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Theorem List for Metamath Proof Explorer - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnegeqd 9301 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)

Theoremnfnegd 9302 Deduction version of nfneg 9303. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfneg 9303 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 9304 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.)

Theoremnegex 9305 A negative is a set. (Contributed by NM, 4-Apr-2005.)

Theoremsubcl 9306 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremnegcl 9307 Closure law for negative. (Contributed by NM, 6-Aug-2003.)

Theoremsubf 9308 Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremsubadd 9309 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremsubadd2 9310 Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsubsub23 9311 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)

Theorempncan 9312 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempncan2 9313 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)

Theorempncan3 9314 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)

Theoremnpcan 9315 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddsubass 9316 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddsub 9317 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremsubadd23 9318 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)

Theoremaddsub12 9319 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremaddsubeq4 9321 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremsubid 9322 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubid1 9323 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnpncan 9324 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremnppcan 9325 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)

Theoremnppcan3 9326 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremsubcan2 9327 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremsubeq0 9328 If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)

Theoremnpncan2 9329 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)

Theoremsubsub2 9330 Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnncan 9331 Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremsubsub 9332 Law for double subtraction. (Contributed by NM, 13-May-2004.)

Theoremnppcan2 9333 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)

Theoremsubsub3 9334 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)

Theoremsubsub4 9335 Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsub32 9336 Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)

Theoremnnncan 9337 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)

Theoremnnncan1 9338 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremnnncan2 9339 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)

Theoremnpncan3 9340 Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theorempnpcan 9341 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempnpcan2 9342 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)

Theorempnncan 9343 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremppncan 9344 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)

Theoremaddsub4 9345 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)

Theoremsubadd4 9346 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)

Theoremsub4 9347 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)

Theoremneg0 9348 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)

Theoremnegid 9349 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)

Theoremnegsub 9350 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 9351 Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegneg 9352 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 9353 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegcon1 9354 Negative contraposition law. (Contributed by NM, 9-May-2004.)

Theoremnegcon2 9355 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)

Theoremnegeq0 9356 A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubcan 9357 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegsubdi 9358 Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremnegdi 9359 Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremnegdi2 9360 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)

Theoremnegsubdi2 9361 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)

Theoremneg2sub 9362 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)

Theoremrenegcli 9363 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 9365 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremresubcli 9364 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremrenegcl 9365 Closure law for negative of reals. The weak deduction theorem dedth 3781 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 9363, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.)

Theoremresubcl 9366 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)

Theoremnegreb 9367 The negative of a real is real. (Contributed by NM, 11-Aug-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theorempeano2rem 9368 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)

Theoremnegcli 9369 Closure law for negative. (Contributed by NM, 26-Nov-1994.)

Theoremnegidi 9370 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)

Theoremnegnegi 9371 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 9372 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)

Theoremsubid1i 9373 Identity law for subtraction. (Contributed by NM, 29-May-1999.)

Theoremnegne0bi 9374 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)

Theoremnegrebi 9375 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)

Theoremnegne0i 9376 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)

Theoremsubcli 9377 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theorempncan3i 9378 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)

Theoremnegsubi 9379 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 9380 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)

Theoremsubeq0i 9381 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)

Theoremneg11i 9382 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)

Theoremnegcon1i 9383 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)

Theoremnegcon2i 9384 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)

Theoremnegdii 9385 Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremnegsubdii 9386 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)

Theoremnegsubdi2i 9387 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)

Theoremsubaddi 9388 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremsubsub23i 9391 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)

Theoremaddsubassi 9392 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)

Theoremsubcani 9394 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremsubcan2i 9395 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theorempnncani 9396 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)

Theoremaddsub4i 9397 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.)

Theorem0reALT 9398 0 is a real number. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnegcld 9399 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubidd 9400 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)

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