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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 | ltm1 8901 | A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.) |
         
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| Theorem | lem1 8902 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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| Theorem | letrp1 8903 | 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 8904 | A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.) |
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| Theorem | recgt0 8905 | 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 8906 |
Infer that a multiplicand is positive from a positive multiplier and
positive product. See prodgt0 8907 for the same theorem with
replaced by the weaker condition
. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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| Theorem | prodgt0 8907 | 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 8908 | 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 8909 | 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 8910 | 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 8911 | 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 8912 | 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 8913 | 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 8914 | 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 8915 | Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.) |
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| Theorem | lemul12b 8916 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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| Theorem | lemul12a 8917 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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| Theorem | mulgt1 8918 | The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) |
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| Theorem | ltmulgt11 8919 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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| Theorem | ltmulgt12 8920 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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| Theorem | lemulge11 8921 | Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.) |
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| Theorem | lemulge12 8922 | Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.) |
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| Theorem | ltdiv1 8923 | 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 8924 | 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 8925 | Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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| Theorem | ge0div 8926 | Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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| Theorem | divgt0 8927 | The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.) |
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| Theorem | divge0 8928 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.) |
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| Theorem | ltmuldiv 8929 | '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 8930 | 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | ltdivmul 8931 | 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | ledivmul 8932 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
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| Theorem | ltdivmul2 8933 | 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | lt2mul2div 8934 | 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.) |
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| Theorem | ledivmul2 8935 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
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| Theorem | lemuldiv 8936 | 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.) |
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| Theorem | lemuldiv2 8937 | 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.) |
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| Theorem | ltrec 8938 | 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 8939 | 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 8940 | Lemma for lt2msq 8941. (Contributed by Mario Carneiro, 27-May-2016.) |
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| Theorem | lt2msq 8941 | 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 8942 | Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.) |
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| Theorem | ltrec1 8943 | Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | lerec2 8944 | Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | ledivdiv 8945 | Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.) |
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| Theorem | lediv2 8946 | 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 8947 | Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.) |
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| Theorem | lediv23 8948 | Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.) |
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| Theorem | lediv12a 8949 | Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.) |
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| Theorem | lediv2a 8950 | 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 8951 | 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 8952 | 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 8953 | 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 8954 | Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.) |
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| Theorem | recreclt 8955 |
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 8956 | 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 8957 | 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 8958 | 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 8959* | 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 8960 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
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| Theorem | recgt0i 8961 | 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 8962 | 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 8963 | 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 8964 | 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 8965 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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| Theorem | divge0i 8966 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.) |
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| Theorem | ltreci 8967 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
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| Theorem | lereci 8968 | The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.) |
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| Theorem | lt2msqi 8969 | The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.) |
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| Theorem | le2msqi 8970 | The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.) |
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| Theorem | msq11i 8971 | The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) |
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| Theorem | divgt0i2i 8972 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltrecii 8973 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
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| Theorem | divgt0ii 8974 | The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
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| Theorem | ltmul1i 8975 | 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 8976 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltmuldivi 8977 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) |
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| Theorem | ltmul2i 8978 | 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 8979 | 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 8980 | 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 8981 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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| Theorem | ltdiv23ii 8982 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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| Theorem | ltmul1ii 8983 | 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 8984 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltp1d 8985 | A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lep1d 8986 | A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | ltm1d 8987 | A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lem1d 8988 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | recgt0d 8989 | 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 8990 | The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | mulgt1d 8991 | 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 8992 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemulge12d 8993 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemul1ad 8994 | 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 8995 | 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 8996 | Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemul12ad 8997 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemul12bd 8998 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | mulle0r 8999 | Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.) |
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| Theorem | lbreu 9000* | If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.) |
   
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