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 8310. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  -u A  =/=  -u B )
 
Theoremnegne0d 8296 The negative of a nonzero number is nonzero. See also negap0d 8618 which is similar but for apart from zero rather than not equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  -u A  =/=  0 )
 
Theoremnegrebd 8297 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  -u A  e.  RR )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremsubcld 8298 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  -  B )  e.  CC )
 
Theorempncand 8299 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  -  B )  =  A )
 
Theorempncan2d 8300 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  -  A )  =  B )
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