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Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | mvllmulapd 8801 | Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.) |
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| Theorem | rerecapb 8802* | A real number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 18-Jan-2025.) |
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| Theorem | ltp1 8803 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
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| Theorem | lep1 8804 | A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.) |
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| Theorem | ltm1 8805 | A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.) |
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| Theorem | lem1 8806 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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| Theorem | letrp1 8807 | A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.) |
![/\](wedge.gif "/\"){kind=small}
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| Theorem | p1le 8808 | A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.) |
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| Theorem | recgt0 8809 | 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.) |
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| Theorem | prodgt0gt0 8810 |
Infer that a multiplicand is positive from a positive multiplier and
positive product. See prodgt0 8811 for the same theorem with
replaced by the weaker condition
. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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| Theorem | prodgt0 8811 | Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
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| Theorem | prodgt02 8812 | Infer that a multiplier is positive from a nonnegative multiplicand and positive product. (Contributed by NM, 24-Apr-2005.) |
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| Theorem | prodge0 8813 | Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
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| Theorem | prodge02 8814 | Infer that a multiplier is nonnegative from a positive multiplicand and nonnegative product. (Contributed by NM, 2-Jul-2005.) |
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| Theorem | ltmul2 8815 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) |
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| Theorem | lemul2 8816 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 16-Mar-2005.) |
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| Theorem | lemul1a 8817 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 21-Feb-2005.) |
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| Theorem | lemul2a 8818 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) |
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| Theorem | ltmul12a 8819 | Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.) |
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| Theorem | lemul12b 8820 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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| Theorem | lemul12a 8821 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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| Theorem | mulgt1 8822 | The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) |
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| Theorem | ltmulgt11 8823 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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| Theorem | ltmulgt12 8824 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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| Theorem | lemulge11 8825 | Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.) |
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| Theorem | lemulge12 8826 | Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.) |
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| Theorem | ltdiv1 8827 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
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| Theorem | lediv1 8828 | Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | gt0div 8829 | Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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| Theorem | ge0div 8830 | Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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| Theorem | divgt0 8831 | The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.) |
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| Theorem | divge0 8832 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.) |
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| Theorem | ltmuldiv 8833 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
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| Theorem | ltmuldiv2 8834 | 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | ltdivmul 8835 | 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | ledivmul 8836 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
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| Theorem | ltdivmul2 8837 | 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | lt2mul2div 8838 | 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.) |
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| Theorem | ledivmul2 8839 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
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| Theorem | lemuldiv 8840 | 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.) |
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| Theorem | lemuldiv2 8841 | 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.) |
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| Theorem | ltrec 8842 | The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
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| Theorem | lerec 8843 | 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.) |
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| Theorem | lt2msq1 8844 | Lemma for lt2msq 8845. (Contributed by Mario Carneiro, 27-May-2016.) |
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| Theorem | lt2msq 8845 | Two nonnegative numbers compare the same as their squares. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
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| Theorem | ltdiv2 8846 | Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.) |
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| Theorem | ltrec1 8847 | Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | lerec2 8848 | Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | ledivdiv 8849 | Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.) |
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| Theorem | lediv2 8850 | Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.) |
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| Theorem | ltdiv23 8851 | Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.) |
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| Theorem | lediv23 8852 | Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.) |
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| Theorem | lediv12a 8853 | Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.) |
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| Theorem | lediv2a 8854 | Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) |
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| Theorem | reclt1 8855 | The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.) |
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| Theorem | recgt1 8856 | The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.) |
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| Theorem | recgt1i 8857 | The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.) |
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| Theorem | recp1lt1 8858 | Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.) |
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| Theorem | recreclt 8859 |
Given a positive number , construct a new positive number less than
both and 1.
(Contributed by NM, 28-Dec-2005.)
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| Theorem | le2msq 8860 | The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
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| Theorem | msq11 8861 | 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.) |
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| Theorem | ledivp1 8862 | 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.) |
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| Theorem | squeeze0 8863* | If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.) |
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| Theorem | ltp1i 8864 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
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| Theorem | recgt0i 8865 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.) |
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| Theorem | recgt0ii 8866 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.) |
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| Theorem | prodgt0i 8867 | Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 15-May-1999.) |
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| Theorem | prodge0i 8868 | Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) |
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| Theorem | divgt0i 8869 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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| Theorem | divge0i 8870 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.) |
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| Theorem | ltreci 8871 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
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| Theorem | lereci 8872 | The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.) |
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| Theorem | lt2msqi 8873 | The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.) |
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| Theorem | le2msqi 8874 | The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.) |
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| Theorem | msq11i 8875 | The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) |
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| Theorem | divgt0i2i 8876 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltrecii 8877 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
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| Theorem | divgt0ii 8878 | The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
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| Theorem | ltmul1i 8879 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltdiv1i 8880 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltmuldivi 8881 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) |
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| Theorem | ltmul2i 8882 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) |
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| Theorem | lemul1i 8883 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.) |
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| Theorem | lemul2i 8884 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.) |
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| Theorem | ltdiv23i 8885 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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| Theorem | ltdiv23ii 8886 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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| Theorem | ltmul1ii 8887 | 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.) |
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| Theorem | ltdiv1ii 8888 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltp1d 8889 | A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lep1d 8890 | A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | ltm1d 8891 | A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lem1d 8892 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | recgt0d 8893 | The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | divgt0d 8894 | The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | mulgt1d 8895 | The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemulge11d 8896 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemulge12d 8897 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemul1ad 8898 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemul2ad 8899 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | ltmul12ad 8900 | Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
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