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Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltsubadd 7501 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltsubadd2 7502 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)

Theoremlesubadd 7503 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlesubadd2 7504 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)

Theoremltaddsub 7505 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)

Theoremltaddsub2 7506 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)

Theoremleaddsub 7507 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)

Theoremleaddsub2 7508 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)

Theoremsuble 7509 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)

Theoremlesub 7510 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltsub23 7511 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)

Theoremltsub13 7512 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)

Theoremle2add 7513 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.)

Theoremlt2add 7514 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.)

Theoremltleadd 7515 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)

Theoremleltadd 7516 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)

Theoremaddgt0 7517 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgegt0 7518 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgtge0 7519 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddge0 7520 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltaddpos 7521 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)

Theoremltaddpos2 7522 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)

Theoremltsubpos 7523 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremposdif 7524 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)

Theoremlesub1 7525 Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlesub2 7526 Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltsub1 7527 Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltsub2 7528 Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt2sub 7529 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremle2sub 7530 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremltneg 7531 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.)

Theoremltnegcon1 7532 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)

Theoremltnegcon2 7533 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremleneg 7534 Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlenegcon1 7535 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)

Theoremlenegcon2 7536 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)

Theoremlt0neg1 7537 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremlt0neg2 7538 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)

Theoremle0neg1 7539 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)

Theoremle0neg2 7540 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)

Theoremaddge01 7541 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)

Theoremaddge02 7542 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)

Theoremadd20 7543 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.)

Theoremsubge0 7544 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsuble0 7545 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremleaddle0 7546 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.)

Theoremsubge02 7547 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)

Theoremlesub0 7548 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmullt0 7549 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)

Theorem0le1 7550 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremleidi 7551 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)

Theoremgt0ne0i 7552 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)

Theoremgt0ne0ii 7553 Positive implies nonzero. (Contributed by NM, 15-May-1999.)

Theoremaddgt0i 7554 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddge0i 7555 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgegt0i 7556 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddgt0ii 7557 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremadd20i 7558 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)

Theoremltnegi 7559 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)

Theoremlenegi 7560 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)

Theoremltnegcon2i 7561 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)

Theoremlesub0i 7562 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltaddposi 7563 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)

Theoremposdifi 7564 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)

Theoremltnegcon1i 7565 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)

Theoremlenegcon1i 7566 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)

Theoremsubge0i 7567 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)

Theoremltadd1i 7568 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)

Theoremleadd1i 7569 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)

Theoremleadd2i 7570 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)

Theoremltsubaddi 7571 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremlesubaddi 7572 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltsubadd2i 7573 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)

Theoremlesubadd2i 7574 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)

Theoremltaddsubi 7575 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)

Theoremlt2addi 7576 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremle2addi 7577 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)

Theoremgt0ne0d 7578 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0ne0d 7579 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theoremleidd 7580 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0neg1d 7581 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0neg2d 7582 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle0neg1d 7583 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle0neg2d 7584 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddgegt0d 7585 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddgt0d 7586 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge0d 7587 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegd 7588 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegd 7589 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegcon1d 7590 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegcon2d 7591 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegcon1d 7592 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegcon2d 7593 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddposd 7594 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddpos2d 7595 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubposd 7596 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremposdifd 7597 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge01d 7598 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge02d 7599 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge0d 7600 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

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