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Theorem List for Metamath Proof Explorer - 9701-9800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdivdir 9701 Distribution of division over addition. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivcan3 9702 A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivcan4 9703 A cancellation law for division. (Contributed by NM, 8-Feb-2005.)

Theoremdiv11 9704 One-to-one relationship for division. (Contributed by NM, 20-Apr-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdivid 9705 A number divided by itself is one. (Contributed by NM, 1-Aug-2004.)

Theoremdiv0 9706 Division into zero is zero. (Contributed by NM, 14-Mar-2005.)

Theoremdiv1 9707 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremdiveq1 9708 Equality in terms of unit ratio. (Contributed by Stefan O'Rear, 27-Aug-2015.)

Theoremdivneg 9709 Move negative sign inside of a division. (Contributed by NM, 17-Sep-2004.)

Theoremdivsubdir 9710 Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.)

Theoremrecrec 9711 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 9712 Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremrec11r 9713 Mutual reciprocals. (Contributed by Paul Chapman, 18-Oct-2007.)

Theoremdivmuldiv 9714 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 1-Aug-2004.)

Theoremdivdivdiv 9715 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.)

Theoremdivcan5 9716 Cancellation of common factor in a ratio. (Contributed by NM, 9-Jan-2006.)

Theoremdivmul13 9717 Swap the denominators in the product of two ratios. (Contributed by NM, 3-May-2005.)

Theoremdivmul24 9718 Swap the numerators in the product of two ratios. (Contributed by NM, 3-May-2005.)

Theoremdivmuleq 9719 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theoremrecdiv 9720 The reciprocal of a ratio. (Contributed by NM, 3-Aug-2004.)

Theoremdivcan6 9721 Cancellation of inverted fractions. (Contributed by NM, 28-Dec-2007.)

Theoremdivdiv32 9722 Swap denominators in a division. (Contributed by NM, 2-Aug-2004.)

Theoremdivcan7 9723 Cancel equal divisors in a division. (Contributed by Jeff Hankins, 29-Sep-2013.)

Theoremdmdcan 9724 Cancellation law for division and multiplication. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Fan Zheng, 3-Jul-2016.)

Theoremdivdiv1 9725 Division into a fraction. (Contributed by NM, 31-Dec-2007.)

Theoremdivdiv2 9726 Division by a fraction. (Contributed by NM, 27-Dec-2008.)

Theoremrecdiv2 9727 Division into a reciprocal. (Contributed by NM, 19-Oct-2007.)

Theoremddcan 9728 Cancellation in a double division. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremdivadddiv 9729 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 9730 Subtraction of two ratios. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 2-May-2016.)

Theoremconjmul 9731 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 9732 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremredivcl 9733 Closure law for division of reals. (Contributed by NM, 27-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremeqneg 9734 A number equal to its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremeqnegd 9735 A complex number equals its negative iff it is zero. Deduction form of eqneg 9734. (Contributed by David Moews, 28-Feb-2017.)

Theoremeqnegad 9736 If a complex number equals its own negative, it is zero. One-way deduction form of eqneg 9734. (Contributed by David Moews, 28-Feb-2017.)

Theoremdiv2neg 9737 Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012.)

Theoremdivneg2 9738 Move negative sign inside of a division. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremrecclzi 9739 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremrecne0zi 9740 The reciprocal of a nonzero number is nonzero. (Contributed by NM, 14-May-1999.)

Theoremrecidzi 9741 Multiplication of a number and its reciprocal. (Contributed by NM, 14-May-1999.)

Theoremdiv1i 9742 A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.)

Theoremeqnegi 9743 A number equal to its negative is zero. (Contributed by NM, 29-May-1999.)

Theoremreccli 9744 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremrecidi 9745 Multiplication of a number and its reciprocal. (Contributed by NM, 9-Feb-1995.)

Theoremrecreci 9746 A number is equal to the reciprocal of its reciprocal. Theorem I.10 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)

Theoremdividi 9747 A number divided by itself is one. (Contributed by NM, 9-Feb-1995.)

Theoremdiv0i 9748 Division into zero is zero. (Contributed by NM, 12-Aug-1999.)

Theoremdivclzi 9749 Closure law for division. (Contributed by NM, 7-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivcan1zi 9750 A cancellation law for division. (Contributed by NM, 2-Oct-1999.)

Theoremdivcan2zi 9751 A cancellation law for division. (Contributed by NM, 10-Aug-1999.)

Theoremdivreczi 9752 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 11-Oct-1999.)

Theoremdivcan3zi 9753 A cancellation law for division. (Eliminates a hypothesis of divcan3i 9760 with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004.)

Theoremdivcan4zi 9754 A cancellation law for division. (Contributed by NM, 12-Oct-1999.)

Theoremrec11i 9755 Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.)

Theoremdivcli 9756 Closure law for division. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivcan2i 9757 A cancellation law for division. (Contributed by NM, 9-Feb-1995.)

Theoremdivcan1i 9758 A cancellation law for division. (Contributed by NM, 18-May-1999.)

Theoremdivreci 9759 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by NM, 9-Feb-1995.)

Theoremdivcan3i 9760 A cancellation law for division. (Contributed by NM, 16-Feb-1995.)

Theoremdivcan4i 9761 A cancellation law for division. (Contributed by NM, 18-May-1999.)

Theoremdivne0i 9762 The ratio of nonzero numbers is nonzero. (Contributed by NM, 9-Feb-1995.)

Theoremrec11ii 9763 Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.)

Theoremdivasszi 9764 An associative law for division. (Contributed by NM, 12-Aug-1999.)

Theoremdivmulzi 9765 Relationship between division and multiplication. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivdirzi 9766 Distribution of division over addition. (Contributed by NM, 31-Jul-2004.)

Theoremdivdiv23zi 9767 Swap denominators in a division. (Contributed by NM, 15-Sep-1999.)

Theoremdivmuli 9768 Relationship between division and multiplication. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremdivdiv32i 9769 Swap denominators in a division. (Contributed by NM, 15-Sep-1999.)

Theoremdivassi 9770 An associative law for division. (Contributed by NM, 15-Feb-1995.)

Theoremdivdiri 9771 Distribution of division over addition. (Contributed by NM, 16-Feb-1995.)

Theoremdiv23i 9772 A commutative/associative law for division. (Contributed by NM, 3-Sep-1999.)

Theoremdiv11i 9773 One-to-one relationship for division. (Contributed by NM, 20-Aug-2001.)

Theoremdivmuldivi 9774 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 16-Feb-1995.)

Theoremdivmul13i 9775 Swap denominators of two ratios. (Contributed by NM, 6-Aug-1999.)

Theoremdivadddivi 9776 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by NM, 21-Feb-1995.)

Theoremdivdivdivi 9777 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 22-Feb-1995.)

Theoremrerecclzi 9778 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremrereccli 9779 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremredivclzi 9780 Closure law for division of reals. (Contributed by NM, 9-May-1999.)

Theoremredivcli 9781 Closure law for division of reals. (Contributed by NM, 9-May-1999.)

Theoremdiv1d 9782 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremreccld 9783 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecne0d 9784 The reciprocal of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecidd 9785 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecid2d 9786 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecrecd 9787 A number is equal to the reciprocal of its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdividd 9788 A number divided by itself is one. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv0d 9789 Division into zero is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcld 9790 Closure law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan1d 9791 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan2d 9792 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivrecd 9793 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivrec2d 9794 Relationship between division and reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan3d 9795 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan4d 9796 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq0d 9797 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1d 9798 Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1ad 9799 The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 9708. Generalization of diveq1d 9798. (Contributed by David Moews, 28-Feb-2017.)

Theoremdiveq0ad 9800 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 9688. (Contributed by David Moews, 28-Feb-2017.)

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