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Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem0cnALT 7201 Alternate proof of 0cn 7019. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnegeu 7202* 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.)

Theoremsubval 7203* Value of subtraction, which is the (unique) element such that . (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremnegeq 7204 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)

Theoremnegeqi 7205 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)

Theoremnegeqd 7206 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)

Theoremnfnegd 7207 Deduction version of nfneg 7208. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfneg 7208 Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbnegg 7209 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.)

Theoremsubcl 7210 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremnegcl 7211 Closure law for negative. (Contributed by NM, 6-Aug-2003.)

Theoremnegicn 7212 is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)

Theoremsubf 7213 Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremsubadd 7214 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremsubadd2 7215 Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsubsub23 7216 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)

Theorempncan 7217 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempncan2 7218 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)

Theorempncan3 7219 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)

Theoremnpcan 7220 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddsubass 7221 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddsub 7222 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremsubadd23 7223 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)

Theoremaddsub12 7224 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremaddsubeq4 7226 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theorempncan3oi 7227 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 7288 and pncan 7217, 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 7323. (Contributed by David A. Wheeler, 11-Oct-2018.)

Theoremmvrraddi 7228 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)

Theoremmvlladdi 7229 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)

Theoremsubid 7230 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubid1 7231 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnpncan 7232 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremnppcan 7233 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)

Theoremnnpcan 7234 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.)

Theoremnppcan3 7235 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremsubcan2 7236 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremsubeq0 7237 If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)

Theoremnpncan2 7238 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)

Theoremsubsub2 7239 Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnncan 7240 Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremsubsub 7241 Law for double subtraction. (Contributed by NM, 13-May-2004.)

Theoremnppcan2 7242 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)

Theoremsubsub3 7243 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)

Theoremsubsub4 7244 Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsub32 7245 Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)

Theoremnnncan 7246 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)

Theoremnnncan1 7247 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremnnncan2 7248 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)

Theoremnpncan3 7249 Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theorempnpcan 7250 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempnpcan2 7251 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)

Theorempnncan 7252 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremppncan 7253 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)

Theoremaddsub4 7254 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)

Theoremsubadd4 7255 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)

Theoremsub4 7256 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)

Theoremneg0 7257 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)

Theoremnegid 7258 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)

Theoremnegsub 7259 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.)

Theoremsubneg 7260 Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegneg 7261 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.)

Theoremneg11 7262 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegcon1 7263 Negative contraposition law. (Contributed by NM, 9-May-2004.)

Theoremnegcon2 7264 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)

Theoremnegeq0 7265 A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubcan 7266 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegsubdi 7267 Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremnegdi 7268 Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremnegdi2 7269 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)

Theoremnegsubdi2 7270 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)

Theoremneg2sub 7271 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)

Theoremrenegcl 7272 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)

Theoremrenegcli 7273 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 7272 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremresubcli 7274 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremresubcl 7275 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)

Theoremnegreb 7276 The negative of a real is real. (Contributed by NM, 11-Aug-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theorempeano2cnm 7277 "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.)

Theorempeano2rem 7278 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)

Theoremnegcli 7279 Closure law for negative. (Contributed by NM, 26-Nov-1994.)

Theoremnegidi 7280 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)

Theoremnegnegi 7281 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.)

Theoremsubidi 7282 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)

Theoremsubid1i 7283 Identity law for subtraction. (Contributed by NM, 29-May-1999.)

Theoremnegne0bi 7284 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)

Theoremnegrebi 7285 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)

Theoremnegne0i 7286 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)

Theoremsubcli 7287 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theorempncan3i 7288 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)

Theoremnegsubi 7289 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.)

Theoremsubnegi 7290 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)

Theoremsubeq0i 7291 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)

Theoremneg11i 7292 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)

Theoremnegcon1i 7293 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)

Theoremnegcon2i 7294 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)

Theoremnegdii 7295 Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremnegsubdii 7296 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)

Theoremnegsubdi2i 7297 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)

Theoremsubaddi 7298 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)