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Theorem List for Intuitionistic Logic Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulneg1i 7401 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulneg2i 7402 Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmul2negi 7403 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubdii 7404 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)

Theoremsubdiri 7405 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)

Theoremmuladdi 7406 Product of two sums. (Contributed by NM, 17-May-1999.)

Theoremmulm1d 7407 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulneg1d 7408 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulneg2d 7409 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul2negd 7410 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubdid 7411 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubdird 7412 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmuladdd 7413 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulsubd 7414 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulsubfacd 7415 Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)

3.3.4  Ordering on reals (cont.)

Theoremltadd2 7416 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltadd2i 7417 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)

Theoremltadd2d 7418 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd2dd 7419 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltletrd 7420 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)

Theoremlelttrdi 7421 If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.)

Theoremgt0ne0 7422 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt0ne0 7423 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremltadd1 7424 Addition to both sides of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremleadd1 7425 Addition to both sides of 'less than or equal to'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremleadd2 7426 Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)

Theoremltsubadd 7427 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltsubadd2 7428 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)

Theoremlesubadd 7429 '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 7430 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)

Theoremltaddsub 7431 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)

Theoremltaddsub2 7432 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)

Theoremleaddsub 7433 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)

Theoremleaddsub2 7434 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)

Theoremsuble 7435 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)

Theoremlesub 7436 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltsub23 7437 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)

Theoremltsub13 7438 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)

Theoremle2add 7439 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 7440 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 7441 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)

Theoremleltadd 7442 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)

Theoremaddgt0 7443 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgegt0 7444 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgtge0 7445 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddge0 7446 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltaddpos 7447 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)

Theoremltaddpos2 7448 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)

Theoremltsubpos 7449 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremposdif 7450 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)

Theoremlesub1 7451 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 7452 Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltsub1 7453 Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltsub2 7454 Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt2sub 7455 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremle2sub 7456 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremltneg 7457 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 7458 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)

Theoremltnegcon2 7459 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremleneg 7460 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 7461 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)

Theoremlenegcon2 7462 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)

Theoremlt0neg1 7463 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremlt0neg2 7464 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)

Theoremle0neg1 7465 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)

Theoremle0neg2 7466 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)

Theoremaddge01 7467 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)

Theoremaddge02 7468 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)

Theoremadd20 7469 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 7470 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsuble0 7471 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremleaddle0 7472 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 7473 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)

Theoremlesub0 7474 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmullt0 7475 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)

Theorem0le1 7476 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremleidi 7477 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)

Theoremgt0ne0i 7478 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)

Theoremgt0ne0ii 7479 Positive implies nonzero. (Contributed by NM, 15-May-1999.)

Theoremaddgt0i 7480 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddge0i 7481 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgegt0i 7482 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddgt0ii 7483 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremadd20i 7484 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)

Theoremltnegi 7485 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)

Theoremlenegi 7486 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)

Theoremltnegcon2i 7487 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)

Theoremlesub0i 7488 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltaddposi 7489 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)

Theoremposdifi 7490 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)

Theoremltnegcon1i 7491 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)

Theoremlenegcon1i 7492 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)

Theoremsubge0i 7493 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)

Theoremltadd1i 7494 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)

Theoremleadd1i 7495 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)

Theoremleadd2i 7496 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)

Theoremltsubaddi 7497 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremlesubaddi 7498 '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 7499 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)

Theoremlesubadd2i 7500 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)

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