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Theorem List for Intuitionistic Logic Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltsubadd 8301 '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 8302 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶)))

Theoremlesubadd 8303 '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 8304 '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 8305 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵)))

Theoremltaddsub2 8306 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶𝐵 < (𝐶𝐴)))

Theoremleaddsub 8307 '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 8308 '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 8309 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) ≤ 𝐶 ↔ (𝐴𝐶) ≤ 𝐵))

Theoremlesub 8310 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 8311 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) < 𝐶 ↔ (𝐴𝐶) < 𝐵))

Theoremltsub13 8312 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < (𝐵𝐶) ↔ 𝐶 < (𝐵𝐴)))

Theoremle2add 8313 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 8314 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 8315 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 8316 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 8317 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 8318 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 8319 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 8320 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 8321 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))

Theoremltaddpos2 8322 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐴 + 𝐵)))

Theoremltsubpos 8323 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 8324 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))

Theoremlesub1 8325 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 8326 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 8327 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 8328 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 8329 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 8330 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 8331 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 8332 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴))

Theoremltnegcon2 8333 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < -𝐵𝐵 < -𝐴))

Theoremleneg 8334 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 8335 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴𝐵 ↔ -𝐵𝐴))

Theoremlenegcon2 8336 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ -𝐵𝐵 ≤ -𝐴))

Theoremlt0neg1 8337 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 8338 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ -𝐴 < 0))

Theoremle0neg1 8339 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
(𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))

Theoremle0neg2 8340 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
(𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0))

Theoremaddge01 8341 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 8342 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 8343 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 8344 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴))

Theoremsuble0 8345 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴𝐵) ≤ 0 ↔ 𝐴𝐵))

Theoremleaddle0 8346 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 8347 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ (𝐴𝐵) ≤ 𝐴))

Theoremlesub0 8348 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 8349 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 8350 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
0 ≤ 1

Theoremltordlem 8351* Lemma for eqord1 8352. (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))

Theoremeqord1 8352* A strictly increasing real function on a subset of is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝐶𝐴 = 𝑀)    &   (𝑥 = 𝐷𝐴 = 𝑁)    &   𝑆 ⊆ ℝ    &   ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐴 < 𝐵))       ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 = 𝐷𝑀 = 𝑁))

Theoremeqord2 8353* 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))

Theoremleidi 8354 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℝ       𝐴𝐴

Theoremgt0ne0i 8355 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℝ       (0 < 𝐴𝐴 ≠ 0)

Theoremgt0ne0ii 8356 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 ≠ 0

Theoremaddgt0i 8357 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 8358 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 8359 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 8360 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 8361 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 8362 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 8363 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵 ↔ -𝐵 ≤ -𝐴)

Theoremltnegcon2i 8364 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < -𝐵𝐵 < -𝐴)

Theoremlesub0i 8365 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 8366 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (0 < 𝐴𝐵 < (𝐵 + 𝐴))

Theoremposdifi 8367 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴))

Theoremltnegcon1i 8368 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴)

Theoremlenegcon1i 8369 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (-𝐴𝐵 ↔ -𝐵𝐴)

Theoremsubge0i 8370 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴)

Theoremltadd1i 8371 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 8372 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 8373 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 8374 '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 8375 '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 8376 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶))

Theoremlesubadd2i 8377 '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 8378 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵))

Theoremlt2addi 8379 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 8380 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 8381 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑 → 0 < 𝐴)       (𝜑𝐴 ≠ 0)

Theoremlt0ne0d 8382 Something less than zero is not zero. Deduction form. See also lt0ap0d 8518 which is similar but for apartness. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 < 0)       (𝜑𝐴 ≠ 0)

Theoremleidd 8383 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑𝐴𝐴)

Theoremlt0neg1d 8384 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 8385 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (0 < 𝐴 ↔ -𝐴 < 0))

Theoremle0neg1d 8386 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))

Theoremle0neg2d 8387 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0))

Theoremaddgegt0d 8388 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 8389 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 8390 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 8391 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 8392 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 8393 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 8394 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → -𝐴 < 𝐵)       (𝜑 → -𝐵 < 𝐴)

Theoremltnegcon2d 8395 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < -𝐵)       (𝜑𝐵 < -𝐴)

Theoremlenegcon1d 8396 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 8397 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 8398 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 8399 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 8400 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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