HomeTheorem List (p. 79 of 102)
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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 | div12apd 7801 | A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.) |
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| Theorem | div23apd 7802 | A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.) |
| #
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| Theorem | divdirapd 7803 | Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.) |
# 
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| Theorem | divsubdirapd 7804 | Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.) |
# 
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| Theorem | div11apd 7805 | One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.) |
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| Theorem | divmuldivapd 7806 | Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.) |
 # #
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| Theorem | rerecclapd 7807 | Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.) |
 # 
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| Theorem | redivclapd 7808 | Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.) |
| #
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| Theorem | mvllmulapd 7809 | Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.) |
| #
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| Theorem | ltp1 7810 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
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| Theorem | lep1 7811 | A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.) |
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| Theorem | ltm1 7812 | A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.) |
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| Theorem | lem1 7813 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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| Theorem | letrp1 7814 | 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 7815 | A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.) |
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| Theorem | recgt0 7816 | 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 7817 |
Infer that a multiplicand is positive from a positive multiplier and
positive product. See prodgt0 7818 for the same theorem with
replaced by the weaker condition
. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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| Theorem | prodgt0 7818 | 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 7819 | 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 7820 | 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 7821 | 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 7822 | 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 7823 | 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 7824 | 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 7825 | 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 7826 | Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.) |
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| Theorem | lemul12b 7827 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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| Theorem | lemul12a 7828 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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| Theorem | mulgt1 7829 | The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) |
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| Theorem | ltmulgt11 7830 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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| Theorem | ltmulgt12 7831 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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| Theorem | lemulge11 7832 | Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.) |
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| Theorem | lemulge12 7833 | Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.) |
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| Theorem | ltdiv1 7834 | 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 7835 | 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 7836 | Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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| Theorem | ge0div 7837 | Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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| Theorem | divgt0 7838 | The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.) |
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| Theorem | divge0 7839 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.) |
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| Theorem | ltmuldiv 7840 | '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 7841 | 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | ltdivmul 7842 | 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | ledivmul 7843 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
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| Theorem | ltdivmul2 7844 | 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | lt2mul2div 7845 | 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.) |
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| Theorem | ledivmul2 7846 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
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| Theorem | lemuldiv 7847 | 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.) |
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| Theorem | lemuldiv2 7848 | 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.) |
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| Theorem | ltrec 7849 | 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 7850 | 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 7851 | Lemma for lt2msq 7852. (Contributed by Mario Carneiro, 27-May-2016.) |
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| Theorem | lt2msq 7852 | 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 7853 | Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.) |
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| Theorem | ltrec1 7854 | Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | lerec2 7855 | Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | ledivdiv 7856 | Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.) |
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| Theorem | lediv2 7857 | 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 7858 | Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.) |
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| Theorem | lediv23 7859 | Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.) |
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| Theorem | lediv12a 7860 | Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.) |
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| Theorem | lediv2a 7861 | 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 7862 | 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 7863 | 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 7864 | 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 7865 | Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.) |
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| Theorem | recreclt 7866 |
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 7867 | 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 7868 | 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 7869 | 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 7870* | 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 7871 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
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| Theorem | recgt0i 7872 | 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 7873 | 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 7874 | 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 7875 | 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 7876 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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| Theorem | divge0i 7877 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.) |
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| Theorem | ltreci 7878 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
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| Theorem | lereci 7879 | The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.) |
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| Theorem | lt2msqi 7880 | The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.) |
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| Theorem | le2msqi 7881 | The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.) |
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| Theorem | msq11i 7882 | The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) |
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| Theorem | divgt0i2i 7883 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltrecii 7884 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
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| Theorem | divgt0ii 7885 | The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
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| Theorem | ltmul1i 7886 | 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 7887 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltmuldivi 7888 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) |
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| Theorem | ltmul2i 7889 | 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 7890 | 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 7891 | 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 7892 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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| Theorem | ltdiv23ii 7893 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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| Theorem | ltmul1ii 7894 | 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 7895 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltp1d 7896 | A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lep1d 7897 | A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | ltm1d 7898 | A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lem1d 7899 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | recgt0d 7900 | 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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