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Theorem List for Metamath Proof Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdivassi 9801 An associative law for division. (Contributed by NM, 15-Feb-1995.)

Theoremdivdiri 9802 Distribution of division over addition. (Contributed by NM, 16-Feb-1995.)

Theoremdiv23i 9803 A commutative/associative law for division. (Contributed by NM, 3-Sep-1999.)

Theoremdiv11i 9804 One-to-one relationship for division. (Contributed by NM, 20-Aug-2001.)

Theoremdivmuldivi 9805 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by NM, 16-Feb-1995.)

Theoremdivmul13i 9806 Swap denominators of two ratios. (Contributed by NM, 6-Aug-1999.)

Theoremdivadddivi 9807 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by NM, 21-Feb-1995.)

Theoremdivdivdivi 9808 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 22-Feb-1995.)

Theoremrerecclzi 9809 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremrereccli 9810 Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.)

Theoremredivclzi 9811 Closure law for division of reals. (Contributed by NM, 9-May-1999.)

Theoremredivcli 9812 Closure law for division of reals. (Contributed by NM, 9-May-1999.)

Theoremdiv1d 9813 A number divided by 1 is itself. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremreccld 9814 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecne0d 9815 The reciprocal of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecidd 9816 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecid2d 9817 Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecrecd 9818 A number is equal to the reciprocal of its reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdividd 9819 A number divided by itself is one. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv0d 9820 Division into zero is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcld 9821 Closure law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan1d 9822 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan2d 9823 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivrecd 9824 Relationship between division and reciprocal. Theorem I.9 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivrec2d 9825 Relationship between division and reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan3d 9826 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan4d 9827 A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq0d 9828 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1d 9829 Equality in terms of unit ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1ad 9830 The quotient of two complex numbers is one iff they are equal. Deduction form of diveq1 9739. Generalization of diveq1d 9829. (Contributed by David Moews, 28-Feb-2017.)

Theoremdiveq0ad 9831 A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 9719. (Contributed by David Moews, 28-Feb-2017.)

Theoremdivne1d 9832 If two complex numbers are unequal, their quotient is not one. Contrapositive of diveq1d 9829. (Contributed by David Moews, 28-Feb-2017.)

Theoremdivne0bd 9833 A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivnegd 9834 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivneg2d 9835 Move negative sign inside of a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv2negd 9836 Quotient of two negatives. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivne0d 9837 The ratio of nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecdivd 9838 The reciprocal of a ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecdiv2d 9839 Division into a reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan6d 9840 Cancellation of inverted fractions. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremddcand 9841 Cancellation in a double division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrec11d 9842 Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuld 9843 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv32d 9844 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv13d 9845 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdiv32d 9846 Swap denominators in a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan5d 9847 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan5rd 9848 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdivcan7d 9849 Cancel equal divisors in a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdmdcand 9850 Cancellation law for division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdmdcan2d 9851 Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017.)

Theoremdivdiv1d 9852 Division into a fraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdiv2d 9853 Division by a fraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul2d 9854 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul3d 9855 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivassd 9856 An associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv12d 9857 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv23d 9858 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdird 9859 Distribution of division over addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivsubdird 9860 Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv11d 9861 One-to-one relationship for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuldivd 9862 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul13d 9863 Swap denominators of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul24d 9864 Swap the numerators in the product of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivadddivd 9865 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivsubdivd 9866 Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuleqd 9867 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdivdivd 9868 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1bd 9869 If two complex numbers are equal, their quotient is one. One-way deduction form of diveq1 9739. Converse of diveq1d 9829. (Contributed by David Moews, 28-Feb-2017.)

Theoremdiv2sub 9870 Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremdiv2subd 9871 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 9870. (Contributed by David Moews, 28-Feb-2017.)

Theoremrereccld 9872 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremredivcld 9873 Closure law for division of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubrec 9874 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)

Theoremsubreci 9875 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)

Theoremsubrecd 9876 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)

5.3.7  Ordering on reals (cont.)

Theoremelimgt0 9877 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 15-May-1999.)

Theoremelimge0 9878 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 30-Jul-1999.)

Theoremltp1 9879 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremlep1 9880 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)

Theoremltm1 9881 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)

Theoremlem1 9882 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremletrp1 9883 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)

Theoremp1le 9884 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)

Theoremrecgt0 9885 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremprodgt0 9886 Infer that a multiplicand is positive from a nonnegative muliplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremprodgt02 9887 Infer that a multiplier is positive from a nonnegative muliplicand and positive product. (Contributed by NM, 24-Apr-2005.)

Theoremprodge0 9888 Infer that a multiplicand is nonnegative from a positive muliplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremprodge02 9889 Infer that a multiplier is nonnegative from a positive muliplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremltmul1a 9890 Lemma for ltmul1 9891. Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltmul1 9891 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltmul2 9892 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)

Theoremlemul1 9893 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)

Theoremlemul2 9894 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)

Theoremlemul1a 9895 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.)

Theoremlemul2a 9896 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremltmul12a 9897 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)

Theoremlemul12b 9898 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremlemul12a 9899 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremmulgt1 9900 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)

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