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

Theoremltsub2dd 9401 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd1dd 9402 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd2dd 9403 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub1dd 9404 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub2dd 9405 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremle2addd 9406 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle2subd 9407 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltleaddd 9408 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleltaddd 9409 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2addd 9410 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2subd 9411 Adding both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)

Theorem1le1 9412 . Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

5.3.5  Reciprocals

Theoremixi 9413 times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremrecextlem1 9414 Lemma for recex 9416. (Contributed by Eric Schmidt, 23-May-2007.)

Theoremrecextlem2 9415 Lemma for recex 9416. (Contributed by Eric Schmidt, 23-May-2007.)

Theoremrecex 9416* Existence of reciprocal of nonzero complex number. (Contributed by Eric Schmidt, 22-May-2007.)

Theoremmulcand 9417 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulcan2d 9418 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulcanad 9419 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand 9417. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulcan2ad 9420 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d 9418. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulcan 9421 Cancellation law for multiplication (full theorem form). Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulcan2 9422 Cancellation law for multiplication. (Contributed by NM, 21-Jan-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulcani 9423 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.)

Theoremmul0or 9424 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulne0b 9425 The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremmulne0 9426 The product of two nonzero numbers is nonzero. (Contributed by NM, 30-Dec-2007.)

Theoremmulne0i 9427 The product of two nonzero numbers is nonzero. (Contributed by NM, 15-Feb-1995.)

Theoremmuleqadd 9428 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)

Theoremreceu 9429* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremmulnzcnopr 9430 Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.)

Theoremmsq0i 9431 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by NM, 28-Jul-1999.)

Theoremmul0ori 9432 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 7-Oct-1999.)

Theoremmsq0d 9433 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul0ord 9434 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulne0bd 9435 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulne0d 9436 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulne0bad 9437 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9436 and consequence of mulne0bd 9435. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulne0bbd 9438 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9436 and consequence of mulne0bd 9435. (Contributed by David Moews, 28-Feb-2017.)

5.3.6  Division

Syntaxcdiv 9439 Extend class notation to include division.

Definitiondf-div 9440* Define division. Theorem divmuli 9530 relates it to multiplication, and divcli 9518 and redivcli 9543 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)

Theorem1div0 9441 You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)

Theoremdivval 9442* Value of division: the (unique) element such that . This is meaningful only when is nonzero. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivmul 9443 Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivmul2 9444 Relationship between division and multiplication. (Contributed by NM, 7-Feb-2006.)

Theoremdivmul3 9445 Relationship between division and multiplication. (Contributed by NM, 13-Feb-2006.)

Theoremdivcl 9446 Closure law for division. (Contributed by NM, 21-Jul-2001.) (Proof shortened by Mario Carneiro, 17-Feb-2014.)

Theoremreccl 9447 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremdivcan2 9448 A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremdivcan1 9449 A cancellation law for division. (Contributed by NM, 5-Jun-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremdiveq0 9450 A ratio is zero iff the numerator is zero. (Contributed by NM, 20-Apr-2006.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremdivne0b 9451 The ratio of nonzero numbers is nonzero. (Contributed by NM, 2-Aug-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivne0 9452 The ratio of nonzero numbers is nonzero. (Contributed by NM, 28-Dec-2007.)

Theoremrecne0 9453 The reciprocal of a nonzero number is nonzero. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremrecid 9454 Multiplication of a number and its reciprocal. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremrecid2 9455 Multiplication of a number and its reciprocal. (Contributed by NM, 22-Jun-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivrec 9456 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremdivrec2 9457 Relationship between division and reciprocal. (Contributed by NM, 7-Feb-2006.)

Theoremdivass 9458 An associative law for division. (Contributed by NM, 2-Aug-2004.)

Theoremdiv23 9459 A commutative/associative law for division. (Contributed by NM, 2-Aug-2004.)

Theoremdiv32 9460 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv13 9461 A commutative/associative law for division. (Contributed by NM, 22-Apr-2005.)

Theoremdiv12 9462 A commutative/associative law for division. (Contributed by NM, 30-Apr-2005.)

Theoremdivdir 9463 Distribution of division over addition. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivcan3 9464 A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivcan4 9465 A cancellation law for division. (Contributed by NM, 8-Feb-2005.)

Theoremdiv11 9466 One-to-one relationship for division. (Contributed by NM, 20-Apr-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivid 9467 A number divided by itself is one. (Contributed by NM, 1-Aug-2004.)

Theoremdiv0 9468 Division into zero is zero. (Contributed by NM, 14-Mar-2005.)

Theoremdiv1 9469 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdiveq1 9470 Equality in terms of unit ratio. (Contributed by Stefan O'Rear, 27-Aug-2015.)

Theoremdivneg 9471 Move negative sign inside of a division. (Contributed by NM, 17-Sep-2004.)

Theoremdivsubdir 9472 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)

Theoremrecrec 9473 A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 26-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremrec11 9474 Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremrec11r 9475 Mutual reciprocals. (Contributed by Paul Chapman, 18-Oct-2007.)

Theoremdivmuldiv 9476 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.)

Theoremdivdivdiv 9477 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.)

Theoremdivcan5 9478 Cancellation of common factor in a ratio. (Contributed by NM, 9-Jan-2006.)

Theoremdivmul13 9479 Swap the denominators in the product of two ratios. (Contributed by NM, 3-May-2005.)

Theoremdivmul24 9480 Swap the numerators in the product of two ratios. (Contributed by NM, 3-May-2005.)

Theoremdivmuleq 9481 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremrecdiv 9482 The reciprocal of a ratio. (Contributed by NM, 3-Aug-2004.)

Theoremdivcan6 9483 Cancellation of inverted fractions. (Contributed by NM, 28-Dec-2007.)

Theoremdivdiv32 9484 Swap denominators in a division. (Contributed by NM, 2-Aug-2004.)

Theoremdivcan7 9485 Cancel equal divisors in a division. (Contributed by Jeff Hankins, 29-Sep-2013.)

Theoremdmdcan 9486 Cancellation law for division and multiplication. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Fan Zheng, 3-Jul-2016.)

Theoremdivdiv1 9487 Division into a fraction. (Contributed by NM, 31-Dec-2007.)

Theoremdivdiv2 9488 Division by a fraction. (Contributed by NM, 27-Dec-2008.)

Theoremrecdiv2 9489 Division into a reciprocal. (Contributed by NM, 19-Oct-2007.)

Theoremddcan 9490 Cancellation in a double division. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdivadddiv 9491 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 2-May-2016.)

Theoremdivsubdiv 9492 Subtraction of two ratios. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)

Theoremconjmul 9493 Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 12-Nov-2006.)

Theoremrereccl 9494 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremredivcl 9495 Closure law for division of reals. (Contributed by NM, 27-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremeqneg 9496 A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremeqnegd 9497 A complex number equals its negative iff it is zero. Deduction form of eqneg 9496. (Contributed by David Moews, 28-Feb-2017.)

Theoremeqnegad 9498 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 9496. (Contributed by David Moews, 28-Feb-2017.)

Theoremdiv2neg 9499 Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012.)

Theoremdivneg2 9500 Move negative sign inside of a division. (Contributed by Mario Carneiro, 15-Sep-2014.)

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