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Theorem List for Metamath Proof Explorer - 11501-11600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlesub0 11501 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))
 
Theoremmulge0 11502 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 · 𝐵))
 
Theoremmullt0 11503 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))
 
Theoremmsqgt0 11504 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.)
((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 < (𝐴 · 𝐴))
 
Theoremmsqge0 11505 A square is nonnegative. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)
(𝐴 ∈ ℝ → 0 ≤ (𝐴 · 𝐴))
 
Theorem0lt1 11506 0 is less than 1. Theorem I.21 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.)
0 < 1
 
Theorem0le1 11507 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
0 ≤ 1
 
Theoremrelin01 11508 An interval law for less than or equal. (Contributed by Scott Fenton, 27-Jun-2013.)
(𝐴 ∈ ℝ → (𝐴 ≤ 0 ∨ (0 ≤ 𝐴𝐴 ≤ 1) ∨ 1 ≤ 𝐴))
 
Theoremltordlem 11509* Lemma for ltord1 11510. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝐶𝐴 = 𝑀)    &   (𝑥 = 𝐷𝐴 = 𝑁)    &   𝑆 ⊆ ℝ    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐴 < 𝐵))       ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑀 < 𝑁))
 
Theoremltord1 11510* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝐶𝐴 = 𝑀)    &   (𝑥 = 𝐷𝐴 = 𝑁)    &   𝑆 ⊆ ℝ    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐴 < 𝐵))       ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑀 < 𝑁))
 
Theoremleord1 11511* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝐶𝐴 = 𝑀)    &   (𝑥 = 𝐷𝐴 = 𝑁)    &   𝑆 ⊆ ℝ    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐴 < 𝐵))       ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶𝐷𝑀𝑁))
 
Theoremeqord1 11512* A strictly increasing real function on a subset of is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝐶𝐴 = 𝑀)    &   (𝑥 = 𝐷𝐴 = 𝑁)    &   𝑆 ⊆ ℝ    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐴 < 𝐵))       ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 = 𝐷𝑀 = 𝑁))
 
Theoremltord2 11513* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝐶𝐴 = 𝑀)    &   (𝑥 = 𝐷𝐴 = 𝑁)    &   𝑆 ⊆ ℝ    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐵 < 𝐴))       ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑁 < 𝑀))
 
Theoremleord2 11514* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝐶𝐴 = 𝑀)    &   (𝑥 = 𝐷𝐴 = 𝑁)    &   𝑆 ⊆ ℝ    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐵 < 𝐴))       ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶𝐷𝑁𝑀))
 
Theoremeqord2 11515* A strictly decreasing real function on a subset of is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝐶𝐴 = 𝑀)    &   (𝑥 = 𝐷𝐴 = 𝑁)    &   𝑆 ⊆ ℝ    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐵 < 𝐴))       ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 = 𝐷𝑀 = 𝑁))
 
Theoremwloglei 11516* Form of wlogle 11517 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.)
((𝑧 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))    &   ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))    &   (𝜑𝑆 ⊆ ℝ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜃)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)       ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝜒)
 
Theoremwlogle 11517* If the predicate 𝜒(𝑥, 𝑦) is symmetric under interchange of 𝑥, 𝑦, then "without loss of generality" we can assume that 𝑥𝑦. (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.)
((𝑧 = 𝑥𝑤 = 𝑦) → (𝜓𝜒))    &   ((𝑧 = 𝑦𝑤 = 𝑥) → (𝜓𝜃))    &   (𝜑𝑆 ⊆ ℝ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝜒𝜃))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑥𝑦)) → 𝜒)       ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝜒)
 
Theoremleidi 11518 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℝ       𝐴𝐴
 
Theoremgt0ne0i 11519 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℝ       (0 < 𝐴𝐴 ≠ 0)
 
Theoremgt0ne0ii 11520 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 ≠ 0
 
Theoremmsqgt0i 11521 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.)
𝐴 ∈ ℝ       (𝐴 ≠ 0 → 0 < (𝐴 · 𝐴))
 
Theoremmsqge0i 11522 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
𝐴 ∈ ℝ       0 ≤ (𝐴 · 𝐴)
 
Theoremaddgt0i 11523 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 11524 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 11525 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 11526 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 11527 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 11528 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 11529 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵 ↔ -𝐵 ≤ -𝐴)
 
Theoremltnegcon2i 11530 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < -𝐵𝐵 < -𝐴)
 
Theoremmulge0i 11531 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → 0 ≤ (𝐴 · 𝐵))
 
Theoremlesub0i 11532 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 11533 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (0 < 𝐴𝐵 < (𝐵 + 𝐴))
 
Theoremposdifi 11534 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴))
 
Theoremltnegcon1i 11535 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴)
 
Theoremlenegcon1i 11536 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (-𝐴𝐵 ↔ -𝐵𝐴)
 
Theoremsubge0i 11537 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴)
 
Theoremltadd1i 11538 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 11539 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 11540 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 11541 '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 11542 '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 11543 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶))
 
Theoremlesubadd2i 11544 '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 11545 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵))
 
Theoremlt2addi 11546 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 11547 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 11548 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑 → 0 < 𝐴)       (𝜑𝐴 ≠ 0)
 
Theoremlt0ne0d 11549 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 < 0)       (𝜑𝐴 ≠ 0)
 
Theoremleidd 11550 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴𝐴)
 
Theoremmsqgt0d 11551 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ 0)       (𝜑 → 0 < (𝐴 · 𝐴))
 
Theoremmsqge0d 11552 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → 0 ≤ (𝐴 · 𝐴))
 
Theoremlt0neg1d 11553 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 11554 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (0 < 𝐴 ↔ -𝐴 < 0))
 
Theoremle0neg1d 11555 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
 
Theoremle0neg2d 11556 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0))
 
Theoremaddgegt0d 11557 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))
 
Theoremaddgtge0d 11558 Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → 0 < (𝐴 + 𝐵))
 
Theoremaddgt0d 11559 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 11560 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))
 
Theoremmulge0d 11561 The product of two 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 11562 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 11563 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 11564 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → -𝐴 < 𝐵)       (𝜑 → -𝐵 < 𝐴)
 
Theoremltnegcon2d 11565 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < -𝐵)       (𝜑𝐵 < -𝐴)
 
Theoremlenegcon1d 11566 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 11567 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 11568 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 11569 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 11570 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))
 
Theoremposdifd 11571 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))
 
Theoremaddge01d 11572 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ 𝐵𝐴 ≤ (𝐴 + 𝐵)))
 
Theoremaddge02d 11573 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ 𝐵𝐴 ≤ (𝐵 + 𝐴)))
 
Theoremsubge0d 11574 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴))
 
Theoremsuble0d 11575 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((𝐴𝐵) ≤ 0 ↔ 𝐴𝐵))
 
Theoremsubge02d 11576 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (0 ≤ 𝐵 ↔ (𝐴𝐵) ≤ 𝐴))
 
Theoremltadd1d 11577 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶)))
 
Theoremleadd1d 11578 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)))
 
Theoremleadd2d 11579 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)))
 
Theoremltsubaddd 11580 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) < 𝐶𝐴 < (𝐶 + 𝐵)))
 
Theoremlesubaddd 11581 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐶 + 𝐵)))
 
Theoremltsubadd2d 11582 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶)))
 
Theoremlesubadd2d 11583 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴𝐵) ≤ 𝐶𝐴 ≤ (𝐵 + 𝐶)))
 
Theoremltaddsubd 11584 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵)))
 
Theoremltaddsub2d 11585 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 + 𝐵) < 𝐶𝐵 < (𝐶𝐴)))
 
Theoremleaddsub2d 11586 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 + 𝐵) ≤ 𝐶𝐵 ≤ (𝐶𝐴)))
 
Theoremsubled 11587 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐴𝐵) ≤ 𝐶)       (𝜑 → (𝐴𝐶) ≤ 𝐵)
 
Theoremlesubd 11588 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ≤ (𝐵𝐶))       (𝜑𝐶 ≤ (𝐵𝐴))
 
Theoremltsub23d 11589 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐴𝐵) < 𝐶)       (𝜑 → (𝐴𝐶) < 𝐵)
 
Theoremltsub13d 11590 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐵𝐶))       (𝜑𝐶 < (𝐵𝐴))
 
Theoremlesub1d 11591 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐴𝐶) ≤ (𝐵𝐶)))
 
Theoremlesub2d 11592 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴𝐵 ↔ (𝐶𝐵) ≤ (𝐶𝐴)))
 
Theoremltsub1d 11593 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴𝐶) < (𝐵𝐶)))
 
Theoremltsub2d 11594 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐶𝐵) < (𝐶𝐴)))
 
Theoremltadd1dd 11595 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴 + 𝐶) < (𝐵 + 𝐶))
 
Theoremltsub1dd 11596 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴𝐶) < (𝐵𝐶))
 
Theoremltsub2dd 11597 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐶𝐵) < (𝐶𝐴))
 
Theoremleadd1dd 11598 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))
 
Theoremleadd2dd 11599 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))
 
Theoremlesub1dd 11600 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ≤ (𝐵𝐶))
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