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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 | div23apd 8701 | A commutative/associative law for division. (Contributed by Jim Kingdon, 2-Mar-2020.) |
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| Theorem | divdirapd 8702 | Distribution of division over addition. (Contributed by Jim Kingdon, 2-Mar-2020.) |
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| Theorem | divsubdirapd 8703 | Distribution of division over subtraction. (Contributed by Jim Kingdon, 2-Mar-2020.) |
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| Theorem | div11apd 8704 | One-to-one relationship for division. (Contributed by Jim Kingdon, 2-Mar-2020.) |
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| Theorem | divmuldivapd 8705 | Multiplication of two ratios. (Contributed by Jim Kingdon, 30-Jul-2021.) |
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| Theorem | rerecclapd 8706 | Closure law for reciprocal. (Contributed by Jim Kingdon, 29-Feb-2020.) |
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| Theorem | redivclapd 8707 | Closure law for division of reals. (Contributed by Jim Kingdon, 29-Feb-2020.) |
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| Theorem | diveqap1bd 8708 | If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8578. Converse of diveqap1d 8671. (Contributed by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.) |
| # | ||
| Theorem | div2subap 8709 | Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.) |
![/\](wedge.gif "/\"){kind=small}
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| Theorem | div2subapd 8710 | Swap subtrahend and minuend inside the numerator and denominator of a fraction. Deduction form of div2subap 8709. (Contributed by David Moews, 28-Feb-2017.) |
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| Theorem | subrecap 8711 | Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.) |
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| Theorem | subrecapi 8712 | Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.) |
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| Theorem | subrecapd 8713 | Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017.) |
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| Theorem | mvllmulapd 8714 | Move LHS left multiplication to RHS. (Contributed by Jim Kingdon, 10-Jun-2020.) |
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| Theorem | ltp1 8715 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
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| Theorem | lep1 8716 | A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.) |
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| Theorem | ltm1 8717 | A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.) |
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| Theorem | lem1 8718 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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| Theorem | letrp1 8719 | A transitive property of 'less than or equal' and plus 1. (Contributed by NM, 5-Aug-2005.) |
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| Theorem | p1le 8720 | A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.) |
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| Theorem | recgt0 8721 | 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 8722 |
Infer that a multiplicand is positive from a positive multiplier and
positive product. See prodgt0 8723 for the same theorem with
replaced by the weaker condition
. (Contributed by Jim
Kingdon, 29-Feb-2020.)
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| Theorem | prodgt0 8723 | 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 8724 | 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 8725 | 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 8726 | 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 8727 | 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 8728 | 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 8729 | 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 8730 | 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 8731 | Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.) |
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| Theorem | lemul12b 8732 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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| Theorem | lemul12a 8733 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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| Theorem | mulgt1 8734 | The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) |
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| Theorem | ltmulgt11 8735 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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| Theorem | ltmulgt12 8736 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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| Theorem | lemulge11 8737 | Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.) |
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| Theorem | lemulge12 8738 | Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.) |
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| Theorem | ltdiv1 8739 | 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 8740 | 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 8741 | Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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| Theorem | ge0div 8742 | Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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| Theorem | divgt0 8743 | The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.) |
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| Theorem | divge0 8744 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.) |
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| Theorem | ltmuldiv 8745 | '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 8746 | 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | ltdivmul 8747 | 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | ledivmul 8748 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
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| Theorem | ltdivmul2 8749 | 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | lt2mul2div 8750 | 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.) |
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| Theorem | ledivmul2 8751 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
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| Theorem | lemuldiv 8752 | 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.) |
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| Theorem | lemuldiv2 8753 | 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.) |
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| Theorem | ltrec 8754 | 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 8755 | 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 8756 | Lemma for lt2msq 8757. (Contributed by Mario Carneiro, 27-May-2016.) |
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| Theorem | lt2msq 8757 | 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 8758 | Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.) |
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| Theorem | ltrec1 8759 | Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | lerec2 8760 | Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | ledivdiv 8761 | Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.) |
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| Theorem | lediv2 8762 | 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 8763 | Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.) |
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| Theorem | lediv23 8764 | Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.) |
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| Theorem | lediv12a 8765 | Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.) |
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| Theorem | lediv2a 8766 | 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 8767 | 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 8768 | 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 8769 | 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 8770 | Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.) |
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| Theorem | recreclt 8771 |
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 8772 | 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 8773 | 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 8774 | 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 8775* | 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 8776 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
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| Theorem | recgt0i 8777 | 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 8778 | 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 8779 | 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 8780 | 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 8781 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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| Theorem | divge0i 8782 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.) |
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| Theorem | ltreci 8783 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
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| Theorem | lereci 8784 | The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.) |
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| Theorem | lt2msqi 8785 | The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.) |
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| Theorem | le2msqi 8786 | The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.) |
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| Theorem | msq11i 8787 | The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) |
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| Theorem | divgt0i2i 8788 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltrecii 8789 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
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| Theorem | divgt0ii 8790 | The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
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| Theorem | ltmul1i 8791 | 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 8792 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltmuldivi 8793 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) |
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| Theorem | ltmul2i 8794 | 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 8795 | 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 8796 | 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 8797 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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| Theorem | ltdiv23ii 8798 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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| Theorem | ltmul1ii 8799 | 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 8800 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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