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

Theoremdivcan6d 9801 Cancellation of inverted fractions. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremddcand 9802 Cancellation in a double division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrec11d 9803 Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuld 9804 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv32d 9805 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv13d 9806 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdiv32d 9807 Swap denominators in a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan5d 9808 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivcan5rd 9809 Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremdivcan7d 9810 Cancel equal divisors in a division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdmdcand 9811 Cancellation law for division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdmdcan2d 9812 Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017.)

Theoremdivdiv1d 9813 Division into a fraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdiv2d 9814 Division by a fraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul2d 9815 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul3d 9816 Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivassd 9817 An associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv12d 9818 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv23d 9819 A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdird 9820 Distribution of division over addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivsubdird 9821 Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiv11d 9822 One-to-one relationship for division. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuldivd 9823 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul13d 9824 Swap denominators of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmul24d 9825 Swap the numerators in the product of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivadddivd 9826 Addition of two ratios. Theorem I.13 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivsubdivd 9827 Subtraction of two ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivmuleqd 9828 Cross-multiply in an equality of ratios. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdivdivdivd 9829 Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremdiveq1bd 9830 If two complex numbers are equal, their quotient is one. One-way deduction form of diveq1 9700. Converse of diveq1d 9790. (Contributed by David Moews, 28-Feb-2017.)

Theoremdiv2sub 9831 Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremdiv2subd 9832 Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2sub 9831. (Contributed by David Moews, 28-Feb-2017.)

Theoremrereccld 9833 Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremredivcld 9834 Closure law for division of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubrec 9835 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)

Theoremsubreci 9836 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)

Theoremsubrecd 9837 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)

5.3.7  Ordering on reals (cont.)

Theoremelimgt0 9838 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 15-May-1999.)

Theoremelimge0 9839 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 30-Jul-1999.)

Theoremltp1 9840 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremlep1 9841 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)

Theoremltm1 9842 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)

Theoremlem1 9843 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremletrp1 9844 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)

Theoremp1le 9845 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)

Theoremrecgt0 9846 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 9847 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 9848 Infer that a multiplier is positive from a nonnegative muliplicand and positive product. (Contributed by NM, 24-Apr-2005.)

Theoremprodge0 9849 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 9850 Infer that a multiplier is nonnegative from a positive muliplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremltmul1a 9851 Lemma for ltmul1 9852. 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 9852 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 9853 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 9854 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)

Theoremlemul2 9855 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)

Theoremlemul1a 9856 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.)

Theoremlemul2a 9857 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremltmul12a 9858 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)

Theoremlemul12b 9859 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremlemul12a 9860 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremmulgt1 9861 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)

Theoremltmulgt11 9862 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremltmulgt12 9863 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremlemulge11 9864 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)

Theoremlemulge12 9865 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremltdiv1 9866 Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremlediv1 9867 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)

Theoremgt0div 9868 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremge0div 9869 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremdivgt0 9870 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)

Theoremdivge0 9871 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)

Theoremltmuldiv 9872 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltmuldiv2 9873 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremltdivmul 9874 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremledivmul 9875 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

TheoremledivmulOLD 9876 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremltdivmul2 9877 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)

Theoremlt2mul2div 9878 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)

Theoremledivmul2 9879 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

Theoremledivmul2OLD 9880 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremlemuldiv 9881 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremlemuldiv2 9882 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremltrec 9883 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremlerec 9884 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt2msq1 9885 Lemma for lt2msq 9886. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2msq 9886 Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltdiv2 9887 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)

Theoremltdiv2OLD 9888 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremltrec1 9889 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)

Theoremlerec2 9890 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)

Theoremledivdiv 9891 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)

Theoremlediv2 9892 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)

Theoremltdiv23 9893 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)

Theoremlediv23 9894 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)

Theoremlediv12a 9895 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)

Theoremlediv2a 9896 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremreclt1 9897 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecgt1 9898 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)

Theoremrecgt1i 9899 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecp1lt1 9900 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)

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