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Theorem List for Metamath Proof Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulge0 9501 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulge0OLD 9502 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmullt0 9503 The product of two negative numbers is positive. (Contributed by Jeffrey Hankins, 8-Jun-2009.)

Theoremmsqgt0 9504 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by NM, 6-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmsqge0 9505 A square is nonnegative. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)

Theorem0lt1 9506 0 is less than 1. Theorem I.21 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.)

Theorem0le1 9507 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremltordlem 9508* Lemma for ltord1 9509. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremltord1 9509* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremleord1 9510* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremeqord1 9511* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremltord2 9512* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremleord2 9513* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremeqord2 9514* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremwloglei 9515* Form of wlogle 9516 where both sides of the equivalence are proven rather than showing that they are equivalent to each other. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremwlogle 9516* If the predicate is symmetric under interchange of , then "without loss of generality" we can assume that . (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.)

Theoremleidi 9517 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)

Theoremgt0ne0i 9518 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)

Theoremgt0ne0ii 9519 Positive implies nonzero. (Contributed by NM, 15-May-1999.)

Theoremmsqgt0i 9520 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmsqge0i 9521 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgt0i 9522 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddge0i 9523 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgegt0i 9524 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddgt0ii 9525 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremadd20i 9526 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)

Theoremltnegi 9527 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)

Theoremlenegi 9528 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)

Theoremltnegcon2i 9529 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)

Theoremmulge0i 9530 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)

Theoremlesub0i 9531 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltaddposi 9532 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)

Theoremposdifi 9533 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)

Theoremltnegcon1i 9534 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)

Theoremlenegcon1i 9535 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)

Theoremsubge0i 9536 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)

Theoremltadd1i 9537 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)

Theoremleadd1i 9538 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)

Theoremleadd2i 9539 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)

Theoremltsubaddi 9540 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremlesubaddi 9541 '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 9542 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)

Theoremlesubadd2i 9543 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)

Theoremltaddsubi 9544 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)

Theoremlt2addi 9545 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremle2addi 9546 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)

Theoremgt0ne0d 9547 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0ne0d 9548 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theoremleidd 9549 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmsqgt0d 9550 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmsqge0d 9551 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0neg1d 9552 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0neg2d 9553 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle0neg1d 9554 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle0neg2d 9555 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddgegt0d 9556 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddgt0d 9557 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge0d 9558 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulge0d 9559 The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegd 9560 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegd 9561 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegcon1d 9562 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegcon2d 9563 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegcon1d 9564 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegcon2d 9565 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddposd 9566 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddpos2d 9567 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubposd 9568 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremposdifd 9569 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge01d 9570 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge02d 9571 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge0d 9572 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsuble0d 9573 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge02d 9574 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd1d 9575 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleadd1d 9576 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleadd2d 9577 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubaddd 9578 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubaddd 9579 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubadd2d 9580 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubadd2d 9581 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddsubd 9582 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddsub2d 9583 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremleaddsub2d 9584 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubled 9585 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubd 9586 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub23d 9587 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub13d 9588 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesub1d 9589 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesub2d 9590 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub1d 9591 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub2d 9592 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd1dd 9593 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltsub1dd 9594 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltsub2dd 9595 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd1dd 9596 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd2dd 9597 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub1dd 9598 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub2dd 9599 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremle2addd 9600 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)

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