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

Theoremsubrecd 9801 Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.)

5.3.7  Ordering on reals (cont.)

Theoremelimgt0 9802 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 15-May-1999.)

Theoremelimge0 9803 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 30-Jul-1999.)

Theoremltp1 9804 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremlep1 9805 A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.)

Theoremltm1 9806 A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.)

Theoremlem1 9807 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremletrp1 9808 A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.)

Theoremp1le 9809 A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.)

Theoremrecgt0 9810 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 9811 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 9812 Infer that a multiplier is positive from a nonnegative muliplicand and positive product. (Contributed by NM, 24-Apr-2005.)

Theoremprodge0 9813 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 9814 Infer that a multiplier is nonnegative from a positive muliplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremltmul1a 9815 Lemma for ltmul1 9816. 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 9816 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 9817 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 9818 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)

Theoremlemul2 9819 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.)

Theoremlemul1a 9820 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.)

Theoremlemul2a 9821 Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremltmul12a 9822 Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)

Theoremlemul12b 9823 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremlemul12a 9824 Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.)

Theoremmulgt1 9825 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)

Theoremltmulgt11 9826 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremltmulgt12 9827 Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.)

Theoremlemulge11 9828 Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.)

Theoremlemulge12 9829 Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremltdiv1 9830 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 9831 Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.)

Theoremgt0div 9832 Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremge0div 9833 Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.)

Theoremdivgt0 9834 The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.)

Theoremdivge0 9835 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.)

Theoremltmuldiv 9836 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltmuldiv2 9837 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremltdivmul 9838 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.)

Theoremledivmul 9839 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

TheoremledivmulOLD 9840 '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 9841 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.)

Theoremlt2mul2div 9842 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)

Theoremledivmul2 9843 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.)

Theoremledivmul2OLD 9844 '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 9845 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremlemuldiv2 9846 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.)

Theoremltrec 9847 The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremlerec 9848 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 9849 Lemma for lt2msq 9850. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2msq 9850 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 9851 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.)

Theoremltdiv2OLD 9852 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 9853 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)

Theoremlerec2 9854 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)

Theoremledivdiv 9855 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)

Theoremlediv2 9856 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)

Theoremltdiv23 9857 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)

Theoremlediv23 9858 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)

Theoremlediv12a 9859 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)

Theoremlediv2a 9860 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremreclt1 9861 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecgt1 9862 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)

Theoremrecgt1i 9863 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecp1lt1 9864 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)

Theoremrecreclt 9865 Given a positive number , construct a new positive number less than both and 1. (Contributed by NM, 28-Dec-2005.)

Theoremle2msq 9866 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmsq11 9867 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremledivp1 9868 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)

Theoremsqueeze0 9869* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)

Theoremltp1i 9870 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremrecgt0i 9871 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremrecgt0ii 9872 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremprodgt0i 9873 Infer that a multiplicand is positive from a nonnegative muliplier and positive product. (Contributed by NM, 15-May-1999.)

Theoremprodge0i 9874 Infer that a multiplicand is nonnegative from a positive muliplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremdivgt0i 9875 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremdivge0i 9876 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)

Theoremltreci 9877 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremlereci 9878 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)

Theoremlt2msqi 9879 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremle2msqi 9880 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)

Theoremmsq11i 9881 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)

Theoremdivgt0i2i 9882 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremltrecii 9883 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremdivgt0ii 9884 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltmul1i 9885 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)

Theoremltdiv1i 9886 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)

Theoremltmuldivi 9887 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)

Theoremltmul2i 9888 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)

Theoremlemul1i 9889 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)

Theoremlemul2i 9890 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)

Theoremltdiv23i 9891 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)

Theoremledivp1i 9892 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)

Theoremltdivp1i 9893 Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)

Theoremltdiv23ii 9894 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)

Theoremltmul1ii 9895 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)

Theoremltdiv1ii 9896 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)

Theoremltp1d 9897 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlep1d 9898 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltm1d 9899 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlem1d 9900 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)

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