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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | lemul1 10401 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.) |
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| Theorem | lemul2 10402 | 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 10403 | Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by NM, 21-Feb-2005.) |
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| Theorem | lemul2a 10404 | 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 10405 | Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.) |
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| Theorem | lemul12b 10406 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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| Theorem | lemul12a 10407 | Comparison of product of two nonnegative numbers. (Contributed by NM, 22-Feb-2008.) |
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| Theorem | mulgt1 10408 | The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) |
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| Theorem | ltmulgt11 10409 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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| Theorem | ltmulgt12 10410 | Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
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| Theorem | lemulge11 10411 | Multiplication by a number greater than or equal to 1. (Contributed by NM, 17-Dec-2005.) |
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| Theorem | lemulge12 10412 | Multiplication by a number greater than or equal to 1. (Contributed by Paul Chapman, 21-Mar-2011.) |
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| Theorem | ltdiv1 10413 | 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 10414 | 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 10415 | Division of a positive number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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| Theorem | ge0div 10416 | Division of a nonnegative number by a positive number. (Contributed by NM, 28-Sep-2005.) |
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| Theorem | divgt0 10417 | The ratio of two positive numbers is positive. (Contributed by NM, 12-Oct-1999.) |
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| Theorem | divge0 10418 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999.) |
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| Theorem | mulge0b 10419 | A condition for multiplication to be nonnegative. (Contributed by Scott Fenton, 25-Jun-2013.) |
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| Theorem | mulle0b 10420 | A condition for multiplication to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.) |
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| Theorem | mulsuble0b 10421 | A condition for multiplication of subtraction to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.) |
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| Theorem | ltmuldiv 10422 | '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 10423 | 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | ltdivmul 10424 | 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
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| Theorem | ledivmul 10425 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
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| Theorem | ledivmulOLD 10426 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
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| Theorem | ltdivmul2 10427 | 'Less than' relationship between division and multiplication. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | lt2mul2div 10428 | 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.) |
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| Theorem | ledivmul2 10429 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
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| Theorem | ledivmul2OLD 10430 | 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) (Proof modification is discouraged.) (New usage is discouraged.) |
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| Theorem | lemuldiv 10431 | 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.) |
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| Theorem | lemuldiv2 10432 | 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006.) |
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| Theorem | ltrec 10433 | 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 10434 | 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 10435 | Lemma for lt2msq 10436. (Contributed by Mario Carneiro, 27-May-2016.) |
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| Theorem | lt2msq 10436 | 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 10437 | Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.) |
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| Theorem | ltdiv2OLD 10438 | 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.) |
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| Theorem | ltrec1 10439 | Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | lerec2 10440 | Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.) |
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| Theorem | ledivdiv 10441 | Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.) |
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| Theorem | lediv2 10442 | 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 10443 | Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.) |
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| Theorem | lediv23 10444 | Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.) |
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| Theorem | lediv12a 10445 | Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.) |
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| Theorem | lediv2a 10446 | 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 10447 | 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 10448 | 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 10449 | 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 10450 | Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.) |
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| Theorem | recreclt 10451 |
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 10452 | 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 10453 | 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 10454 | 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 10455* | 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 10456 | A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.) |
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| Theorem | recgt0i 10457 | 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 10458 | 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 10459 | 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 10460 | 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 10461 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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| Theorem | divge0i 10462 | The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.) |
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| Theorem | ltreci 10463 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
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| Theorem | lereci 10464 | The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.) |
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| Theorem | lt2msqi 10465 | The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.) |
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| Theorem | le2msqi 10466 | The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.) |
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| Theorem | msq11i 10467 | The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) |
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| Theorem | divgt0i2i 10468 | The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltrecii 10469 | The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
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| Theorem | divgt0ii 10470 | The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.) |
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| Theorem | ltmul1i 10471 | 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 10472 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltmuldivi 10473 | 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.) |
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| Theorem | ltmul2i 10474 | 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 10475 | 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 10476 | 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 10477 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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| Theorem | ledivp1i 10478 | 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.) |
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| Theorem | ltdivp1i 10479 | Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.) |
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| Theorem | ltdiv23ii 10480 | Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.) |
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| Theorem | ltmul1ii 10481 | 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 10482 | Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.) |
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| Theorem | ltp1d 10483 | A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lep1d 10484 | A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | ltm1d 10485 | A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lem1d 10486 | A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | recgt0d 10487 | 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 10488 | The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | mulgt1d 10489 | 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 10490 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemulge12d 10491 | Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemul1ad 10492 | 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 10493 | 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 10494 | Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemul12ad 10495 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | lemul12bd 10496 | Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.) |
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| Theorem | fimaxre 10497* | A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
    
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| Theorem | fimaxre2 10498* | A nonempty finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.) |
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| Theorem | fimaxre3 10499* | A nonempty finite set of real numbers has a maximum (image set version). (Contributed by Mario Carneiro, 13-Feb-2014.) |
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| Theorem | lbreu 10500* | 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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