Type | Label | Description |
Statement |
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Theorem | ltsub23 8401 |
'Less than' relationship between subtraction and addition. (Contributed
by NM, 4-Oct-1999.)
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Theorem | ltsub13 8402 |
'Less than' relationship between subtraction and addition. (Contributed
by NM, 17-Nov-2004.)
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Theorem | le2add 8403 |
Adding both sides of two 'less than or equal to' relations. (Contributed
by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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Theorem | lt2add 8404 |
Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol]
p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario
Carneiro, 27-May-2016.)
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Theorem | ltleadd 8405 |
Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
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Theorem | leltadd 8406 |
Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
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Theorem | addgt0 8407 |
The sum of 2 positive numbers is positive. (Contributed by NM,
1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | addgegt0 8408 |
The sum of nonnegative and positive numbers is positive. (Contributed by
NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | addgtge0 8409 |
The sum of nonnegative and positive numbers is positive. (Contributed by
NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | addge0 8410 |
The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM,
17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
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|
Theorem | ltaddpos 8411 |
Adding a positive number to another number increases it. (Contributed by
NM, 17-Nov-2004.)
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|
Theorem | ltaddpos2 8412 |
Adding a positive number to another number increases it. (Contributed by
NM, 8-Apr-2005.)
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|
Theorem | ltsubpos 8413 |
Subtracting a positive number from another number decreases it.
(Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon,
19-Nov-2011.)
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|
Theorem | posdif 8414 |
Comparison of two numbers whose difference is positive. (Contributed by
NM, 17-Nov-2004.)
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|
Theorem | lesub1 8415 |
Subtraction from both sides of 'less than or equal to'. (Contributed by
NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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|
Theorem | lesub2 8416 |
Subtraction of both sides of 'less than or equal to'. (Contributed by NM,
29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
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|
Theorem | ltsub1 8417 |
Subtraction from both sides of 'less than'. (Contributed by FL,
3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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|
Theorem | ltsub2 8418 |
Subtraction of both sides of 'less than'. (Contributed by NM,
29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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|
Theorem | lt2sub 8419 |
Subtracting both sides of two 'less than' relations. (Contributed by
Mario Carneiro, 14-Apr-2016.)
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Theorem | le2sub 8420 |
Subtracting both sides of two 'less than or equal to' relations.
(Contributed by Mario Carneiro, 14-Apr-2016.)
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|
Theorem | ltneg 8421 |
Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20.
(Contributed by NM, 27-Aug-1999.) (Proof shortened by Mario Carneiro,
27-May-2016.)
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|
Theorem | ltnegcon1 8422 |
Contraposition of negative in 'less than'. (Contributed by NM,
8-Nov-2004.)
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Theorem | ltnegcon2 8423 |
Contraposition of negative in 'less than'. (Contributed by Mario
Carneiro, 25-Feb-2015.)
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|
Theorem | leneg 8424 |
Negative of both sides of 'less than or equal to'. (Contributed by NM,
12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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|
Theorem | lenegcon1 8425 |
Contraposition of negative in 'less than or equal to'. (Contributed by
NM, 10-May-2004.)
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Theorem | lenegcon2 8426 |
Contraposition of negative in 'less than or equal to'. (Contributed by
NM, 8-Oct-2005.)
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Theorem | lt0neg1 8427 |
Comparison of a number and its negative to zero. Theorem I.23 of
[Apostol] p. 20. (Contributed by NM,
14-May-1999.)
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Theorem | lt0neg2 8428 |
Comparison of a number and its negative to zero. (Contributed by NM,
10-May-2004.)
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Theorem | le0neg1 8429 |
Comparison of a number and its negative to zero. (Contributed by NM,
10-May-2004.)
|
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|
Theorem | le0neg2 8430 |
Comparison of a number and its negative to zero. (Contributed by NM,
24-Aug-1999.)
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|
Theorem | addge01 8431 |
A number is less than or equal to itself plus a nonnegative number.
(Contributed by NM, 21-Feb-2005.)
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|
Theorem | addge02 8432 |
A number is less than or equal to itself plus a nonnegative number.
(Contributed by NM, 27-Jul-2005.)
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Theorem | add20 8433 |
Two nonnegative numbers are zero iff their sum is zero. (Contributed by
Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro,
27-May-2016.)
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|
Theorem | subge0 8434 |
Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof
shortened by Mario Carneiro, 27-May-2016.)
|
    

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Theorem | suble0 8435 |
Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof
shortened by Mario Carneiro, 27-May-2016.)
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Theorem | leaddle0 8436 |
The sum of a real number and a second real number is less then the real
number iff the second real number is negative. (Contributed by Alexander
van der Vekens, 30-May-2018.)
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|
Theorem | subge02 8437 |
Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
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Theorem | lesub0 8438 |
Lemma to show a nonnegative number is zero. (Contributed by NM,
8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
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Theorem | mullt0 8439 |
The product of two negative numbers is positive. (Contributed by Jeff
Hankins, 8-Jun-2009.)
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Theorem | 0le1 8440 |
0 is less than or equal to 1. (Contributed by Mario Carneiro,
29-Apr-2015.)
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Theorem | ltordlem 8441* |
Lemma for eqord1 8442. (Contributed by Mario Carneiro,
14-Jun-2014.)
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Theorem | eqord1 8442* |
A strictly increasing real function on a subset of is
one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised
by Jim Kingdon, 20-Dec-2022.)
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Theorem | eqord2 8443* |
A strictly decreasing real function on a subset of is one-to-one.
(Contributed by Mario Carneiro, 14-Jun-2014.)
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Theorem | leidi 8444 |
'Less than or equal to' is reflexive. (Contributed by NM,
18-Aug-1999.)
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Theorem | gt0ne0i 8445 |
Positive means nonzero (useful for ordering theorems involving
division). (Contributed by NM, 16-Sep-1999.)
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Theorem | gt0ne0ii 8446 |
Positive implies nonzero. (Contributed by NM, 15-May-1999.)
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Theorem | addgt0i 8447 |
Addition of 2 positive numbers is positive. (Contributed by NM,
16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | addge0i 8448 |
Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM,
28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | addgegt0i 8449 |
Addition of nonnegative and positive numbers is positive. (Contributed
by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
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Theorem | addgt0ii 8450 |
Addition of 2 positive numbers is positive. (Contributed by NM,
18-May-1999.)
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|
Theorem | add20i 8451 |
Two nonnegative numbers are zero iff their sum is zero. (Contributed by
NM, 28-Jul-1999.)
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Theorem | ltnegi 8452 |
Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20.
(Contributed by NM, 21-Jan-1997.)
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Theorem | lenegi 8453 |
Negative of both sides of 'less than or equal to'. (Contributed by NM,
1-Aug-1999.)
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Theorem | ltnegcon2i 8454 |
Contraposition of negative in 'less than'. (Contributed by NM,
14-May-1999.)
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Theorem | lesub0i 8455 |
Lemma to show a nonnegative number is zero. (Contributed by NM,
8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
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Theorem | ltaddposi 8456 |
Adding a positive number to another number increases it. (Contributed
by NM, 25-Aug-1999.)
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Theorem | posdifi 8457 |
Comparison of two numbers whose difference is positive. (Contributed by
NM, 19-Aug-2001.)
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Theorem | ltnegcon1i 8458 |
Contraposition of negative in 'less than'. (Contributed by NM,
14-May-1999.)
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Theorem | lenegcon1i 8459 |
Contraposition of negative in 'less than or equal to'. (Contributed by
NM, 6-Apr-2005.)
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|
Theorem | subge0i 8460 |
Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
|
  
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Theorem | ltadd1i 8461 |
Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20.
(Contributed by NM, 21-Jan-1997.)
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Theorem | leadd1i 8462 |
Addition to both sides of 'less than or equal to'. (Contributed by NM,
11-Aug-1999.)
|
  
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Theorem | leadd2i 8463 |
Addition to both sides of 'less than or equal to'. (Contributed by NM,
11-Aug-1999.)
|
  
    |
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Theorem | ltsubaddi 8464 |
'Less than' relationship between subtraction and addition. (Contributed
by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon,
19-Nov-2011.)
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Theorem | lesubaddi 8465 |
'Less than or equal to' relationship between subtraction and addition.
(Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon,
19-Nov-2011.)
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Theorem | ltsubadd2i 8466 |
'Less than' relationship between subtraction and addition. (Contributed
by NM, 21-Jan-1997.)
|
  
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Theorem | lesubadd2i 8467 |
'Less than or equal to' relationship between subtraction and addition.
(Contributed by NM, 3-Aug-1999.)
|
  
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Theorem | ltaddsubi 8468 |
'Less than' relationship between subtraction and addition. (Contributed
by NM, 14-May-1999.)
|
  
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Theorem | lt2addi 8469 |
Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20.
(Contributed by NM, 14-May-1999.)
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Theorem | le2addi 8470 |
Adding both side of two inequalities. (Contributed by NM,
16-Sep-1999.)
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Theorem | gt0ne0d 8471 |
Positive implies nonzero. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | lt0ne0d 8472 |
Something less than zero is not zero. Deduction form. See also
lt0ap0d 8608 which is similar but for apartness.
(Contributed by David
Moews, 28-Feb-2017.)
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Theorem | leidd 8473 |
'Less than or equal to' is reflexive. (Contributed by Mario Carneiro,
27-May-2016.)
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Theorem | lt0neg1d 8474 |
Comparison of a number and its negative to zero. Theorem I.23 of
[Apostol] p. 20. (Contributed by Mario
Carneiro, 27-May-2016.)
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|
Theorem | lt0neg2d 8475 |
Comparison of a number and its negative to zero. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | le0neg1d 8476 |
Comparison of a number and its negative to zero. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | le0neg2d 8477 |
Comparison of a number and its negative to zero. (Contributed by Mario
Carneiro, 27-May-2016.)
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Theorem | addgegt0d 8478 |
Addition of nonnegative and positive numbers is positive.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | addgtge0d 8479 |
Addition of positive and nonnegative numbers is positive.
(Contributed by Asger C. Ipsen, 12-May-2021.)
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|
Theorem | addgt0d 8480 |
Addition of 2 positive numbers is positive. (Contributed by Mario
Carneiro, 27-May-2016.)
|
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|
Theorem | addge0d 8481 |
Addition of 2 nonnegative numbers is nonnegative. (Contributed by
Mario Carneiro, 27-May-2016.)
|
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Theorem | ltnegd 8482 |
Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | lenegd 8483 |
Negative of both sides of 'less than or equal to'. (Contributed by
Mario Carneiro, 27-May-2016.)
|
     
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Theorem | ltnegcon1d 8484 |
Contraposition of negative in 'less than'. (Contributed by Mario
Carneiro, 27-May-2016.)
|
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Theorem | ltnegcon2d 8485 |
Contraposition of negative in 'less than'. (Contributed by Mario
Carneiro, 27-May-2016.)
|
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Theorem | lenegcon1d 8486 |
Contraposition of negative in 'less than or equal to'. (Contributed
by Mario Carneiro, 27-May-2016.)
|
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Theorem | lenegcon2d 8487 |
Contraposition of negative in 'less than or equal to'. (Contributed
by Mario Carneiro, 27-May-2016.)
|
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Theorem | ltaddposd 8488 |
Adding a positive number to another number increases it. (Contributed
by Mario Carneiro, 27-May-2016.)
|
     
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Theorem | ltaddpos2d 8489 |
Adding a positive number to another number increases it. (Contributed
by Mario Carneiro, 27-May-2016.)
|
     
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Theorem | ltsubposd 8490 |
Subtracting a positive number from another number decreases it.
(Contributed by Mario Carneiro, 27-May-2016.)
|
       
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Theorem | posdifd 8491 |
Comparison of two numbers whose difference is positive. (Contributed by
Mario Carneiro, 27-May-2016.)
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Theorem | addge01d 8492 |
A number is less than or equal to itself plus a nonnegative number.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | addge02d 8493 |
A number is less than or equal to itself plus a nonnegative number.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | subge0d 8494 |
Nonnegative subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
     
 
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Theorem | suble0d 8495 |
Nonpositive subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
       
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Theorem | subge02d 8496 |
Nonnegative subtraction. (Contributed by Mario Carneiro,
27-May-2016.)
|
     
 
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Theorem | ltadd1d 8497 |
Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20.
(Contributed by Mario Carneiro, 27-May-2016.)
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Theorem | leadd1d 8498 |
Addition to both sides of 'less than or equal to'. (Contributed by
Mario Carneiro, 27-May-2016.)
|
         
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Theorem | leadd2d 8499 |
Addition to both sides of 'less than or equal to'. (Contributed by
Mario Carneiro, 27-May-2016.)
|
         
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Theorem | ltsubaddd 8500 |
'Less than' relationship between subtraction and addition. (Contributed
by Mario Carneiro, 27-May-2016.)
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