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Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnegidd 9401 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegnegd 9402 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegeq0d 9403 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegne0bd 9404 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegcon1d 9405 Contraposition law for unary minus. Deduction form of negcon1 9353. (Contributed by David Moews, 28-Feb-2017.)

Theoremnegcon1ad 9406 Contraposition law for unary minus. One-way deduction form of negcon1 9353. (Contributed by David Moews, 28-Feb-2017.)

Theoremneg11ad 9407 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 9352. Generalization of neg11d 9423. (Contributed by David Moews, 28-Feb-2017.)

Theoremnegned 9408 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 9423. (Contributed by David Moews, 28-Feb-2017.)

Theoremnegne0d 9409 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegrebd 9410 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsubcld 9411 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempncand 9412 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempncan2d 9413 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempncan3d 9414 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnpcand 9415 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnncand 9416 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegsubd 9417 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubnegd 9418 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubeq0d 9419 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubne0d 9420 Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremsubeq0ad 9421 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 9327. Generalization of subeq0d 9419. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubne0ad 9422 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 9420. Contrapositive of subeq0bd 9463. (Contributed by David Moews, 28-Feb-2017.)

Theoremneg11d 9423 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegdid 9424 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegdi2d 9425 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegsubdid 9426 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegsubdi2d 9427 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremneg2subd 9428 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubaddd 9429 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubadd2d 9430 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsubassd 9431 Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsubd 9432 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubadd23d 9433 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsub12d 9434 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnpncand 9435 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnppcand 9436 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnppcan2d 9437 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnppcan3d 9438 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsubd 9439 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsub2d 9440 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsub3d 9441 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsub4d 9442 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsub32d 9443 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnncand 9444 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnncan1d 9445 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnncan2d 9446 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnpncan3d 9447 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempnpcand 9448 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempnpcan2d 9449 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempnncand 9450 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremppncand 9451 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubcand 9452 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubcan2d 9453 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)

Theoremsubcanad 9454 Cancellation law for subtraction. Deduction form of subcan 9356. Generalization of subcand 9452. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubneintrd 9455 Introducing subtraction on both sides of a statement of nonequality. Contrapositive of subcand 9452. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubcan2ad 9456 Cancellation law for subtraction. Deduction form of subcan2 9326. Generalization of subcan2d 9453. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubneintr2d 9457 Introducing subtraction on both sides of a statement of nonequality. Contrapositive of subcan2d 9453. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddsub4d 9458 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubadd4d 9459 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsub4d 9460 Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorem2addsubd 9461 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsubeq4d 9462 Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubeq0bd 9463 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 9421. Converse of subeq0d 9419. Contrapositive of subne0ad 9422. (Contributed by David Moews, 28-Feb-2017.)

Theoremrenegcld 9464 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremresubcld 9465 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)

5.3.3  Multiplication

Theoremmuladd 9466 Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremsubdi 9467 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)

Theoremsubdir 9468 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)

Theoremine0 9469 The imaginary unit is not zero. (Contributed by NM, 6-May-1999.)

Theoremmulneg1 9470 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmulneg2 9471 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)

Theoremmulneg12 9472 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)

Theoremmul2neg 9473 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremsubmul2 9474 Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)

Theoremmulm1 9475 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)

Theoremmulsub 9476 Product of two differences. (Contributed by NM, 14-Jan-2006.)

Theoremmulsub2 9477 Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremmulm1i 9478 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)

Theoremmulneg1i 9479 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulneg2i 9480 Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmul2negi 9481 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubdii 9482 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)

Theoremsubdiri 9483 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)

Theoremmuladdi 9484 Product of two sums. (Contributed by NM, 17-May-1999.)

Theoremmulm1d 9485 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulneg1d 9486 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulneg2d 9487 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul2negd 9488 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubdid 9489 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubdird 9490 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmuladdd 9491 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulsubd 9492 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)

5.3.4  Ordering on reals (cont.)

Theoremgt0ne0 9493 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt0ne0 9494 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremltadd1 9495 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremleadd1 9496 Addition to both sides of 'less than or equal to'. (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremleadd2 9497 Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)

Theoremltsubadd 9498 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltsubadd2 9499 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)

Theoremlesubadd 9500 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

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