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Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubdid 8201 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (𝐴 · (𝐵𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)))

Theoremsubdird 8202 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶)))

Theoremmuladdd 8203 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) + ((𝐴 · 𝐷) + (𝐶 · 𝐵))))

Theoremmulsubd 8204 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ((𝐴𝐵) · (𝐶𝐷)) = (((𝐴 · 𝐶) + (𝐷 · 𝐵)) − ((𝐴 · 𝐷) + (𝐶 · 𝐵))))

Theoremmulsubfacd 8205 Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 · 𝐵) − 𝐵) = ((𝐴 − 1) · 𝐵))

4.3.4  Ordering on reals (cont.)

Theoremltadd2 8206 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))

Theoremltadd2i 8207 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))

Theoremltadd2d 8208 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵)))

Theoremltadd2dd 8209 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐶 + 𝐴) < (𝐶 + 𝐵))

Theoremltletrd 8210 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴 < 𝐶)

Theoremltaddneg 8211 Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))

Theoremltaddnegr 8212 Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐴 + 𝐵) < 𝐵))

Theoremlelttrdi 8213 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.)
(𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ))    &   (𝜑𝐵𝐶)       (𝜑 → (𝐴 < 𝐵𝐴 < 𝐶))

Theoremgt0ne0 8214 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0)

Theoremlt0ne0 8215 A number which is less than zero is not zero. See also lt0ap0 8435 which is similar but for apartness. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ≠ 0)

Theoremltadd1 8216 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 8217 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 8218 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 8219 '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 8220 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶)))

Theoremlesubadd 8221 '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 8222 '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 8223 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵)))

Theoremltaddsub2 8224 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶𝐵 < (𝐶𝐴)))

Theoremleaddsub 8225 '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 8226 '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 8227 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) ≤ 𝐶 ↔ (𝐴𝐶) ≤ 𝐵))

Theoremlesub 8228 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 8229 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐵) < 𝐶 ↔ (𝐴𝐶) < 𝐵))

Theoremltsub13 8230 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < (𝐵𝐶) ↔ 𝐶 < (𝐵𝐴)))

Theoremle2add 8231 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 8232 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 8233 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 8234 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 8235 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 8236 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 8237 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 8238 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 8239 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))

Theoremltaddpos2 8240 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐴 + 𝐵)))

Theoremltsubpos 8241 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 8242 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴)))

Theoremlesub1 8243 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 8244 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 8245 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 8246 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 8247 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 8248 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 8249 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 8250 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴))

Theoremltnegcon2 8251 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < -𝐵𝐵 < -𝐴))

Theoremleneg 8252 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 8253 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴𝐵 ↔ -𝐵𝐴))

Theoremlenegcon2 8254 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ -𝐵𝐵 ≤ -𝐴))

Theoremlt0neg1 8255 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 8256 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ -𝐴 < 0))

Theoremle0neg1 8257 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
(𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))

Theoremle0neg2 8258 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
(𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0))

Theoremaddge01 8259 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 8260 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 8261 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 8262 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴))

Theoremsuble0 8263 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴𝐵) ≤ 0 ↔ 𝐴𝐵))

Theoremleaddle0 8264 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 8265 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ (𝐴𝐵) ≤ 𝐴))

Theoremlesub0 8266 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 8267 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 8268 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
0 ≤ 1

Theoremltordlem 8269* Lemma for eqord1 8270. (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 8270* 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 8271* 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 8272 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
𝐴 ∈ ℝ       𝐴𝐴

Theoremgt0ne0i 8273 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
𝐴 ∈ ℝ       (0 < 𝐴𝐴 ≠ 0)

Theoremgt0ne0ii 8274 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
𝐴 ∈ ℝ    &   0 < 𝐴       𝐴 ≠ 0

Theoremaddgt0i 8275 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 8276 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 8277 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 8278 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 8279 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 8280 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 8281 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴𝐵 ↔ -𝐵 ≤ -𝐴)

Theoremltnegcon2i 8282 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < -𝐵𝐵 < -𝐴)

Theoremlesub0i 8283 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 8284 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (0 < 𝐴𝐵 < (𝐵 + 𝐴))

Theoremposdifi 8285 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (𝐴 < 𝐵 ↔ 0 < (𝐵𝐴))

Theoremltnegcon1i 8286 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴)

Theoremlenegcon1i 8287 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (-𝐴𝐵 ↔ -𝐵𝐴)

Theoremsubge0i 8288 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ       (0 ≤ (𝐴𝐵) ↔ 𝐵𝐴)

Theoremltadd1i 8289 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 8290 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 8291 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 8292 '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 8293 '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 8294 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴𝐵) < 𝐶𝐴 < (𝐵 + 𝐶))

Theoremlesubadd2i 8295 '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 8296 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ       ((𝐴 + 𝐵) < 𝐶𝐴 < (𝐶𝐵))

Theoremlt2addi 8297 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 8298 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 8299 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(𝜑 → 0 < 𝐴)       (𝜑𝐴 ≠ 0)

Theoremlt0ne0d 8300 Something less than zero is not zero. Deduction form. See also lt0ap0d 8436 which is similar but for apartness. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 < 0)       (𝜑𝐴 ≠ 0)

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