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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | addgegt0d 11201 | Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 + 𝐵)) | ||
Theorem | addgtge0d 11202 | Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 + 𝐵)) | ||
Theorem | addgt0d 11203 | Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 + 𝐵)) | ||
Theorem | addge0d 11204 | Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) | ||
Theorem | mulge0d 11205 | The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵)) | ||
Theorem | ltnegd 11206 | Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴)) | ||
Theorem | lenegd 11207 | Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) | ||
Theorem | ltnegcon1d 11208 | Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → -𝐴 < 𝐵) ⇒ ⊢ (𝜑 → -𝐵 < 𝐴) | ||
Theorem | ltnegcon2d 11209 | Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < -𝐵) ⇒ ⊢ (𝜑 → 𝐵 < -𝐴) | ||
Theorem | lenegcon1d 11210 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → -𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → -𝐵 ≤ 𝐴) | ||
Theorem | lenegcon2d 11211 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ -𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≤ -𝐴) | ||
Theorem | ltaddposd 11212 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴))) | ||
Theorem | ltaddpos2d 11213 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 < 𝐴 ↔ 𝐵 < (𝐴 + 𝐵))) | ||
Theorem | ltsubposd 11214 | Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 < 𝐴 ↔ (𝐵 − 𝐴) < 𝐵)) | ||
Theorem | posdifd 11215 | Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) | ||
Theorem | addge01d 11216 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵))) | ||
Theorem | addge02d 11217 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 + 𝐴))) | ||
Theorem | subge0d 11218 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)) | ||
Theorem | suble0d 11219 | Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 0 ↔ 𝐴 ≤ 𝐵)) | ||
Theorem | subge02d 11220 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 ≤ 𝐵 ↔ (𝐴 − 𝐵) ≤ 𝐴)) | ||
Theorem | ltadd1d 11221 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶))) | ||
Theorem | leadd1d 11222 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))) | ||
Theorem | leadd2d 11223 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))) | ||
Theorem | ltsubaddd 11224 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 + 𝐵))) | ||
Theorem | lesubaddd 11225 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) | ||
Theorem | ltsubadd2d 11226 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶))) | ||
Theorem | lesubadd2d 11227 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐵 + 𝐶))) | ||
Theorem | ltaddsubd 11228 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵))) | ||
Theorem | ltaddsub2d 11229 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐵 < (𝐶 − 𝐴))) | ||
Theorem | leaddsub2d 11230 | 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐵 ≤ (𝐶 − 𝐴))) | ||
Theorem | subled 11231 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (𝐴 − 𝐵) ≤ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) ≤ 𝐵) | ||
Theorem | lesubd 11232 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ (𝐵 − 𝐶)) ⇒ ⊢ (𝜑 → 𝐶 ≤ (𝐵 − 𝐴)) | ||
Theorem | ltsub23d 11233 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (𝐴 − 𝐵) < 𝐶) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) < 𝐵) | ||
Theorem | ltsub13d 11234 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < (𝐵 − 𝐶)) ⇒ ⊢ (𝜑 → 𝐶 < (𝐵 − 𝐴)) | ||
Theorem | lesub1d 11235 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝐶) ≤ (𝐵 − 𝐶))) | ||
Theorem | lesub2d 11236 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 − 𝐵) ≤ (𝐶 − 𝐴))) | ||
Theorem | ltsub1d 11237 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 − 𝐶) < (𝐵 − 𝐶))) | ||
Theorem | ltsub2d 11238 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 − 𝐵) < (𝐶 − 𝐴))) | ||
Theorem | ltadd1dd 11239 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (𝐴 + 𝐶) < (𝐵 + 𝐶)) | ||
Theorem | ltsub1dd 11240 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) < (𝐵 − 𝐶)) | ||
Theorem | ltsub2dd 11241 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐵) ⇒ ⊢ (𝜑 → (𝐶 − 𝐵) < (𝐶 − 𝐴)) | ||
Theorem | leadd1dd 11242 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)) | ||
Theorem | leadd2dd 11243 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)) | ||
Theorem | lesub1dd 11244 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 − 𝐶) ≤ (𝐵 − 𝐶)) | ||
Theorem | lesub2dd 11245 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 − 𝐵) ≤ (𝐶 − 𝐴)) | ||
Theorem | lesub3d 11246 | The result of subtracting a number less than or equal to an intermediate number from a number greater than or equal to a third number increased by the intermediate number is greater than or equal to the third number. (Contributed by AV, 13-Aug-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → (𝑋 + 𝐶) ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ≤ 𝑋) ⇒ ⊢ (𝜑 → 𝐶 ≤ (𝐴 − 𝐵)) | ||
Theorem | le2addd 11247 | Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐵 ≤ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) | ||
Theorem | le2subd 11248 | Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐵 ≤ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 − 𝐷) ≤ (𝐶 − 𝐵)) | ||
Theorem | ltleaddd 11249 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐶) & ⊢ (𝜑 → 𝐵 ≤ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) | ||
Theorem | leltaddd 11250 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐶) & ⊢ (𝜑 → 𝐵 < 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) | ||
Theorem | lt2addd 11251 | Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐶) & ⊢ (𝜑 → 𝐵 < 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) | ||
Theorem | lt2subd 11252 | Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 𝐶) & ⊢ (𝜑 → 𝐵 < 𝐷) ⇒ ⊢ (𝜑 → (𝐴 − 𝐷) < (𝐶 − 𝐵)) | ||
Theorem | possumd 11253 | Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (0 < (𝐴 + 𝐵) ↔ -𝐵 < 𝐴)) | ||
Theorem | sublt0d 11254 | When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) < 0 ↔ 𝐴 < 𝐵)) | ||
Theorem | ltaddsublt 11255 | Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐶 ↔ ((𝐴 + 𝐵) − 𝐶) < 𝐴)) | ||
Theorem | 1le1 11256 | One is less than or equal to one. (Contributed by David A. Wheeler, 16-Jul-2016.) |
⊢ 1 ≤ 1 | ||
Theorem | ixi 11257 | i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ (i · i) = -1 | ||
Theorem | recextlem1 11258 | Lemma for recex 11260. (Contributed by Eric Schmidt, 23-May-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + (i · 𝐵)) · (𝐴 − (i · 𝐵))) = ((𝐴 · 𝐴) + (𝐵 · 𝐵))) | ||
Theorem | recextlem2 11259 | Lemma for recex 11260. (Contributed by Eric Schmidt, 23-May-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + (i · 𝐵)) ≠ 0) → ((𝐴 · 𝐴) + (𝐵 · 𝐵)) ≠ 0) | ||
Theorem | recex 11260* | Existence of reciprocal of nonzero complex number. (Contributed by Eric Schmidt, 22-May-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℂ (𝐴 · 𝑥) = 1) | ||
Theorem | mulcand 11261 | Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | mulcan2d 11262 | Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | mulcanad 11263 | Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand 11261. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ≠ 0) & ⊢ (𝜑 → (𝐶 · 𝐴) = (𝐶 · 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | mulcan2ad 11264 | Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d 11262. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ≠ 0) & ⊢ (𝜑 → (𝐴 · 𝐶) = (𝐵 · 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | mulcan 11265 | Cancellation law for multiplication (full theorem form). Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | mulcan2 11266 | Cancellation law for multiplication. (Contributed by NM, 21-Jan-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | mulcani 11267 | Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ 𝐶 ≠ 0 ⇒ ⊢ ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵) | ||
Theorem | mul0or 11268 | If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0))) | ||
Theorem | mulne0b 11269 | The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ (𝐴 · 𝐵) ≠ 0)) | ||
Theorem | mulne0 11270 | The product of two nonzero numbers is nonzero. (Contributed by NM, 30-Dec-2007.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 · 𝐵) ≠ 0) | ||
Theorem | mulne0i 11271 | The product of two nonzero numbers is nonzero. (Contributed by NM, 15-Feb-1995.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐴 ≠ 0 & ⊢ 𝐵 ≠ 0 ⇒ ⊢ (𝐴 · 𝐵) ≠ 0 | ||
Theorem | muleqadd 11272 | Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = (𝐴 + 𝐵) ↔ ((𝐴 − 1) · (𝐵 − 1)) = 1)) | ||
Theorem | receu 11273* | Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 17-Feb-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) | ||
Theorem | mulnzcnopr 11274 | Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.) |
⊢ ( · ↾ ((ℂ ∖ {0}) × (ℂ ∖ {0}))):((ℂ ∖ {0}) × (ℂ ∖ {0}))⟶(ℂ ∖ {0}) | ||
Theorem | msq0i 11275 | A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by NM, 28-Jul-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ ((𝐴 · 𝐴) = 0 ↔ 𝐴 = 0) | ||
Theorem | mul0ori 11276 | If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 7-Oct-1999.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0)) | ||
Theorem | msq0d 11277 | A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐴) = 0 ↔ 𝐴 = 0)) | ||
Theorem | mul0ord 11278 | If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0))) | ||
Theorem | mulne0bd 11279 | The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 ≠ 0 ∧ 𝐵 ≠ 0) ↔ (𝐴 · 𝐵) ≠ 0)) | ||
Theorem | mulne0d 11280 | The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) ≠ 0) | ||
Theorem | mulcan1g 11281 | A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) = (𝐴 · 𝐶) ↔ (𝐴 = 0 ∨ 𝐵 = 𝐶))) | ||
Theorem | mulcan2g 11282 | A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ (𝐴 = 𝐵 ∨ 𝐶 = 0))) | ||
Theorem | mulne0bad 11283 | A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 11280 and consequence of mulne0bd 11279. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 · 𝐵) ≠ 0) ⇒ ⊢ (𝜑 → 𝐴 ≠ 0) | ||
Theorem | mulne0bbd 11284 | A factor of a nonzero complex number is nonzero. Partial converse of mulne0d 11280 and consequence of mulne0bd 11279. (Contributed by David Moews, 28-Feb-2017.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 · 𝐵) ≠ 0) ⇒ ⊢ (𝜑 → 𝐵 ≠ 0) | ||
Syntax | cdiv 11285 | Extend class notation to include division. |
class / | ||
Definition | df-div 11286* | Define division. Theorem divmuli 11382 relates it to multiplication, and divcli 11370 and redivcli 11395 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divval 11288 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.) |
⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) | ||
Theorem | 1div0 11287 | You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that ∅ is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.) |
⊢ (1 / 0) = ∅ | ||
Theorem | divval 11288* | Value of division: if 𝐴 and 𝐵 are complex numbers with 𝐵 nonzero, then (𝐴 / 𝐵) is the (unique) complex number such that (𝐵 · 𝑥) = 𝐴. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) | ||
Theorem | divmul 11289 | Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 17-Feb-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴)) | ||
Theorem | divmul2 11290 | Relationship between division and multiplication. (Contributed by NM, 7-Feb-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵))) | ||
Theorem | divmul3 11291 | Relationship between division and multiplication. (Contributed by NM, 13-Feb-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐶))) | ||
Theorem | divcl 11292 | Closure law for division. (Contributed by NM, 21-Jul-2001.) (Proof shortened by Mario Carneiro, 17-Feb-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℂ) | ||
Theorem | reccl 11293 | Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) | ||
Theorem | divcan2 11294 | A cancellation law for division. (Contributed by NM, 3-Feb-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐵 · (𝐴 / 𝐵)) = 𝐴) | ||
Theorem | divcan1 11295 | A cancellation law for division. (Contributed by NM, 5-Jun-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴) | ||
Theorem | diveq0 11296 | A ratio is zero iff the numerator is zero. (Contributed by NM, 20-Apr-2006.) (Revised by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵) = 0 ↔ 𝐴 = 0)) | ||
Theorem | divne0b 11297 | The ratio of nonzero numbers is nonzero. (Contributed by NM, 2-Aug-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 ≠ 0 ↔ (𝐴 / 𝐵) ≠ 0)) | ||
Theorem | divne0 11298 | The ratio of nonzero numbers is nonzero. (Contributed by NM, 28-Dec-2007.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ≠ 0) | ||
Theorem | recne0 11299 | The reciprocal of a nonzero number is nonzero. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ≠ 0) | ||
Theorem | recid 11300 | Multiplication of a number and its reciprocal. (Contributed by NM, 25-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 · (1 / 𝐴)) = 1) |
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