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Theorem List for Intuitionistic Logic Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlt0ne0 7401 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ≠ 0)
 
Theoremltadd1 7402 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.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶)))
 
Theoremleadd1 7403 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.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)))
 
Theoremleadd2 7404 Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)))
 
Theoremltsubadd 7405 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) < 𝐶𝐴 < (𝐶 + 𝐵)))
 
Theoremltsubadd2 7406 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶)))
 
Theoremlesubadd 7407 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐶 + 𝐵)))
 
Theoremlesubadd2 7408 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐵 + 𝐶)))
 
Theoremltaddsub 7409 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵)))
 
Theoremltaddsub2 7410 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶𝐵 < (𝐶𝐴)))
 
Theoremleaddsub 7411 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶𝐴 ≤ (𝐶𝐵)))
 
Theoremleaddsub2 7412 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶𝐵 ≤ (𝐶𝐴)))
 
Theoremsuble 7413 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) ≤ 𝐶 ↔ (𝐴𝐶) ≤ 𝐵))
 
Theoremlesub 7414 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ (𝐵𝐶) ↔ 𝐶 ≤ (𝐵𝐴)))
 
Theoremltsub23 7415 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) < 𝐶 ↔ (𝐴𝐶) < 𝐵))
 
Theoremltsub13 7416 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < (𝐵𝐶) ↔ 𝐶 < (𝐵𝐴)))
 
Theoremle2add 7417 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.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴𝐶𝐵𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)))
 
Theoremlt2add 7418 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.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷)))
 
Theoremltleadd 7419 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶𝐵𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷)))
 
Theoremleltadd 7420 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴𝐶𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷)))
 
Theoremaddgt0 7421 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵))
 
Theoremaddgegt0 7422 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵))
 
Theoremaddgtge0 7423 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 ≤ 𝐵)) → 0 < (𝐴 + 𝐵))
 
Theoremaddge0 7424 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 + 𝐵))
 
Theoremltaddpos 7425 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))
 
Theoremltaddpos2 7426 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐴 + 𝐵)))
 
Theoremltsubpos 7427 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ (𝐵𝐴) < 𝐵))
 
Theoremposdif 7428 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))
 
Theoremlesub1 7429 Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴𝐵 ↔ (𝐴𝐶) ≤ (𝐵𝐶)))
 
Theoremlesub2 7430 Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴𝐵 ↔ (𝐶𝐵) ≤ (𝐶𝐴)))
 
Theoremltsub1 7431 Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴𝐶) < (𝐵𝐶)))
 
Theoremltsub2 7432 Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶𝐵) < (𝐶𝐴)))
 
Theoremlt2sub 7433 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶𝐷 < 𝐵) → (𝐴𝐵) < (𝐶𝐷)))
 
Theoremle2sub 7434 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴𝐶𝐷𝐵) → (𝐴𝐵) ≤ (𝐶𝐷)))
 
Theoremltneg 7435 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.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
 
Theoremltnegcon1 7436 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴))
 
Theoremltnegcon2 7437 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < -𝐵𝐵 < -𝐴))
 
Theoremleneg 7438 Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐵 ↔ -𝐵 ≤ -𝐴))
 
Theoremlenegcon1 7439 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴𝐵 ↔ -𝐵𝐴))
 
Theoremlenegcon2 7440 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ -𝐵𝐵 ≤ -𝐴))
 
Theoremlt0neg1 7441 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
(𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴))
 
Theoremlt0neg2 7442 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ -𝐴 < 0))
 
Theoremle0neg1 7443 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
(𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
 
Theoremle0neg2 7444 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
(𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0))
 
Theoremaddge01 7445 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵𝐴 ≤ (𝐴 + 𝐵)))
 
Theoremaddge02 7446 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵𝐴 ≤ (𝐵 + 𝐴)))
 
Theoremadd20 7447 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.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
 
Theoremsubge0 7448 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴))
 
Theoremsuble0 7449 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴𝐵) ≤ 0 ↔ 𝐴𝐵))
 
Theoremleaddle0 7450 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.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐴𝐵 ≤ 0))
 
Theoremsubge02 7451 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ (𝐴𝐵) ≤ 𝐴))
 
Theoremlesub0 7452 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 ≤ 𝐴𝐵 ≤ (𝐵𝐴)) ↔ 𝐴 = 0))
 
Theoremmullt0 7453 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
(((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (𝐴 · 𝐵))
 
Theorem0le1 7454 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
0 ≤ 1
 
Theoremleidi 7455 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℝ       𝐴𝐴
 
Theoremgt0ne0i 7456 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℝ       (0 < 𝐴𝐴 ≠ 0)
 
Theoremgt0ne0ii 7457 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 ≠ 0
 
Theoremaddgt0i 7458 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 + 𝐵))
 
Theoremaddge0i 7459 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → 0 ≤ (𝐴 + 𝐵))
 
Theoremaddgegt0i 7460 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 + 𝐵))
 
Theoremaddgt0ii 7461 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   0 < 𝐴    &   0 < 𝐵       0 < (𝐴 + 𝐵)
 
Theoremadd20i 7462 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
 
Theoremltnegi 7463 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵 ↔ -𝐵 < -𝐴)
 
Theoremlenegi 7464 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵 ↔ -𝐵 ≤ -𝐴)
 
Theoremltnegcon2i 7465 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < -𝐵𝐵 < -𝐴)
 
Theoremlesub0i 7466 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴𝐵 ≤ (𝐵𝐴)) ↔ 𝐴 = 0)
 
Theoremltaddposi 7467 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (0 < 𝐴𝐵 < (𝐵 + 𝐴))
 
Theoremposdifi 7468 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴))
 
Theoremltnegcon1i 7469 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴)
 
Theoremlenegcon1i 7470 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (-𝐴𝐵 ↔ -𝐵𝐴)
 
Theoremsubge0i 7471 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴)
 
Theoremltadd1i 7472 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶))
 
Theoremleadd1i 7473 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (𝐴𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))
 
Theoremleadd2i 7474 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (𝐴𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))
 
Theoremltsubaddi 7475 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) < 𝐶𝐴 < (𝐶 + 𝐵))
 
Theoremlesubaddi 7476 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐶 + 𝐵))
 
Theoremltsubadd2i 7477 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶))
 
Theoremlesubadd2i 7478 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐵 + 𝐶))
 
Theoremltaddsubi 7479 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵))
 
Theoremlt2addi 7480 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   𝐷 ∈ ℝ       ((𝐴 < 𝐶𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷))
 
Theoremle2addi 7481 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   𝐷 ∈ ℝ       ((𝐴𝐶𝐵𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))
 
Theoremgt0ne0d 7482 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑 → 0 < 𝐴)       (𝜑𝐴 ≠ 0)
 
Theoremlt0ne0d 7483 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 < 0)       (𝜑𝐴 ≠ 0)
 
Theoremleidd 7484 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴𝐴)
 
Theoremlt0neg1d 7485 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 < 0 ↔ 0 < -𝐴))
 
Theoremlt0neg2d 7486 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (0 < 𝐴 ↔ -𝐴 < 0))
 
Theoremle0neg1d 7487 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
 
Theoremle0neg2d 7488 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0))
 
Theoremaddgegt0d 7489 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 < 𝐵)       (𝜑 → 0 < (𝐴 + 𝐵))
 
Theoremaddgt0d 7490 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 < 𝐵)       (𝜑 → 0 < (𝐴 + 𝐵))
 
Theoremaddge0d 7491 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → 0 ≤ (𝐴 + 𝐵))
 
Theoremltnegd 7492 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
 
Theoremlenegd 7493 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ -𝐵 ≤ -𝐴))
 
Theoremltnegcon1d 7494 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → -𝐴 < 𝐵)       (𝜑 → -𝐵 < 𝐴)
 
Theoremltnegcon2d 7495 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < -𝐵)       (𝜑𝐵 < -𝐴)
 
Theoremlenegcon1d 7496 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → -𝐴𝐵)       (𝜑 → -𝐵𝐴)
 
Theoremlenegcon2d 7497 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 ≤ -𝐵)       (𝜑𝐵 ≤ -𝐴)
 
Theoremltaddposd 7498 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))
 
Theoremltaddpos2d 7499 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < 𝐴𝐵 < (𝐴 + 𝐵)))
 
Theoremltsubposd 7500 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 < 𝐴 ↔ (𝐵𝐴) < 𝐵))
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