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Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlenegcon2d 9601 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddposd 9602 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddpos2d 9603 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubposd 9604 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremposdifd 9605 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge01d 9606 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge02d 9607 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge0d 9608 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsuble0d 9609 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge02d 9610 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd1d 9611 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleadd1d 9612 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleadd2d 9613 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubaddd 9614 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubaddd 9615 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubadd2d 9616 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubadd2d 9617 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddsubd 9618 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddsub2d 9619 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremleaddsub2d 9620 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubled 9621 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubd 9622 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub23d 9623 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub13d 9624 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesub1d 9625 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesub2d 9626 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub1d 9627 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub2d 9628 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd1dd 9629 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltsub1dd 9630 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltsub2dd 9631 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd1dd 9632 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd2dd 9633 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub1dd 9634 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub2dd 9635 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremle2addd 9636 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle2subd 9637 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltleaddd 9638 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleltaddd 9639 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2addd 9640 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2subd 9641 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)

Theorem1le1 9642 . Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

5.3.5  Reciprocals

Theoremixi 9643 times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremrecextlem1 9644 Lemma for recex 9646. (Contributed by Eric Schmidt, 23-May-2007.)

Theoremrecextlem2 9645 Lemma for recex 9646. (Contributed by Eric Schmidt, 23-May-2007.)

Theoremrecex 9646* Existence of reciprocal of nonzero complex number. (Contributed by Eric Schmidt, 22-May-2007.)

Theoremmulcand 9647 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 9648 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulcanad 9649 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand 9647. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulcan2ad 9650 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d 9648. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulcan 9651 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 9652 Cancellation law for multiplication. (Contributed by NM, 21-Jan-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulcani 9653 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.)

Theoremmul0or 9654 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 9655 The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremmulne0 9656 The product of two nonzero numbers is nonzero. (Contributed by NM, 30-Dec-2007.)

Theoremmulne0i 9657 The product of two nonzero numbers is nonzero. (Contributed by NM, 15-Feb-1995.)

Theoremmuleqadd 9658 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)

Theoremreceu 9659* 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 9660 Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.)

Theoremmsq0i 9661 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by NM, 28-Jul-1999.)

Theoremmul0ori 9662 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 9663 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul0ord 9664 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 9665 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulne0d 9666 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulne0bad 9667 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9666 and consequence of mulne0bd 9665. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulne0bbd 9668 A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 9666 and consequence of mulne0bd 9665. (Contributed by David Moews, 28-Feb-2017.)

5.3.6  Division

Syntaxcdiv 9669 Extend class notation to include division.

Definitiondf-div 9670* Define division. Theorem divmuli 9760 relates it to multiplication, and divcli 9748 and redivcli 9773 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)

Theorem1div0 9671 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 9672* 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 9673 Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivmul2 9674 Relationship between division and multiplication. (Contributed by NM, 7-Feb-2006.)

Theoremdivmul3 9675 Relationship between division and multiplication. (Contributed by NM, 13-Feb-2006.)

Theoremdivcl 9676 Closure law for division. (Contributed by NM, 21-Jul-2001.) (Proof shortened by Mario Carneiro, 17-Feb-2014.)

Theoremreccl 9677 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremdivcan2 9678 A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremdivcan1 9679 A cancellation law for division. (Contributed by NM, 5-Jun-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremdiveq0 9680 A ratio is zero iff the numerator is zero. (Contributed by NM, 20-Apr-2006.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremdivne0b 9681 The ratio of nonzero numbers is nonzero. (Contributed by NM, 2-Aug-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivne0 9682 The ratio of nonzero numbers is nonzero. (Contributed by NM, 28-Dec-2007.)

Theoremrecne0 9683 The reciprocal of a nonzero number is nonzero. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremrecid 9684 Multiplication of a number and its reciprocal. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremrecid2 9685 Multiplication of a number and its reciprocal. (Contributed by NM, 22-Jun-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivrec 9686 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 9687 Relationship between division and reciprocal. (Contributed by NM, 7-Feb-2006.)

Theoremdivass 9688 An associative law for division. (Contributed by NM, 2-Aug-2004.)

Theoremdiv23 9689 A commutative/associative law for division. (Contributed by NM, 2-Aug-2004.)

Theoremdiv32 9690 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv13 9691 A commutative/associative law for division. (Contributed by NM, 22-Apr-2005.)

Theoremdiv12 9692 A commutative/associative law for division. (Contributed by NM, 30-Apr-2005.)

Theoremdivdir 9693 Distribution of division over addition. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivcan3 9694 A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivcan4 9695 A cancellation law for division. (Contributed by NM, 8-Feb-2005.)

Theoremdiv11 9696 One-to-one relationship for division. (Contributed by NM, 20-Apr-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivid 9697 A number divided by itself is one. (Contributed by NM, 1-Aug-2004.)

Theoremdiv0 9698 Division into zero is zero. (Contributed by NM, 14-Mar-2005.)

Theoremdiv1 9699 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdiveq1 9700 Equality in terms of unit ratio. (Contributed by Stefan O'Rear, 27-Aug-2015.)

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